BAIL 2006

International Conference onBoundary and Interior Layers

Gottingen,24th - 28th July, 2006

L. Prandtl

Book of Abstracts

Organized by:Georg-August Universitat Gottingen

Deutsches Zentrum fur Luft- und Raumfahrt (DLR)

Sponsored by:Land Niedersachsen,

Graduiertenkolleg 1023Gesellschaft fur Angewandte Mathematik und Mechanik (GAMM)

German Academic Exchange Service (DAAD)Georg-August Universitat Gottingen

Deutsches Zentrum fur Luft- und Raumfahrt (DLR)

GreetingIt gives me great pleasure to welcome all participants to the BAIL 2006 conference inGoettingen, Germany. This conference is the latest in a long line of international confer-ences on Boundary and Interior Layers, that have been held in many parts of the world.The first three conferences were held in Dublin, Ireland in 1980, 1982 and 1984. Thesewere followed by conferences in Novosibirsk, USSR (1986), Shanghai, China (1988),Copper Mountain, USA (1992) and Beijing, China (1994).

After a gap of some years, an international steering committee was formed to advise onthe creation of a new series of BAIL conferences. This committee made a positive recom-mendation, which has resulted in three further BAIL conferences being held, specifically:BAIL 2002 in Perth, Australia, BAIL 2004 in Toulouse, France and this year’s BAIL 2006in Goettingen, Germany.

It is heartening to see how well BAIL 2006 is supported. Indeed, the number of partic-ipants appears to be growing with each successive conference in the new series. I amdelighted to thank the members of the International Steering Committee for their goodadvice over the years. It is a particular pleasure also to thank the local organizers andtheir teams for the hard work involved in the organization of each of these conferences.Only someone who has organized such a conference is aware of just how much work isinvolved! I believe that the success of the BAIL 2006 organizers, especially in makingthis event attractive to a large number of participants, is an important milestone in thedevelopment of this series of conferences. On behalf of all participants I thank the orga-nizers and, in addition, I wish everyone a productive and enjoyable meeting.

John J H Miller

Dublin, IrelandJuly 2006

BAIL conferences

• BAIL 2004, Toulouse, France

• BAIL 2002 , Perth, Australia

• BAIL VII, Beijing, China (1994)

• BAIL VI, Colorado, USA (1992)

• BAIL V, Shanghai, China (1988)

• BAIL IV, Novosibirsk, USSR (1986)

• BAIL III, Dublin, Ireland (1984)

• BAIL II, Dublin, Ireland (1982)

• BAIL I, Dublin, Ireland (1980)

3 BAIL 2006

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6 BAIL 2006

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C.H

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(14:

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W.S

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(14:

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sym

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(16:

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7 BAIL 2006

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(SL)

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8 BAIL 2006

Room

School-L

ab

(SL)

Room

MPI

Tuesd

ay,

25

July

9:15

-10

:15

Ple

nary

Talk

:

M.

Sty

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apr

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10:1

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nsf

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M.H

ollin

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puta

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oftu

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funct

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(10:

45-1

1:10

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hkin

a,

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Wagner:

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inTurb

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Ther

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Con

vect

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(11:

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ori

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hi :

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and

Kin

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Appro

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(11:

35-1

2:00

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Min

isym

pos

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:J.M

aubac

h,I.V

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lish

chev

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nad

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the

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12:3

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Bre

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9 BAIL 2006

Room

School-L

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(SL)

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MPI

14:0

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auss

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(14:

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aubac

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att

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ka:M

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(15:

35-1

6:00

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M.

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ii:

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ente

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rangi

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ased

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Coff

eebre

ak

10 BAIL 2006

Room

School-L

ab

(SL)

Room

MPI

16:4

5-18

:00

Min

isym

pos

ium

:R

.H

artm

ann,P.H

oust

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euveline:O

na

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kin

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pec

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men

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ppro

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pre

s-

sible

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w(1

6:45

-17:

10)

F.

Nata

f,G

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apin

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pplica

tion

ofth

eSm

ith

Fac

-

tori

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onto

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ain

Dec

ompos

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ethods

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kes

Equat

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(17:

10-1

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)

A.

Cangia

ni,

E.H

.G

eorg

oulis,

M.

Jense

n:

Con

tinuou

s-D

isco

nti

nuou

sFin

ite

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men

tM

ethods

for

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vect

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iffusi

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roble

ms

(17:

35-1

8:00

):

20:0

0-21

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Public

Talk

:

Gers

ten:

Vom

Koch

topf

bis

zum

Fußb

alls

pie

l:E

pi-

soden

zuder

wel

twei

ten

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kung

der

Got

tinge

r

Str

omungs

fors

cher

(in

germ

an)

11 BAIL 2006

Room

School-L

ab

(SL)

Room

MPI

Wednesd

ay,

26

July

9:15

-10

:15

Ple

nary

Talk

:

W.

Wall:

Var

iation

alM

ultis

cale

Met

hods

for

inco

m-

pres

sibl

eflow

s.

10:1

5-10

:45

Coff

eeB

reak

10:4

5-12

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Ses

sion

:N

um

eric

alM

ethods

3

F.A

liza

rd,

J.-C

h.

Robin

et:

Tw

o-dim

ensi

onal

tem

-

por

alm

odes

innon

par

alle

lflow

s(1

0:45

-11:

10)

Q.

Ye:

Num

eric

alsi

mula

tion

oftu

rbule

nt

bou

ndar

y

for

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12 BAIL 2006

Room

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14:0

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13 BAIL 2006

Room

School-L

ab

(SL)

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MPI

Thurs

day,

27

July

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14 BAIL 2006

Room

School-L

ab

(SL)

Room

MPI

15:1

0-16

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Min

isym

pos

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ss,

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Madden:

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hes

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tim

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action

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usi

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avath

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num

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inco

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ms

15 BAIL 2006

Room

School-L

ab

(SL)

Room

MPI

Fri

day,

28

July

9:15

-10

:30

Min

isym

pos

ium

:D

.D

as,T

.K.Sen

gupta

M.H

.B

usc

hm

ann

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and

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Savic

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sof

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mix

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11:0

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isym

pos

ium

:D

.D

as,T

.K.Sen

gupta

T.K

.Sengupta

,A

.K

am

esw

ara

Rao:

Spat

io-

tem

por

algr

owin

gw

aves

inbou

ndar

y-lay

ers

by

Bro

n-

wic

hco

nto

ur

inte

gral

met

hod

A.

Nayak,

D.D

as :

Thre

e-dim

ensi

onal

Tem

por

al

Inst

ability

ofU

nst

eady

Pip

eFlo

w

J.H

uss

ong,N

.B

leie

r,V

.I.V

.R

am

:T

he

stru

cture

ofth

ecr

itic

alla

yer

ofa

swir

ling

annula

rflow

intr

ansi

-

tion

12:2

5-12

:40

Clo

sing

Ses

sion

12:4

0-14

:00

Lunch

16 BAIL 2006

Contents

Greetings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Plenary Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

KL. GERSTEN:Vom Kochtopf bis zum Fußballspiel (Public Talk) . . . . . . . . . 3

P. HOUSTON:Discontinuous Galerkin Finite Element Methods for CFD: A Pos-teriori Error Estimation and Adaptivity . . . . . . . . . . . . . . 4

L. LESSHAFFT, P. HUERRE, P. SAGAUT, M. TERRACOL:Dynamics of Hot Jets: A Numerical and Theoretical Study . . . . 5

M. STYNES:Convection-diffusion problems, SDFEM/SUPG and a priori meshes 6

V. GRAVEMEIER, S. LENZ, W.A. WALL:Variational Multiscale Methods for incompressible flows . . . . . 8

Minisymposia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13MS: N. Kopteva, M. Stynes, E. O’Riordan . . . . . . . . . . . . . . . . . . . . 13

R.K. DUNNE, E. O’RIORDAN, M.M. TURNER:A singular perturbation problem arising in the modelling of plasmasheaths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

G.I. SHISHKIN:A posteriori adapted meshes in the approximation of singularlyperturbed quasilinear parabolic convection-diffusion equations . . 16

W. LAYTON, I. STANCULESCU:Numerical Analysis of Approximate Deconvolution Models ofTurbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

H. WANG:A Component-Based Eulerian-Lagrangian Formulation for Com-positional Flow in Porous Media . . . . . . . . . . . . . . . . . . 19

MS: P. Houston, R. Hartmann . . . . . . . . . . . . . . . . . . . . . . . . . . . 21R. HARTMANN:

Discontinuous Galerkin methods for compressible flows: higherorder accuracy, error estimation and adaptivity . . . . . . . . . . 22

17 BAIL 2006

CONTENTS

V. HEUVELINE:On a new refinement strategy for adaptive hp finite element method 24

J.A. MACKENZIE, A. NICOLA:A Discontinuous Galerkin Moving Mesh Method for Hamilton-Jacobi Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 25

S. PEROTTO:Layer Capturing via Anisotropic Mesh Adaption . . . . . . . . . 27

R. SCHNEIDER, P. JIMACK:Anisotropic mesh adaption based on a posteriori estimates andoptimisation of node positions . . . . . . . . . . . . . . . . . . . 28

MS: Debopam Das, Tapan Sengupta . . . . . . . . . . . . . . . . . . . . . . . 31M.H. BUSCHMANN, M. GAD-EL-HAK:

Turbulent Boundary Layers: Reality and Myth . . . . . . . . . . 32A. NAYAK, D. DAS:

Three-dimesnional Temporal Instability of Unsteady Pipe Flow . 35J. HUSSONG, N. BLEIER, V.V. RAM:

The structure of the critical layer of a swirling annular flow intransition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

T.K. SENGUPTA, A. KAMESWARA RAO:Spatio-temporal growing waves in boundary-layers by Bromwichcontour integral method . . . . . . . . . . . . . . . . . . . . . . 39

L. SAVIC, H. STEINRUCK:Asymptotic Analysis of the mixed convection flow past a hori-zontal plate near the trailing edge . . . . . . . . . . . . . . . . . 41

MS: G.I. Shiskin, P. Hemker . . . . . . . . . . . . . . . . . . . . . . . . . . . 43D. BRANLEY, A.F. HEGARTY, H. PURTILL, G.I. SHISHKIN:

A Schwarz method for a convection-diffusion problem with a cor-ner singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

T. LINSS, N. MADDEN:Layer-adapted meshes for time-dependent reaction-diffusion . . . 47

G.I. SHISHKIN:Grid Approximation of Parabolic Equations with Nonsmooth Ini-tial Condition in the Presence of Boundary Layers of DifferentTypes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

L.P. SHISHKINA, G.I. SHISHKIN:A Difference Scheme of Improved Accuracy for a QuasilinearSingularly Perturbed Elliptic Convection-Diffusion Equation inthe Case of the Third-Kind Boundary Condition . . . . . . . . . . 49

I.V. TSELISHCHEVA, G.I. SHISHKIN:Domain Decomposition Method for a Semilinear Singularly Per-turbed Elliptic Convection-Diffusion Equation with ConcentratedSources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

18 BAIL 2006

CONTENTS

S. HEMAVATHI, S. VALARMATHI:A parameter-uniform numerical method for a system of singularlyperturbed ordinary differential equations . . . . . . . . . . . . . . 53

MS: J. Maubach, I, Tselishcheva . . . . . . . . . . . . . . . . . . . . . . . . . 55A.F. HEGARTY, S. SIKWILA, G.I. SHISKIN:

An adaptive method for the numerical solution of an elliptic con-vection diffusion problem . . . . . . . . . . . . . . . . . . . . . 56

M. ANTHONISSEN, I. SEDYKH, J. MAUBACH:A Convergence Proof of Local Defect Correction for Convection-Diffusion Problems . . . . . . . . . . . . . . . . . . . . . . . . . 57

J. MAUBACH:On the Difference between Left and Right Preconditioning forConvection Dominated Convection-Diffusion Problems . . . . . . 58

S. LI, L.P. SHISHKINA, G.I. SHISHKIN:Parameter-Uniform Method for a Singularly Perturbed ParabolicEquation Modelling the Black-Scholes equation in the Presenceof Interior and Boundary Layers . . . . . . . . . . . . . . . . . . 59

A.I. ZADORIN:Numerical Method for the Blasius Equation on an Infinite Interval 61

P. ZEGELING:An Adaptive Grid Method for the Solar Coronal Loop Model . . . 63

Contributed Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . 67F. ALIZARD, J.-CH. ROBINET:

Two-dimensional temporal modes in nonparallel flows . . . . . . 67TH. ALRUTZ, T. KNOPP:

Near-wall grid adaptation for wall functions . . . . . . . . . . . . 69TH. APEL, G. MATTHIES:

A family of non-conforming finite elements of arbitrary order forthe Stokes problem on anisotropic quadrilateral meshes . . . . . . 71

M. BAUSE:Aspects of SUPG/PSPG and GRAD-DIV Stabilized Finite Ele-ment Approximation of Compressible Viscous Flow . . . . . . . 72

L. BOGUSLAWSKI:Sheare Stress Distribution on Sphere Surface at Different InflowTurbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

A.CANGIANI, E.H.GEORGOULIS, M. JENSEN:Continuous-Discontinuous Finite Element Methods for Convection-Diffusion Problems . . . . . . . . . . . . . . . . . . . . . . . . . 75

C. CLAVERO, J.L. GRACIA, F. LISBONA:A second order uniform convergent method for a singularly per-turbed parabolic system of reaction-diffusion type . . . . . . . . . 77

19 BAIL 2006

CONTENTS

B. EISFELD:Computation of complex compressible aerodynamic flows with aReynolds stress turbulence model . . . . . . . . . . . . . . . . . 79

A. FIROOZ, M. GADAMI:Turbulence Flow for NACA 4412 in Unbounded Flow and GroundEffect with Different Turbulence Models and Two Ground Con-ditions: Fixed and Moving Ground Conditions . . . . . . . . . . 81

S.A. GAPONOV, G.V. PETROV, B.V. SMORODSKY:Boundary layer intercation with external disturbances . . . . . . . 85

Z.HAMMOUCH:Similarity solutions of a power-law non-Newtonian laminar bound-ary layer flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

M. HAMOUDA, R. TEMAM:Boundary layers for the Navier-Stokes equations : asymptoticanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

M. HOLLING, H. HERWIG:Computation of turbulent natural convection at vertical walls us-ing new wall functions . . . . . . . . . . . . . . . . . . . . . . . 91

A.-M. IL’IN, B.I. SULEIMANOV:The coefficients of inner asymptotic expansions for solutions ofsome singular boundary value problems . . . . . . . . . . . . . . 93

W.S. ISLAM, V.R. RAGHAVAN:Numerical Simulation of High Sub-critical Reynolds Number FlowPast a Circular Cylinder . . . . . . . . . . . . . . . . . . . . . . 96

D. KACHUMA, I. SOBEY:Fast waves during transient flow in an asymmetric channel . . . . 98

A. KAUSHIK, K.K. SHARMA:A Robust Numerical Approach for Singularly Perturbed Time De-layed Parabolic Partial Differential Equations . . . . . . . . . . . 100

P. KNOBLOCH:On methods diminishing spurious oscillations in finite elementsolutions of convection-diffusion equations . . . . . . . . . . . . 102

T. KNOPP:Model-consistent universal wall-functions for RANS turbulencemodelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

J.-S. LEU, J.-Y. JANG, Y.-C. CHOU:Evaporating cooling of liquid film along an inclined plate coverdwith a porous layer . . . . . . . . . . . . . . . . . . . . . . . . . 106

V.D. LISEYKIN, Y.V. LIKHANOVA, D.V. PATRAKHIN, I.A. VASEVA:Application of boundary layer-type functions to comprehensivegrid generation codes . . . . . . . . . . . . . . . . . . . . . . . . 108

20 BAIL 2006

CONTENTS

G. LUBE:A stabilized finite element method with anisotropic mesh refine-ment for the Oseen equations . . . . . . . . . . . . . . . . . . . . 109

H. LUDEKE:Detached Eddy Simulation of Supersonic Shear Layer Wake Flows 111

K. MANSOUR:Boundary Layer Solution For laminar flow through a Looselycurved Pipe by Using Stokes Expansion . . . . . . . . . . . . . . 113

G. MATTHIES, L. TOBISKA:Mass conservation of finite element methods for coupled flow-transport problems . . . . . . . . . . . . . . . . . . . . . . . . . 115

J. MAUSS, J. COUSTEIX:Global Interactive Boundary Layer (GIBL) for a Channel . . . . . 116

O. MIERKA, D. KUZMIN:On the implementation of turbulence models in incompressibleflow solvers based on a finite element discretization . . . . . . . . 118

K. MORINISHI:Rarefied Gas Boundary Layer Predicted with Continuum and Ki-netic Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . 120

A. NASTASE:Qualitative Analysis of the Navier-Stokes Solutions in Vicinity oftheir Critical Lines . . . . . . . . . . . . . . . . . . . . . . . . . 122

F. NATAF, G. RAPIN:Application of the Smith Factorization to Domain DecompositionMethods for the Stokes equations . . . . . . . . . . . . . . . . . 124

N. NEUSS:Numerical approximation of boundary layers for rough boundaries 126

M.A. OLSHANSKII:An Augmented Lagrangian based solver for the low-viscosity in-compressible flows . . . . . . . . . . . . . . . . . . . . . . . . . 128

N. PARUMASUR, J. BANASIAK, J.M. KOZAKIEWICZ:Numerical and Asymptotic Analysis of Singularly Perturbed PDEsof Kinetic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 130

B. RASUO:On Boundary Layer Control in Two-Dimensional Transonic WindTunnels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

H.-G. ROOS:A Comparison of Stabilization Methods for Convection-Diffusion-Reaction Problems on Layer-Adapted Meshes . . . . . . . . . . . 134

H.-G. ROOS, H. ZARIN:Discontinuous Galerkin stabilization for convection-diffusion prob-lems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

21 BAIL 2006

B. SCHEICHL, A. KLUWICK:On Turbulent Marginal Separation: How the Logarithmic Law ofthe Wall is Superseded by the Half-Power Law . . . . . . . . . . 136

O. SHISHKINA, C. WAGNER:Boundary and Interior Layers in Turbulent Thermal Convection . 138

M. STYNES, L. TOBISKA:Using rectangular Qp elements in the SDFEM for a convection-diusion problem with a boundary layer . . . . . . . . . . . . . . 140

P. SVACEK:Numerical Approximation of Flow Induced Airfoil Vibrations . . 141

N.V. TARASOVA:Full asymptotic analysis of the Navier-Stokes equations in theproblems of gas flows over bodies with large Reynolds number . . 143

C.H. TAI, C.-Y. CHAO, J.-C. LEONG, Q.-S. HONG:Effects of golf ball dimple configuration on aerodynamics, trajec-tory, and acoustics . . . . . . . . . . . . . . . . . . . . . . . . . 145

H. TIAN:Uniformly Convergent Numerical Methods for Singularly Per-turbed Delay Differential Equations . . . . . . . . . . . . . . . . 147

A.I. TOLSTYKH, M.V. LIPAVSKII, E.N. CHIGEREV:Highly accurate 9th-order schemes and their applications to DNSof thin shear layer instability . . . . . . . . . . . . . . . . . . . . 149

M. VASILIEV:About unsteady Boundary Layer on a dihedral angle . . . . . . . 150

A.E.P. VELDMANN:High-order symmetry-preserving discretization on strongly stretchedgrids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Z.-H. YANG, Y.-Z. LI, Y. ZHU:Application of Bifurcation Method to Computing Numerical So-lutions of Lane-Emden Equation . . . . . . . . . . . . . . . . . . 154

Q. YE:Numerical simulation of turbulent boundary for stagnation-flowin the spray-painting process . . . . . . . . . . . . . . . . . . . . 156

Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

22 BAIL 2006

Plenary Presentations

Öffentlicher Vortrag

am 25. Juli 2006 – 20:00 Uhr im DLR_School_Lab - Bunsenstraße 10 – 37073 Göttingen

im Rahmen der International Conference Boundary and Interior Layers 2006 (BAIL 2006)

Kontakt: DLR, Dr. H. J. Heinemann 0551- 709 2108, hajo.heinemann@dlr.de Universität Göttingen, Prof. Dr. G. Lube 0551- 39 4503, lube@math.uni-goettingen.de

Vom Kochtopf bis zum Fußballspiel Episoden zu den weltweiten Wirkungen der Göttinger Strömungsforschung

Professor Dipl. Math. Dr.-Ing. Dr.-Ing. E.h. Klaus Gersten Ruhr-Universität Bochum

Im Vortrag werden unter anderem folgende Fragen erörtert:

Als Modell für ein Herz mit defekten Herzklappen entwickelte der Arzt Dr. Liebau in den fünfziger Jahren eine ventillose Pumpe. Weshalb funktionierte die Pumpe?

Wie lässt sich die Bananenflanke beim Fußballspiel physikalisch erklären?

Kann die Kraftfahrzeug-Aerodynamik von der Flugzeug-Aerodynamik lernen?

Wie kann der Besitzer eines Automobils mit Schrägheck leicht feststellen, ob sein Fahrzeug gute aerodynamische Eigenschaften besitzt?

Warum sind bestimmte bei der Umströmung von Körpern auftretende Strömungsstrukturen je nach Anwendung erwünscht (Überschallflugzeug) oder unerwünscht (Automobil)?

Welches für die Göttinger Strömungsforschung charakteristische Konzept liegt allen bisher genannten Fragen zugrunde?

Zur Person:

Professor Gersten, Jahrgang 1929, studierte Mathematik und Physik an der TH Braunschweig. Als Assistent arbeitete er am Institut für Strömungsmechanik der TH Braunschweig unter der Leitung von Professor Hermann Schlichting. Nach seiner Promotion zum Dr.-Ing. leitete er die Abteilung Theoretische Aerodynamik und wurde anschließend Stellvertretender Direktor des Institutes für Aerodynamik der Deutschen Forschungsanstalt für Luftfahrt (DFL) Braunschweig. Nach seiner Habilitation für das Fach Strömungsmechanik lehrte er dreißig Jahre als Ordentlicher Professor an der Ruhr-Universität Bochum. Während dieser Zeit war er als Visiting Professor an der University of Rio de Janeiro, Brazil, der Nagoya University, Japan und an der University of Arizona, Tucson, USA tätig. 1992 wurde er zum Dr.-Ing. E.h. der Universität Essen ernannt. Er ist Mitglied der Österreichischen Akademie der Wissenschaften in Wien. 2003 wurde Professor Gersten der Ludwig-Prandtl-Ring, die höchste Auszeichnung im Bereich der Flugwissenschaften, von der Deutschen Gesellschaft

für Luft- und Raumfahrt Lilienthal-Oberth e.V. (DGLR) verliehen.

KL. GERSTEN: Vom Kochtopf bis zum Fußballspiel (Public Talk)

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%Speaker: GERSTEN, KL. 3 BAIL 2006

Discontinuous Galerkin Finite Element Methods for CFD: A Posteriori Error Estimation and

Adaptivity

Paul Houston

School of Mathematical Sciences, University of Nottingham, UK.

In recent years there has been considerable interest in the mathematical design andpractical application of nonconforming finite element methods that are based on discon-tinuous piecewise polynomial approximation spaces; such approaches are referred to asdiscontinuous Galerkin (DG) methods. The main advantages of these methods lie in theirconservation properties, their ability to treat a wide range of problems within the sameunified framework, and their great flexibility in the mesh-design. Indeed, DG methods caneasily handle non-matching grids and non-uniform, even anisotropic, polynomial approx-imation degrees, which makes them ideally suited for application within adaptive finiteelement software.

In this talk we present an overview of some recent developments concerning the a pos-

teriori error analysis and adaptive mesh design of h– and hp–version DG finite elementmethods for the numerical approximation of second–order elliptic boundary value prob-lems. In particular, we consider the derivation of computable upper and lower boundson the error measured in terms of an appropriate (mesh–dependent) energy norm. Theproofs of the upper bounds are based on rewriting the method in a non-consistent man-ner using polynomial lifting operators and employing an appropriate decomposition resultfor the underlying discontinuous spaces. Applications to the numerical approximationof second–order linear elliptic problems, including Poisson’s equation, Stokes equations,nearly–incompressible elasticity, and the time harmonic eddy current problem, as wellas second–order quasilinear boundary value problems, which typically arise in the mod-elling of non-Newtonian flows, will be considered. Numerical experiments confirming thereliability and efficiency of the proposed a posteriori error bounds within an automaticmesh refinement algorithm employing both local mesh subdivision and local polynomialenrichment will be presented.

This research has been carried out in collaboration with Dominik Schotzau (Univer-sity of British Columbia), Thomas Wihler (University of Minnesota), and Ilaria Perugia(University of Pavia).

References

[1 ] P. Houston, I. Perugia, and D. Schotzau. An posteriori error indicator for dis-continuous Galerkin discretizations of H(curl)–elliptic partial differential equations.Submitted to IMA J. Numer. Anal.

[2 ] P. Houston, D. Schotzau, and T. Wihler. Energy norm a posteriori error estimationfor mixed discontinuous Galerkin approximations of the Stokes problem. J. Sci.

Comp., 22(1):357–380, 2005.

[3 ] P. Houston, D. Schotzau, and T. Wihler. An hp-adaptive mixed discontinuousGalerkin FEM for nearly incompressible linear elasticity. Comput. Methods Appl.

Mech. Engrg., (to appear).

[4 ] P. Houston, D. Schotzau, and T. P. Wihler. Energy norm a posteriori error esti-mation of hp-adaptive discontinuous Galerkin methods for elliptic problems. Math.

Models Methods Appl. Sci., (to appear).

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P. HOUSTON: Discontinuous Galerkin Finite Element Methods for CFD: A PosterioriError Estimation and Adaptivity

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%Speaker: HOUSTON, P. 4 BAIL 2006

BAIL 2006

DYNAMICS OF HOT JETS: A NUMERICAL AND THEORETICAL STUDY

Lutz Lesshafft1.2, Patrick Huerre1, Pierre Sagaut3 and Marie Terracol2

1Laboratoire d'Hydrodynamique (LadHyX), CNRS – École Polytechnique, F-91128 Palaiseau, France2ONERA, Department of CFD and Aeroacoustics, 29 avenue de la Division Leclerc, F-92322Châtillon, France3Laboratoire de Modélisation en Mécanique, Université Pierre et Marie Curie, Boîte 162, 4 placeJussieu, F-75252 Paris cedex 05, France

Since the experiments of Monkewitz, Bechert, Barsikow & Lehmann (1990), sufficiently hot circularjets have been known to give rise to self-sustained synchronized oscillations induced by a locallyabsolutely unstable region. Numerical simulations (Lesshafft, Huerre, Sagaut & Terracol 2005, 2006)have been carried out in order to determine if such synchronized states correspond to a nonlinearglobal mode of the underlying basic flow, as predicted in the context of Ginzburg-Landau amplitudeevolution equations by Couairon & Chomaz (1997, 1999), Pier, Huerre, Chomaz & Couairon (1998)and Pier, Huerre & Chomaz (2001). In the presence of a pocket of absolute instability embeddedwithin a convectively unstable jet, global oscillations are generated by a steep nonlinear front locatedat the upstream station of marginal absolute instability. The global frequency is given, within 10%accuracy, by the absolute frequency at the front location. For jet flows displaying absolutely unstableinlet conditions, global instability is observed to arise if the streamwise extent of the absolutelyunstable region is sufficiently large: While local absolute instability sets in for ambient-to-jettemperature ratios S < 0.453, global modes only appear for S < 0.325. In agreement with theoreticalpredictions, the selected frequency near the onset of global instability coincides with the absolutefrequency at the inlet, provided that the ratio of jet radius R to shear layer momentum thickness θ issufficiently small (R/ θ ∼ 10, thick shear layers). For thinner shear layers (R/ θ ∼ 25), the numericallydetermined global frequency gradually departs from the inlet absolute frequency.

References

Couairon, A. & Chomaz, J.-M. 1997 Absolute and convective instabilities, front velocities and globalmodes in nonlinear systems. Physica D 108, 236-276Couairon, A. & Chomaz, J.-M. 1999 Fully nonlinear global modes in slowly varying flows. Phys.Fluids 11, 3688-3703.Lesshafft, L., Huerre, P., Sagaut, P. & Terracol, M. 2005 Global modes in hot jets, absolute /convective instabilities and acoustic feedback. 10 pages 11th AIAA / CEAS Aeroacoustics Conference,Monterey, USA, May 23-25, 2005.Lesshaft, L., Huerre, P., Sagaut, P. & Terracol, M. 2006 Nonlinear global modes in hot jets. J. FluidMech., in press.Monkewitz, P. A., Bechert, D. W., Barsikow, B & Lehmann, B. 1990 Self-excited oscillations andmixing in a heated round jet. J. Fluid Mech. 213, 611-639.

L. LESSHAFFT, P. HUERRE, P. SAGAUT, M. TERRACOL: Dynamics of Hot Jets: ANumerical and Theoretical Study

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%Speaker: HUERRE, P. 5 BAIL 2006

Convection-diffusion problems, SDFEM/SUPG

and a priori meshes

Martin Stynes∗

Abstract

Linear convection-diffusion problems will be briefly described (cf. [15]). The nature andstructure of their solutions will be examined, including the main features of exponentialand characteristic/parabolic layers. Next, Shishkin meshes will be described and discussed[4, 11, 15]; these piecewise-uniform meshes are suited to the numerical solution of convection-diffusion problems with boundary layers — yet they do not fully resolve these layers.

Then the Streamline Diffusion Finite Element Method (SDFEM), which is also knownas the Streamline-Upwinded Petrov-Galerkin method (SUPG), will be introduced. Sinceits inception [6] in 1979, this method has been the subject of a huge number of theoreticalanalyses and numerical investigations that continue to this day; see the references in [12, 13,15]. The main body of the talk is a comprehensive survey of the application of the methodto convection-diffusion problems, including discussions of its strengths and weaknesses, andpresenting recent theoretical results. In particular the following topics will be addressed:

• stability and the choice of SDFEM parameter [1, 5, 12]

• quasioptimality [3, 14]

• accuracy on general meshes [9, 18]

• accuracy on Shishkin meshes [5, 10, 16, 17]

• variants of SDFEM:(i) nonconforming spaces [7](ii) nonlinear shock-capturing modifications of SDFEM [2, 8]

— the references given here are only a subset of those available.

References

[1] J. E. Akin and T. E. Tezduyar. Calculation of the advective limit of the SUPG stabilizationparameter for linear and higher-order elements. Comput. Methods Appl. Mech. Engrg.,193:1909–1922, 2004.

[2] E. Burman and A. Ern. Stabilized Galerkin approximation of convection-diffusion-reactionequations: discrete maximum principle and convergence. Math. Comp., 74:1637–1652 (elec-tronic), 2005.

[3] L. Chen and J. Xu. An optimal streamline diffusion finite element method for a singularlyperturbed problem. In Z.C Shi, Z. Chen, T. Tang, and D. Yu, editors, Recent Advances inAdaptive Computation, volume 383 of Contemporary Mathematics, pages 236–246. Ameri-can Mathematical Society, 2005.

∗Mathematics Department, National University of Ireland, Cork, Ireland

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M. STYNES: Convection-diffusion problems, SDFEM/SUPG and a priori meshes

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%Speaker: STYNES, M. 6 BAIL 2006

[4] P.A. Farrell, A.F. Hegarty, J.J. Miller, E. O’Riordan, and G.I. Shishkin. Robust Computa-tional Techniques for Boundary Layers. Chapman & Hall/CRC, Boca Raton, 2000.

[5] S. Franz and T. Linß. Superconvergence analysis of Galerkin FEM and SDFEM for ellipticproblems with characteristic layers. Technical Report MATH-NM-03-2006, Institut furNumerische Mathematik, Technische Universitat Dresden, 2006.

[6] T.J.R. Hughes and A.N. Brooks. A multidimensional upwind scheme with no crosswinddiffusion. In T.J.R. Hughes, editor, Finite Element Methods for Convection DominatedFlows, volume 34 of AMD. ASME, New York, 1979.

[7] P. Knobloch and L. Tobiska. The Pmod1 element: a new nonconforming finite element for

convection-diffusion problems. SIAM J. Numer. Anal., 41:436–456 (electronic), 2003.

[8] T. Knopp, G. Lube, and G. Rapin. Stabilized finite element methods with shock capturingfor advection-diffusion problems. Comput. Methods Appl. Mech. Engrg., 191:2997–3013,2002.

[9] N. Kopteva. How accurate is the streamline-diffusion FEM inside characteristic (boundaryand interior) layers? Comput. Methods Appl. Mech. Engrg., 193:4875–4889, 2004.

[10] T. Linß and M. Stynes. Numerical methods on Shishkin meshes for linear convection-diffusion problems. Comput. Methods Appl. Mech. Engrg., 190:3527–3542, 2001.

[11] J.J.H. Miller, E. O’Riordan, and G.I. Shishkin. Solution of singularly perturbed problemswith ε-uniform numerical methods–introduction to the theory of linear problems in one andtwo dimensions. World Scientific, Singapore, 1996.

[12] H.-G. Roos, M. Stynes, and L. Tobiska. Numerical methods for singularly perturbed differ-ential equations, volume 24 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 1996.

[13] H.-G. Roos, M. Stynes, and L. Tobiska. Numerical methods for singularly perturbed differ-ential equations. Springer Series in Computational Mathematics. Springer-Verlag, Berlin,Second edition, (to appear).

[14] G. Sangalli. Quasi optimality of the SUPG method for the one-dimensional advection-diffusion problem. SIAM J. Numer. Anal., 41:1528–1542 (electronic), 2003.

[15] M. Stynes. Steady-state convection-diffusion problems. Acta Numer., 14:445–508, 2005.

[16] M. Stynes and L. Tobiska. The SDFEM for a convection-diffusion problem with a boundarylayer: optimal error analysis and enhancement of accuracy. SIAM J. Numer. Anal., 41:1620–1642, 2003.

[17] M. Stynes and L. Tobiska. Using rectangular Qp elements in the SDFEM for a convection-diffusion problem with a boundary layer. Technical Report 08-2006, Faculty of Mathematics,Otto-von-Guericke-Universitat, Magdeburg, 2006.

[18] G. Zhou. How accurate is the streamline diffusion finite element method? Math. Comp.,66:31–44, 1997.

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M. STYNES: Convection-diffusion problems, SDFEM/SUPG and a priori meshes

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%Speaker: STYNES, M. 7 BAIL 2006

Variational Multiscale Methods for incompressible flows

V. Gravemeier, S. Lenz & W.A. Wall

Chair for Computational MechanicsTechnische Universitat Munchen

Boltzmannstr. 15, D-85747 Garching b. Munchenhttp://www.lnm.mw.tum.de

(vgravem,lenz,wall)@lnm.mw.tum.de

1. Introduction

The numerical simulation of incompressible flows governed by the Navier-Stokes equations re-quires to deal with subgrid phenomena. Particularly in turbulent flows, the scale spectra arenotably widened and need to be handled adequately to get a reasonable numerical solution.Separating the complete scale range into subranges enables a different treatment of any of thesesubranges. In this talk, we will present the general framework of a two- and a three-scale sepa-ration of the incompressible Navier-Stokes equations based on the variational multiscale methodas proposed in [Hughes et al. (2000)] and [Collis (2001)]. In a two-scale separation, resolvedand unresolved scales are distinguished, and in a three-scale separation, large resolved scales,small resolved scales, and unresolved scales are differentiated. After having presented the gen-eral framework, three different approaches to numerical realizations will be addressed (i.e., aglobal, a local, and a new residual-based approach). A comprehensive overview may be foundin [Gravemeier (2006b)].

2. The variational multiscale framework

A variational form of the incompressible Navier-Stokes equations reads

BNS (v, q;u, p) = (v, f)Ω

∀ (v, q) ∈ Vup (1)

where Vup denotes the combined form of the weighting function spaces for velocity and pressurein the sense that Vup := Vu × Vp.

In a three-scale separation, which will be focused on in the present abstract, the solutionfunctions are separated as

u = u + u′ + u, p = p + p′ + p, (2)

where (·), (·)′, and (·) denote the large resolved, small resolved, and unresolved scales, respec-tively. The weighting functions are separated accordingly. Due to the linearity of the weightingfunctions, the variational equation (1) may now be decomposed into a system of three variationalequations reading

BNS

(

v, q;u + u′ + u, p + p′ + p

)

= (v, f)Ω

∀ (v, q) ∈ Vup (3)

BNS

(

v′, q′;u + u

′ + u, p + p′ + p)

=(

v′, f

)

Ω∀

(

v′, q′

)

∈ V′

up (4)

BNS

(

v, q;u + u′ + u, p + p′ + p

)

= (v, f)Ω

∀ (v, q) ∈ Vup (5)

Furthermore, it is assumed that the direct influence of the unresolved scales on the large resolvedscales is close to zero, relying on a clear separation of the large-scale space and the space ofunresolved scales. This leads to a simplified equation system.

1

V. GRAVEMEIER, S. LENZ, W.A. WALL: Variational Multiscale Methods for incom-pressible flows

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%Speaker: WALL, W.A. 8 BAIL 2006

It is not intended to resolve anything called unresolved a priori. Taking into account theeffect of the unresolved scales onto the small scales is the only desire. Some alternatives lendthemselves for this purpose, but the focus in this talk will be on the subgrid viscosity approachas a usual and well-established way of taking into account the effect of unresolved scales inthe traditional LES. In the variational multiscale approach to LES, the subgrid viscosity termdirectly acts only on the small resolved scales. Indirect influence on the large resolved scales,however, is ensured due to the coupling of the large- and the small-scale equations.

3. Practical methods

At this stage, it should be pointed out that the variational multiscale method is a theoreticalframework for the separation of scales. Thus, it is essential to develop corresponding practicalimplementations by incorporating the variational multiscale framework into a specific numericalmethod. For such practical methods, it is crucial that the aforementioned separation of thedifferent scale groups is actually achieved. Implementations of the three-scale separation havealready been done within continuous Galerkin finite element methods, discontinuous Galerkinfinite element methods, finite volume methods, finite difference methods, and spectral meth-ods. According to the numerical treatment of the smaller of the resolved scales, a global (see,e.g., [Gravemeier (2006a)]) and a local (see, e.g., [Gravemeier et al. (2005)]) approach may bedistinguished.

Recently [Calo (2005)], a new residual-based approach has been developed where only re-solved and unresolved scales are distinguished (i.e., a two-scale separation). This new con-cept approxmiates the unresolved scales by analytical expressions. It may be considered anadvanced stabilized method which takes into account the nonlinear nature of the Navier-Stokes-equations. Our results obtained with the residual-based approach are about to be published in[Lenz & Wall (2006)].

All three approaches (i.e., the global, the local, and the residual-based approach) will bepresented together with numerical results in this talk.

References

[Calo (2005)] Calo, V.M. 2005 Residual-based Multiscale Turbulence Modeling: Finite VolumeSimulations of Bypass Transition. PhD thesis, Department of Civil and EnvironmentalEngineering, Stanford University.

[Collis (2001)] Collis, S.S. 2001 Monitoring unresolved scales in multiscale turbulence model-ing. Phys. Fluids 13, 1800-1806.

[Gravemeier (2006a)] Gravemeier, V. 2006 Scale-separating operators for variational multi-scale large eddy simulation of turbulent flows. J. Comp. Phys. 212, 400-435.

[Gravemeier (2006b)] Gravemeier, V. 2006 The variational multiscale method for laminar andturbulent flow. Arch. Comp. Meth. Engrg. , in press.

[Gravemeier et al. (2005)] Gravemeier, V., Wall, W.A. and Ramm, E. 2005 Large eddysimulation of turbulent incompressible flows by a three-level finite element method Int. J.

Numer. Meth. Fluids 48, 1067-1099.

[Hughes et al. (2000)] Hughes, T. J. R., Mazzei, L. & Jansen, K. E. 2000 Large eddysimulation and the variational multiscale method. Comput. Visual. Sci. 3, 47-59.

[Lenz & Wall (2006)] Lenz, S. and Wall, W.A. 2006 A residual-based variational multiscalemethod and its application to turbulent channel flow. To be submitted to Theoret. Comput.

Fluid Dyn.

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V. GRAVEMEIER, S. LENZ, W.A. WALL: Variational Multiscale Methods for incom-pressible flows

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%Speaker: WALL, W.A. 9 BAIL 2006

Minisymposia

N. Kopteva, M. Stynes, E. O’RiordanRobust methods for nonlinear singularly perturbed differential equations

This minisymposium is concerned with nonlinear singularly perturbeddifferential equations such as semilinear reaction-diffusion problems,quasilinear parabolic convection-diffusion equations, flows in porousmedia, and the modelling of catalytic chemical reactions and of turbu-lence. The talks deal with methods that are ”robust” for such problems,i.e., that yield accurate approximations of the solution for a broad rangeof values of the singular perturbation parameter. This includes numer-ical methods for which numerical experiments demonstrate robustnessfor a wide range of values of the parameter, even though no theoreticalproof of convergence exists.

Speaker:

• Eugene O’Riordan: A singular perturbation problem arising in the modelling ofplasma sheaths

• Grigory I. Shishkin: A posteriori adapted meshes in the approximation of singularlyperturbed quasilinear parabolic convection-diffusion equations

• Iuliana Stanculescu: Numerical Analysis of Approximate Deconvolution Modelsof Turbulence

• Hong Wang: An Eulerian-Langrangian Formulation for Compositional Flow inPorous Media

A singular perturbation problem arising in the

modelling of plasma sheaths ∗

R. K. Dunne†, E. O’Riordan‡ and M. M. Turner§

Abstract

Consider the interaction of a flowing plasma with a planar Langmuir probe [2]. As-sume that the plasma flows in the plane of the probe surface and that the plasma consistssolely of positive ions with density n+ and electrons with density ne. Downstream ofthe probe, the ions are moving with a velocity of u = (ui, uv + uF ) and (u0, uF ) isthe flow velocity of the ions upstream of the probe. We wish to consider the influenceof the probe on the flow of the ions. Let X be the horizontal distance to the right ofthe probe and Y the distance along the probe (from the tip of the probe). Assuminga collision-less plasma and that the ions are cold, the continuity equations for the iondensity and momentum are [2]

∂n+

∂t+∇ · (n+u) = 0

m+n+

(∂u∂t

+ (u · ∇)u)

= en+E

where E = (Ex, Ey) = −∇φ is the electric field and m+ is the mass of the ions. Ourinterest is in the steady state case and if uF >> uv, we disregard terms involving uv.Hence this system is approximated with the system

∂

∂X(n+ui) + uF

∂n+

∂Y= 0

ui∂ui

∂X+ uF

∂ui

∂Y=

e

m+EX

where (if EX >> EY ) then the electric field is determined from solving Poisson’s equa-tion

−∂EX

∂X=

∂2φ

∂X2=

e(n+ − ne)ε0

.

Since me << m+ and we assume that the electrostatic potential φ tends to zero asone moves away from the probe, the electron density ne is approximately related to theelectrostatic potential φ as

ne = n0exp(− eφ

kTe)

∗This research was supported in part by the National Center for Plasma Science and Technology Ireland.†School of Mathematical Sciences, Dublin City University, Ireland‡School of Mathematical Sciences, Dublin City University, Ireland§School of Physical Sciences, Dublin City University, Ireland.

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R.K. DUNNE, E. O’RIORDAN, M.M. TURNER: A singular perturbation problemarising in the modelling of plasma sheaths

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%Speaker: O’RIORDAN, E. 14 BAIL 2006

where Te is the electron temperature, k is Boltzmann’s constant, ε0 is the permittivityof free space and e is the electron charge.

In a similar fashion to the scaling of the variables used in [3], we introduce the non-dimensional independent variables x, y and the non-dimensional dependent variablesn, u and φ, which are defined as follows:

n =n+

n0u =

ui

csφ =

eφ

kTe, x =

X

Ly =

Y cs

uF L

where the ion sound speed cs and the electron Debyre length λD are defined by

c2s =

kTe

m+, λ2

D =ε0kTe

n0e2.

The length L is a distance sufficiently far from the probe so that the effect of the probeon the plasma at this distance is negligible. Introduce the small parameter

ε =λD

L.

After reformulating the problem with the above transformations and formulatingsuitable boundary and initial conditions we propose to examine the related mathemat-ical problem :

Find (u(x, y), n(x, y), φ(x, y)) which satisfy the following system of differential equa-tions in the domain (x, y) ∈ (0, 1)× (0, T ]

∂n

∂y+

∂(nu)∂x

= 0, (x, y) ∈ [0, 1)× (0, T ]

∂u

∂y+ u

∂u

∂x= −∂φ

∂x, (x, y) ∈ [0, 1)× (0, T ]

ε2 ∂2φ

∂x2= eφ − n, (x, y) ∈ (0, 1)× (0, T ]

subject to the following set of boundary and initial conditions

φ(0, y) = −A, φ(1, y) = 0, y ≥ 0 φ(x, 0) = φ0(x), 0 ≤ x ≤ 1n(x, 0) = 1, 0 ≤ x ≤ 1; ny(1, y) = −(nux)(1, y), y ≥ 0u(x, 0) = u0, 0 ≤ x ≤ 1; uy(1, y) = −φx(1, y), y ≥ 0.

The parameters u0 and A are assumed to be known and the initial condition φ0(x) ischosen so that ε2φ′′

0(x) = eφ0(x) − 1, φ0(0) = −A, φ0(1) = 0.Due to the presence of the singular perturbation parameter ε, layers or sheaths

appear in the solutions. The system is discretized using simple upwinding and a specialpiecewise-uniform Shishkin-type mesh [1]. Numerical results are presented to displaythe robustness of the numerical algorithm with respect to ε.

References

[1] P. A. Farrell, A.F. Hegarty, J. J. H. Miller, E. O’Riordan and G. I. Shishkin, RobustComputational Techniques for Boundary Layers, Chapman and Hall/CRC Press,Boca Raton, U.S.A., (2000).

[2] M. A. Lieberman and A. J. Lichtenberg, Principles of plasma discharges and ma-terials processing, Wiley and sons, (1994).

[3] H. Liu and M. Slemrod, KDV Dynamics in the plasma-sheath transition, Appl.Math. Lett., 17 (2004) 401-419.

2

R.K. DUNNE, E. O’RIORDAN, M.M. TURNER: A singular perturbation problemarising in the modelling of plasma sheaths

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%Speaker: O’RIORDAN, E. 15 BAIL 2006

A posteriori adapted meshes in the approximation

of singularly perturbed quasilinear parabolic

convection-diffusion equations ∗

Grigory I. Shishkin

A Dirichlet problem on a segment for a quasilinear parabolicconvection-diffusion equation with a small (perturbation) param-eter ε multiplying the highest derivative is considered. For thisproblem, a solution of the classical finite difference scheme on auniform mesh converges only under the condition h ε, whereh is the step-size of the space mesh; moreover, the order of con-vergence in x is O

(εN−1

), where N + 1 is the number of nodes

in the uniform mesh with respect to x.

To improve the accuracy of the approximate solution, we ap-ply a posteriori sequential procedure of grid refinement in thesubdomains that are defined by the gradient of solutions of inter-mediate discrete problems. The correction of the grid solutionsis performed only on these subdomains, where uniform meshesare used. We construct a difference scheme that converges ”al-most ε-uniformly”, i.e., with an error weakly depending on theparameter ε.

The convergence rate of the constructed scheme isO(ε−ν N−1

1 +N−1/21 +N−1

0

),

where N1 + 1 and N0 + 1 are the numbers of mesh points withrespect to x and t, respectively, ν is an arbitrary number from(0, 1]. Thus, the scheme on a posteriori adapted meshes con-verges under the condition N−1 εν , which is essentially weakerin comparison with the scheme on uniform meshes.

∗This research was supported in part by the Russian Foundation for Basic Research(grant No 04-01-00578, 04–01–89007–NWO a), by the Dutch Research Organisation NWOunder grant No 047.016.008 and by the Boole Centre for Research in Informatics, NationalUniversity of Ireland, Cork.

2

G.I. SHISHKIN: A posteriori adapted meshes in the approximation of singularlyperturbed quasilinear parabolic convection-diffusion equations

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%Speaker: SHISHKIN, G.I. 16 BAIL 2006

A scheme on a posteriori adapted meshes in the case of alinear problem is considered in [1].

References

[1] G.I. Shishkin, A posteriori adapted (to the solution gra-dient) grids in the approximation of singularly perturbedconvection-diffusion equations, Vychisl. Tekhnol. (Compu-tational Technologies), 6 (1), 72-87 (2001) (in Russian).

3

G.I. SHISHKIN: A posteriori adapted meshes in the approximation of singularlyperturbed quasilinear parabolic convection-diffusion equations

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%Speaker: SHISHKIN, G.I. 17 BAIL 2006

Numerical Analysis of Approximate Deconvolution Models of

Turbulence

William Layton∗ and Iuliana Stanculescu†

Abstract

If the NSE are averaged with a local, spacial, convolution type filter the resultingsystem is not closed due to the term g ∗ (uu). A deconvolution operator GN is onewhich satisfies:

u = GN (g ∗ u) + O(δ2N+2) (0.1)

where δ is the filter width. This yields the closure method:

g ∗ (uu) = g ∗ (GN (g ∗ u)GN (g ∗ u)) + O(δ2N+2) (0.2)

We will review several solutions to the ill-possed deconvolution problem, presentan ”optimal” deconvolution procedure and present numerical analysis and numericalexperiments with it.

∗Department of Mathematics, University of Pittsburgh, Pittsburgh, PA,15260, U.S.A.; email:wjl@pitt.edu, www.math.pitt.edu/˜wjl

†Department of Mathematics, University of Pittsburgh, Pittsburgh, PA,15260, U.S.A.; email:ius1@pitt.edu, www.math.pitt.edu/˜ius1

1

W. LAYTON, I. STANCULESCU: Numerical Analysis of Approximate DeconvolutionModels of Turbulence

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%Speaker: STANCULESCU, I. 18 BAIL 2006

A Component-Based Eulerian-Lagrangian Formulationfor Compositional Flow in Porous Media

Hong Wang ∗

Introduction Compositional models describe the simultaneous transport of multiple compo-nents flowing in coexisting phases in porous media [1, 4]. Because each component can transferbetween different phases, the mass of each phase or a component within a particular phase isno longer conserved. Instead, the total mass of each component among all the phases must beconserved, leading to strongly coupled systems of transient nonlinear advection-diffusion equa-tions. These equations are closely coupled to a set of constraining equations, which are stronglynonlinear, implicit functions of phase pressure, temperature, and composition. These equationsneed to be solved in all spatial cells at each iterative step of each time step via thermodynamicflash calculation. In industrial applications, upwind methods have commonly been used to sta-bilize the numerical approximations [1, 4]. However, these methods often generate excessivenumerical dispersion and serious spurious effects due to grid orientation.

Eulerian-Lagrangian methods symmetrize the transport equations and generate accuratenumerical solutions even if large time steps and coarse spatial grids are used. They have demon-strated excellent performance in the numerical simulations of single-phase flow [3, 5] and immis-cible two phase flow [2]. However, there exist serious mathematical and numerical difficulties inthe development of Eulerian-Lagrangian methods for multiphase multicomponent compositionalflow. We present a component-based Eulerian-Lagrangian formulation for compositional flow,which can be used by many Eulerian-Lagrangian methods.

Numerical experiments We simulate the transport of Methane, Propane, and n-Hexane,flowing in coexisting liquid and vapor phases in a horizontal porous medium reservoir Ω =(0, 1000)× (0, 1000) ft2 with a thickness of 1 ft over a time period of 20 years. An injection wellis located at the upper-right corner of Ω with a volumetric injection rate of Q = 15 ft3/day. Aproduction well is located at the lower-left corner with a production rate of Q = −15 ft3/day.The porosity φ = 0.1. The permeability is 60 md. The effect of capillary pressure is neglected.The relative permeability kr,l = (sl)2 and kr,v = (1− sl)2. The initial reservoir pressure is 2100psia and the reservoir temperature is 350oK. The composition of the resident fluid is cMethane

= 0.5, cPropane = 0.2, and cn−Hexane = 0.3, which is in liquid phase at the given temperatureand pressure. The composition of the injected fluid is cMethane = 0.8, cPropane = 0.15, andcn−Hexane = 0.05, which is in vapor phase.

We use a uniform coarse spatial grid of ∆x = ∆y = 25 ft. We use a time step of ∆tel = 1year for the Eulerian-Lagrangian method, and of ∆tup = 2 days for the upwind method that isthe largest possible. In Figure 1(a) we present the plots of the overall mole fraction cMethane

generated by the Eulerian-Lagrangian method at t = 5 and 20 years. In Figure 1(b) we presentthe plots of the normalized molar amount nv

Methane = ρvsvcvMethane/(ρ cMethane), where ρv and

sv are density and saturation of the vapor phase and ρ is bulk molar density of the fluid mixture.Note that nv

i represents the fraction of ci in the vapor phase. Thus, nvi = 1 or 0 indicates that

component i stays in the vapor phase or liquid phase completely. 0 < nvi < 1 implies that

component i is present in both phases. Finally, the plots of the overall mole fraction of Methaneand of the normalized molar amount of Methane in vapor phase, which are generated by theupwind method at t = 5 and 20 years, are presented in Figures 1(c) and 1(d).

∗Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208

1

H. WANG: A Component-Based Eulerian-Lagrangian Formulation for CompositionalFlow in Porous Media

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%Speaker: WANG, H. 19 BAIL 2006

Figure 1: The ELLAM ((a)-(b)) and upwind ((c)-(d)) simulation at 5 and 20 years.

(a) The overall mole fraction of Methane (b) The molar amount of Methane in vapor

(c) The overall mole fraction of Methane (d) The molar amount of Methane in vapor

Discussion The numerical results show that the Eulerian-Lagrangian formulation generatesstable and accurate solutions (overall mole fractions) that have preserved physically reasonablemoving steep fronts, even if a large time step of ∆tel = 1 year is used. With the (largestpossible) fine time step ∆tup = 2 days, the upwind method generates qualitatively similar (butmuch more diffusive) overall mole fraction. We observe a similar comparison in terms of thenormalized molar amounts in the vapor and liquid phases.

It is instructive to look at the computational efficiency. Although the Eulerian-Lagrangianmethod uses more CPU time than the upwind method in solving the mass balance equations,both simulators have to perform the same computations on the pressure system and the thermo-dynamic flash calculations which consume a larger portion of the CPU time per iterative stepat each time step. These results indicate that the Eulerian-Lagrangian simulator uses less thantwice the CPU time an upwind simulator uses per time step. Therefore, the Eulerian-Lagrangianmethod generates accurate and stable solutions with steeper fronts using much less CPU time.

References

[1] H. Aziz and A. Settari, Petroleum Reservoir Simulation, Applied Science Publishers, 1979.

[2] M.S. Espedal and R.E. Ewing, Characteristic Petrov-Galerkin sub-domain methods for two-phase immiscible flow, Comput. Meth. Appl. Mech. Engrg., 64, (1987) 113–135.

[3] R.E. Ewing, T.F. Russell, and M.F. Wheeler, Simulation of miscible displacement usingmixed methods and a modified method of characteristics, SPE 12241, (1983), 71–81.

[4] R. Helmig, Multiphase Flow and Transport Processes in the Subsurface, Springer Verlag,Berlin, 1997.

[5] H. Wang, D. Liang, R.E. Ewing, S.L. Lyons, and G. Qin, An ELLAM-MFEM solutiontechnique for compressible fluid flows in porous media with point sources and sinks, J. Comput.Phys., 159, (2000) 344–376.

H. WANG: A Component-Based Eulerian-Lagrangian Formulation for CompositionalFlow in Porous Media

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%Speaker: WANG, H. 20 BAIL 2006

P. Houston, R. HartmannSelf-adaptive Methods for PDEs

Many processes in science and engineering are formulated in terms ofpartial differential equations. Typically, for problems of practical in-terest, the underlying analytical solution exhibits localised phenomenasuch as boundary and interior layers and corner and edge singularities,for example, and their numerical approximation presents a challengingcomputational task. Indeed, in order to resolve such localised features,in an accurate and efficient manner, it is essential to exploit so-calledself-adaptive methods.

Such approaches are typically based on a posteriori error estimates forthe underlying discretization method in terms of local quantities, suchas local residuals, computed from the discrete solution. Over the lastfew years, there have been significant developments within this field interms of both rigorous a posteriori error analysis, as well as the subse-quent design of optimal meshes. In this minisymposium, a number ofrecent developments, such as the design of high-order and hp-adaptivefinite element methods will be considered, as well as anisotropic meshadaptation and mesh movement.

Speaker:

• Ralf Hartmann: Discontinuous Galerkin methods for compressible flows: higherorder accuracy, error estimation and adaptivity

• Vincent Heuveline: On a new refinement strategy for adaptive hp finite elementmethod

• John Mackenzie: A Discontinuous Galerkin Moving Mesh Method for Hamilton-Jacobi Equations

• Simona Perotto: Layer Capturing via Anisotropic Mesh Adaption

• Rene Schneider: Anisotropic mesh adaption based on a posteriori estimates andoptimisation of node positions

Discontinuous Galerkin methods for compressible flows:

higher order accuracy, error estimation and adaptivity

Ralf Hartmann

Institute of Aerodynamics and Flow Technology, German Aerospace Center (DLR),Lilienthalplatz 7, 38108 Braunschweig, Germany

The Discontinuous Galerkin (DG) method for the compressible Euler equations is extendedto the symmetric interior penalty (SIP)DG method for the compressible Navier-Stokes equations,see Hartmann & Houston [2]. Shock-capturing is used to avoid overshoots near shocks, see Hart-mann [1]. The nonlinear equations are solved using a fully implicit solver. We demonstrate theaccuracy of higher order DG discretizations with respect to the approximation of aerodynam-ical force coefficients and to the approximation of viscous boundary layers, see Figure 1 andHartmann & Houston [2].

Finally, we demonstrate the use of a posteriori error estimation and goal-oriented (alsocalled weighted-residual-based or adjoint-based) adaptive mesh refinement for subsonic flows,Hartmann & Houston [3], and for supersonic compressible flows, see Figure 2 and Hartmann [1].

References

[1] R. Hartmann. Adaptive discontinuous Galerkin methods with shock-capturing for the com-pressible Navier-Stokes equations. Int. J. Numer. Meth. Fluids, 2006. To appear.

[2] R. Hartmann and P. Houston. Symmetric interior penalty DG methods for the compressibleNavier–Stokes equations I: Method formulation. Int. J. Num. Anal. Model., 3(1):1–20, 2006.

[3] R. Hartmann and P. Houston. Symmetric interior penalty DG methods for the compressibleNavier–Stokes equations II: Goal–oriented a posteriori error estimation. Int. J. Num. Anal.

Model., 3(2):141–162, 2006.

1

R. HARTMANN: Discontinuous Galerkin methods for compressible flows: higher orderaccuracy, error estimation and adaptivity

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%Speaker: HARTMANN, R. 22 BAIL 2006

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0.022

0.024

10000 100000

reference cdp DG(3), global refinementDG(2), global refinementDG(1), global refinement

cdp

number of elements

0.0001

0.001

0.01

10000 100000

DG(1), global refinementDG(2), global refinementDG(3), global refinement

number of elements

|cdp−

0.0222875|

(a) (b)

Figure 1: Subsonic laminar flow around the NACA0012 airfoil at M = 0.5, α = 0 and Re = 5000:Convergence of the pressure induced drag cdp under global refinement for DG(p), p = 1, 2, 3: (a)cdp versus number of elements; (b) Error in cdp (reference cdp − cdp) versus number of elements.For more detail cf. Hartmann & Houston [2].

-8

-4

0

4

8

-4 0 4 8-8

-4

0

4

8

-4 0 4 8

(a) (b)

Figure 2: Supersonic laminar flow around the NACA0012 airfoil at M = 1.2, α = 0 andRe = 1000: (a) Residual-based refined mesh of 17670 elements with 282720 degrees of freedomand |Jcdp

(u) − Jcdp(uh)| = 1.9 · 10−3 ; (b) Adjoint-based refined mesh for cdp: mesh of 10038

elements with 160608 degrees of freedom and |Jcdp(u) − Jcdp

(uh)| = 1.6 · 10−4. For more detailcf. Hartmann [1].

2

R. HARTMANN: Discontinuous Galerkin methods for compressible flows: higher orderaccuracy, error estimation and adaptivity

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%Speaker: HARTMANN, R. 23 BAIL 2006

On a new refinement strategy for adaptive hp finiteelement method

V. Heuveline

University Karlsruhe (TH)

Institute for Applied Mathematics II

vincent.heuveline@math.uni-karlsruhe.de

Abstract

We consider finite element methods with varying meshsize h as well as varyingpolynomial degree p. Such methods have been proven to show exponentially fastconvergence in some classes of partial differential equations if an adequate distribu-tion of h− and p−refinement is chosen. In order to find hp−refinement strategiesthat show up automaticaly with optimal complexity, it is a first step to establishconvergent adaptive algorithms. We develop a strategy that automatically constructa solution adapted approximation space by combining local h− and p− refinementand that can be proven for the 1d case to be convergent with a linear rate. Thisconstruction is based on an a posteriori error estimate with respect to the errorin the energy norm. We then extend the proposed approach to the 2d and 3dcase. Numerical experiments as well as implementation issues are considered inthat framework.

1

V. HEUVELINE: On a new refinement strategy for adaptive hp finite element method

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%Speaker: HEUVELINE, V. 24 BAIL 2006

A Discontinuous Galerkin Moving Mesh Method for

Hamilton-Jacobi Equations

J.A. Mackenzie and A. Nicola

Department of Mathematics

University of Strathclyde

26 Richmond St, Glasgow

G1 1XH, U.K.

jam@maths.strath.ac.uk

Abstract

1 Introduction

In this talk we consider the adaptive numerical solution of Hamilton-Jacobi (HJ) equations

φt + H(φx1, . . . , φxd

) = 0, φ(x, 0) = φ0(x), (1)

where x = (x1, . . . , xd) ∈ IRd, t > 0. HJ equations arise in many practical areas suchas differential games, mathematical finance, image enhancement and front propagation. Itis well known that solutions of (1) are Lipschitz continuous but derivatives can becomediscontinuous even if the initial data is smooth. Since generalised solutions are not unique, aselection principle is required to pick out the physically relevant solution. For HJ equationsthe most commonly used condition is the vanishing viscosity condition which requires that thecorrect solution should be the vanishing viscosity limit of smooth solutions of correspondingviscous problems. The notion of viscosity solutions was introduced by Crandall and Lions[5], where the questions of existence, uniqueness and stability of solutions were addressed.

Crandall and Lions were also the first to study numerical approximations of (1) andintroduced the important class of monotone methods which were shown to converge to theviscosity solution [4]. However, monotonic schemes are well known to be at most first-orderaccurate.

There is a close relation between HJ equations and hyperbolic conservation laws. Withthis in mind, it not surprising to find that many of the numerical methods used to solve HJequations are motivated by conservative finite difference or finite volume methods for conser-vation laws. Methods that have been proposed include high-order essentially nonoscillatory(ENO) schemes, weighted ENO schemes and high resolution central schemes.

An increasingly popular approach to solve hyperbolic conservation laws is the discontin-uous Galerkin (DG) finite element method [2], [3]. Recently, Hu and Shu [7] proposed a DGmethod to solve HJ equations by first rewriting (1) as a system of conservation laws

(wi)t + (H(w))xi= 0, i = 1, . . . , d, w(x, 0) = ∇φ0(x), (2)

where w = ∇φ. The usual DG formulation would be obtained if w belonged to a spaceof piecewise polynomials. However, we note that wi, i = 1, . . . , d are not independent dueto the restriction that w = ∇φ. In [7] a least squares procedure was used to enforce thiscondition. More recently it was shown that it is possible to enforce the gradient conditionusing a smaller solution space [12]. Theoretical analysis of the accuracy and stability of themethod was performed in [11].

One of the often cited advantages of DG methods is that since the numerical solution isnot continuous across inter-element boundaries then, in theory, this makes solution adaptivestrategies much easier to implement. This has lead to the development of a number ofadaptive methods based on hp-refinement strategies for hyperbolic conservation laws [1] [6].

1

J.A. MACKENZIE, A. NICOLA: A Discontinuous Galerkin Moving Mesh Method forHamilton-Jacobi Equations

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%Speaker: MACKENZIE, J.A. 25 BAIL 2006

An alternative adaptive strategy that has worked well for time dependent problems isto use moving meshes. A useful way to construct an adaptive moving mesh is to regard itas the image of a uniform mesh covering a computational domain under a time-dependenttransformation that clusters mesh elements towards areas where improved spatial resolutionis required. The transformation is often found through a variational formulation where themapping is the minimiser of a functional involving properties of the mesh and the solution(see e.g. [10]). To improve the stability and smooth the evolution of the moving meshRussell, Huang and coworkers [9] [8] found it useful to obtain the mapping as the solution ofparabolic moving mesh PDEs which are modified gradient flow equations for the minimisationof a suitable mesh functional.

For problems where solution discontinuities exist the correct choice of an adaptivitycriterion, or monitor function, is problematic. It is not unusual to find in the literature thatmany grid adaptation criteria are singular at solution discontinuities. To prevent singularitiesproducing degenerate meshes either some form of smoothing procedure is employed or aregularised functional is used in the variational formulation.

The aim of this talk is to consider the use of the DG method of Hu and Shu [7] to solveHJ equation using a moving mesh method based on the solution of MMPDEs. The governingequation are transformed to include the effect of the movement of the mesh and this is donein such a way that the conservation properties of the original equation are not lost. Theadaptive mesh is driven by a monitor function which is shown to be non-singular in thepresence of solution discontinuities. To produce an acceptable mesh we smooth the monitorfunction before it is used to drive the adaptive procedure. Numerical examples in one andtwo dimensions are presented to demonstrate the effectiveness of the adaptive procedure..

References

[1] K.S. Bey and J.T. Oden. hp-version discontinuous Galerkin methods for hyperbolicconservation laws. Comput. Methods Appl. Mech. Engrg., 133:259–286, 1996.

[2] B. Cockburn, G.E. Karniadakis, and C.W. Shu. Discontinuous Galerkin methods: The-

ory, Computation and Applications. Springer, 2000.

[3] B. Cockburn and C.W. Shu. The Runge-Kutta discontinuous Galerkin finite elementmethod for conservation laws V: Multidimensional systems. Journal of Computational

Physics, 141:199–224, 1998.

[4] M.G. Crandall and P.L. Lions. Two Approximations of Solutions of Hamilton-JacobiEquations. Mathematics of Computation, 43(167):1–19, 1984.

[5] M.J. Crandall and P.L. Lions. Viscosity Solutions of Hamilton-Jacobi Equations. Trans-

actions of the American Mathematical Society, 277(1):1–42, 1983.

[6] P. Houston, B. Senior, and E. Suli. hp-discontinuous Galerkin finite element methodsfor hyperbolic problems: analysis and adaptivity. Int. J. Numer. Methods in Fluids,40:153–169, 2002.

[7] C. Hu and C.W. Shu. A Discontinuous Galerkin Finite Element Method for Hamilton-Jacobi Equations. SIAM Journal on Scientific Computing, 21(2):666–690, 1999.

[8] W. Huang. Practical aspects of formulation and solution of moving mesh partial differ-ential equations. Journal of Computational Physics, 171:753–775, 2001.

[9] W. Huang and R. D. Russell. Moving mesh strategy based upon a gradient flow equationfor two-dimensional problems. SIAM Journal on Scientific Computing, 20:998–1015,1999.

[10] P. Knupp and S. Steinberg. Foundations of Grid Generation. CRC Press, Boca Raton,1994.

[11] O. Lepsky, C. Hu, and C. W. Shu. Analysis of the discontinuous Galerkin method forHamilton-Jacobi equations. Appl. Numer. Math., 33:423–434, 2000.

[12] F. Li and C.W. Shu. Reinterpretation and simplified implementation of a discontinuousGalerkin method for Hamilton-Jacobi equations. Applied Mathematics Letters, 18:1204–1209, 2005.

J.A. MACKENZIE, A. NICOLA: A Discontinuous Galerkin Moving Mesh Method forHamilton-Jacobi Equations

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%Speaker: MACKENZIE, J.A. 26 BAIL 2006

Layer Capturing via Anisotropic Mesh Adaption

Simona Perotto

MOX - Dipartimento di Matematica “F. Brioschi”Politecnico di Milano, Via Bonardi 9

I-20133 Milano, Italysimona.perotto@mate.polimi.it

The numerical approximation of many problems in Computational Fluid Dynamics (CFD) leadsoften to deal with solutions exhibiting directional features, namely great variations along certaindirections and less significant changes along the other ones. This is the case of boundary andinternal layers typical, for instance, of the advection-diffusion and Navier-Stokes equations aswell as of shocks in the case of Euler equations. In view of an efficient numerical approach tothis kind of problems, it turns out advisable to resort to a suitably oriented and deformed (i.e.,anisotropic) computational mesh, matching the local directional features of the solution. Fora fixed solution accuracy, a considerable reduction of the number of the degrees of freedom isshown in the presence of an anisotropic grid, besides a sharper capture of the solution features.

In this regard, we have introduced a theoretically sound anisotropic framework moving fromthe spectral properties of the standard affine map between the reference and the general meshelement [2]. As first step, we have derived suitable anisotropic interpolation error estimates forpiecewise linear finite elements. Then, in view of an anisotropic mesh adaption driven by an aposteriori error estimator, these anisotropic estimates have been merged with the standard dual-based approach proposed by R. Becker and R. Rannacher in [1]. Thus the final outcome of ouranalysis consists of an automatic tool able to properly orient and stretch the mesh elements sothat a goal-functional of the exact solution, representing a quantity of interest, is approximatedwithin a user-defined tolerance [3, 4].

In this communication the leading ideas of our anisotropic framework are presented togetherwith some numerical results associated with the goal-oriented a posteriori analysis.

References

[1] R. Becker and R. Rannacher, “A feed-back approach to error control in finite elementmethods: basic analysis and examples”, East-West J. Numer. Math., 4(4), 237–264 (1996).

[2] L. Formaggia and S. Perotto, “New anisotropic a priori error estimates”, Numer. Math.,89, 641–667 (2001).

[3] L. Formaggia and S. Perotto, “Anisotropic error estimates for elliptic problems”, Numer.

Math., 94, 67–92 (2003).

[4] L. Formaggia, S. Micheletti and S. Perotto, “Anisotropic mesh adaptation in ComputationalFluid Dynamics: application to the advection-diffusion-reaction and the Stokes problems.”,Appl. Numer. Math., 51 (4), 511–533 (2004).

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S. PEROTTO: Layer Capturing via Anisotropic Mesh Adaption

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%Speaker: PEROTTA, S. 27 BAIL 2006

Anisotropic mesh adaption based on a posteriori estimates and

optimisation of node positions

René Schneider∗ & Peter Jimack†

Abstract

Introduction

Efficient numerical approximation of solution features like boundary or interior layers by meansof the finite element method requires the use of layer adapted meshes. Anisotropic meshes, likefor example Shishkin meshes, allow the most efficient approximation of these highly anisotropicsolution features. However, application of this approach relies on a priori analysis on the thick-ness, position and stretching direction of the layers. If it is impossible to obtain this informationa priori, as it is often the case for problems with interior layers of unknown position for example,automatic mesh adaption based on a posteriori error estimates or error indicators is essential inorder to obtain efficient numerical approximations.

Historically the majority of work on automatic mesh adaption used locally uniform refine-ment, splitting each element into smaller elements of similar shape. This procedure is clearly notsuitable to produce anisotropically refined meshes. The resulting meshes are over-refined in atleast one spatial direction, rendering the approach far less efficient than that of the anisotropicmeshes based on a priori analysis.

Automatic anisotropic mesh adaption is an area of active research, e.g. [2, 1]. Here we presenta new approach to this problem, based upon using not only an a posteriori error estimate toguide the mesh refinement, but its sensitivities with respect to the positions of the nodes in themesh as well.

Outline of the approach

The underlying idea is to use techniques from mathematical optimisation to minimise the esti-mated error by moving the positions of the nodes in the mesh appropriately. This basic idea isof course not new, but the approach taken to realise it is.

A key ingredient is the utilisation of the discrete adjoint technique to evaluate the sensitivitiesof an error estimate J = J(uh(s), s) with respect to the node positions s, where uh = uh(s)denotes the solution of the discretised PDE, R(uh, s) = 0, which depends upon the node positionss. The sensitivities

DJ

Ds=

∂J

∂uh

∂uh

∂s+

∂J

∂s

=∂J

∂s−ΨT ∂R

∂s, (1)

[

∂R

∂uh

]T

Ψ =∂J

∂uh

, (2)

∗Chemnitz University of Technology, Germany†School of Computing, University of Leeds, UK

R. SCHNEIDER, P. JIMACK: Anisotropic mesh adaption based on a posteriori esti-mates and optimisation of node positions

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%Speaker: SCHNEIDER, R. 28 BAIL 2006

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

y

initial mesh

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

y

final mesh for eps=1.0e−03

(a) (b)

Figure 1: Initial (a) and adapted (b) meshes for model problem (3) with a boundary layer aroundthe square hole

are thereby evaluated according to (1), utilising the adjoint solution Ψ which is defined by (2).This way, DJ/Ds can be evaluated without computing ∂uh/∂s first, reducing the number ofequation systems to be solved from O(dim(s)) to just two, independent of dim(s). As thenumber of nodes can easily be larger than one hundred even in extremely coarse meshes (if thedomain geometry is complicated), this approach is of fundamental importance to make gradientbased optimisation methods feasible for this type of problem. Fast optimisation algorithms likeBFGS-type methods can be applied to obtain significant reductions in the error estimate J aftera few optimisation steps. The aim of this procedure is to provide a mesh with problem adaptedanisotropic elements, which may then be used as a good basis for further locally uniform adaptivemesh refinement.

Numerical results

To demonstrate feasibility of the approach it is applied to a number of model problems. For thepurpose of this abstract we consider a reaction diffusion equation,

−∆u + 1

ε2 u = 1

ε2 in Ω := (−1, 1)2 \ (−1

5, 1

5)2

subject to u = 0 on ΓD :=[

−1

5, 1

5

]2\

(

−1

5, 1

5

)2

∂u

∂n= 0 on ΓN := [−1, 1]2 \ (−1, 1)2 .

(3)

Figure 1 shows an initial mesh for this problem and an adapted one for ε = 10−3. Concentrationof the elements in the boundary layer which forms around the hole at the centre of the domain isclearly visible, and significantly increased aspect ratios in the boundary layer may be observed.For more detail on the approach and the example we refer to [3].

References

[1] L. Formaggia, S. Micheletti, and S. Perotto. Anisotropic mesh adaptation in computational fluid dy-namics: Application to the advection-diffusion-reaction and the Stokes problems. Applied Numerical

Mathematics, 51(4):511–533, December 2004.

[2] G. Kunert. Toward anisotropic mesh construction and error estimation in the finite element method.Numerical Methods for Partial Differential Equations, 18(5):625–648, 2002.

[3] R. Schneider and P.K. Jimack. Toward anisotropic mesh adaption based upon sensitivity of a posterioriestimates. School of Computing Research Report Series 2005.03, University of Leeds, http://www.comp.leeds.ac.uk/research/pubs/reports/2005/2005_03.pdf, 2005.

R. SCHNEIDER, P. JIMACK: Anisotropic mesh adaption based on a posteriori esti-mates and optimisation of node positions

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%Speaker: SCHNEIDER, R. 29 BAIL 2006

Debopam Das, Tapan Sengupta:Asymptotic Methods for Laminar and Turbulent boundary Layers

The minisymposium will have contributions dealing with lami-nar/turbulent flows- their morphology and structures. Specifically, re-sults would be presented that discuss the very concept of turbulentboundary layers. Also, detailed flow structures arising in mixed con-vection problem involving velocity and thermal boundary/ interior lay-ers will be a focus of another contribution. This session will also havecontributions dealing specifically with temporal and/or spatio-temporalgrowth of waves in boundary/ interior layers. For example, existenceof spatio-temporal growth of waves in boundary layer is established forthe first time that is related non-orthogonal modes proposed earlier forCouette and channel flows. The focus will be on receptivity and stabil-ity of equilibrium steady flows as well as unsteady bi-directional pipeflows that identifies non-axisymmetric modes.

Speaker:

• Matthias H. Buschman: Turbulent Boundary Layers: Reality and Myth (Key-NoteLecture)

• Debopam Das: Three-dimensional Temporal Instability of Unsteady Pipe Flow.

• Venkatesa I. Vasanta Ram: The structure of the critical layer of a swirling annularflow in transition.

• T.K. Sengupta: Spatio-temporal growing waves in boundary-layers by Bromwichcontour integral method

• Herbert Steinrueck: Asymptotic Analysis of the mixed convection flow past a hori-zontal plate near the trailing edge.

M. H. Buschmann & M. Gad-el-Hak, Abstract for BAIL 2006

International Conference on Boundary and Interior Layers—Computational & Asymptotic Methods (BAIL 2006)

Abstract of keynote lecture for the

Minisymposium on Asymptotic Methods for Laminar and Turbulent Boundary Layers

Turbulent Boundary Layers: Reality and Myth

Matthias H. Buschmann Privatdozent, Institut für Strömungsmechanik, Technische Universität Dresden, Dresden, Germany

Mohamed Gad-el-Hak

Caudill Professor and Chair of Mechanical Engineering, Virginia Commonwealth University, Richmond, VA 23284-3015, USA

Hundred years after Ludwig Prandtl’s fundamental lecture on boundary layer theory,1 the mean-velocity profile and the shear-stress distribution of the seemingly simplest case of wall-bounded flow, the zero-pressure-gradient turbulent boundary layer (ZPG TBL), still appears to be terra incognita. Even less is known about confined and semi-confined flows undergoing pressure gradients, such as pipe and channel flows and wall-bounded flows approaching pressure-driven separation. The problem is of course related to the lack of analytical solutions to the instantaneous, nonlinear Navier–Stokes equations that govern the stochastic dependent variables of almost all turbulent flows. What little we know about turbulence comes from experiments and heuristic modeling, not first-principles solutions. (Direct numerical simulations provide first-principles integration of the instantaneous Navier–Stokes equations, but are at present limited to modest Reynolds numbers and simple geometries.)

Consider a two-dimensional, isothermal, incompressible, steady flow over a body of length L. The fluid may have constant density ρ and constant dynamic viscosity µ. Assuming that the characteristic velocity is U, we write the Reynolds number as

Re=!UL µ (1)

It was Prandtl’s genius that discovered that, at sufficiently high Reynolds number, a thin shear layer exists close to the body. The thickness of this layer δ is much smaller than L. Due to the strong streamwise velocity gradient normal to the wall, even a small viscosity such as that for air or water can cause considerable viscous shear stress, ! =µ "u "y . Originally Prandtl called this layer therefore Reibungsschicht, which literally translates to friction layer. Introducing the transformation

x!

, y!

,u!

,v!

, p!

( )! L x,Re"

1

2 L y,U u,Re"

1

2 U v,#U2

p$

%&'

() (2)

1 Prandtl, L., “Über Flüssigkeitsbewegung bei sehr kleiner Reibung,” Proc. Third Int. Math. Cong., pp. 484–491, Heidelberg, Germany, 1904.

M.H. BUSCHMANN, M. GAD-EL-HAK: Turbulent Boundary Layers: Reality andMyth

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%Speaker: BUSCHMANN, M.H. 32 BAIL 2006

M. H. Buschmann & M. Gad-el-Hak, Abstract for BAIL 2006

into the Navier-Stokes equations and taking the limit Re → ∞, leads to Prandtl’s boundary layer equations

uu

x+ vu

y= ! p

x+ u

yy;

0 = p

y;

u

x+ v

y= 0 (3)

The boundary conditions are

y = 0 : u = v = 0 and

y ! " : u ! U

ex( ) (4)

The task is now to find physically appropriate solution for eqn. (3).

What goals do we have when solving eqn. (3)? One of the main objectives is to find self-similar solutions. Boundary layers are self-similar when normalization can be found so that data of different physical realizations (e.g., experiments in different wind tunnels, profiles at different downstream positions within one experiment) can be collapsed within one single curve. Examples are the mean velocity profiles of the Blasius’ laminar zero-pressure-gradient boundary layer and the fully-developed turbulent flow in pipes.

Which physical problems do we face when solving eqn. (3)? We neither know if for a certain type of wall-bounded flow a transformation as searched for in the first question exists in general, nor do we know what the transformation parameters are. The physically appropriate transformation is a non-dimensionalization that is much more than simply changing the coordinates. The crucial issue is top choose the scaling based on the physics of the problem. At a minimum, the scale basis has to satisfy two criteria. It should consist of characteristic parameters and represent problem-intrinsic scales. The foundation of dimensional analysis is the Π-theorem formulated by Buckingham.2

Here xi denote the n variables of the system having m dimensions and Δi are the non-dimensional similarity parameters of the problem. 2 Buckingham, E., “On physically similar systems: illustrations of the use of dimensional equations,” Phys. Rev., 2. Ser., vol. 4, pp. 1119–1126, 1914.

( )u x, y

U

( )1 2 nf x ,x ,...x 0= ( )1 2 n mF , ,... 0! ! !

"=

Π-theorem

Find proper scaling parameters

Δ and U

( )u x, y ( )u x, y

( )u x, y y

y

!

1x

3x

2x

M.H. BUSCHMANN, M. GAD-EL-HAK: Turbulent Boundary Layers: Reality andMyth

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%Speaker: BUSCHMANN, M.H. 33 BAIL 2006

M. H. Buschmann & M. Gad-el-Hak, Abstract for BAIL 2006

Which mathematical techniques do we have to solve eqn. (3)? Basically we have analytical, numerical, and asymptotic methods and combination of those.3 In this paper, we will focus on recent advances in analytical and asymptotic approaches. During the mid 1990s, a new debate on the aforementioned subject arose. Caused by new unconventional approaches questioning one of the cornerstones of modern fluid mechanics—the logarithmic law of turbulent boundary layers—several new scalings were developed. In conjunction with these theoretical investigations, high-quality experiments in zero-pressure-gradient turbulent boundary layers and turbulent pipe and channel flows were undertaken. In general, the physical picture of wall-bounded flow is now much more complex than was thought a decade ago. However, the physical picture seems to be also more controversial than ever before. Which of these new approaches will survive and contribute substantially to fluid mechanics in the future is still open.

The present talk will discuss four main schools of thought, which can be summarized as follows:

(1) Standard Logarithmic Overlap Layer Above a certain critical Reynolds number, a pure logarithmic region exists in the

mean-velocity profile of ZPG TBL and channel and pipe flows. The parameters of this logarithmic law are completely Reynolds number invariant.

(2) Power Law Based On Similarity Assumption An inner and an outer power law describe the overlap layer of ZPG TBL and

channel and pipe flows. Both laws exhibit Reynolds number dependent parameters, and never achieve an asymptotic state.

(3) Asymptotic Invariance Principle Full similarity solutions have to be searched separately for the inner and outer

equations. Because in the limit the outer boundary layer equations are independent of Reynolds number, the properly scaled profiles for the outer region must also be independent of Reynolds number.

(4) Higher-Order Asymptotic Matching Based on asymptotic matching, a higher-order approach with respect to Reynolds

number and the wall-normal coordinate can be derived. In the limit of infinite Reynolds number, the classical logarithmic law is recovered. At finite Re, however, the similarity laws for both the mean and higher-order statistics are Reynolds-number dependent.

The presentation will summarize, analyze and critique the recent theoretical and experimental investigations of a variety wall-bounded flows. Based on that analysis, open issues in the field will be highlighted.

3 Gersten, K., and Herwig, H., Strömungsmechanik, Fundamentals and Advances in the Engineering Science, Verlag Vieweg, 1992.

Numericalmethods

Analytical methods

Asymptotic methods

M.H. BUSCHMANN, M. GAD-EL-HAK: Turbulent Boundary Layers: Reality andMyth

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%Speaker: BUSCHMANN, M.H. 34 BAIL 2006

Three-dimensional Temporal Instability of Unsteady Pipe Flow By

Avinash Nayak and Debopam Das Department of Aerospace Engineering

Indian Institute of Technology Kanpur, India Introduction:

In this paper we present temporal three-dimensional linear stability analysis of unsteady bi-directional laminar flow in a duct. For a known but arbitrary volume flow rate, analytical solution of one-dimensional unsteady flow through a pipe has been obtained by Das & Arakeri (1998, 2000). Generalized form of this solution and the solution for unsteady flow through annular space between two concentric pipes has been obtained by Nayak (2005). Prior to this study, solutions that are available in the literature were for known pressure gradient (Womersley 1955) and Uchida 1956). Das (1998) and Das & Arakeri (1998) have observed the transitional nature of this flow in a pipe and channel where the velocity profiles are inflectional and have reverse flow near wall. In their experiments, for flows with average volume flow rate varying like a trapezoidal function with time, shows that both axisymmetric and non-axisymmetric mode of disturbance grows. In this paper through linear stability analysis of three dimensional disturbances some of the observed experimental facts are explained such as growth of helical modes for certain cases. Linear stability analysis: The disturbance stream-function is assumed as ( ) ( ) ( ) ( ) [ ]θα inctzirprwrvrupwvu +−= )(exp,,,,,, . (1) Here, u, v and w are radial, azimuthal and axial perturbation velocity components respectively and p is the fluctuating pressure. The disturbance quantities are substituted in perturbed Navier-Stokes equations in cylindrical coordinates and the governing equations after simplifications are,

)1(0ReRe111Re2

11111)Re(3321

333212)Re(11

223

2

22222

2

2

4322

432222

2

−−−−−−=′′−′′−⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡+⎥⎦

⎤⎢⎣⎡ +′⎥

⎦

⎤⎢⎣

⎡′+−+

⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡ −′+⎥⎦

⎤⎢⎣⎡ −′+′′⎥

⎦

⎤⎢⎣

⎡−+++⎥⎦

⎤⎢⎣⎡ −′+′′−′′′−

⎥⎦⎤

⎢⎣⎡ −′+′′−′′′+−−⎥

⎦

⎤⎢⎣

⎡−++

+−′+′′

WuiWuivr

inur

uWirn

vr

vr

inur

ur

ucWirnv

rv

rv

rv

rin

ur

ur

ur

ur

uvrinucWi

rnu

ru iv

ααααα

αααα

α

ααα

( )

( ) )2(01Re111Re

11111212Re11

222

2

2

4322432222

2

−−−−−−−=′⎟⎠⎞

⎜⎝⎛+

⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡+⎥⎦

⎤⎢⎣⎡ +′⎥

⎦

⎤⎢⎣

⎡−+++

⎥⎦⎤

⎢⎣⎡ +′−′′+⎥⎦

⎤⎢⎣⎡ +′−′′+′′′−+⎥

⎦

⎤⎢⎣

⎡−++

+−′+′′

Wur

nvr

inur

ucWirn

rin

vr

vr

vr

nur

ur

ur

ur

inurinvcWi

rnv

rv

ααααα

αααα

22

These equations are solved using the finite difference technique and it is written in the form of . The complex eigenvalues , are obtained for a particular value of [ ] [ ]( ) 0=

⎭⎬⎫

⎩⎨⎧

−vu

BcA c α and

Re, using MATLAB. In these two sets of equations u has the highest 4th derivative while has the highest 3

vrd derivative. Thus 7 boundary conditions are required to solve these two coupled

equations. Following are the boundary conditions given for different values of . nn

R=0 (center-line) 0=u 0=v 0=′u

0 R=1 (at wall) 0=u 0=v 0=′u

1 R=0 (center-line) 0=+ ivu 0=′u 0=′v

1

0lim −

→∝ n

rru

A. NAYAK, D. DAS: Three-dimesnional Temporal Instability of Unsteady Pipe Flow

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%Speaker: DAS, D. 35 BAIL 2006

R=1 (at wall) 0=u 0=v 0=′u

R=0 (center-line) 0=u 0=v 0=′+′ viu 023 =′′+′′ viu

2 R=1 (at wall) 0=u 0=v 0=′u

R=0 (center-line) 0=u 0=v 0=′u 0=′v

>2 R=1 (at wall) 0=u 0=v 0=′u

Table. 1 Boundary conditions for perturbation at wall and the centerline

For n=0, the two equations decouple to give a fourth order equation in and a second order equation in v . For n=1, the seventh condition is when (Batchelor & Gill 1962).

u1−n∝ ru 0lim →r

4.4 Results and Discussions

We have considered a case (case2 of Das & Arakeri 1998) in which a perceptible wave was observed through flow visualization at time t= . During this period the piston has stopped after a

trapezoidal motion and the profile considered for analysis is at time

pt

( 22 41 tttt p −+= ) where, is

the time when piston has stopped. The neutral stability curves are shown in figure 1. It is observed that critical Reynolds number for n=1 mode is minimum (=398). Figure 2 shows the corresponding flow visualization picture of Das & Arakeri (1998). The position of vortex wave at 90

2t

0 phases between top and bottom portion of pipe is visible in figure 2. Hence the helical mode with n=1 is most unstable in this case. As the neutral curve for n=0 mode is close to n=1 mode for some cases n=0 mode might be most unstable mode. Symmetric modes has been observed as most unstable in other cases experimentally (Das & Arakeri 1998 case 3 ). The stability analysis will be shown in the final paper for these cases, but the neutral curves (figure1) itself indicate the possibility of growth of such axisymmetric modes. The calculated wave length is 22.3≈

δλ

matches with the wavelength observed in experiment, the value of which is approximately 3.0.

Reynolds number (Re)

alph

a ( α

)

Neutral Stability Curve for n 0, 1 & 2

200 300 400 500 600 700 800 900 1000 1100 12001

2

3

4

5

6

7

8n=0n=1n=2

Fig.1 Neutral Stability Curve for different Fig. 2 Flow viz. picture of Das & Arakeri (case2) nREFERENCES

1. Batchelor, G. K. & Gill, A. E. 1962. J. Fluid Mech. 14, 529. 2. Das, D. 1998 Ph.D Thesis, Department of Mechanical Engg, Indian Institute of Science, Bangalore. 3. Das, D. & Arakeri, J. H. 1998. J. Fluid Mech. 374, 251-283. 4. Das, D. & Arakeri, J. H. 2000 Unsteady Jl. Applied Mech. 67, 274-281. 5. Nayak, A. 2005 MTech report Dept of Aerospace Engg, Indian Institute of Technology, Kanpur. 6. Uchida, S. 1956 Z. Angrew. Math. Phys. 7, 403-422. 7. Weinbaum, S. & Parker, K. 1975. J. Fluid Mech. 69, 729-752.

A. NAYAK, D. DAS: Three-dimesnional Temporal Instability of Unsteady Pipe Flow

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%Speaker: DAS, D. 36 BAIL 2006

The structure of the critical layer of a swirling annularflow in transition

J. Hussong, N. Bleier and V. Vasanta RamRuhr-Universitat Bochum, D-44780 Bochum, Germany

Subject of our paper falls under the heading of transition of the swirling flowin an annulus. Two transition mechanisms with some fundamental differences,which we call the Taylor and the Tollmien-Schlichting mechanisms, are generallyin competition with each other in this flow. The salient difference referred tois the presence of a critical layer in the Tollmien-Schlichting mechanism. Incontrast, such a critical layer is absent in the Taylor mechanism. The focus ofattention in our work is the modification swirl causes to the structure of thiscritical layer in annular flow.

The characteristic feature of the critical layer is that the propagation veloc-ity of the disturbance wave is the same as the local velocity in the basic flow.Mathematically, this results in the equations for the propagation of small am-plitude disturbances exhibiting a turning point, i.e. the coefficient of a crucialterm in the governing differential equations crossing a zero value (see eg. [5],[6], [2], [3], [7]). For this reason viscous effects gain importance in the criticallayer which may be regarded as an internal layer in the flow. The viscous effectshave to be taken properly into account in the critical layer in order to arrive atthe stability characteristics of the flow of interest undergoing transition.

Our basic flow is the fully developed flow with swirl in the uniform annulargap between concentric circular cylinders. In the flow in this geometrical con-figuration, swirl comes into existence when the axial pressure gradient drivingthe flow acts in conjunction with a rotation of a cylinder about its own axis.We restrict our attention for the present to the case when the outer cylinder isset in rotation at an angular velocity Ωa and the inner cylinder is pulled axiallyat a translational velocity Vwi. Both of these flow boundary conditions tend toraise the critical Reynolds number of the flow. The geometrical and flow pa-rameters in our problem are, in a self-explanatory notation: the transverse cur-vature parameter εR = Ra−Ri

Ra+Ri, the Reynolds number Re = Uref x (Ra−Ri)

2 ν with

Uref x = (Ra−Ri)2

8µ

∣∣dPG

dx

∣∣, the swirl parameter Sa = Uref ϕ

Uref xwith Urefϕ = ΩaRa and

the translational wall velocity parameter Twi = Vwi

Uref x.

We have approached the transition problem in this flow configuration byconducting a modal analysis of the dynamics of the propagation of disturbancesin the basic flow in question, starting from small amplitude disturbances forwhich the governing equations may be linearised. The dispersion relationshipfor this linearized problem may formally be written as

F(εR, Re, Sa, Twi, λx, nϕ, ω) = 0,

J. HUSSONG, N. BLEIER, V.V. RAM: The structure of the critical layer of a swirlingannular flow in transition

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%Speaker: RAM, V.V. 37 BAIL 2006

where ω is the frequency, λx the disturbance wave number in the axial directionand nϕ the corresponding quantity in azimuthal direction which has to be aninteger. nϕ = 0 represents an axisymmetric disturbance whereas nϕ 6= 0 cor-responds to helical disturbances. We have solved the linearised equations forthe disturbance numerically by the spectral collocation method using Matlabover a range of parameters εR, Sa and Twi to determine the dependence of thecritical Reynolds number on the parameters Sa and Twi from which the wavevelocity and the location of the critical layer at this Reynolds number have beenobtained. A sample of the computational results is presented in Fig. 1.

0 0.1 0.2 0.3 0.40.26

0.27

0.28

0.29

0.3

c =

om

ega

/ lam

bda x

Sa

0 0.1 0.2 0.3 0.40.22

0.24

0.26

0.28

0.3c

= o

meg

a / l

ambd

a x

Twi

Figure 1: wave velocity for Twi = 0.1 (left) and Sa = 0.1 (right)

An asymptotic analysis of the critical layer, conducted along the same linesas for the corresponding case in planar flow (see eg. [1], [4]), brings out the fol-lowing points as the outstanding features of the critical layer under the influenceof swirl:

• Swirl exerts no influence on the scale of thickness of the critical layerthrough its axisymmetric disturbance mode, nϕ = 0

• For the nonaxisymmetric modes, nϕ 6= 0, the thickness of the criticallayer retains the same scale as in the case of classical planar flow , viz.O(λxRe)−

13 , as long as the product of the transverse curvature and swirl

parameters, εRSa, remains small to within O(Re−13 )

Results obtained from this asymptotic analysis of the critical layer are setagainst the computational results outlined above.

References[1] Drazin, Reid, Hydrodynamic stability, Cambridge Univ. Pr., 1982.

[2] Hinch, E. J., Perturbation Methods, Cambridge Univ. Pr., 1992.

[3] Holmes, M. H., Introduction to Perturbation Methods, 20 in the Series Texts in Appliedmathematics, Springer, 1995.

[4] Maslowe, S. A., Critical layers in shear flows. In Annual Review of Fluid Mechanics, 18,1986, 405-432.

[5] Nayfeh, A. H., Perturbation Methods, John Wiley, 1973.

[6] O Malley Jr., R. E., Introduction to Singular Perturbations, 14 in the Series Applied Math-ematics and Mechanics, Academic Press, 1974.

[7] Verhulst, F., Methods and Applications of Singular Perturbations, 50 in the Series Textsin Applied mathematics, Springer, 2000.

2

J. HUSSONG, N. BLEIER, V.V. RAM: The structure of the critical layer of a swirlingannular flow in transition

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%Speaker: RAM, V.V. 38 BAIL 2006

Spatio-temporal growing waves in boundary-layers by Bromwi h ontour integral methodT. K. Sengupta and A. Kameswara RaoDepartment of Aerospa e Engineering, I.I.T. Kanpur, U.P. 208016,IndiaIt is well-known that unstable, zero pressure gradient, boundary-layer dis-play spatial growth, even when this is posed as a spa e-time dependent problem[1. For some internal ows, it has often been onje tured that while the owdisplays linear stability by normal mode analysis, transition o urs due to su-perposition of non-normal de aying modes that an display very high transientenergy growth [2. In the present work, we show by Bromwi h ontour integralmethod of omplete spatio-temporal approa h [1,3 that su h transient growthof energy also takes pla e for at plate boundary-layer, when all the normalmodes display spatial stability.We investigate the response of a zero pressure gradient boundary layer ex- ited by a harmoni sour e at the wall- at a ir ular frequen y 0. If we representthe disturban e stream fun tion by, (x; y; t) = Z ZBr (; ; y)ei(xt)dd (1)where Br indi ate the Bromwi h ontours followed in evaluating the above in-tegral in the omplex and plane - 0 appearing via the wall boundary ondition [1. The mathemati al basis of Bromwi h ontour integral is given in[1,3, and is one of the elegant ways of treating full spatio-temporal problems.The posed problem is solved by linearizing the Navier-Stokes equation in thespe tral planes and expressing it as the Orr-Sommerfeld equation given by,(U =)(00 2) U 00 = 1iRe(iv 2200 + 4) (2)where U(y) is the mean ow and the Reynolds number is based on displa e-ment thi kness. Here, a Blasius boundary layer is onsidered whose existingspatial normal modes have been evaluated by grid-sear h te hnique employing ompound matrix method [1 for Re = 1000 and 0 = 0:05 and 0:15 and givenin Table-1. For these parameter ombinations, existing spatial modes are alldamped- one set orresponding to 0 = 0:05 (identied as 1 and 2) are belowthe neutral urve and the other set orresponding to 0 = 0:15 (identied as 3-5in Table 1).Eq. (2) is solved along the Bromwi h ontours: (A) 20 r 20;i =0:001 and (B) 1 r 1;i = 0:02- su h that Causality prin iple is notviolated and waves travel in the orre t dire tion. Obtained 's are used tore onstru t the disturban e eld in Eq. (1) and the velo ity eld are as plottedin gure 1 for the ase of 0 = 0:15. While the table shows all three spatialmodes as stable, the ontour integral results in gure 1 show (i) a lo al solution;(ii) a de aying wave and (iii) a temporally growing wave-pa ket. After suÆ ienttime, the de aying wave orresponds to mode-5 and the pa ket orresponds to

T.K. SENGUPTA, A. KAMESWARA RAO: Spatio-temporal growing waves inboundary-layers by Bromwich contour integral method

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%Speaker: SENGUPTA, T.K. 39 BAIL 2006

mode-4 of table 1. Mode- 3 is never visible due to its extremely large de ayrate. In gure 2, inverse Lapla e transform of the data is shown for the signalat t = 801:1, that learly idenitifes the modes- 4 and 5 and others that arisedue to intera tions of modes that show up as the spatio-temporal propagationof the signal.In the nal paper, we will present analysis for the lo al solution and detailed omparison for the asymptoti part of the solution omprising of dierent modesand their intera tions emphasizing the need to perform spatio-temporal analysisinstead of temporal or spatial theories.Referen es:[1 T.K. Sengupta et al. Physi s Fluids, 6(3), 1213 (1994)[2 L.N. Trefethen et al., S ien e 261, 578 (1993)[3 B. Van der Pol, H. Bremmer, Operational Cal ulus Based on Two-sidedLapla e Integral, Cambridge Univ. Press (1959)Table 1:

β0 αr αi

1) 0.0621 413 0.0696 5940.05

2) 0.1607 670 0.0015 2063) 0.1894 256 0.3226 357

0.15 4) 0.2728 701 0.1675 5855) 0.3940 036 0.0104 936

x

u′

0 100 200 300 400

-0.002

0

0.002t = 110.0

x

u′

0 100 200 300 400

-0.002

0

0.002t = 411.6

5 4

x

u′

0 100 200 300 400

-0.002

0

0.002t = 637.7

5 4

x

u′

0 100 200 300 400

-0.002

0

0.002t = 801.1

5 4

Figure 1: Streamwise disturbance velocity plotted as a function of x at dif-ferent t for β0 = 0.15, Re = 1000 at y = 0.278δ∗

α

φ′

0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6 t = 801.13 4 5

Figure 2: FFT of streamwise disturbance velocity plotted as a function of α

T.K. SENGUPTA, A. KAMESWARA RAO: Spatio-temporal growing waves inboundary-layers by Bromwich contour integral method

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Asymptotic Analysis of the mixed convection flow past a horizontal plate

near the trailing edge

Lj. Savic and H. Steinruck

Institute of Fluid Mechanics and Heat TransferVienna University of Technology

Resselgasse 3, 1040 Vienna, Austriaherbert.steinrueck@tuwien.ac.at

1. Introduction

The flow near the trailing edge of a horizontal heated plate which is aligned under a small angleof attack φ to the oncoming parallel flow with velocity U∞ in the limit of large Reynolds Reand large Grashof Gr = gβ∆TL3/ν2 number will be investigated (see figure 1). As usual βand ν denote the isobaric expansion coefficient and the kinematic viscosity, respectively. Thedifference between the plate temperature and the temperature of the oncoming fluid is ∆T andL is the length of the plate. A measure for the influence of the buoyancy onto the boundarylayer flow along a horizontal plate is the buoyancy parameter K = GrRe−5/2 as defined in [4]).

PSfrag replacements

g

x

y

−1 0

φ

U∞ = 1, θ∞ = 0

θp = 1

K

Re−1/2

Figure 1: Mixed convection flow past a horizontal plate.

The starting point of the analysis are the Navier Stokes equations for an incompressible fluidusing Bousinesq’s approximation to take buoyancy forces into account and the energy equation.Additionally to the above mentioned dimensionless parameters the Prandtl number Pr, whichis assumed to be of order one, and the angle of attack φ enter the problem.

To analyze the flow behavior near the trailing edge in the frame work of interacting boundarylayers ([1, 2] it turns out that the buoyancy parameter K and the angle of attack φ have to be ofthe order Re−1/4. Thus we define the reduced buoyancy parameter κ = K Re1/4 and the reducedinclination parameter λ = φK

√Re. We note that the magnitude of φ is not only dictated by

the trailing edge analysis, but it is also a consequence of the analysis of the far field (see [5]). Aswe will see the inclination parameter λ will play no role in the trailing edge analysis. Only forpositive values of λ an outer potential flow field exists [5]. We remark that in case of symmetricflow conditions (upper side of the plate heated, lower side cooled) the interaction mechanismwould allow K to be larger, namely of order Re−1/8. However, the scope of the present paperwill be limited to the flow near the trailing edge.

2. Asymptotic structure of the solution

For the analysis of the flow field near the trailing edge the velocities, the pressure and thetemperature are decomposed into a symmetric and anti-symmetric part. To leading order forthe symmetric part the classical triple deck problem [1, 2] is obtained. For the anti-symmetricpart (difference of the flow quantities above and below the plate) a new interaction mechanismis obtained. The difference of the displacement thicknesses between the upper and lower side

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L. SAVIC, H. STEINRUCK: Asymptotic Analysis of the mixed convection flow past ahorizontal plate near the trailing edge

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%Speaker: STEINRUCK, H. 41 BAIL 2006

of the plate ∆A interacts not only via the potential flow in the upper deck but also via thehydrostatic pressure in the main deck with the pressure ∆p in the lower deck. The interactionlaw for the pressure difference between the upper and lower side can be written in the form

∆A′(x(3)) +√

3ash(x(3))(

x(3))1/3

=

−[

1

π

∫ 0

−∞

∆p(ξ) + 2as|ξ|1/3

x(3) − ξdξ − 1

π

∫

∞

−∞

As(ξ) − ash(ξ)|ξ|1/3

x(3) − ξdξ

]

Where As(x(3)) is the displacement thickness obtained for the classical trailing edge problem

[3, 1] and x(3) is the coordinate in the lower, main and upper deck parallel to the plate. Here as

is a constant and h(.) the heayside function. It turns out that the difference pressure ∆p has adiscontinuity at the trailing edge. Thus new sub-layers (in the main and lower deck) to resolvethis discontinuity are introduced.

In the following table we give an overview of the most important layers needed for theasymptotic analysis. We introduce the following notation according to the stretching factor asa power of the Reynolds number: x(α) = Reα/8x and y(β) = Reβ/8y.

α β flow region

0 0 potential flow region0 4 boundary layer and wake3 3 upper deck3 4 main deck3 5 lower deck4 4 main deck - trailing edge5 5 lower deck - trailing edge

Table 1: Scales of the different flow regions

The new results of the analysis will be the local behavior of the difference pressure at thetrailing edge. For both of the newly introduced sub-layers elliptic equations for the local pressurevariation can be derived and will be solved numerically. Thus on triple deck scales there will bea flow around the trailing edge. Thus the perturbation of the classical triple deck problem bysmall buoyancy effects behaves quite differently than by small angles of attack [3].

References

[1] K. Stewartson, On the flow near the trailing edge of flat plate, Mathematika, 16, 106-121(1969).

[2] A.F. Messiter, Boundary layer flow near the trailing edge of flat plate, SIAM J. Appl. Math.,18, 241-257, (1970).

[3] R. Chow and R. E. Melnik, Numerical Solutions of Triple-Deck Equations for Laminar

Trailing Edge Stall, Grumman Research Dept., Report RE-526J (1976).

[4] W. Schneider and M. G. Wasel, Breakdown of the boundary-layer approximation for mixedconvection flow above a horizontal plate, Int. J. Heat Transfer, 28, 2307-2313 (1985).

[5] Lj. Savic and H. Steinruck, Mixed convection flow past a horizontal plate, Theoretical and

Applied Mechanics, 32, 1-19 (2005).

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L. SAVIC, H. STEINRUCK: Asymptotic Analysis of the mixed convection flow past ahorizontal plate near the trailing edge

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%Speaker: STEINRUCK, H. 42 BAIL 2006

G.I. Shiskin, P. HemkerRobust Methods for Problems with Layer Phenomena and Additional Singularities

The minisymposium will be concerned with singularly perturbedmultiscale problems having additional singularities. A complicatedgeometry or unboundedness of the domain and/or the lack of sufficientsmoothness (or compatibility) of the problem data may result insingular solutions that have their own specific scales, besides bound-ary/interior layers. We intend to examine techniques for constructingnumerical methods that converge parameter-uniformly (in the maxi-mum norm).

The following research aspects will be also considered: (i) As a rule,such parameter-uniformly convergent numerical methods have too loworder of uniform convergence, which restricts their applicability inpractice. With this respect, methods how to increase the accuracy ofparameter-uniformly convergent numerical methods will be considered.(ii) When standard numerical methods, for example, domain decompo-sition methods are used to find solutions of parameter-uniformly conver-gent discrete approximations, the decomposition errors of the discretesolutions and the number of iterations required to solve the discreteproblem depend on the perturbation parameter and grow when it tendsto zero. We will consider decomposition methods preserving the prop-erty of parameter-uniform convergence. Domain decomposition andlocal defect correction techniques allows us to reduce the constructionof robust numerical methods for multiscale problems to locally robustmethods for monoscale problems on the specific subdomains. Other as-pects and applications will be also under consideration. Problems forpartial differential equations with different types of boundary and inte-rior layers will be considered. To construct special numerical methods,fitted meshes, which are a priori and a posteriori condensing in the layerregions, are used.

Speaker:

• Deirdre Branley: A Schwarz method for a convection-diffusion problem with acorner singularity

• Thorsten Linss: Layer-adapted meshes for time-dependent reaction-diffusion

• Grigory I. Shishkin: Grid Approximation of Parabolic Equations with Nonsmooth

Initial Condition in the Presence of Boundary Layers of Different Types

• Lidia P. Shishkina: A Difference Scheme of Improved Accuracy for a Quasilin-ear Singularly Perturbed Elliptic Convection-Diffusion Equation in the Case of theThird-Kind Boundary Condition

• Irina V. Tselishcheva: Domain Decomposition Method for a Semilinear SingularlyPerturbed Elliptic Convection-Diffusion Equation with Concentrated Sources

• S. Valarmathi: A parameter-uniform numerical method for a system of singularlyperturbed ordinary differential equations

A Schwarz method for a convection-diffusion problem with a cornersingularity.∗

Deirdre Branley1, Alan F. Hegarty1, Helen Purtill1 and Grigory I. Shishkin2.

1 Department of Mathematics and Statistics, University of Limerick, Plassey, Limerick, Ireland.deirdre.branley@ul.ie, alan.hegarty@ul.ie, helen.purtill@ul.ie,

2 Institute of Mathematics and Mechanics, Ural Branch of Russian Academy of Sciences16 S. Kovalevskaya Street, Ekaterinburg 620219, Russia

shishkin@imm.uran.ru

We are concerned with two dimensional steady state convection-diffusion problems withsingular outflow boundary conditions. It is well known that, where the boundary conditionsare sufficiently smooth and compatible, such problems can be solved with uniform accuracywith respect to the small parameter ε using a standard finite difference operator on specialpiece-wise uniform meshes [1, 2]. Where the outflow boundary data are only weakly regular andcompatible, parameter-uniform solutions may also be obtained by this method [2]. However,orders of convergence are relatively small and pointwise errors relatively large in this case.

Numerical methods for singularly perturbed problems comprising domain decomposition andSchwarz iterative technique have been examined by a number of authors, for example in [1], [3],[4] and [5]. In particular, MacMullen et al. [5] constructed a parameter-uniform Schwarz methodfor singularly perturbed linear convection-diffusion problems in two dimensions with sufficientlysmooth and compatible boundary data. We examine experimentally the performance of suchmethods extended to the class of singularly perturbed convection-diffusion problems with moregeneral boundary conditions described below.

We consider problems of the form

Lu ≡ ε∆uε + a(x, y).∇u = f

in a domain Ω, the unit square, with Dirichlet boundary conditions, where all components of aare strictly positive. Such problems exhibit regular layers along the outflow boundaries, as wellas a corner boundary layer at the outflow boundary corner. We deal with outflow boundaryconditions, where the first derivatives are not compatible at the outflow boundary corner. Weimplement domain decomposition methods to isolate the neighbourhood of the singularity, alongwith a Schwarz iterative technique, with the aim of developing a Schwarz method to produceparameter-uniformly accurate solutions on the whole domain in the oresence of such a singularity.

References

[1] J.J.H. Miller, E.O’Riordan, G.I. Shishkin, Fitted Numerical Methods for Singular Pertur-bation Problems, World Scientific, Singapore, 1996.

[2] P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E. O’Riordan, G.I. Shishkin, Robust Computa-tional Techniques for Boundary Layers, Chapman & Hall, CRC, Boca Raton, FL, 2000.

[3] J.J.H. Miller, E. O’Riordan, G.I. Shishkin, S. Wang, A parameter-uniform Schwarz methodfor a singularly-perturbed reaction-diffusion problem with an interior layer, Appl. Num.Math., 35 (2000), 323-337.

∗ This research was supported in part by the Irish Research Council for Science, Engineering and Technologyand by the Russian Foundation for Basic Research under grant No. 04-01-00578.

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D. BRANLEY, A.F. HEGARTY, H. PURTILL, G.I. SHISHKIN: A Schwarz methodfor a convection-diffusion problem with a corner singularity

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%Speaker: BRANLEY, D. 45 BAIL 2006

[4] H. MacMullen, J.J.H. Miller, E. O’Riordan, G.I. Shishkin, A second-order parameter uni-form overlapping Schwarz method for reaction-diffusion problems with boundary layers, J.Comput. & Appl. Math., 130 (2001), 231-244.

[5] H. MacMullen, E. O’Riordan, G.I. Shishkin, The convergence of classical Schwarz methodsapplied to convection-diffusion problems with regular boundary layers, Appl. Num. Math.,43 (2002), 297-313.

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D. BRANLEY, A.F. HEGARTY, H. PURTILL, G.I. SHISHKIN: A Schwarz methodfor a convection-diffusion problem with a corner singularity

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%Speaker: BRANLEY, D. 46 BAIL 2006

Layer-adapted meshes for time-dependent reaction-diffusion

Torsten Linß1, Niall Madden2

1 Institut fur Numerische Mathematik, Technische Universitat Dresdene-mail: torsten.linss@tu-dresden.de

2 Department of Mathematics, National University of Ireland Galwaye-mail: niall.madden@math.nuigalway.ie

ABSTRACT

We consider singularly perturbed reaction-diffusion problems of the type

ut + Lεu = f in (0, 1)× (0, T ],

where Lεv := −ε2vxx + rv, subject to boundary conditions

u(0, t) = γ0(t), u(1, t) = γ1(t) in (0, T ]

and initial condition

u(·, 0) = u0 in (0, 1)

where 0 < ε 1, r(x) > ρ2 > 0 for x ∈ [0, 1]. The nature of the differential equation changes whenε → 0 giving rise to boundary layers that require special attention in the design of numerical methods,in particular local refinement of the meshes used.

We study an inverse monotone difference scheme on arbitrary meshes. An maximum-norm error boundis derived that allows easy classification of various layer-adapted meshes proposed in the literature.

For example, this general result implies

‖u− U‖∞≤ C

τ + N−2 ln2 N for Shishkin meshes,

τ + N−2 for Bakhvalov meshes,

where τ is the maximal time-step size and N the number of mesh points used in the space discretization.

T. LINSS, N. MADDEN: Layer-adapted meshes for time-dependent reaction-diffusion

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%Speaker: LINSS, T. 47 BAIL 2006

Grid Approximation of Parabolic Equations with Nonsmooth Initial

Condition in the Presence of Boundary Layers of Different Types ∗

Grigory I. Shishkin

Institute of Mathematics and Mechanics,Ural Branch of Russian Academy of Sciences,

Yekaterinburg 620219, Russia

shishkin@imm.uran.ru

A Dirichlet problem for a singularly perturbed parabolic equation with the perturbationvector-parameter ε, ε = (ε1, ε2), is considered on a semiaxis. The highest derivative of the

equation and also the first derivative with respect to x contain respectively the parametersε1 and ε2, which take arbitrary values in the half-open interval (0, 1] and the segment [−1, 1].

Depending on the parameter ε2, the type of the equation may be reaction-diffusion or convection-diffusion. The first order derivative of the initial function has a discontinuity of the first kind atthe point x0.

For small values of the parameter ε1, a boundary layer appears in a neighbourhood of thelateral part of the domain boundary. Depending on the ratio between the parameters ε1 and

ε2, these layers may be regular, parabolic or hyperbolic (characteristic scales of these boundarylayers also depend on the ratio between ε1 and ε2).

In a neighbourhood of the set Sγ , that is, the characteristic of the reduced equation outgoing

from the point (x0, 0), the parabolic transient layer arises.

Using the method of piecewise uniform meshes condensing in a neighbourhood of the layer,

we construct a special difference scheme that converges ε-uniformly.

Numerical methods for problems with different types of boundary layers for elliptic convection-diffusion equations in the case of sufficiently smooth boundary data are studied, e.g., in [1].

References

[1] G.I. Shishkin, “Grid approximation of a singularly perturbed elliptic equations with con-vective terms in the presence of various boundary layers”, Comput. Maths. Math. Phys.,

45 (1), 104–119 (2005).

∗This research was supported in part by the Russian Foundation for Basic Research (grants No. 04-01-00578,04–01–89007–NWO a), by the Dutch Research Organisation NWO under grant No. 047.016.008 and by the BooleCentre for Research in Informatics, National University of Ireland, Cork.

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G.I. SHISHKIN: Grid Approximation of Parabolic Equations with Nonsmooth InitialCondition in the Presence of Boundary Layers of Different Types

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%Speaker: SHISHKIN G.I. 48 BAIL 2006

A Difference Scheme of Improved Accuracy for a Quasilinear

Singularly Perturbed Elliptic Convection-Diffusion Equation

in the Case of the Third Kind Boundary Condition ∗

Lidia P. Shishkina and Grigory I. Shishkin

Institute of Mathematics and Mechanics,

Ural Branch of Russian Academy of Sciences,Yekaterinburg 620219, Russia

Lida@convex.ru and shishkin@imm.uran.ru

A boundary value problem for a quasilinear singularly perturbed elliptic convection-diffusion

equation on a strip is considered. The third kind boundary condition admitting both Dirichletand Neumann conditions is given on the domain boundary. For small values of the perturbation

parameter ε, a boundary layer arises in a neighbourhood of the outflow part of the boundary. Forsuch a problem, the base (nonlinear) difference scheme constructed by classical approximations

of the problem on piecewise uniform meshes condensing in the layer converges ε-uniformly withan order of accuracy not higher than 1.

Our aim is for this boundary value problem to construct grid approximations that converge

ε-uniformly with an order of convergence close to two.

Using the Richardson technique, we construct a (nonlinear) scheme that converges ε-uniformly

at the rate O(

N−2

1ln2

N1+N−2

2

)

, where N1+1 is the number of nodes in the mesh with respect

to x1 and N2 +1 is the number of mesh points on a unit interval along the x2-axis. Based on the

nonlinear Richardson scheme, a linearized iterative scheme is constructed where the nonlinearterm is computed using the unknown function taken at the previous iteration. The solution ofthis iterative scheme converges to the solution of the nonlinear Richardson scheme at the rate of

a geometry progression ε-uniformly with respect to the number of iterations. Thus, the numberof iterations required for solving the problem (as well as the accuracy of the resulting solution)

is independent of the parameter ε.

The use of lower and upper solutions of the linearized iterative Richardson scheme as astopping criterion allows us during the computational process to define a current iteration under

which the same ε-uniform accuracy of the solution is achieved as for the nonlinear Richardsonscheme. To construct the improved scheme, the technique developed in [1]–[4] for a Dirichlet

problem is applied.

References

[1] G.I. Shishkin, “The method of increasing the accuracy of solutions of difference schemes

for parabolic equations with a small parameter at the highest derivative”, USSR Comput.

Maths. Math. Phys., 24 (6), 150–157 (1984).

[2] G.I. Shishkin, ”Finite-difference approximations of singularly perturbed elliptic equations”,

Comp. Math. Math. Phys., 38 (12), 1909–1921 (1998).

[3] G.I. Shishkin and L.P. Shishkina, “A high-order Richardson method for a quasilinear singu-

larly perturbed elliptic reaction-diffusion equation”, Differential Equations, 41 (7), 1030–1039 (2005).

∗This research was supported in part by the Russian Foundation for Basic Research (grants No. 04-01-00578,04–01–89007–NWO a) and by the Dutch Research Organisation NWO under grant No. 047.016.008.

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L.P. SHISHKINA, G.I. SHISHKIN: A Difference Scheme of Improved Accuracy for aQuasilinear Singularly Perturbed Elliptic Convection-Diffusion Equation in the Case ofthe Third-Kind Boundary Condition

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[4] L. Shishkina and G. Shishkin, “The discrete Richardson method for semilinear parabolic sin-gularly perturbed convection-diffusion equations”, in: Proceedings of the 10th International

Conference “Mathematical Modelling and Analysis” 2005 and 2nd International Conference“Computational Methods in Applied Mathematics”, R. Ciegis ed., Vilnius, “Technika”,2005, pp. 259–264.

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L.P. SHISHKINA, G.I. SHISHKIN: A Difference Scheme of Improved Accuracy for aQuasilinear Singularly Perturbed Elliptic Convection-Diffusion Equation in the Case ofthe Third-Kind Boundary Condition

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%Speaker: SHISHKINA, L.P. 50 BAIL 2006

Domain Decomposition Method for a Semilinear Singularly Perturbed

Elliptic Convection-Diffusion Equation with Concentrated Sources ∗

Irina V. Tselishcheva and Grigory I. Shishkin

Institute of Mathematics and Mechanics, Ural Branch of Russian Academy of Sciences16 S. Kovalevskaya Street, Ekaterinburg 620219, Russia

tsi@imm.uran.ru, shishkin@imm.uran.ru

For singularly perturbed boundary value problems in a composed domain (in particular,with concentrated sources) whose solution has several singularities such as boundary and inte-

rior layers, it is of keen interest to construct a parameter-uniform numerical method based ona domain decomposition technique so that each subdomain in the decomposition contains no

more than a single singularity. Because of the thin layers, standard numerical methods appliedto problems of this type yield unsatisfactorily large errors for small values of the singular per-

turbation parameter ε. The fact that the partial differential equation is nonlinear complicatesthe solution process.

We develop monotone linearized schemes based on an overlapping Schwarz method for asemilinear singularly perturbed elliptic convection-diffusion equation on a compound strip in thepresence of concentrated sources acting inside the domain. We first study a special (base) scheme

comprising a standard finite difference operator on a piecewise-uniform fitted mesh and anoverlapping domain decomposition scheme constructed on the basis of the former that converge ε-

uniformly at the rates O(

N−1

1lnN1 + N−1

2

)

and O(

N−1

1ln N1 + N−1

2+ qt

)

, respectively. HereN1 +1 and N2 +1 are the number of mesh points in x1 and the minimal number of mesh points

in x2 on a unit interval, q < 1 is the common ratio of a geometric progression, independent of ε,t is the iteration count. For these nonlinear schemes we construct monotone linearized schemes

of the same ε-uniform accuracy, in which the nonlinear term is computed from the unknownfunction taken at the previous iteration.

The linearized schemes are monotone, which admits to construct their upper and lowersolutions. We apply the technique of upper and lower solutions to find a posteriori the (optimal)number of iterations T in the linearized scheme for which the accuracy of its solution is the same

(up to a constant factor) as that for the base scheme, where T = O (ln (min [N1, N2])) (see also[1] for a reaction-diffusion problem). Thus, the number of required iterations is independent of

ε. With respect to total computational costs, the iterative method is close to a solution methodfor linear problems, since the number of iterations is only weakly depending on the number of

mesh points used. The linearized iterative schemes inherit the ε-uniform rate of convergence ofthe nonlinear schemes.

The decomposition schemes can be computed sequentially and in parallel (so that the suprob-lems on the overlapping subdomains are solved independently of each other).

Note that schemes of the overlapping domain decomposition method were considered earlierby the authors in [2] for linear problems and in [3, 4] for nonlinear problems.

References

[1] G.I. Shishkin and L.P. Shishkina, “A high-order Richardson method for a quasilinear singu-

larly perturbed elliptic reaction-diffusion equation”, Differential Equations, 41 (7), 1030–1039 (2005).

∗ This research was supported in part by the Russian Foundation for Basic Research (grants No. 04-01-00578,04–01–89007–NWO a) and by the Dutch Research Organisation NWO grant No. 047.016.008.

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I.V. TSELISHCHEVA, G.I. SHISHKIN: Domain Decomposition Method for a Semi-linear Singularly Perturbed Elliptic Convection-Diffusion Equation with ConcentratedSources

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%Speaker: TSELISHCHEVA, I.V. 51 BAIL 2006

[2] G.I. Shishkin and I.V. Tselishcheva, “Parallel methods for solving singularly perturbedboundary value problems for elliptic equations”, (in Russian) Mat. Model., 8 (3), 111–127

(1996).

[3] I.V. Tselishcheva and G.I. Shishkin, “Monotone domain decomposition schemes for a sin-

gularly perturbed semilinear elliptic reaction-diffusion equation with Robin boundary con-ditions”, in: Proceedings of the 10th International Conference “Mathematical Modelling

and Analysis” 2005 and 2nd International Conference “Computational Methods in AppliedMathematics”, R. Ciegis ed., Vilnius, “Technika”, 2005, pp. 251–258.

[4] I.V. Tselishcheva and G.I. Shishkin, “Monotone domain decomposition schemes for aquasilinear singularly perturbed elliptic convection-diffusion equation with a concentrated

source”, in: Grid Methods for Boundary Value Problems and Applications, Proceedings ofthe VI-th All-Russian Workshop, Kazan, Kazan State University, 2005, pp. 251-255 (in

Russian).

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I.V. TSELISHCHEVA, G.I. SHISHKIN: Domain Decomposition Method for a Semi-linear Singularly Perturbed Elliptic Convection-Diffusion Equation with ConcentratedSources

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%Speaker: TSELISHCHEVA, I.V. 52 BAIL 2006

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S. HEMAVATHI, S. VALARMATHI: A parameter-uniform numerical method for asystem of singularly perturbed ordinary differential equations

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%Speaker: VALARMATHI, S. 53 BAIL 2006

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S. HEMAVATHI, S. VALARMATHI: A parameter-uniform numerical method for asystem of singularly perturbed ordinary differential equations

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%Speaker: VALARMATHI, S. 54 BAIL 2006

J. Maubach, I, TselishchevaRobust Numerical Methods for Problems with Layer Phenomena and Applications

Numerical modeling of many processes and physical phenomena leadsto boundary value problems for PDEs having non-smooth solutions withsingularities of thin layer type. Among them are convection-dominatedconvection-diffusion problems, Navier-Stokes equations and boundary-layer equations at high Reynolds number, the drift-diffusion equationsof semiconductor device simulation, flow problems with lift, drag, tran-sition and interface phenomena, phenomena in plasma fluid dynamics,mathematical models for the spreading of pollutants, combustion, shockhydrodynamics or transport in porous media and other related prob-lems. The solutions of these problems contain thin boundary and inte-rior layers, shocks, discontinuities, shear layers, or current sheets, etc.The singular behaviour of the solution in such local structures gener-ally gives rise to difficulties in the numerical solution of the problem inquestion by traditional methods on uniform meshes and requires the useof highly accurate discretization methods and adaptive grid refinementtechniques. The problem of resolving layers, which is of great practicalimportance, is still not solved satisfactorily for a wide class of problemswith layer phenomena and applications, which the minisymposium isconcerned with.

Speaker:

• Alan Hegarty: An adaptive method for the numerical solution of an elliptic convec-tion diffusion problem

• Joseph Maubach: A Convergence Proof of Local Defect Correction for Convection-Diffusion Problems

• Joseph Maubach: On the Difference between Left and Right Preconditioning forConvection Dominated Convection-Diffusion Problems

• Lidia P. Shishkina: Parameter-Uniform Method for a Singularly Perturbed ParabolicEquation Modelling the Black-Scholes equation in the Presence of Interior andBoundary Layers

• Alexander I. Zadorin: Numerical Method for the Blasius Equation on an InfiniteInterval

• Paul Zegeling: An Adaptive Grid Method for the Solar Coronal Loop Model

An adaptive method for the numerical solution of an elliptic convectiondiffusion problem∗

Alan F. Hegarty1, Stephen Sikwila1 and Grigory I. Shishkin2.

1 Department of Mathematics and Statistics, University of Limerick, Plassey, Limerick, Ireland.alan.hegarty@ul.ie, stephen.sikwila@ul.ie,

2 Institute of Mathematics and Mechanics, Ural Branch of Russian Academy of Sciences16 S. Kovalevskaya Street, Ekaterinburg 620219, Russia

shishkin@imm.uran.ru

A two dimensional elliptic linear convection diffusion problem:

Pε :

ε∆uε + a.∇uε + d(x, y)uε = f(x, y), (x, y) ∈ Ω;uε = g, (x, y) ∈ Γ,

(1)

is considered, where the diffusion parameter 0 < ε ≤ 1 is typically small. The use of spe-cial piecewise uniform meshes appropriately condensed in the boundary layer regions togetherwith montone finite difference operators is well known to result in numerical methods whichare uniformly convergent with respect to ε (see e.g., [1]). Such methods depend on a prioriknowledge of the location of any boundary layers. Since such information may not be availablefor more complicated problems, it is of interest to examine whether robust numerical results canbe obtained for the model problem (1), without using the a priori information.

An adaptive technique is presented, based on piecewise uniform meshes, where the locationof the transition point between the coarse and fine meshes is computed iteratively. This is anextension of the one dimensional method presented in [2] and [3].

Numerical experiments are presented which indicate that the computational solutions ob-tained are uniformly in ε convergent and also that the number of iterations required does notincrease significantly for small ε.

References

[1] P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E. O’Riordan, G.I. Shishkin, Robust Computa-tional Techniques for Boundary Layers, Chapman & Hall, CRC, Boca Raton, FL, 2000.

[2] S. Sikwila, A.F. Hegarty and G.I. Shishkin, Novel robust adaptive techniques for the nu-merical solution of convection diffusion problem, Proceedings of BAIL 2004 Conference,Toulouse, July 2004, ONERA, 2004.

[3] S. Sikwila, Novel robust layer resolving adaptive mesh methods for convection diffusionproblems in one and two dimesnions, Ph.D. thesis, University of Limerick, 2005.

∗This work was supported by the Russian Foundation for Basic Research under grant No. 04-01-00578.

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A.F. HEGARTY, S. SIKWILA, G.I. SHISKIN: An adaptive method for the numericalsolution of an elliptic convection diffusion problem

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%Speaker: HEGARTY, A.F. 56 BAIL 2006

A Convergence Proof of Local Defect Correction

for Convection–Diffusion Problems

M. Anthonissen1, I. Sedykh2 and J. Maubach1

1 Department of Mathematics and Computer Science, Eindhoven University of Technology,Eindhoven, The Netherlands

m.j.h.anthonissen@tue.nl, j.m.l.maubach@tue.nl

2 Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia

Examples of partial differential equations with solutions that are rapidly varying functionsof the spatial or temporal coordinates appear e.g. in combustion, shock hydrodynamics ortransport in porous media. For boundary value problems with solutions that have one or a fewsmall regions with high activity, a fine grid is needed in regions with high activity, whereas acoarser grid would suffice in the rest of the domain. Rather than using a truly nonuniform grid,we study a method called Local Defect Correction (LDC) that is based on local uniform gridrefinement.

Several properties hold for the LDC fixed point iteration for convection-diffusion equations intwo dimensions. We study the convergence behavior of the LDC method as an iterative processand derive an upper bound for the norm of the iteration matrix for the linear two-dimensionalconvection-diffusion equation with constant coefficients on the unit square. The research resultscan be extended to the case of non-constant coefficients.

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M. ANTHONISSEN, I. SEDYKH, J. MAUBACH: A Convergence Proof of LocalDefect Correction for Convection-Diffusion Problems

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%Speaker: MAUBACH, J. 57 BAIL 2006

On the Difference between Left and Right Preconditioning for Convection

Dominated Convection–Diffusion Problems

Joseph Maubach

Department of Mathematics and Computer Science, Eindhoven University of Technology,Eindhoven, The Netherlandsj.m.l.maubach@tue.nl

For convection dominated convection-diffusion problems in several space dimensions dis-cretized with finite differences on a locally fine tensor–grid, an approximate inverse opera-tor Gn is calculated from the linear operator An. The spectral condition numbers κ2(GnAn)and κ2(AnGn) estimated numerically are different: κ2(GnAn)

.= O(log h−1) and κ2(AnGn)

.=

O(h−1). This is corroborated by a proof for the one-dimensional case for a modified approximateinverse operator Gn.

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J. MAUBACH: On the Difference between Left and Right Preconditioning for Convec-tion Dominated Convection-Diffusion Problems

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%Speaker: MAUBACH, J. 58 BAIL 2006

Parameter–Uniform Method for a Singularly Perturbed Parabolic Equation

Modelling the Black–Scholes equation in the Presence of Interior and

Boundary Layers ∗

Shuiying Li1, Lidia P. Shishkina2 and Grigory I. Shishkin2

1 Department of Computational Science,National University of Singapore,

Singapore 117543scip1075@nus.edu.sg

2 Institute of Mathematics and Mechanics,Ural Branch of Russian Academy of Sciences,

Yekaterinburg 620219, RussiaLida@convex.ru and shishkin@imm.uran.ru

Solutions of regular parabolic equations with nonsmooth initial data are of limited smooth-

ness themselves (in a neighbourhood of the nonsmoothness of the data). When the equationis singularly perturbed, its solution has singularities such as initial, boundary and/or interior

layers with own scales. These problems belong to the class of singularly perturbed multiscaleproblems. The multiscale behaviour of solutions complicates the construction and study ofefficient patameter-uniform numerical methods.

A singularly perturbed parabolic convection-diffusion equation with the second derivativemultiplied by a singular perturbation parameter ε, which ranges in (0, 1], arises when we model

the Black-Scholes equation upon a European call option (see, e.g., [1]). There are a few singu-larities in the problem, i.e., a single discontinuity of the first derivative of the initial condition

at a point x0, the unbounded growth of the initial condition at infinity and the unboundeddomain in space. The solution of this initial value problem grows exponentially without bound

as x → ∞. Thus, we deal with a singularly perturbed multiscale problem that has differenttypes of singularities.

In this presentation we consider a boundary value problem (in a bounded domain) for thissingularly perturbed parabolic convection-diffusion equation with an additional singularity gen-erated by the discontinuity of the first derivative of the initial function. For such a problem,

singularities such as a boundary and an interior layer with own specific scales arise. For smallvalues of the parameter ε, the interior layer due to nonsmooth initial data appears in a neigh-

bourhood of the characteristic of the reduced equation passing through the point (x0, 0), andthe regular boundary layer appears in a neighbourhood of the outflow boundary through which

the convective flux leaves the domain.By using the singularity splitting method and piecewise uniform meshes condensing in the

boundary layer, we construct a special difference scheme that allows us to approximate ε-uniformly the solution of the problem under consideration, as well as the first derivative of

the solution. The corresponding theoretical and numerical results are provided in the paper.The description of the technique applied in this study can be found in [2]. Preliminary resultsfor a similar problem with no boundary layer were given in [3].

∗This research was supported in part by the Russian Foundation for Basic Research (grant No 04-01-00578,04–01–89007–NWO a) and by the Dutch Research Organization NWO under grant No 047.016.008.

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S. LI, L.P. SHISHKINA, G.I. SHISHKIN: Parameter-Uniform Method for a Singu-larly Perturbed Parabolic Equation Modelling the Black-Scholes equation in the Presenceof Interior and Boundary Layers

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%Speaker: SHISHKINA, L.P. 59 BAIL 2006

References

[1] J.J.H. Miller, G.I. Shishkin, “Robust numerical methods for the singularly perturbed Black-

Scholes equation”, in: Proceedings of the Conference on Applied Mathematics and ScientificComputing, Springer, Dordrecht, 2005, pp. 95–105.

[2] G.I. Shishkin, “Grid approximation of parabolic convection-diffusion equations with piece-wise smooth initial conditions”, Doklady Akad. Nauk, 405 (1), 1–4 (2005); transl. in Doklady

Mathematics, 72 (3) (2005).

[3] Li Shuiying, D.B. Creamer, G.I. Shishkin, “Discrete approximations of a singularly per-turbed Black-Scholes equation with nonsmooth initial data”, in: An International Confer-

ence on Boundary and Interior Layers — Computational and Asymptotic Methods BAIL2004, ONERA, Toulouse, 5th-9th July, 2004, pp. 441-446.

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S. LI, L.P. SHISHKINA, G.I. SHISHKIN: Parameter-Uniform Method for a Singu-larly Perturbed Parabolic Equation Modelling the Black-Scholes equation in the Presenceof Interior and Boundary Layers

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%Speaker: SHISHKINA, L.P. 60 BAIL 2006

Numerical Method for the Blasius Equation on an Infinite Interval ∗

A.I. Zadorin

Omsk Branch of Sobolev Institute of Mathematics,

Siberian Branch of Russian Academy of Sciences, Omsk, Russiazadorin@iitam.omsk.net.ru

The Blasius equation on a semi-infinite interval is considered. This problem can be considered

as a model problem for constructing a numerical method for problems in unbounded domains.The Blasius problem have been investigated in many papers. For example, G.I. Shishkin [1]studied the asymptotic behavior of differential and difference solutions to get a difference scheme

with a finite number of mesh points for a sufficiently long interval. We apply the method ofseparation of the set of solutions that satisfy the limit boundary condition at infinity to transform

the problem to a problem on a finite interval (see, e.g., [2, 3]). Difficulties are concerned withthe nonlinearity of the differential equation and with an unbounded coefficient multiplying the

second derivative. We propose to make a transformation of the independent variable to avoidthe last problem.

So, consider the Blasius problem

u′′′(x) + u(x)u′′(x) = 0,

u(0) = 0, u′(0) = 0, limx→∞

u′(x) = 1.(1)

Let u(x) = v(x) + x, w(x) = v′(x). Then problem (1) take the form

v′(x) = w(x), v(0) = 0,

w′′(x) + [v(x) + x]w′(x) = 0, w(0) = −1, limx→∞

w(x) = 0.(2)

To transfer the limit condition at infinity to a finite point, consider the linear problem

εu′′(x) + [a(x) + x]u′(x) = f(x),

u(0) = A, limx→∞

u(x) = 0.(3)

Suppose that there is a unique solution of (3), ε > 0, ∃ limx→∞

a(x), limx→∞

f(x) = 0. To avoid the

difficulty with unboundedness of the coefficient multiplying the first derivative, we use the new

variable t = x2/2. Then problem (3) can be written in the form

εu′′(t) + b(t)u′(t) = F (t), u(0) = A, limt→∞

u(t) = 0, (4)

where

b(t) =a(√

2t)√2t

+ 1 +1

2t, F (t) = f(

√2t)/(2t).

By the next equation we can restrict the set of solutions of the differential equation (4) satisfying

the limit condition at infinity:εu′(t) + g(t)u(t) = β(t), (5)

∗Supported by the Russian Foundation for Basic Research under grant No. 04-01-00578.

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A.I. ZADORIN: Numerical Method for the Blasius Equation on an Infinite Interval

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%Speaker: ZADORIN, A.I. 61 BAIL 2006

where g(t), β(t) are solutions of auxiliary singular Cauchy problems and can be found as seriesin (2t)−0.5 with a given accuracy. We can use equation (5) as the exact boundary condition for

a finite interval. Then we return to the variable x and get the exact restriction of problem (3)to a finite interval as follows:

εu′′(x) + [a(x) + x]u′(x) = f(x),

u(0) = A,ε

Lu′(L) + g(L)u(L) = β(L).

(6)

Return to problem (2). Consider the iterative method

v′n(x) = wn−1(x), vn(0) = 0,

w′′n(x) + [vn(x) + x]w′

n(x) = 0, wn(0) = −1, lim

x→∞

wn(x) = 0.(7)

It is proved that method (7) has the property of convergence. At each iteration, we can transformproblem (7) to a problem for a finite interval, as it was done for a linear problem. One can use

a difference scheme to solve the problem obtained on a finite interval. Theoretical results areconfirmed by results of numerical experiments.

References

[1] G.I. Shishkin, “Grid approximation of the solution to the Blasius Equation and of its Deriv-atives”, Computational Mathematics and Mathematical Physics, 41 (1), 37–54 (2001).

[2] A.A. Abramov, N.B. Konyukhova, “Transfer of admissible boundary conditions from a sin-gular point of linear ordinary differential equations”, Sov. J. Numer.Anal. Math. Modelling,

1, (4), 245–265 (1986).

[3] A.I. Zadorin, “The transfer of the boundary condition from the infinity for the numericalsolution of second order equations with a small parameter” (in Russian), Siberian Journal

of Numerical Mathematics, 2 (1), 21–36 (1999).

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A.I. ZADORIN: Numerical Method for the Blasius Equation on an Infinite Interval

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%Speaker: ZADORIN, A.I. 62 BAIL 2006

An Adaptive Grid Method for the Solar Coronal Loop Model

Paul Zegeling

Department of Mathematics, Faculty of Science, Utrecht University, Utrecht, The Netherlandszegeling@math.uu.nl

Many interesting phenomena in plasma fluid dynamics can be described within the frameworkof magnetohydrodynamics. Numerical studies in plasma flows frequently involve simulationswith highly varying spatial and temporal scales. As a consequence, numerical methods onuniform grids are inefficient to be used, since (too) many grid points are needed to resolve thespatial structures, such as boundary and internal layers, shocks, discontinuities, shear layers, orcurrent sheets. For the efficient study of these phenomena, adaptive grid methods are neededwhich automatically track and spatially resolve one or more of these structures. An interestingapplication within this framework can be found in and around our sun. There are several casesfor which we can expect steep boundary and internal layers (see figure 1).

The first one deals with the expulsion of the magnetic flux by eddies in a solar magneticfield model. This model addresses the role of the magnetic field in a convecting plasma and thedistortion of the field by cellular convection patterns for various (small) values of the resistivity(magnetic diffusion coefficient). This situation is of importance around convection cells justbelow the solar photosphere. Steep boundary and internal layers are formed when the magneticinduction reaches a steady-state configuration.

In this presentation we discuss another phenomenon that takes place a little bit farther fromthe solar interior, viz., in the solar corona. It is known that the temperature gradually decreasesfrom the center of the sun down to values of around 104 degrees Kelvin at the foot of the corona(see figure 2). From that point, however, it surprisingly increases dramatically again up to severalmillions of degrees Kelvin forming a non-trivial transition zone (boundary layer) between thephotosphere and the chromosphere. Moreover, because of the extreme temperatures, the solarcorona is highly structured with closed magnetic structures which are generally known as coronalloops. It can be derived that the temperature T and pressure distribution P in the loop as afunction of a mass-coordinate z satisfy the following PDE model:

5

2

P

T

∂T

∂t−

∂P

∂t= ε

P

T

∂

∂z(T

3

2 P∂T

∂z) + EH − P 2χ(T ), (1)

where P (z, t) = P0(t) − µz, EH is a heating function, χ(T ) the radiative loss function andε a small parameter representing the thermal conductivity in the loop. Near the base of theloop there are two adjacent boundary layers where the temperature increases very quickly whenmoving upward in the loop; in these thin layers the pressure in nearly constant. We will examinethe nature of this special boundary layer via the theory of significant degenerations and also interms of a dynamical system of the steady-state of PDE (1) in which a non-trivial saddle-centerconnection occurs. A complicating factor is the fact that we also need to take into account theso-called loop-condition:

L = 2MR

∫1

0

T

Pdz = constant, (2)

with gasconstant R, total mass in the loop M and (half) looplength L.To support and confirm the theory, we have applied an adaptive grid technique, based on

an equidistribution principle with additional smoothing properties, to numerically simulate theforming of the thin boundary layer.

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P. ZEGELING: An Adaptive Grid Method for the Solar Coronal Loop Model

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%Speaker: ZEGELING, P. 63 BAIL 2006

Coronal loops

Convection cells

Figure 1: Convection cells and coronal loops in, and on top of, the sun in the form of magneticfield lines.

Figure 2: The temperature distribution and the steep transition zone in the outer layers of thesun.

References

[1] P.A. Davidson, An introduction to magneto-hydrodynamics, Cambridge Univ. Press (2001).

[2] J.W. Pakkert, P.C.H. Martens and F. Verhulst, The thermal stability of coronal loops by

nonlinear diffusion asymptotics, Astron. Astrophys., 179, 285-293 (1987).

[3] F. Verhulst and P.A. Zegeling, An asymptotic-numerical approach to the coronal loop prob-

lem, Math. Meth. in the Appl. Sciences, 13, 431-439 (1990).

[4] N.O. Weiss, The expulsion of magnetic flux by eddies, Proc. of Roy. Soc. A, 293, 310-328(1966).

[5] P.A. Zegeling, On resistive MHD models with adaptive moving meshes, J. of ScientificComputing, 24, 263-284 (2005).

[6] P.A. Zegeling and R. Keppens, Adaptive method of lines for magnetohydrodynamic PDE

models, Chapter 4, 117-137, in Adaptive method of lines, Chapman & Hall/CRC Press(2001).

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P. ZEGELING: An Adaptive Grid Method for the Solar Coronal Loop Model

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%Speaker: ZEGELING, P. 64 BAIL 2006

Contributed Presentations

Two-dimensional temporal modes in nonparallel flows

F. Alizard and J.-Ch. Robinet

SINUMEF Laboratory, ENSAM-PARIS 151, Boulevard de l’Hopital, 75013 PARIS, FRANCEFrederic.Alizard@paris.ensam.fr,

1. Introduction

To describe the evolution of a two-dimensional wavepacket in flow such as growing boundarylayer, the classical stability approach is based on the assumption of locally parallel or weakly nonparallel base flow. These approach was used by Gaster (1982) to characterize the spatio-temporaldynamic of a perturbation in a boundary layer. However this approach could failed if a wavelength of any perturbation is larger than a characteristic length of the spatial inhomogeneity ofthe base flow. Consequently a more general eigenvalue problem was developed by some authorsas Lin & Malik (1995), Theofilis et al. (2000) and Erhenstein & Gallaire (2005) where two spatialdirections are inhomogeneous. In this abstract we will focus on two-dimensional modes for aconvectively unstable attached boundary layer. After the description of the numerical method,preliminary results on temporal linear stability modes depending on two space directions arecomputed for a boundary layer flow along a flat plat.

2. Generalities and Basic Flow

At the conference we will be interested exclusively in computations of convective instabilitiesin nonparallel flows. Two types of flows will be studied: a flat plate boundary-layer withoutpressure gradient, shown as a preliminary result and a separated boundary-layer. The two-dimensional Navier-Stokes equations for incompressible fluids in the stream function-vorticityformulation are considered:

∂ω

∂t+ u

∂ω

∂x+ v

∂ω

∂y=

1Re

(∂2ω

∂x2+∂2ω

∂y2

)and ∆ψ = ω, (1)

where ω and ψ are the vorticity and the stream function respectively. System (1) is closedby classical boundary conditions on ψ and ω (Briley (1971)). A second order finite differencesscheme was used for the vorticity transport equation as well as the poisson equation of streamfunction. An A.D.I algorithm has been employed to solve the transport equation and the poissonequation. Preliminary results, shown on part 4, are realized on an attached boundary layer atRe=610, with a grid (450x200).

3. Linearized Perturbed Flow and Numerical Procedure

The proposed stability analysis is based on the classical perturbations technique where theinstantaneous flow (q) is the superposition of the basic flow (Q), data of this problem, andunknown fluctuations (q): q(x, y, t) = Q(x, y)+ q(x, y)exp(−iΩt), where Ω is the circular globalfrequency of the fluctuation. The two-dimensional generalized eigenvalue problem is obtainedby the linearized Navier Stokes equations:

div(u) = 0 and[∆2d

Re−U.grad

]u− gradU.u− gradp + iΩu = 0, (2)

1

F. ALIZARD, J.-CH. ROBINET: Two-dimensional temporal modes in nonparallelflows

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%Speaker: ALIZARD, F. 67 BAIL 2006

Figure 1: Discrete global spectrum from a con-vectively instable boundary layer. The realpart of the eigenvalue is represented on the X-axis and the imaginary part on the Y-axis.

Figure 2: Real part of the horizontal velocityeigenfunction of a discrete eigenvalue from thespectrum ( Fig. 1).

with U(x, y) = [U, V ]t and u(x, y) = [u, v]t. The boundary conditions on velocity disturbancesand constraints on pressure (T. N. Phillips & G. W. Roberts (1993), C. Canuto et al. (1988))complete the eigenvalue value problem. The partial differential stability equations (2) are dis-cretized using an algorithm based on the collocation method based on Chebyshev Gauss-Lobattogrid. The algebraic eigenvalue problem (A− ΩB)X = 0 is solved by the QZ algorithm.

4. Two-Dimensional Temporal Modes, results and perspectives

The figure 1 represents a global spectrum of the boundary layer. The stable discrete eigenvaluesappear in concordance with the fact a boundary layer is globally stable. These discrete valuesrepresent spatio-temporal convective modes as it can been shown fig. 2 (structure similar to aspatial exponential growth). At the conference, this analysis will be also applied on a separatedincompressible boundary layer which is not absolutely unstable but only convectively unstable.

References

[1] M. Gaster , “The development of a two-dimensional wavepacket in a growing boundarylayer ”, Proc. R. Soc. London, 3, 317–332 (1982).

[2] Lin R.S. and Malik M.R , “On the stability of attachment-line boundary layers. ”, J. FluidMech., 3,239–255 (1995).

[3] Theofilis V. Hein S and Dallmann U , “On the origins of unsteadiness and three dimension-ality in a laminar separation bubble ”, Proc. R. Soc. London, 3,3229–3246 (2000).

[4] U. Ehrenstein and F. Gallaire , “On two dimensional temporal modes in spatially evolvingopen flows: the flat-plate boundary layer ”, J. Fluid Mech., (2005).

[5] Briley WR., “A numerical study of laminar separation bubbles using the Navier-Stokesequations.”, J. Fluid Mech., 3,713–736 (1971).

[6] Timothy N. Phillips and Gareth W.Roberts, “The Treatment of Spurious Pressure Modes inSpectral Incompressible Flow Calculations.”, Journal of Computational Physics, 3,150–164(1993).

[7] C. Canuto and M. Y. Hussaini and A. Quarteroni and T. A. Zang, “Spectral Methods inFluids Dynamics”, Springer, (1988).

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F. ALIZARD, J.-CH. ROBINET: Two-dimensional temporal modes in nonparallelflows

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%Speaker: ALIZARD, F. 68 BAIL 2006

Near-wall grid adaptation for wall-functions

Th. Alrutz, T. Knopp

Institute of Aerodynamics and Flow TechnologyDLR (German Aerospace Center)

Bunsenstr. 10, 37073 Gottingen, GermanyThomas.Alrutz@dlr.de

A near-wall grid adaptation method for RANS turbulence modelling using wall-functions is pro-posed. Universal wall functions allow a considerable acceleration of the flow solver, but their predictionsmay deviate from the results with near-wall integration in flow situations which are beyond the underly-ing modelling assumptions of wall-functions. In aerodynamic flows these include(i) stagnation points(flow impingement) and subsequent transition from laminar to turbulent flow,(ii) large surface curvaturein conjunction with strong pressure gradients (as can be observed around leading edge of an airfoil),(iii)strong adverse pressure gradients leading to separation, and (iv) regions of separated flow. Boundarylayer theory shows that the deviation from the universal wall-law becomes more significant as the wall-distance of the first node above the wall is increased. But close agreement with the universal wall-lawcan be achived if the first node is close enough to the wall.To reduce the modelling error of universal wall-functions,a near-wall grid-adaptation technique is pro-posed. Regions of strong surface curvature, large pressuregradients and recirculating flow are de-tected by a flow and geometry based sensor. Then in critical regions, the nodes of the prismatic near-wall grid are shifted towards the wall so that a user-specified target value fory+(1) is ensured, wherey+(1) = y(1)u= with wall-distance of the first nodey(1), friction velocityu and viscosity. Thisapproach is applied to a transonic airfoil flow with shock induced separation [5] and to a subsonic high-lift airfoil close to stall [6]. Thereby, the predictions around the leading edge (suction peak) and of theseparation point and the recirculation region can be improved significantly.

x/c

y+

0 0.25 0.5 0.75 10

20

40

60

80

100 y+(1) = 20y+(1) = 40y+(1) = 60

x/c

y+

0 0.25 0.5 0.75 10

20

40

60

y+(1) = 20y+(1) = 40y+(1) = 60

Figure 1: RAE2822 case 10 [5]: Distribution ofy+(1) for Menter SST model [4] without adaptation(left) and withy+-adaptation of the structured (prismatic) near-wall grid (right).

References

[1] T. Knopp, T. Alrutz and D. Schwamborn, “A grid and flow adaptive wall-function method for RANSturbulence modelling”,Journal of Computational Physics(submitted).

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TH. ALRUTZ, T. KNOPP: Near-wall grid adaptation for wall functions

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%Speaker: ALRUTZ, TH 69 BAIL 2006

x/c

c f

0 0.25 0.5 0.75 1

0

0.002

0.004

0.006 exp.low-Rey+(1) = 1y+(1) = 20y+(1) = 40y+(1) = 60cf = 0

x/c

c f

0 0.25 0.5 0.75 1

0

0.002

0.004

0.006 exp.low-Rey+(1) = 1y+(1) = 20y+(1) = 40y+(1) = 60cf = 0

Figure 2: RAE2822 case 10 [5]: Prediction for skin friction coefficient f for Spalart-Allmaras-Edwardsmodel [3] without adaptation (left) and withy+-adaptation of the prismatic near-wall grid (right).

x/c

c p

0 0.05 0.1 0.15

-4.2

-3.8

-3.4

-3

exp.low-Rey+(1) = 12y+(1) = 24y+(1) = 40y+(1) = 80

y+(1) Ny xsep= xsep= SA-Edwards Menter SST

low-Re 33 0.771 0.8661 33 0.770 0.8664 28 0.755 0.8637 26 0.759 0.86812 24 0.761 0.86124 21 0.787 0.861 (0.864)50 19 (0.771) 0.881 (0.873)70 17 (0.788) 0.903 (0.867)

Figure 3: A-airfoil [6]: Detail of pressure coefficient p for SST model [4] on adapted grid (left). Right:Prediction of the separation point without adaptation and with y+-adaptation (values in brackets).

[2] Th. Alrutz, “Hybrid grid adaptation in TAU”,In: MEGAFLOW - Numerical flow simulation foraircraft design, Notes on Numerical Fluid Mechanics and Multidisciplinary Design. N. Kroll andJ.K. Fassbender, Eds., (2005).

[3] J.R. Edwards and S. Chandra, “Comparison of eddy viscosity-transport turbulence models for three-dimensional, shock separated flowfields”,AIAA Journal, 34, 756–763 (1996).

[4] F.R. Menter, “Zonal two equationk/! turbulence models for aerodynamic flows”,AIAA Paper1993-2906, (1993).

[5] P. H. Cook, M. A. McDonald and M.C.P. Firmin, “Aerofoil RAE 2822 - Pressure distributions andboundary layer and wake measurements”,AGARD Advisory Report AR-138, A6.1-A6.77 (1979).

[6] Ch. Gleyzes, “Operation decrochage - Resultats de la 2eme campagne d’essaisa F2 – Mesures depression et velocimetrie laser”,RT-DERAT 55/5004 DN, ONERA, (1989).

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TH. ALRUTZ, T. KNOPP: Near-wall grid adaptation for wall functions

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%Speaker: ALRUTZ, TH 70 BAIL 2006

A family of non-conforming finite elements of arbitrary order for the Stokes

problem on anisotropic quadrilateral meshes

Thomas Apel1 and Gunar Matthies2

1 Institut fur Mathematik und Bauinformatik, Universitat der Bundeswehr Munchen2 Fakultat fur Mathematik, Ruhr-Universitat Bochum,

The solution of the Stokes problem in polygonal or polyhedral domains shows in general asingular behaviour near corners and edges of the domain. Both edge singularities and layersare anisotropic phenomena since the solution changes only slightly in one direction while thederivatives in the perpendicular direction(s) are large. These anisotropic behaviour can be wellapproximated on anisotropic triangulations.

Let Ω ⊂ R2 be a bounded polygonal domain. We consider the Stokes problem

−4u +∇p = f in Ω,

div u = 0 in Ω,

u = 0 on ∂Ω,

where u and p are the velocity and the pressure, respectively, while f is a given force.We solve this problem by using finite element methods on triangulations with special prop-

erties. Let the domain Ω be partitioned by an admissible triangulation which consists of shaperegular and isotropic macro-cells. Each macro-cell is further refined by applying admissiblepatches which are adapted to boundary layers and corner singularities, respectively. For adetailed description of such meshes, we refer to [1].

We are interested in solving the Stokes problem by non-conforming finite element methodsof higher order. To this end, we use the families with finite element pairs of arbitrary orderwhich were given recently in [2]. Each pair consists of a non-conforming space of order r forapproximating the velocity and a discontinuous, piecewise polynomial pressure approximationof order r − 1.

For the stability of finite element methods for solving the Stokes problem and its relatives, itis necessary that the discrete spaces for the velocity and the pressure fulfil an inf-sup condition.In order to get error estimates with constants which are independent of the aspect ratio of theunderlying mesh, it is important that on the one hand the inf-sup constant is independent ofthe aspect ratio and that on the other hand the approximation error and the consistency errorcan be bounded by expressions whose constants don’t depend on the aspect ratio.

Considering the families given in [2], we will show that we obtain for one family optimalerror estimates on anisotropic meshes, i.e., the constant are independent of the aspect ratio.The other families give only error estimates with constant which depend on the aspect ratio,i.e., the constants blow up with increasing aspect ratio.

We will show by means of numerical results how the inf-sup constant behaves on specialfamilies of triangulation with increasing aspect ratio.

References

[1] Th. Apel, S. Nicaise, The inf-sup condition for low order elements on anisotropic meshes,CALCOLO, 41, 89–113 (2004).

[2] G. Matthies, Inf-sup stable nonconforming finite elements of higher order on quadrilater-als and hexahedra. Bericht Nr. 373, Fakultat fur Mathematik, Ruhr-Universitat Bochum(2005).

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TH. APEL, G. MATTHIES: A family of non-conforming finite elements of arbitraryorder for the Stokes problem on anisotropic quadrilateral meshes

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%Speaker: MATTHIES, G. 71 BAIL 2006

Aspects of SUPG/PSPG and GRAD-DIV StabilizedFinite Element Approximation of Compressible Viscous Flow

Markus Bause

Institut fur Angewandte Mathematik, Universitat Erlangen-NurnbergMartensstr. 3, 91058 Erlangenbause@am.uni-erlangen.de

In this contribution various aspects of a theoretical analysis and numerical study ofSUPG/PSPG and grad-div stabilized finite element approximations (cf. [3]) of steadyand unsteady compressible isothermal viscous flow are addressed. After temporal dis-cretization of the time variable by means of the implicit Euler method, the equations ofcompressible viscous flow are solved by an iteration between a generalized Oseen problemfor the velocity and a hyperbolic transport equation for the perturbation from the meandensity (cf. [1]). Such a splitting-type approach seems attractive for computations becauseit offers a reduction to simpler problems for which highly refined numerical methods eitherare known or can be built from existing discretization techniques for similar equations,and because of the guidance that can be drawn from an existence theory based on it.In the case of steady motions of a compressible viscous gas, decribed by the equations

∇ · (ρv) = 0 , ρv · ∇v − µ∆v − (λ+ µ)∇∇ · v +∇p = ρf ,

p = kρ , v|∂Ω = 0 ,∫

Ω ρ dx = M

the iteration for solving this system reads as: Let ρ = 1 + σ. Put v0 = 0, σ0 = 0. Forgiven vn, σn compute vn+1, σn+1 by

(i) solving the generalized Oseen-system

∇ · vn+1 = −∇ · (σnvn) ,

(1 + σn)vn · ∇vn+1 + 12∇ · ((1 + σn)vn)vn+1 − µ∆vn+1 +∇πn+1 = (1 + σn)f ,

vn+1|∂Ω = 0,∫

Ω πn+1 dx = 0

(ii) and, then, solving the hyperbolic transport equation

kσn+1 + (λ+ 2µ)∇ · (σn+1vn+1) = πn+1 − µ∇ · vn+1 .

For the approximation of the separated Oseen problem, SUPG/PSPG and grad-div stabi-lized higher order finite techniques based on LBB-stable elements are used and analyzed.The transport equation is discretized by SUPG stabilized finite element methods. Errorestimates are provided. Numerical results for realistic steady and unsteady benchmarkproblems (driven cavity flow, flow over a backward facing step and DFG-benchmark) aregiven for a large scale of Reynolds numbers.

References

[1] M. Bause, J. G. Heywood, A. Novotny and M. Padula, On some approximation schemesfor steady compressible viscous flow, J. Math. Fluid Mech., 5 (2003), pp. 201–230.

[2] M. Bause, Stabilized finite element approximation of compressible viscous flow, to appear, 2006.[3] T. Gelhard et al., Stabilized finite element schemes with LBB-stable elements for incompressible

flow, J. Comput. Appl. Math. 177 (2005), pp. 243–267.

M. BAUSE: Aspects of SUPG/PSPG and GRAD-DIV Stabilized Finite ElementApproximation of Compressible Viscous Flow

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BAIL 2006

SHEARE STRESS DISTRIBUTION ON SPHERE SURFACE AT DIFFERNT INFLOW

TURBULENCE

L. Bogusławski

Poznań University of Technology, Chair of Thermal Engineering, 60 965 Poznań, Poland;

e-mail: leon.boguslawski@put.poznan.pl

Momentum and heat transfer processes on surfaces are sensitive on intensity of turbulence of flow

above surface. Descriptions of share stress or heat transfer distributions usually assume certain level of

intensity of turbulence of free flow which overflows surface. When structure of flow is formed as

developed flow for typical channels one can assume that turbulence level and structure of turbulent

flow is repeated. In such case detailed knowledge of flow turbulence is not necessary because

Reynolds number indicated average flow similarity and similarity of turbulence by the way.

Unfortunately for most technical applications level of turbulence and its structure can vary in wide

borders. More over this level is difficult to prediction based on channel geometry especially when any

promoters of turbulence occur. Experimental data indicate that increase of turbulence intensity cause

increasing heat and momentum transfer coefficients even when average flow velocity does not change.

To estimate influence of turbulence on local distribution of shear stress a sphere was chosen as the

simplest, repeatable geometry. Flow was generated by a free round jet. Level of turbulence, in the jet

axis, change from about 0.5% near the nozzle outlet till about 20% far away from the nozzle.

Changing of the average flow velocity at the nozzle outlet it is possible to keep constant value of

velocity at different distances from the nozzle outlet. Turbulent fluctuations of a flow velocity were

measured by means of a constant temperature anemometer. The local shear stress distribution on

sphere surface was measured by used a surface sensor connected to the constant temperature

anemometer as well.

For low level of turbulence the shear stress distribution was similar to literature data. Increase of

turbulence cause increase of a local value of shear stress. The local share stress distributions and its

turbulent fluctuations for two chosen turbulence level are presented in figure 1 as an example. The

Reynolds number of average flow was the some for both cases. Increasing of inflow turbulence level

cause increasing of local, average shear stress distributions and equalizing distribution of ‘rms’ of

turbulent fluctuations on rather high level.

Figure 1. Distribution of the local average shear stress and its turbulent fluctuations on the sphere

surface at different inflow turbulence level.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 30 60 90 120 150 180

φ

o

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

' rms/

' orm

s

d = 0.03 m, Re = 3 104, Tu = 2.5 %

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 30 60 90 120 150 180

φ

o

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

' ms/

' orm

s

d = 0.03 m, Re = 3 104, Tu = 12.3 %

L. BOGUSLAWSKI: Sheare Stress Distribution on Sphere Surface at Different InflowTurbulence

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%Speaker: BOGUSLAWSKI, L. 73 BAIL 2006

External turbulence of inflow jet increasing flow turbulence near sphere surface where flow

accelerate. This cause increase of local shear stress and its fluctuations. Increasing of shear stress

transfer fluctuations indicated that turbulence of external flow intensify momentum processes near the

wall of sphere. In flow deceleration zone of flow >90o for both presented in figure 1 turbulence levels

the time average value and its fluctuations rise in this some way. Both runs near parallel. For

acceleration zone of flow near sphere ( ~60o

) shear stress reach maximum value. In this some region

of flow the flow acceleration reduced turbulent fluctuations of momentum transfer generated by

external flow turbulence.

Distributions for different level of turbulence without average velocity change will be presented and

discussed. For such conditions only influence of turbulence are indicated.

Analysis of the power spectrum of turbulent fluctuations of shear stress on sphere surfaces for

different flow intensity at chosen locations will be presented and will be compared with the power

spectrum of inflow turbulence.

References

[1] S.Whitaker. Forced convection heat transfer correlation for flow in pipes, past flat plates, single

cylinders, single spheres and flow in packed beds and tube bundles, J. of AICHE, vol. 18, 361-

371, 1972.

[2] L. Bogusławski. Losses of heat from sphere surface at different inflow conditions (in polish),

XVI Thermodinamics Conference, Kołobrzeg , vol. 1, 135-140, 1996.

[3] L. Bogusławski. Measurements technique of stagnation point heat transfer and its

fluctuations by means of a constant temperature sensor, Proceedings of Turbulent Heat

Transfer Conference, 1-6, San Diego, May 1996.

[4] H. Giedt. Trans. ASME, 71, 375, 1949

L. BOGUSLAWSKI: Sheare Stress Distribution on Sphere Surface at Different InflowTurbulence

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%Speaker: BOGUSLAWSKI, L. 74 BAIL 2006

Continuous-Discontinuous Finite Element Methods for

Convection-Diffusion Problems

Andrea Cangiani, Emmanuil H. Georgoulis and Max Jensen

February 14, 2006

Abstract

Standard (conforming) finite element approximations of convection-dominated convection-diffusionproblems often exhibit poor stability properties that manifest themselves as non-physical oscillationspolluting the numerical solution. Various techniques have been proposed for the stabilisation of finiteelement methods (FEMs) for convection-diffusion problems, see for example, Morton [12] and Roos,Stynes and Tobiska [14] for a complete survey. Common techniques are Petrov-Galerkin methods,like the streamline upwind Petrov-Galerkin (SUPG) method introduced by Hughes and Brooks [8],exponential fitting [13], ad hoc meshing, like graded meshes [15] and Shishkin type meshes [11], andadaptive mesh refinement (see, e.g., [5] and [6]). More recently, the residual-free bubble method ofBrezzi et al [2], [1], [4] and the variational multiscale method of Hughes and co-authors [9],[10].

During the last decade, families of discontinuous Galerkin finite element methods (DGFEMs)have been proposed for the numerical solution of convection-diffusion problems, due to many at-tractive properties they exhibit. In particular, DGFEMs admit good stability properties, they offerflexibility in the mesh design (irregular meshes are admissible) and in the imposition of boundaryconditions (Dirichlet boundary conditions are weakly imposed), and they are increasingly popularin the context of hp-adaptive algorithms.

The above mentioned attractive features of DGFEMs come at the price of the higher numberof degrees of freedom used. For instance, for piece-wise linear approximations in d-dimensions,DGFEMs require 2d times more degrees of freedom than conforming FEMs and their stabilisedvariants. The relative difference in the number of degrees of freedom reduces when consideringhigher order local polynomial degree in the approximations; nevertheless, the conforming FEMsalways require less degrees of freedom than their DGFEM counterparts.

The issue of the number of degrees of freedom required by DGFEMs has recently been addressedby T.J.R. Hughes and co-workers in [7], where the Multiscale Discontinuous Galerkin (MDG) finiteelement method framework is introduced. MDG uses local, element-wise problems to develop atransformation between the degrees of freedom of the discontinuous space and a related, smaller,continuous space. The transformation enables a direct construction of the global matrix problemin terms of the degrees of freedom of the continuous space. It is proved that the method has theaccuracy and stability of the original DGFEM. The drawback of this approach is the computationaloverhead of the solution of the local element-wise problems.

We propose a numerical scheme for linear convection-diffusion problems which couples continuousand discontinuous Galerkin finite elements (CG-DG) in a different manner. Depending on thelocal variation of the solution, the scheme locally uses either the computationally less expensivecontinuous finite elements or the computationally more costly but also more stable discontinuousGalerkin discretisation. Given that good stability properties are required near features of almost(d − 1)-dimensional character (such as boundary and/or interior layers) discontinuous Galerkindiscretisation is only used in the neighborhoods of such features, whereas standard conforming FEMis used away from the layers. Thus, the increased overhead from the use of DGFEMs is balanced bytheir minimal use, limited to the neighborhoods of layers.

This work introduces the continuous-discontinuous Galerkin (CG-DG) finite element method,and presents the first results in the analysis of this approach. In particular, we derive an a-priorierror analysis of the CG-DG blending technique, along with numerical experiments that evaluatethe accuracy and efficiency of the CG-DG approach in practice.

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A.CANGIANI, E.H.GEORGOULIS, M. JENSEN: Continuous-Discontinuous FiniteElement Methods for Convection-Diffusion Problems

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References

[1] Brezzi, F., Marini, D., and Suli, E. Residual-free bubbles for advection-diffusion problems:the general error analysis. Numer. Math. 85, 1 (2000), 31–47.

[2] Brezzi, F., and Russo, A. Choosing bubbles for advection-diffusion problems. Math. ModelsMethods Appl. Sci. 4, 4 (1994), 571–587.

[3] Brooks, A. N., and Hughes, T. J. R. Streamline upwind/Petrov-Galerkin formulationsfor convection dominated flows with particular emphasis on the incompressible Navier-Stokesequations. Comput. Methods Appl. Mech. Engrg. 32, 1-3 (1982), 199–259. FENOMECH ’81,Part I (Stuttgart, 1981).

[4] Cangiani, A., and Suli, E. Enhanced RFB method. Numer. Math. 101(2) (2005), 273–308.

[5] Eriksson, K., Estep, D., Hansbo, P., and Johnson, C. Introduction to adaptive meth-ods for differential equations. In Acta numerica, 1995, Acta Numer. Cambridge Univ. Press,Cambridge, 1995, pp. 105–158.

[6] Giles, M. B., and Suli, E. Adjoint methods for PDEs: a posteriori error analysis andpostprocessing by duality. In Acta numerica, 2002, vol. 11 of Acta Numer. 2002, pp. 145–236.

[7] Hughes, Thomas J. R.and Scovazzi, G., Bochev, P. B., and Buffa, A. A multi-scale discontinuous Galerkin method with the computational structure of a continuous galerkinmethod. ICES Report 05-16 (2005).

[8] Hughes, T. J. R., and Brooks, A. A multidimensional upwind scheme with no crosswinddiffusion. In Finite element methods for convection dominated flows (Papers, Winter Ann.Meeting Amer. Soc. Mech. Engrs., New York, 1979), vol. 34 of AMD. Amer. Soc. Mech. Engrs.(ASME), New York, 1979, pp. 19–35.

[9] Hughes, T. J. R., Feijoo, G. R., Mazzei, L., and Quincy, J.-B. The variational multi-scale method—a paradigm for computational mechanics. Comput. Methods Appl. Mech. Engrg.166, 1-2 (1998), 3–24.

[10] Hughes, T. J. R., and Stewart, J. R. A space-time formulation for multiscale phenomena.J. Comput. Appl. Math. 74, 1-2 (1996), 217–229.

[11] Madden, N., and Stynes, M. Efficient generation of shishkin meshes in solving convection-diffusion problems. Preprint of the Department of Mathematics, University College, Cork,Ireland no. 1995-2 (1995).

[12] Morton, K. W. Numerical solution of convection-diffusion problems, vol. 12 of Applied Math-ematics and Mathematical Computation. Chapman & Hall, London, 1996.

[13] O’Riordan, E., and Stynes, M. A globally uniformly convergent finite element method fora singularly perturbed elliptic problem in two dimensions. Math. Comp. 57, 195 (1991), 47–62.

[14] Roos, H.-G., Stynes, M., and Tobiska, L. Numerical Methods for Singularly PerturbedDifferential Equations. Springer-Verlag, Berlin, 1996. Convection-diffusion and flow problems.

[15] Schwab, C., and Suri, M. The p and hp versions of the finite element method for problemswith boundary layers. Math. Comp. 65, 216 (1996), 1403–1429.

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A.CANGIANI, E.H.GEORGOULIS, M. JENSEN: Continuous-Discontinuous FiniteElement Methods for Convection-Diffusion Problems

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A second order uniform convergent method for a singularly perturbedparabolic system of reaction–diffusion type∗

C. Clavero, J.L. Gracia and F. Lisbona

Department of Applied MathematicsUniversity of Zaragoza. Spain

clavero@unizar.es, jlgracia@unizar.es, lisbona@unizar.es

Abstract

In this work we are interested in solving singularly perturbed parabolic boundary value problemsgiven by

L~ε~u ≡∂~u

∂t+ Lx,~ε~u = ~f, (x, t) ∈ Q = Ω× (0, T ] = (0, 1)× (0, T ],

~u(0, t) = ~g0(t), ~u(1, t) = ~g1(t), ∀t ∈ [0, T ],~u(x, 0) = ~0, ∀x ∈ Ω,

(1)

where

Lx,~ε ≡

(−ε1

∂2

∂x2

−ε2∂2

∂x2

)+ A, A =

(a11(x) a12(x)a21(x) a22(x)

),

with 0 < ε1 ≤ ε2 ≤ 1. The components of the functions ~g0(t), ~g1(t), the right hand side~f(x, t) = (f1(x, t), f2(x, t))T and the matrix A are supposed smooth enough. The positivitycondition

ai1 + ai2 ≥ αi > 0, aii > 0, i = 1, 2, aij ≤ 0 if i 6= j, (2)

is satisfied by matrix A. Otherwise, we consider the transformation ~v(x, t) = ~u(x, t)e−α0t withα0 > 0 sufficiently large so that (2) holds.

This problem is a simple model of the classical linear double–diffusion model for saturatedflow in fractures porous media (Barenblatt system) developed in [1]. The first equation describesthe flow in the matrix and the second one the flow in the fracture system. The coupling termsare given by the matrix

A =(

1 −1−1 1

),

which describes the interchange of fluid between the two systems. The permeabilities ε1 and ε2

in these equations could be very small and also they can have different magnitudes.It is well known that the exact solution of these problems has a multiscale character, i.e.,

there are narrow regions (the boundary layer regions) where the solution has strong gradientsand in the rest of the domain the solution varies smoothly. Therefore, it is necessary to disposeof efficient numerical methods (uniformly convergent methods) to approximate the solutionindependently of the values of the diffusion parameters ε1 and ε2.

Recently some papers study uniform convergent numerical methods to solve singularly per-turbed elliptic linear systems on a special piecewise uniform Shishkin mesh. In the analysis ofthese type of problems the three following different cases can be distinguished depending on therelation between the singular perturbation parameters ε1 and ε2:

[i.] ε1 = ε, ε2 = 1, [ii.] ε1 = ε2 = ε [iii.] ε1, ε2 arbitrary.

∗This research will be partially supported by the project MEC/FEDER MTM 2004-01905 and the DiputacionGeneral de Aragon.

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C. CLAVERO, J.L. GRACIA, F. LISBONA: A second order uniform convergentmethod for a singularly perturbed parabolic system of reaction-diffusion type

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%Speaker: CLAVERO, C. 77 BAIL 2006

In [7] and [6] it was developed a first order uniformly convergent method in cases [ii.] and [iii.]respectively. In [8], [4] and [5] a second order uniformly convergent scheme was obtained forcases [i.], [ii.] and [iii.] respectively.

In [3] a decomposition of the exact solution of problem (1), into its regular and singularcomponents, was given. From this decomposition it follows which is the asymptotic behaviourof each one of these components with respect to the singular perturbation parameters ε1 and ε2.Moreover, in that work a first order in time and second order in space (except by a logarithmicfactor) uniformly convergent method was developed, using the classical Euler and central differ-ences discretizations respectively. In order to increase the order of uniform convergence of thisnumerical scheme, here we replace the Euler scheme by the Crank-Nicolson method in the timediscretization. This method has been used in the framework of singularly perturbed problem; forinstance, in [2] to solve 1D evolutionary problems of convection–diffusion type. Some numericalexperiments will be showed, which illustrate in practice the improvement in the uniform orderof convergence of the new scheme.

Keywords: Singular perturbation, reaction-diffusion problems, uniform convergence, coupledsystem, Shishkin mesh.AMS classification: 65N12, 65N30, 65N06

References

[1] G.I. Barenblatt, I.P. Zheltov and I.N. Kochina, “Basic concepts in the theory of seepage ofhomogeneous liquids in fissured rocks”, J. Appl. Math. and Mech., 24, 1286–1303 (1960).

[2] C. Clavero, J.L. Gracia and J.C. Jorge, “Second order numerical methods for one–dimensional parabolic singularly perturbed problems with regular layers”, Numerical Meth-ods for Partial Differential Equations, 21 149–169 (2005).

[3] J.L. Gracia and F. Lisbona “A uniformly convergent scheme for a system of reaction–diffusion equations”, submitted.

[4] T. Linß and N. Madden, “An improved error estimate for a numerical method for a systemof coupled singularly perturbed reaction–diffusion equations”, Comp. Meth. Appl. Math. 3417–423 (2003).

[5] T. Linß and N. Madden, “Accurate solution of a system of coupled singularly perturbedreaction-diffusion equations”, Computing, 73, 121–133 (2004).

[6] N. Madden and M. Stynes, “A uniformly convergent numerical method for a coupled systemof two singularly perturbed linear reaction-diffusion problems“, IMA J. Numer. Anal., 23,627–644 (2003).

[7] S. Matthews, J.J.H. Miller, E. O’Riordan, G.I. Shishkin, A parameter robust numericalmethod for a system of singularly perturbed ordinary differential equations, in: Analyti-cal and Numerical Methods for Convection–Dominated and Singularly Perturbed Problems(J.J.H. Miller, G.I. Shishkin and L. Vulkov, eds.), Nova Science Publishers, New York, 2000,219–224.

[8] S. Matthews, E. O’Riordan and G.I. Shishkin, “A nunerical method for a system of sin-gularly perturbed reaction–diffusion equations“, J. Comput. Appl. Math., 145, 151–166(2002).

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C. CLAVERO, J.L. GRACIA, F. LISBONA: A second order uniform convergentmethod for a singularly perturbed parabolic system of reaction-diffusion type

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%Speaker: CLAVERO, C. 78 BAIL 2006

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B. EISFELD: Computation of complex compressible aerodynamic flows with aReynolds stress turbulence model

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%Speaker: EISFELD, B. 79 BAIL 2006

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B. EISFELD: Computation of complex compressible aerodynamic flows with aReynolds stress turbulence model

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%Speaker: EISFELD, B. 80 BAIL 2006

Turbulence Flow for NACA 4412 in Unbounded Flow and Ground Effect with Different Turbulence Models and Two Ground Conditions: Fixed and

Moving Ground Conditions

Abdolhamid Firooz, Academic board

(ahfirooz@hamoon.usb.ac.ir)

Mojgan Gadami, B.S. student

(mozhgan_gadami@yahoo.com)

Mechanical Engineering Department, University of Sistan & Baloochestan, Zahedan, Iran.

Abstarct In this paper, the turbulence fluid flow around a

two dimensional 4412 airfoil on different angles of attack near and far from the ground with the RANS (Reynolds averaged Navier-stokes) equations is calculated. Realizable K−ε turbulence model with Enhanced wall treatment and Spalart-Allmaras model are used (Re=2 × 10

6

). Equations are approximated by finite volumes method, and they are solved by segregated method. The second order upwind method is used for the convection term, also for pressure interpolation the PRESTO method is used, and the relation between pressure and velocity with SIMPLEC algorithm is calculated.

The computational domain extended 3C upstream of the leading edge of the airfoil 5C, downstream from the trailing edge, and 4C above the pressure surface. Distance from below the airfoil was defined with H/C where C is chord, and H is ground distance to the trailing edge.

Velocity inlet boundary condition was applied upstream with speed of (U∞=29.215) and outflow boundary condition was applied downstream. The pressure and suction side of the airfoil and above and below's boundaries of domain were defined independently with no slip wall boundary condition. Moving wall with speed of (U∞ =29.215) for above, and fixed or moving wall for below the airfoil was used.

An unstructured mesh arrangement with quadrilateral elements was adopted to map the flow domain in ground effect and unbounded flow. A considerably fine C-type mesh was applied to achieve sufficient resolution of the airfoil surface and boundary layer region. Particular attention was directed to an offset 'inner region' encompassing the airfoil, and also C-type mesh was applied on near the airfoil at above and bottom, which it’s domain depends on the H/C in ground effects condition. Continuing downstream from

leading edge and continuing far from above the airfoil H-type mesh was applied.

Distance from the wall-adjacent cells must be determined by considering the range over which the log-law is valid. The distance is usually measured in the wall unit, y+ (= µρ τ /yu ). By increasing the grid numbers and changing the type of arranging mesh, adapting, around the airfoil a proper y+ value, is obtained, and with this value solution results have good agreement with experimental data. Fig (1), Fig (2).

The aerodynamic characteristic of an airfoil in ground proximity is known to be much different from that of unbounded flow. The condition of the wind tunnel bottom, I. e., moving or fixed relative to the airfoil would influence the performance of the airfoil in ground effect. The presence of boundary layer when air is flowing over bottom of the wind tunnel would be different from the real situation for a flying WIG.

Proper velocity in moving ground boundary condition is considered with moving ground, and boundary layer is considered with fixed ground, and in the moving ground the boundary layer's effect is omitted, and so is the proper velocity in the fix ground.

A grid independence analysis was conducted using some meshes of varying cell number. Each mesh was processed using the Realisable K-ε turbulence model with Enhanced wall treatment and Spalart-Allmaras model. Application of each mesh generally produces accurate predictions of the lift coefficients with comparison to the experimental data. It can be seen that the error associated with the predicated lift coefficient decreases with mesh refinement.

In order to validate the present numerical data the computational results for NACA 4412 in unbounded flow is compared with published numerical results and experimental data. Fig (3).

Fig (4) shows Cp variation on surface of the airfoil at four relative ground high computed (α =8). As the

A. FIROOZ, M. GADAMI: Turbulence Flow for NACA 4412 in Unbounded Flow andGround Effect with Different Turbulence Models and Two Ground Conditions: Fixedand Moving Ground Conditions

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%Speaker: FIROOZ, A. 81 BAIL 2006

wing approaches the ground, the pressure on the pressure side of wing gradually increases due to slow-down of flow in region, while the pressure on the suction side of wing gradually decreases resulting in Lift increase that is regarded as the advantage of the WIG vehicle.

The velocity fields around this section in ground effect with H/C=.2 for two different ground conditions at α =8 are shown in Fig (5) and (6). The different in the velocity field near the wing surface due to the bottom condition differences can not be clearly seen in Fig (5) and (6), but a boundary layer developed on the fix ground can be clearly seen in Fig (6), on the other

hand, for the moving ground with oncoming undisturbed velocity as seen in Fig (6), the velocity decreases with increasing high.

Meanwhile the difference in the Lift simulated by the fixed and moving bottom conditions is negligible but the Drag force simulated by the moving bottom is to some extent larger than that of the fixed one. Also it is concluded that on different angles of attack Lift force of the airfoil increases as it approaches the ground, and the Drag force decreases.

Figure (2): adapted C-grid for unbounded flow (α =5) Figure (1): C-grid for bounded flow ( α =5)

CL vs. angel of attack Figure (3):

(Experimental data ( Abbott, I.H., and von Doenhoff, A.E.,

Theory of Wing Sections, Dover, New York, 1959.)

A. FIROOZ, M. GADAMI: Turbulence Flow for NACA 4412 in Unbounded Flow andGround Effect with Different Turbulence Models and Two Ground Conditions: Fixedand Moving Ground Conditions

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%Speaker: FIROOZ, A. 82 BAIL 2006

Figure (4): Surface pressure distributions for NACA 4412 at the different ground clearance

(α =8, R=2× 106)

Figure (5): Velocity vector field for NACA 4412 in ground effect

(α=8, Re=2×106, H/C=.2, moving ground)

Figure (6): Velocity vector field for NACA 4412 in ground effect

(α =8, Re = 2 × 10 6 , H/C=.2, fix ground )

A. FIROOZ, M. GADAMI: Turbulence Flow for NACA 4412 in Unbounded Flow andGround Effect with Different Turbulence Models and Two Ground Conditions: Fixedand Moving Ground Conditions

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%Speaker: FIROOZ, A. 83 BAIL 2006

EFERENCES

Chawal M. D., Edwards L. C. and Franke M, E 1990 Wind Tunnel Investigation of Wing –in Ground Effect, Journal of Aircraft, Vol 27 ,No.4, pp.289-293

Chorin A. J., 1967 A Numerical Method for Solving Incompressible Viscous Flow Problems, Journal Computational Physics Vol.2 No.2, pp,14-23

Chun H. H., Chang C. H. and Paik K. J.,1999, Longitudinal Stability of a Wing in Ground Effect Craft, J. of The Society of Naval Architects of Korea, Vol.36,No.3,pp.60-70

RINA, London, U., total of 19 papers include machines 2000, Saint Petersburg State Marine Technical University , Russia

Sowdon A., 1995 A Simple Method to remove the Boundary Layer on a Ground Plate, Papers of Ship Research Institute Japan , Vol.32 Vo.2, pp. 53-78

Stinton D. 1998 The Anatomy of the Aeroplan , 2nd Edition published Blacewell Science, pp. 86-92

Staufenbiel, D, 1996 Comment on Aerodynamic Characteristics of a Two Dimensional Airfoil with Ground Effect, Journal of Aircraft

Aerodynamic of Wing-In-Ground Effect Vehicles Rheinisch-West-falische Technische Hochschule, project Rept. 78/1, Aachen, Germany, 1978

Steinbach, D 1996 Comment on Aerodynamic Characteristics of a Two-Dimensional Airfoil with Ground Effect , Journal of Aircraft, Vol.34, No.3, pp.455-456

Thomas J. L., Paulson J,w. and Margason R, J., 1979 Powered Low-Aspect-ratio Wing in Ground Effect (WIG) Aerodynamic Characteristics, NASA TM78793

Turner T. R., 1966 Endless-Belt Technique for Ground Simulation, NASA SP-116

Speziale, C.G., Abid, R. & Anderson, E.C. 1992, .Critical Evaluation of Two-Equation Models for Near-Wall Turbulence., AIAA J., Vol. 30 No. 2, pp. 324-331.

Wilcox, D.C. 1988, .Reassessment of the scale-determining equation for advanced turbulence models., AIAA J., Vol. 26 No. 11, pp. 1299-1310.

Pajayakrit, P. & Kind, R.J. 2000, .Assessment and Modification of Two-Equation Turbulence Models., AIAA J., Vol. 38 No. 6, pp. 955-963.

Gorski, J. & Nguyen, P. 1991, .Navier-Stokes Analysis of Turbulent Boundary Layer Wake for Two-Dimensional Lifting Bodies.,Eighteenth Symposium on Naval Hydrodynamics, Ann Abor, USA, pp. 633-643.

Gersten, K. & Schlichting, H. 2000, .Boundary Layer Theory., 8th Ed., Springer-Verlag, Berlin, Germany.

Abbott, I.H., and von Doenhoff, A.E., Theory of Wing Sections, Dover, New York, 1959.

Akimoto, H. and Kubo, S., 1998 Characteristics Study of Twp-dimensional in Surface Effect by CFD Simulation, Journal of the Society of Naval Architects of Japan, Vol. 184,pp.a7-54

Bagley J. A., 1961 The Pressure Distribution on Two-Dimensional Wings Near Ground, Ministry of Aviation Aeronautical Research Council R&M, NO,3238,40 pages

Chang R, H 2000 Numerical Simulation of Turbulent Flow around Two-Dimensional Wings In Ground Effect Wing Different Ground Boundary Condition, MSc Thesis, Dept of Naval Architecture & ocean Engineering Pusan National University, Korea

A. FIROOZ, M. GADAMI: Turbulence Flow for NACA 4412 in Unbounded Flow andGround Effect with Different Turbulence Models and Two Ground Conditions: Fixedand Moving Ground Conditions

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%Speaker: FIROOZ, A. 84 BAIL 2006

BOUNDARY LAYER INTERACTION WITH EXTERNAL DISTURBANCES

S.A.Gaponov, G.V.Petrov, B.V.Smorodsky

Institute of Theoretical and Applied Mechanics SB RAS, Novosibirsk 630090, Russia; E-mail: gaponov@itam.nsc.ru, Fax:+7(3832)342268

Generation of initial unstable eigen disturbances inside the boundary layer (BL) by external

perturbations is an actual problem nowadays in the investigation of laminar-turbulent transition. Morkovin [1] was the first who has formulated so called BL receptivity problem. Up to now a lot of experimental and theoretical works studying subsonic BL has been performed. The detailed review of these efforts can be found in [2,3]. The knowledge about the supersonic BL is much shorter at the present moment. The existing papers are devoted mainly to the investigation of external acoustic field interaction with a supersonic BL on a smooth flat plate [4-6]. However an oncoming supersonic flow always comprises not only acoustic but also vortical and thermal (entropy) perturbations [7]. This paper presents results concerning irrotational external disturbances with zero damping in the direction of the main flow. Present paper is devoted to the investigation of disturbance excitation in sub- and supersonic BL by external vortical and thermal waves. Interaction of the acoustic waves with the BL on the non-smooth surface has also been investigated and is reported here.Studies were conducted both in an approaching of parallel basic flow and taken in consideration dependence of the main flow in boundary layer on longitudinal coordinate. In the second case were used of an equation of stability offered in [8].

For the acoustic field we distinguish two different kinds of interaction with the BL. In the first case the incidence angle of the acoustic wave onto the BL is finite. In the second case the incidence angle is zero. If the incidence angle is equal to zero, then in the result of the interaction the normal to the-wall disturbance velocity at the BL outer edge is non-zero and it is an additional source of the streamwise acoustic field. Such interaction leads to a very fast amplification of initial wave. For nonzero incidence angles we have performed computations of the reflection coefficient. It is important to note here that in some cases such reflectance is greater than unity. The amplitude of mass flux perturbation inside the BL excited by an external acoustic wave is practically always an order of magnitude larger than the initial wave amplitude in the free stream. Comparison of theoretical results with the experiment [6] leads authors to necessary to consider the problem of diffraction of acoustic waves on a plate leading edge. Using Fourier transformation and the equations of gas dynamics, one can show, that mass flux fluctuations along the plate could be completely determined by means of the jump of normal-to-the-wall disturbance velocity at the plate leading edge. The normal velocity disturbance upstream of the plate leading edge (x ) is determined by the distribution of the mass flux. The obtained formulas give the relations of the mass flux in the region x with its distribution at x . It was shown that intensity of the mass flux fluctuation is reduced with distance from the leading edge and is dependent of the orientation of the incident wave.

0<0> 0<

Tollmien-Schlichting (TS) wave excitation in the supersonic BL by a pair of acoustic waves has been also investigated theoretically. The receptivity coefficient and the range of wave parameters where TS-wave generation takes place have been computed. It was obtained, that at enough large values of a Reynold's number, in narrow range of the acoustic wave numbers, the excited disturbances practically do not differ from an own increasing disturbance. Outside of this range the TS wave is not excited. The parameters of excited disturbances hardly differ from TS wave parameters near a neutral point, the coincidence is observed only at essential move away from neutral area downstream

Beside in the paper investigation of the sound interaction with BL on the flat plate with roughness is consider also. The problem of the disturbance generation by small amplitude spatially periodic wall roughness has been formulated and solved in the linear approximation. The parameters of sound irradiated by the model with roughness in the supersonic flow have been computed. The peculiarities of the disturbances in the case when the angle between the wave fronts and the mean flow is close to the Mach angle have been investigated specially. It has been shown that the amplitude of the disturbance streamline inside the BL is greatly reduced in the near-wall sub-layer. At M mass flux perturbations in the main part of the BL are close to zero also. Considerable reduction of the

1=

S.A. GAPONOV, G.V. PETROV, B.V. SMORODSKY: Boundary layer intercation withexternal disturbances

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%Speaker: GAPONOV, S.A. 85 BAIL 2006

disturbance amplitude takes place for oblique roughness at 1=M where cosχ=M M and is the angle between the roughness wave front and the plate leading edge. At the BL outer edge the

disturbance amplitude is almost zero because it is already damped inside the BL. The problem of nonlinear interactions of the disturbances induced by sound and roughness was investigated. It has been found that the parameter, which defines the ability to the instability wave excitation (receptivity coefficient) is strongly dependent of the acoustics incidence angle and the roughness wave front orientation. Computations show that even at high subsonic external flow velocities the receptivity is strongly dependent also on the direction of sound propagation. At M the instability could be excited by disturbances with the wavelength much smaller than TS wave wavelength. Maximal receptivity corresponds to the case when the angle between the roughness wave front and the mean flow is close to the Mach angle.

χ

1>

The paper presents results of the detailed investigation of the interaction of supersonic BL with external vortical and thermal perturbations. The boundary conditions for the disturbance at the BL outer edge are under special consideration. It has been found that inside the BL the streamwise velocity and mass flux perturbations can exceed much the amplitude of the external vorticity waves. The disturbance excitation rate is reducing with increasing Mach number. The influence of the external thermal wave onto the flow inside the BL is considerably The results of the computations of the intensity and the spatial dimensions of the streamwise streaky structures in the BL are in agreement with measurements [9].

This work has been performed under the financial support of RFBR and the Russian Federation President Council (projects 05-01-00079-a, NSh-9642003.1).

References

1. Morkovin M.V. Critical evaluation of transition from laminar to turbulent shear layer with emphasis on hypersonically trevelling bodies. Tech. Rep. AFFDL 68-149, 1969.

2. Kachanov Yu.S. Physical mechanisms of laminar-boundary-layer transition. Annu.Rev.Fluid Mech., Vol.26, 1994, p.411.

3. Boyko A.V., Greek G.R., Dovgal A.V., Kozlov V.V. Turbulence origin in near-wall flows. Novosibirsk: Science, 1999 (in Russian).

4. Gaponov S.A. On the interaction of a supersonic boundary layer with acoustic disturbances. Thermophysics and Aeromechanics, Vol.2, 3, 1995, pp.181-188.

5. Fedorov A.V., Hohlov A.P. Excitation of unstable modes in a supersonic boundary layer by acoustic waves. Fluid Dynamics, 4, 1991, pp.67-71 (in Russian).

6. Semionov N.V., Kosinov A.D., Maslov A.A. Experimental investigation of supersonic boundary layer receptivity. Transitional boundary layers in aeronautics, edited by R.A.W.M.Henkes and J.L. van Ingen. North-Holland, Amsterdam, 1996, pp.413-420.

7. Kovasznay L.S.G. Turbulence in supersonic flow. J.Aeron.Sci., Vol.20, 10, 1953, pp.657-674. 8. Petrov G.V. New parabolized system of equations of stability of a compressible boundary

layer. J. Appl.Mech. Techn. Phys. Vol.41, 1, 2000, p. 55-61. 9. Westin K.J.A., BakchinovA.A., Kozlov V.V., Alfredsson P.H. Experiments on localized

disturbances in a flat plate boundary layer. Pt 1: The receptivity and evolution of a localized free stream disturbances. Europ. J. Mech., B. Fluids, Vol.17, 6, 1998, pp.823-846.

S.A. GAPONOV, G.V. PETROV, B.V. SMORODSKY: Boundary layer intercation withexternal disturbances

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%Speaker: GAPONOV, S.A. 86 BAIL 2006

Similarity solutions of a power-law non-Newtonian laminar boundary

layer flows

Z. Hammouch

LAMFA, CNRS UMR 6140, Universite de Picardie Jules Verne,

Faculte de Mathematiques et d’Informatique, 33, rue Saint-Leu 80039 Amiens, France

Abstract

A steady–state laminar boundary layer flow, governed by the Ostwald-de Waele power–law model of a non–Newtonian

fluid past a semi-infinite plate is considered. The Blasius method is used to find similarity solutions. Under appropriate

assumptions, partial differential equations are transformed into an autonomous third–order nonlinear degenerate ordinary

differential equation with boundary conditions. We establish the existence of an infinite family of unbounded global

solutions. The asymptotic behavior is also discussed. Some properties of those solutions depend on the power–law index.

Keywords: Boundary–layer; Power–law fluid; Degenerate differential equation; Global existence; Asymptotic behavior; Similarity

solutions.

MSC: 34B15; 34B40; 76D10; 76M55 .

PACS: 47.15, 47.27 Te .

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Z.HAMMOUCH: Similarity solutions of a power-law non-Newtonian laminar boundarylayer flows

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%Speaker: HAMMOUCH, Z. 87 BAIL 2006

Boundary layers for the Navier-Stokes equations : asymptoticanalysis.

M. Hamouda] and R. Temam∗]

∗Laboratoire d’Analyse Numerique, Universite de Paris–Sud, Orsay, France.]The Institute for Scientific Computing and Applied Mathematics,

Indiana University, Bloomington, IN, USA.

Abstract

In this talk, we consider the asymptotic analysis of the solutions of the Navier-Stokes problem, when the viscosity goes to zero; we consider the flow in a channel ofR3, in the non-characteristic boundary case. More precisely, a complete asymptoticexpansion, at all orders, is given in the linear case. For the full nonlinear Navier-Stokessolution, we give a convergence theorem up to order 1, thus improving and simplifyingthe results of [TW].

MSC : 76D05, 76D10, 35C20.

We consider the Navier-Stokes equations in a channel Ω∞ = R2 × (0, h) with a per-meable boundary, making the boundaries z = 0, h, non-characteristic. More precisely wehave

∂uε

∂t− ε∆uε + (uε.∇) uε +∇pε = f, in Ω∞,

div uε = 0, in Ω∞,uε = (0, 0,−U), on Γ∞,uε is periodic in the x and y directions with periods L1, L2,uε|t=0 = u0,

(0.1)

which is equivalent to

∂vε

∂t− ε∆vε − UD3v

ε + (vε.∇) vε +∇pε = f, in Ω∞,

div vε = 0, in Ω∞,vε = 0, on Γ∞,vε is periodic in the x and y directions with periods L1, L2,vε|t=0 = v0.

(0.2)

Here Γ∞ = ∂Ω∞ = R2 × 0, h and we introduce also Ω and Γ :

Ω = (0, L1)× (0, L2)× (0, h), Γ = (0, L1)× (0, L2)× 0, h.

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M. HAMOUDA, R. TEMAM: Boundary layers for the Navier-Stokes equations :asymptotic analysis

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%Speaker: HAMOUDA, M. 88 BAIL 2006

We assume that f and u0 are given functions as regular as necessary in the channelΩ∞, and that U is a given constant; at the price of long technicalities, we can also considerthe case where U is nonconstant everywhere.

Theorem 0.1 For each N ≥ 1, there exists C > 0 and for all k ∈ [0, N ] an explicit given

function θk,ε

such that :

‖vε −N∑

k=0

εk(vk + θk,ε‖L∞(0,T ;L2(Ω)) ≤ C εN+1, (0.3)

‖vε −N∑

k=0

εk(vk + θk,ε‖L2(0,T ;H1(Ω)) ≤ C εN+1/2, (0.4)

where L2(Ω) = (L2(Ω))3, H1(Ω) = (H1(Ω))3, and C denotes a constant which dependson the data (and N) but not on ε. Here vε denotes the solution of the linearized problemof (0.2) and vk the ones of the limit problems (ε = 0) at all orders.

The second result concerns the solution of the full nonlinear problem (0.2) :

Theorem 0.2 For vε solution of the Navier-Stokes problem (0.2), there exist a time T∗ >

0, correctors function θ0,ε

and θ1,ε

explicitly given, and a constant κ > 0 depending on thedata but not on ε, such that :

‖vε − (v0 + θ0,ε

)− ε(v1 + θ1,ε

)‖L∞(0,T∗; L2(Ω)) ≤ κ ε2, (0.5)

‖vε − (v0 + θ0,ε

)− ε(v1 + θ1,ε

)‖L2(0,T∗; H1(Ω)) ≤ κ ε3/2. (0.6)

We recall here the limit problem which is the Euler problem and its solution v0 satisfies :

∂v0

∂t− UD3v

0 + (v0.∇) v0 +∇p0 = f, in Ω∞,

div v0 = 0, in Ω∞,

v03 = 0, on Γ0,

v0 = 0, on Γh,

v0 is periodic in the x and y, directions with periods L1, L2.

(0.7)

It is obvious that we can not expect a convergence result between vε and v0 (in H1(Ω) forexample) and more precisely we are leading with a boundary layer problem close to someparts of the boundary Γ. This is the main object of our work.At the following order v1 is solution of

∂v1

∂t− UD3v

1 + (v1.∇) v0 + (v0.∇) v1 +∇p1

= −∂ϕ0

∂t+ UD3ϕ

0 − (v0.∇) ϕ0 − (ϕ0.∇) v0 + ∆v0, in Ω∞,

div v1 = 0, in Ω∞,

v13 = 0, on Γ0,

v1 = 0, on Γh,

v1 is periodic in the x and y, directions with periods L1, L2.

(0.8)

The function ϕ0 is a known function at this level.

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M. HAMOUDA, R. TEMAM: Boundary layers for the Navier-Stokes equations :asymptotic analysis

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References

[F] K.O. Friedrichs, The mathematical strucure of the boundary layer problem in FluidDynamics (R. von Mises and K.O. Friedrichs, eds), Brown Univ., Providence, RI(reprinted by Springer-Verlag, New York, 1971). pp. 171-174.

[Li2] J. L. Lions, Problemes aux limites dans les equations aux derivees partielles. LesPresses de l’Universite de Montreal, Montreal, Que., 1965. Reedited in [?].

[O] R. E. O’Malley, Singular perturbation analysis for ordinary differential equations.Communications of the Mathematical Institute, Rijksuniversiteit Utrecht, 5. Rijk-suniversiteit Utrecht, Mathematical Institute, Utrecht, 1977.

[TW] R. Temam and X. Wang, Boundary layers associated with incompressible Navier-Stokes equations: the noncharacteristic boundary case. J. Differential Equations 179(2002), no. 2, 647–686.

[VL] M.I. Vishik and L.A. Lyusternik, Regular degeneration and boundary layer for lineardifferential equations with small parameter, Uspekki Mat. Nauk, 12 (1957), 3-122.

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%Speaker: HAMOUDA, M. 90 BAIL 2006

Computation of turbulent natural convection at vertical walls using new wall

functions

M. Holling, H. Herwig

Technical ThermodynamicsHamburg University of Technology

Denickestraße 15, 21073 Hamburg, Germanym.hoelling@tu-harburg.de

h.herwig@tu-harburg.de

Turbulent natural convection at vertical walls has been under investigation in the last decadesbut is still not sufficiently understood. The viscous sublayer, i.e. the region very close to the wall,can be described properly, see e.g. Tsuji and Nagano [1], since there the turbulent fluctuationsare damped by the wall and the governing equations can be solved directly. But in the moreinteresting region in which turbulence dominates, the flow cannot be described properly. Georgeand Capp [2] offer analytical temperature and velocity profiles which have become a kind ofstandard for natural convection. But, it was shown by Versteegh and Nieuwstadt [3] and byHenkes and Hoogendoorn [4] that at least the velocity profile is erroneous. We employ a differentapproach presented in Holling and Herwig [5] to describe the turbulence affected region of theflow field.

The starting point for the analysis is a channel with a hot and a cold wall of infinite extentas used by Versteegh and Nieuwstadt [3] for their DNS study. The governing equations thenreduce to:

0 =∂

∂y

(

a∂T

∂y− v′T ′

)

(1)

0 =∂

∂y

(

ν∂u

∂y− u′v′

)

+ gβ(

T − T0

)

(2)

It is found that the temperature field consists of a viscosity influenced wall layer and a fullyturbulent outer layer. The temperature profile is obtained by matching of gradients betweenthese layers that reveals a logarithmic profile. It is in good agreement with DNS as well asexperimental temperature profiles from various studies.

For the velocity profile a different approach is chosen; the profile is not obtained by matchingof gradients. Instead the momentum equation (2) is rewritten in such a way that the temperatureprofile and the Reynolds stresses are expressed as a function of the wall distance. The Reynoldsstresses are modelled using the eddy viscosity approach. A constant turbulent Prandtl numberis assumed as can be concluded from DNS data. Then the eddy viscosity is directly linked to theturbulent thermal diffusivity and therefore is a linear function of wall distance. Once all termsare expressed as a function of wall distance the momentum equation can be integrated and avelocity profile emerges. This profile is in good agreement with DNS and experimental data.

Straightforward numerical solutions without adequate near wall treatment, like with FLU-ENT 6.2, to reproduce the DNS data of Versteegh and Nieuwstadt [3] for Ra = 5.0 · 106 showthat even with fine grids also in the viscous sublayer only poor agreement can be achieved.Thus, we conclude that it would be diserable to have a new approach and improved near walltreatment.

Therefore, we apply the new universal profiles as wall functions for CFD calculations. Theyare implemented in the two dimensional CAFFA code of Ferziger and Peric [6] that uses the k-ω-turbulence model and the Boussinesq-approximation. The standard k-ω model is used inspite

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0 0.05 0.1320

330

340

350

360

370

380

T in

K

y in m

8×2004×200DNS

0 0.05 0.1−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

u in

m/s

y in m

8×2004×200DNS

Figure 1: Temperature and velocity profile of Versteegh and Nieuwstadt [3] for Ra = 5.0 · 106

compared to results of the modified CAFFA code using two grid sizes.

that it is inadequate for natural convection. Instead of trying to find another modification ofan existing turbulence model, the deficiences could be compensated by modifying the boundaryconditions for k and ω.

The results are again compared to the DNS data of Versteegh and Nieuwstadt [3] and goodagreement is found. Figure 1 shows the temperature and velocity profile for Ra = 5.0 · 106

together with the results of the modified CAFFA code. Two different grid sizes were used anddespite the very few control volumes (8 and 4 CVs) across the channel width the profiles arematched very well. Also the wall gradients, i.e. the Nusselt number and the shear stress, arecorrect within 4 %.

References

[1] T. Tsuji and Y. Nagano, “Characteristics of a turbulent natural convection boundary layeralong a vertical flat plate”, Int. J. Heat Mass Transfer 31, 1723–1734 (1989).

[2] W.K. George and S.P. Capp, “A theory for natural convection turbulent boundary layersnext to heated vertical surfaces”, Int. J. Heat Mass Transfer 22, 813–826 (1979).

[3] T.A.M. Versteegh and F.T.M. Nieuwstadt, “A direct numerical simulation of natural con-vection between two infinite vertical differentially heated walls: scaling laws and wall func-tions”, Int. J. Heat Mass Transfer 42, 3673–3693 (1999).

[4] R.A.W.M. Henkes and C.J. Hoogendoorn, “Numerical determination of wall functions forthe turbulent natural convection boundary layer”, Int. J. Heat Mass Transfer 33, 1087–1097 (1990).

[5] M. Holling and H. Herwig, “Asymptotic analysis of the near wall region of turbulent naturalconvection flows”, J. Fluid Mech. 541, 383–397 (2005).

[6] J.H. Ferziger and M. Peric, “Computational methods for fluid dynamics”, 2nd ed., Springer,

Berlin (1999).

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A.M.Il'in, B.I.SuleimanovThe coecients of inner asymptotic expansions

for solutions of some singular boundary value problems

Two special solutions of ordinary dierential equations

ux + u3 − tu− x = 0, (1)

uxx = u3 − tu− x (2)

are considered. The special solutions are described. They are connected to a wide class ofsolutions of partial dierential equations with small parameter in the case the limit solutionshave points of cusps.

For example consider an equation

ε2(Ux1x1 + Ux2x2) + εbUx1 + f(x1, x2, U) = 0,

where limit equation f(x1, x2, U) = 0 has some roots Uj(x1, x2). In the typical case thisequation has three roots. Let origin be the cusp point: U0(0, 0) = 0 and f(x1, x2, U) =x1 + x2U −U3 + · · · . There is one stable root if x2 < 0 but there are two stable roots andone unstable root if x2 > 0.

In the neighborhood of origin we change variables: if b 6= 0 then

z =x1

ε3/5, t =

x2

ε2/5, w =

U

ε1/5.

In this case the equation for the principal part has the form bwz = w3 − tw − z which isequivalent to equation (1).

If b = 0 thenz =

x1

ε3/4, t =

x2

ε1/2, v =

U

ε1/4

and the equation for the principal part has the form wzz = w3 − tw− z which is equivalentto equation (2).

The main aim is to construct and investigate asymptotics of the solutions to equations(1) and (2) in the entire plane (x,t).

It is natural to expect that solutions will be closed to Whitney fold function H(x, t)

H3 − tH − x = 0 (3)

as x2 + t2 →∞. It is almost truth. But there exist some lines where asymptotics of solutionshave "jumps"at innity. Nevertheless asymptotics for bounded t are suciently simple.

Theorem 1. For every xed t there exist unique monotone increasing solution of thedierential equation (2) and u(x, t)−H(x, t) → 0 as x →∞.

u(x, t) =as x1/3(1 +∞∑

j=2

cj(t)x−j/3), x →∞.

This asymptotic expansion is uniform as |t| < T but it is not uniform at whole plane(x, t).

Theorem 2. For t → −∞ the solution u(x, t) has the following uniform asymptoticexpansion:

u(x, t) = |t|1/2(f(s) +

∞∑j=1

vj(s)

t4j

),

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%Speaker: IL’IN, A.M. 93 BAIL 2006

wheres = x|t|−3/2, f 3(s) + f(s)− s = 0.

If t →∞ asymptotics of the solution u(x, t) is more complicated.Theorem 3. Asymptotic expansion

u(x, t) = t1/2(g(s) +

∞∑j=1

t−4jvj(s) +∞∑

k=0

t−2kwk(ξ))

wheres = x|t|−3/2, g(s)3 − g(s)− s = 0, w0(ξ) = −1 + tanh

ξ√2, ξ = st2

is uniformly valid as t > T > 0. The investigations above about equation (2) were publishedin [1].

New results concern to the rst order Abel equation (1).Asymptotics of the solution of this equation is more complicated than asymptotics for

second order equation.Now we determine H(x, t) as follow:It is solution of the equation (3) as t 6 0.Function H(x, t) is negative continuous solution of the equation (3) as

t > 0, x < xc(t) =2√27

t3/2

and positive continuous solution of the equation (3) as

t > 0, x > xc(t).

The considered solution to the equation (1) is close to that function H(x, t) at innityeverywhere except the narrow domain around the line x = xc(t).

Asymptotics of the Abel equation (1) as t → −∞ and out of the neighborhood of linex = xc(t) is similar to the asymptotics of the equation (2) considered above: u(x, t) =v(s, t)|t|1/2 where s = x |t|−3/2,

v(s, t) =∞∑

k=0

|t|−5k/2v−k (s).

Near the line s = 2√27

in is necessary to investigate two interior layers. The stretchedvariables in the rst interior layer are the following:

z = (s− s∗)t5/3, r = (1√3

+ v(s, t))t5/6.

The stretched variables in the second interior layer are the following:

(z − z0) = ηt−5/6, w = r(z)t−5/6 = (v(s) + 1/√

3)

where z0 is the root of Airy function Ai(−31/6z).Thus we have various asymptotic expansions in the various subdomains of the plane (x,t):

1. u(x, t) =as t1/2

∞∑

k=0

|t|−5k/2v+k (s),

2. u(x, t) =as t1/2[− 1√3

+ t−5/6(r0(z) +

∞∑

j=1

t−5j/6rj(z))]

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A.-M. IL’IN, B.I. SULEIMANOV: The coefficients of inner asymptotic expansions forsolutions of some singular boundary value problems

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and3. u(x, t) =as t1/2[− 1√

3+

∞∑

k=0

t−5k/6wk(ln t, η)].

The coecients of the asymptotic expansions 1. 3. will be constructed and theseexpansions will be matched in the talk.

Bibliography[1] A.M.Il'in and B.I.Suleimanov, "Birth of step-like contrast structures connected with a

cusp catastrophe", Matematicheskii Sbornik 195:12 27-46 (2004); English transl. in Sbornik:Mathematics 195:12 1727-1746 (2004).

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Numerical Simulation of High Sub-critical Reynolds Number Flow Past a Circular Cylinder

Wan Saiful Islam* and Vijay R. Raghavan**

Faculty of Mechanical Engineering Kolej Universiti Teknologi Tun Hussein Onn

86400 Parit Raja, Malaysia *wsaiful@kuittho.edu.my**vijay@kuittho.edu.my

ABSTRACT Few areas in fluid mechanics have received more attention than that of flow past a bluff body. In

particular, flow across a circular cylinder in unconfined and confined flow is a classical problem,

and has been studied experimentally, visually and numerically [1-6]. Although the geometry is

apparently simple, this problem has not yielded to closed form analytical solution except at very

low Reynolds numbers because of the complexity associated with adverse pressure gradients,

separation, eddy shedding, recirculation and reattachment. There are a very large number of

attempts reported in the literature using a variety of numerical approaches viz., finite difference,

finite element and finite volume methods. However, there is room for improvement in the

agreement with experiments that has been obtained hitherto [5,6]. A numerical solution that gives

good agreement is also likely to be useful for benchmarking existing codes and new ones that

may be written. The purpose of the present study is to find a satisfactory solution in the entire

range of sub-critical Reynolds numbers for flow over a circular cylinder.

In earlier attempts to establish benchmark solutions and to obtain agreement with published data

over a range of Reynolds numbers, various turbulence models including large eddy simulation

(LES) had been considered and both 2-D and 3-D had been tried [4-6]. The results were obtained

in the form of general appearance of the wake flow, examination of the velocity magnitudes in

the near-field and far-field, eddy frequencies, Strouhal numbers and detailed local distributions of

pressure. However, most of these authors have carried out their work in the more convenient

W.S. ISLAM, V.R. RAGHAVAN: Numerical Simulation of High Sub-critical ReynoldsNumber Flow Past a Circular Cylinder

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Reynolds number range of 40 to 1000 [2,3]. As one goes to higher Re ranges, results are seen to

deviate progressively more widely from experimental results.

In the present study simulations are performed for unsteady, two-dimensional (2-D) flow past a

circular cylinder in a confined duct with appropriate blockage ratios. At Reynolds numbers as

high as 100,000 the numerical solutions obtained agree remarkably well with experiments, not

only in the global sense in the form of CD, but also locally in terms of pressure distribution. The

paper describes how the agreement was obtained and these results might serve as a benchmark for

validating CFD codes.

REFERENCES 1. Achenbach, E., “Distribution of Local Pressure and Skin Friction Around a Circular

Cylinder in Cross-Flow up to Re = 5×106”, Journal of Fluid Mechanics, 34, 4, 625-639, 1968.

2. Son, J.S. and Hanratty, T.J., “Numerical Solution for the Flow around a Cylinder at

Reynolds Numbers of 40, 200 and 500”, Journal of Fluid Mechanics, 35, 2, 369-386, 1969.

3. Fornberg, B., “Steady Viscous Flow Past a Circular Cylinder up to Reynolds

Number 600”, Journal of Computational Physics, 61, 297-320, 1985. 4. Chou, M.-H. and Huang, W., “Numerical Study of High-Reynolds-Number Flow

Past a Bluff Object”, International Journal for Numerical Methods in Fluids, 23, 711-732, 1996.

5. Selvam, R.P., “Finite Element Modeling of Flow Around a Circular Cylinder using

LES”, Journal of Wind Engineering and Industrial Aerodynamics, 67&68, 129-139, 1997.

6. Breuer, M., “Numerical and Modeling Influences on Large Eddy Simulations for the

Flow Past a Circular Cylinder”, International Journal of Heat and Fluid Flow, 19, 512-521, 1998

W.S. ISLAM, V.R. RAGHAVAN: Numerical Simulation of High Sub-critical ReynoldsNumber Flow Past a Circular Cylinder

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Fast waves during transient flow in an asymmetric channel

Dick Kachuma & Ian Sobey

Oxford University Computing LaboratoryWolfson Building, Parks Road,

Oxford UK OX1 3QDdick.kachuma@comlab.ox.ac.uk

ian.sobey@comlab.ox.ac.uk

Abstract

The use of stepped channels together with unsteady laminar flow provides a powerful mixing mechanismthat is particularly applicable to processes where the fluid contains delicate elements, for example, ap-plications involving mass transfer in blood or in a cell culture. In such channel flows there are parameterregimes where the flow is described by the two-dimensional unsteady Navier–Stokes equations. Sobey(1985) showed both experimentally and numerically that a standing wave of separated regions developedbehind a channel step during oscillatory flow and called the resulting flow a vortex wave. Included in hisexperimental observations were vortex waves of extreme longitudinal extent and he conjectured that thewave formed was, under the correct parameter conditions, virtually undamped in the streamwise direc-tion. We have undertaken calculations in a slightly different parameter region and find that a sequence oftwo events occurs: one is the formation of a vortex wave of finite extent (typically 3-5 vortices alternatingon the two walls behind the step), the second is a subsequent rapidly propagating wave of regular butslightly smaller vortices. The speed of propagation of this second wave is such that its resolution wouldhave been beyond that of the apparatus used in Sobey (1985). In describing these waves we shall refer tothe vortex wave as a V-wave and the second wave as a KH-wave. An example of the waves is illustratedin Figure 1 where contours of a flow are shown.

V-wave KH-wave

Figure 1: Instantaneous streamlines

In order to understand the genesis of KH-waves we have undertaken a study of starting flows inwhich the fluid is accelerated from rest to either steady channel flux or a flux with a small oscillatorycomponent. We have tested two hypotheses: one that the KH-wave results from the evolution of aninviscid rotational core flow that is described by an evolutionary linearised Kortweg-de Vries (KdV)equation. That this might be a plausible hypothesis comes from results by Tutty and Pedley (1994)which show the evolution of waves in solutions of an evolutionary KdV equation that models such a coreflow. The second hypothesis is that the KH-wave results from an Orr-Sommerfeld type instability ofnearly parallel but non-Poiseuille like flow. This has lead us to study stability of a base flow

u0(y) = (1−σy)(1−y2), −1≤ y≤ 1,

whereσ = 0 is Poiseuille flow andσ > 1 indicates reverse flow near one wall.The development of KH-waves has been considered in channels with smoothly varying and sharp

corners and using different numerical methods, as well as with different numerical resolution. The paperis divided into four sections.

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D. KACHUMA, I. SOBEY: Fast waves during transient flow in an asymmetric channel

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(1) We describe the numerical solution of the unsteady Navier–Stokes equations and illustrate thedevelopment of a KH-wave.

(2) We integrate velocities in time to obtain particle paths and use these to help interpret the develop-ment of the flows.

(3) We describe solutions of an evolutionary linearised KdV equation and their interpretation.

(4) We consider solutions of an Orr-Sommerfeld equation as the extent of a reverse flow region isvaried and investigate the consequences of instability in the base flow on subsequent flow devel-opment.

The results we have indicate that it is unlikely that the KH-wave is the result of evolution of aninviscid rotational core flow and instead, that it is more likely the result of a linear instability mechanismdescribed by an Orr–Sommerfeld equation but with growth rates that are orders of magnitude greaterthan those for disturbances to symmetric Poiseuille flow and with instability occuring at relatively lowReynolds number. The complexity of unsteady flows calculated from the full Navier–Stokes equationsis remarkable and it is likely that flows in other parameter regimes are dominated by entirely differentmechanisms.

References

Sobey I.J. (1985).Observation of waves during oscillatory channel flow. J. Fluid Mech., 151:395–426.

Tutty O.R. and Pedley T.J. (1994).Unsteady flow in a nonuniform channel: A model for wave generation.Phys. Fluids, 6:199–208.

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A Robust Numerical Approach for SingularlyPerturbed Time Delayed Parabolic Partial

Differential Equations

Aditya Kaushik† Kapil K. Sharma‡

†Department of Mathematics, Kurukshetra University, Kurukshetra-136 119, India‡MAB, University Bourdauex 1, 351 Cours de Libration F-33405, Cedex Talence, France

[†a-kaushik@lycos.com ‡sharmal@math.u-bordeaux1.fr ]

Summary

In this paper a numerical study for a class of initial boundary value problems forsingularly perturbed unsteady parabolic partial differential equation with variable coef-ficients having time delayed reaction term, is initiated on a rectangular domain. Suchproblems arise in diverse area of science and engineering that take into account not justthe present state of physical systems but also its past history. These models are describedby certain class of functional differential equations often called delay differential equationand from mathematical perspective they are singularly perturbed. In most application inthe life sciences, a delay is introduced when there are some hidden variables and processeswhich are not well understood but are known to cause a time lag. The delay differentialis versatile in mathematical modeling of processes in various application field, where theyprovide the best and sometimes the only realistic simulation of observed phenomena. Thesingularly perturbed differential difference equation with delay arises in general in themodeling of various real life phenomena’s, for instance, in studying heat or mass transferprocess in composite materials with small heat conduction or diffusion, in drift diffusionmodel of semiconductor devices, in fluid flow problems, physiological kinetics, in variousbranches of biosciences and population dynamics, control theory, chemical kinetics, etc.Some modelers ignore the ”lag” effect and use an differential equations model as a sub-stitute for a delay differential equation model. Kuang [1], comments under the heading”Small delays can have large effects” on the dangers that researchers risk if they ignorelags which they think are small. There are inherent qualitative differences between delaydifferential equations and finite systems of differential equations that make such a strategyrisky.The concept of singular perturbation is not new, indeed, it has been a formidable tool inthe solution of some important applied mathematical problems. A singular perturbationis a modification of partial differential equation by adding a multiple ǫ times a higherorder term. In accordance with the informal principle that the behavior of solutions isgoverned primarily by the highest order terms, a solution u

ǫ of the perturbed problem willoften behave analytically quite differently from a solution of the original equation, andexhibits boundary or the transition layers in the outflow boundary reasons when the per-turbation parameter specifying the problem tends to zero [2]. Due to the presence of theperturbation parameter and in particular time delay, the primary mathematical methodsfails to provide the desired results. Thus, the quest for some new numerical techniques

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A. KAUSHIK, K.K. SHARMA: A Robust Numerical Approach for Singularly Per-turbed Time Delayed Parabolic Partial Differential Equations

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to handle the difficulties occurring due to presence of these two parameters and devel-opment of a robust numerical method to solve such type of problem, has found specialrelevance. To sort out both the difficulties, a numerical method consisting of standardfinite difference operator on a uniform mesh is constructed. The first step in this directionconsists of discretizing the time variable with the backward Euler’s method with constanttime step. This produces a set of stationary singularly perturbed semidiscrete problemwhich is further discretized in space using standard finite difference operator on a uniformmesh. An extensive amount of analysis is carried out in order to establish the convergenceand stability of the method proposed. A set of numerical experiment is carried out insupport of the predicted theory and to show the effect of delay on the boundary layerbehavior of the solution for the initial boundary value problem considered which validatescomputationally the theoretical results.

[1] Y. Kuang, Delay differntial equations with applications in population dynamics, Aca-demic Press Inc.,(1993).

[2] H. G. Roos, M. Stynes, and L. Tobiska, Numerical Methods for Singularly PerturbedDi.erential Equa- tions. Convection-Di.usion and Flow Problems, Springer-Verlag, Berlin,1996.

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On methods diminishing spurious oscillationsin finite element solutions of convection–diffusion equations

Petr Knobloch

Charles University, Faculty of Mathematics and Physics, Department of Numerical Mathematics,Sokolovska 83, 186 75 Praha 8, Czech Republic

e-mail: knobloch@karlin.mff.cuni.cz

We discuss the application of the finite element method to thenumerical solution of the scalarconvection–diffusion equation

−ε∆u + b · ∇u = f in Ω, u = ub onΓD, ε∂u

∂n= g onΓN . (1)

HereΩ is a bounded two–dimensional domain with a polygonal boundary ∂Ω, ΓD andΓN are disjointand relatively open subsets of∂Ω satisfyingmeas1(Γ

D) > 0 andΓD ∪ ΓN = ∂Ω, n is the outward unitnormal vector to∂Ω, f is a given outer source of the unknown scalar quantityu, ε > 0 is the constantdiffusivity, b is the flow velocity, andub, g are given functions.

Despite the apparent simplicity of problem (1), its numerical solution is by no means easy if convec-tion is strongly dominant (i.e., ifε ≪ |b|). In this case, the solution of (1) typically possesses interior andboundary layers whose widths are usually significantly smaller than the mesh size and hence the layerscannot be resolved properly. In particular, it is well knownthat the classical Galerkin finite elementdiscretization of (1) is inappropriate in the convection–dominated regime since the discrete solution istypically globally polluted by spurious oscillations. Although, during the last three decades, an exten-sive research has been devoted to the development of methodswhich diminish spurious oscillations inthe discrete solutions of (1), the numerical solution of (1)is still a challenge when convection stronglydominates diffusion. The broad interest in solving problem(1) is caused not only by its actual physicalmeaning but also by the fact that it represents a simple modelproblem for convection–diffusion effectswhich appear in many more complicated problems arising in applications.

Initially, stabilizations of the Galerkin discretizationof (1) imitated upwind finite difference tech-niques. However, like in the finite difference method, the upwind finite element discretizations removethe unwanted oscillations but the accuracy attained is often poor since too much numerical diffusionis introduced. According to our experiences, one of the mostsuccessful upwinding techniques is theimproved Mizukami–Hughes method, see Knobloch [9]. It is a nonlinear Petrov–Galerkin method forconforming linear triangular finite elementsP1 which satisfies the discrete maximum principle on weaklyacute meshes. In contrast with many other upwinding methodsfor P1 elements satisfying the discretemaximum principle, the improved Mizukami–Hughes method adds much less numerical diffusion andprovides rather accurate solutions, cf. Knobloch [10].

One of the most efficient procedures is the streamline upwind/Petrov–Galerkin (SUPG) method de-veloped by Brooks and Hughes [2] which is a higher–order method possessing good stability propertiesachieved by adding artificial diffusion in the streamline direction. Unfortunately, the SUPG methoddoes not preclude spurious oscillations localized in narrow regions along sharp layers. Although theseoscillations are usually small in magnitude, they are not permissible in many applications (e.g., chem-ically reacting flows, free–convection computations, two–equations turbulence models, compressibleflow problems with shocks). Therefore, various terms introducing artificial crosswind diffusion in theneighborhood of layers have been proposed to be added to the SUPG formulation in order to obtain amethod which is monotone or which at least reduces the local oscillations (cf. e.g. [3, 4] and the refer-ences there). This procedure is often referred to as discontinuity capturing (or shock capturing). Thesediscontinuity–capturing methods can be divided into threegroups according to whether the additionalartificial diffusion is isotropic, or orthogonal to streamlines, or based on edge stabilizations. We havedemonstrated forP1 finite elements in [7] that the edge stabilization methods add too much artificialdiffusion. Among the remaining two groups of methods, the best methods forP1 finite elements seem

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P. KNOBLOCH: On methods diminishing spurious oscillations in finite elementsolutions of convection-diffusion equations

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to be the methods of do Carmo and Galeao [5], of Almeida and Silva [1] and slight modifications of themethods of Codina [6] and of Burman and Ern [3], see also John and Knobloch [8].

Our aim is to continue the discussion of properties of the various methods diminishing spuriousoscillations in finite element solutions of (1) mentioned above. We would like to concentrate on threeaspects:

• properties of methods of Mizukami–Hughes type satisfying the discrete maximum principle wherewe mention the improved Mizukami–Hughes method of [9] forP1 finite elements and will discussits generalization to other types of finite elements;

• comparison of the above–mentioned discontinuity–capturing methods for higher order finite ele-ments;

• choice of stabilization parameters where we will present a novel approach to the choice of theSUPG stabilization parameter in outflow boundary layers.

The above described discussion will be performed for the two–dimensional case and the results obtainedin three–dimensions will be mentioned only briefly.

References

[1] R.C. Almeida, R.S. Silva, A stable Petrov–Galerkin method for convection–dominated problems,Comput. Methods Appl. Mech. Eng. 140 (1997) 291–304.

[2] A.N. Brooks, T.J.R. Hughes, Streamline upwind/Petrov–Galerkin formulations for convectiondominated flows with particular emphasis on the incompressible Navier–Stokes equations, Com-put. Methods Appl. Mech. Eng. 32 (1982) 199–259.

[3] E. Burman, A. Ern, Nonlinear diffusion and discrete maximum principle for stabilized Galer-kin approximations of the convection–diffusion–reactionequation, Comput. Methods Appl. Mech.Eng. 191 (2002) 3833–3855.

[4] E.G.D. do Carmo, G.B. Alvarez, A new stabilized finite element formulation for scalar convection–diffusion problems: The streamline and approximate upwind/Petrov–Galerkin method, Com-put. Methods Appl. Mech. Eng. 192 (2003) 3379–3396.

[5] E.G.D. do Carmo, A.C. Galeao, Feedback Petrov–Galerkin methods for convection–dominatedproblems, Comput. Methods Appl. Mech. Eng. 88 (1991) 1–16.

[6] R. Codina, A discontinuity–capturing crosswind–dissipation for the finite element solution of theconvection–diffusion equation, Comput. Methods Appl. Mech. Eng. 110 (1993) 325–342.

[7] V. John, P. Knobloch, A comparison of spurious oscillations at layers diminishing (SOLD) methodsfor convection–diffusion equations: Part I, Preprint Nr. 156, FR 6.1 – Mathematik, Universitat desSaarlandes, Saarbrucken, 2005.

[8] V. John, P. Knobloch, On discontinuity–capturing methods for convection–diffusion equations, sub-mitted to the Proceedings of the conference ENUMATH 2005, Santiago de Compostela, July 18–22,2005.

[9] P. Knobloch, Improvements of the Mizukami–Hughes method for convection–diffusion equa-tions, Preprint No. MATH–knm–2005/6, Faculty of Mathematics and Physics, Charles University,Prague, 2005.

[10] Knobloch, P.: Numerical solution of convection–diffusion equations using upwinding techniquessatisfying the discrete maximum principle. Submitted to the Proceedings of the Czech–JapaneseSeminar in Applied Mathematics 2005, Kuju Training Center,Oita, Japan, September 15–18, 2005.

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P. KNOBLOCH: On methods diminishing spurious oscillations in finite elementsolutions of convection-diffusion equations

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%Speaker: KNOBLOCH, P. 103 BAIL 2006

Model-consistent universal wall-functions for RANS turbulence modelling

T. Knopp

Institute of Aerodynamics and Flow TechnologyDLR (German Aerospace Center)

Bunsenstr. 10, 37073 Gottingen, GermanyTobias.Knopp@dlr.de

A model-consistent universal wall-function method for RANS turbulence modelling [1] is presented,which gives solutions almost independent of the spacing of the first grid node above the wall (denoted byy+(1) in plus-units) and allows a considerable solver acceleration and reduction of memory consump-tions. Model-consistent wall-functions are turbulence model specific, ensuring almost grid-independentpredictions for boundary layer flows at zero pressure gradient, e.g. the flow by Wieghardt [5].

x

Cf

1 2 3 4

0.0025

0.0035

Exp.low Rey+(1) = 1y+(1) = 4y+(1) = 9y+(1) = 17y+(1) = 23y+(1) = 40

x

Cf

1 2 3 4

0.0025

0.0035

Exp.low Re (fine grid)y+(1) = 1y+(1) = 4y+(1) = 9y+(1) = 17y+(1) = 23y+(1) = 40

Figure 1: Skin friction f for equilibrium layer [5]. Almost grid-independent prediction for Spalart-Allmaras (SA) model with Edwards modification [3] (left) andfor Menter baselinek-! model [4] (right).

The underlying approximations, i.e.,(i) one-dimensional boundary-layer flow and(ii) near-wall equilib-rium stress balance by neglecting the streamwise pressure gradient, are assessed by investigation of a flatplate turbulent boundary flow with adverse pressure gradient and separation devised by [2].

x

p+

0 1 20

0.05

0.1

p+ = v / (rho u t3) * dp / dx

p+ = 0station 1 at x 1 = 1.3station 2 at x 2 = 1.6station 3 at x 3 = 1.9station 4 at x 4 = 2.2station 5 at x 5 = 2.4station 6 at x 6 = 2.5station 7 at x 7 = 2.6station 8 at x 8 = 2.7station 9 at x 9 = 2.76station 10 at x 10 = 2.8

y+

u+

100 101 102

5

10

15

20

25

RANS station 4RANS station 6RANS station 7RANS station 8RANS station 9RANS station 10RANS ZPG bou. layer

Figure 2: Pressure gradient parameterp+ = =(u3 )dp=dx (left) and corresponding velocity profilesu+ = u=u vs. y+ = yu= for the adverse pressure gradient flow by Kalitzin et al. [2].

The method is then applied successfully to aerodynamic flowswith separation including a transonic flow

1

T. KNOPP: Model-consistent universal wall-functions for RANS turbulence modelling

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%Speaker: KNOPP, T. 104 BAIL 2006

x/c

c p

0 0.25 0.5 0.75 1

-1

-0.5

0

0.5

1

exp.low-Rey+(1) = 1y+(1) = 20y+(1) = 40y+(1) = 60

x/c

c f

0 0.25 0.5 0.75 1

0

0.002

0.004

0.006 exp.low-Rey+(1) = 1y+(1) = 20y+(1) = 40y+(1) = 60cf = 0

Figure 3: Distribution of p (left) and f (right) for SA-Edwards model [3] for RAE2822 case 10 [6].

x/c

c f

0.25 0.5 0.75

0

0.002

0.004

0.006exp.low-Rey+(1) = 1y+(1) = 4y+(1) = 7y+(1) = 10cf = 0

x/c

c f

0.25 0.5 0.75

0

0.002

0.004

0.006exp.low-Rey+(1) = 1y+(1) = 20y+(1) = 40y+(1) = 80cf = 0

Figure 4: Prediction for f on grids with varying near-wall spacing for SST model [4] forA-airfoil [7].

with shock induced separation [6] and a subsonic highlift airfoil close to stall [7].

References

[1] T. Knopp, T. Alrutz and D. Schwamborn, “A grid and flow adaptive wall-function method for RANSturbulence modelling”,Journal of Computational Physics (submitted).

[2] G. Kalitzin, G. Medic, G. Iaccarino and P. Durbin, “Near-wall behaviour of RANS turbulence mod-els and implications for wall functions”,Journal of Computational Physics, 204, 265-291 (2005).

[3] J.R. Edwards and S. Chandra, “Comparison of eddy viscosity-transport turbulence models for three-dimensional, shock separated flowfields”,AIAA Journal, 34, 756–763 (1996).

[4] F.R. Menter, “Zonal two equationk/! turbulence models for aerodynamic flows”,AIAA Paper1993-2906 (1993).

[5] D. E. Coles and E. A. Hirst (Eds.),Computation of Turbulent Boundary Layers - 1968 AFOSR-IFP-Stanford Conference, Stanford (1969).

[6] P. H. Cook, M. A. McDonald and M.C.P. Firmin, “Aerofoil RAE 2822 - Pressure distributions andboundary layer and wake measurements”,AGARD Advisory Report AR-138, A6.1-A6.77 (1979).

[7] Ch. Gleyzes, “Operation decrochage - Resultats de la 2eme campagne d’essaisa F2 – Mesures depression et velocimetrie laser”,RT-DERAT 55/5004 DN, ONERA (1989).

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T. KNOPP: Model-consistent universal wall-functions for RANS turbulence modelling

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%Speaker: KNOPP, T. 105 BAIL 2006

Evaporating cooling of liquid film along an inclined plate covered with a porous layer

Jin-Sheng Leu

Department of Mechanical Engineering Air Force Institute of Technology

Kaohsiung, Taiwan 82042

Jiin-Yuh Jang* and Yin Chou Department of Mechanical Engineering,

National Cheng-Kung University Tainan, Taiwan 70101

ABSTRACT

The purpose of this work is to evaluate the heat and mass enhancement of liquid film evaporation by covering a porous layer on the plate (as shown in Fig. 1). There is an extensive literature for liquid film evaporating flow based on simplified 1-D and 2-D mathematical models. Wassel and Mills [1] illustrated a 1-D design methodology for a counter-current falling film evaporative cooler. Yan and Soong [2] presented their numerical solution for convective heat and mass transfer along an inclined heated plate with film evaporation with more rigorous treatments of the equations governing the liquid film and liquid-gas interface. The complete two-dimensional boundary layer model for the evaporating liquid and gas flows along an inclined plate was studied recently by Mezaache and Daguenet [3]. Their parametric study focused on the effects of inlet conditions such as gas velocity, liquid mass flow rate and inclined angles and their interaction with both isothermal and heated walls. Zhao [4] studied the coupled heat and mass transfer in a stagnation point flow of air through a heated porous bed with thin liquid film evaporation.

Until now, there seems to be no related theoretical analysis to evaluate the feasibility of

utilizing porous materials for the heat transfer enhancement of falling liquid evaporation. This has motivated the present investigation. The present study analyzes the liquid film evaporation flow along a vertical isothermal plate covered with a thin liquid-saturated porous layer. Liquid and gas streams are approached by two coupled laminar boundary layers. The non-Darcian inertia and boundary effects are included to describe the hydraulic characteristic of the liquid-saturated porous medium. Then, the governing equations (tabulated in Table1) are discretized to a fully implicit difference representation, in which the upwind scheme is used to model the axial convective terms, while second-order central difference schemes are employed for the transverse convection and diffusion terms. Newton linearization procedure is used to linearize the nonlinear terms of governing equation.The numerical solution is obtained by utilizing a fully implicit finite difference method and examined in detail for the effects of porosity ε, porous layer thickness δ, ambient relative humidity φ and Lewis number Le on the average heat and mass transfer performance.

The numerical results conclude that the latent heat flux is the dominant mode for the

present study. The cases for lower ε and δ would produce higher interfacial temperature and mass concentration, and thus enhance the heat and mass transfer performances across the film interface. The influence of ε on the Nusselt number (Nu) and Sherwood number(Sh) is gradually more significant as δ is increased. An applicable range of porous layer thickness δ=0.001~0.005 is suggested for the practical application (as shown in Fig. 2). For the effect of ambient relative humidity φ, it is observed that a lower φ leads to a higher Nu but lower Sh. However, the influence on Nu and Sh appears to be less significant than those of ε and δ. In addition, as the Lewis number is increased(Le>1), a larger heat transfer rate is achieved.

1

* Professor, author to whom correspondence should be addressed Tel: 886-6-2088573 Fax: 886-6-2342232 E-mail: jangjim@mail.ncku.edu.tw

J.-S. LEU, J.-Y. JANG, Y.-C. CHOU: Evaporating cooling of liquid film along aninclined plate coverd with a porous layer

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%Speaker: JANG, J.-Y. 106 BAIL 2006

REFERRENCE

[1]. Wassel, A. T., and Mills, A. F., "Design Methodology for a Counter-current Falling Film Evaporative Condenser". ASME J. of Heat Transfer, Vol. 109, pp.784-787, 1987.

[2] Yan, W. M., and Soong, C. Y., “Convection heat and mass transfer along an inclined heated plate with film evaporation”, Int. J. Heat Mass Transfer, Vol.38, pp.1261-1269, 1995.

[3] Mezaache, E., and Daguenet, M., “Effects of Inlet Conditions on Film Evaporation along an Inclined Plate”, Solar Energy, Vol. 78, pp.535-542, 2005.

[4]. Zhao, T. S., “Coupled Heat and Mass Transfer of a Stagnation Point Flow in a Heated Porous Bed with Liquid Film Evaporation”, Int. J. Heat Mass Transfer, Vol.42, pp.861-872, 1999.

Table 1: The governing equations for liquid film and gas steam regions

2

where the subscript “l”, “g” represents the variables of the liquid and gas stream, respectively. ω, ρ ,ν ,α and D are the mass concentration, density, kinematic viscosity, thermal diffusivity and mass diffusivity of the gas. ε and K is the porosity and permeability of the porous medium, C is the flow inertia parameter.

Liquid film

region

2l

ll

l2

l2

ll u

K

Cu

Ky

ucosg0

ρ−

µ−

∂

∂εµ

+ϕρ=

2l

2

el

lyT

xT

u∂

∂α=

∂∂

Gas stream region

0y

vx

u gg =∂

∂+

∂

∂

2g

2

gg

gg

gy

uy

uv

xu

u∂

∂ν=

∂

∂+

∂

∂

2g

2

gg

gg

gy

Ty

Tv

xT

u∂

∂α=

∂

∂+

∂

∂

2

2

ggy

Dy

vx

u∂

ω∂=

∂ω∂

+∂ω∂

X

Y

porous layer

airliquidfilm

iT aTlTwT

X = 0

Figure 1 Schematic diagram of the physical system

ε0.2 0.4 0.6 0.8 1

100

200

300

400

500

600

Nu

= 0.0010.002

0.005

0.01

0.02

δ

ε0.2 0.4 0.6 0.8 15

10

15

20

25

30

35

Sh

= 0.0010.0020.005

0.01

0.02

δ

(a) (b)

Figure 2 The coupled effects of porosity ε and thickness δ on the average (a) Nusselt and (b)Sherwood numbers with Reg =50000 and φ=70%.

J.-S. LEU, J.-Y. JANG, Y.-C. CHOU: Evaporating cooling of liquid film along aninclined plate coverd with a porous layer

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%Speaker: JANG, J.-Y. 107 BAIL 2006

Liseykin V.D., Likhanova Yu.V, Patrakhin D.V., Vaseva I.A.

Application of boundary layer-type functions to comprehensivegrid generation codes

The paper presents recent results related to the developmentof algorithms and codes for generating both structured and un-structured grids with the use of operator Beltrami [1]. Control ofgrid properties is realized by monitor metrics formulated with theuse of boundary layer-type functions. Formulas for the metricsproviding generation of grids adapting to vector fields, gradients,and/or values of physical quantities are presented. Applicationsof adaptive grids to fluid dynamics and plasma related problemsare demonstrated.

The work over the paper is supported by CRDF (RU-M1-2579) and RFBR (06-01-00205) grants.

[1] V.D. Liseikin ”A Computaional Differential Geometry Approach toGrid Generation”, 2004, Berlin, Springer.

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V.D. LISEYKIN, Y.V. LIKHANOVA, D.V. PATRAKHIN, I.A. VASEVA: Applicationof boundary layer-type functions to comprehensive grid generation codes

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%Speaker: LISEYKIN, V.D. 108 BAIL 2006

Int. Conference on Boundary and Interior LayersBAIL 2006

G. Lube, G. Rapin (Eds)c© University of Gottingen, Germany, 2006

A stabilized finite element method

with anisotropic mesh refinement for the Oseen problem

G. Lube

Institut fur Numerische und Angewandte Mathematik,Georg-August-Universitat Gottingen,

Lotzestrasse 16-18, D-37083 Gottingen, Germanylube@math.uni-goettingen.de

Nonstationary incompressible flow problems can be split into auxiliary problems of Oseentype. We start with a quasi-optimal error estimate of Cea-type for conforming stabilized Galerkinmethods of SUPG/PSPG-type with equal-order interpolation of velocity/pressure on generalmeshes. Then we present some new results for this class of methods on hybrid meshes withstructured anisotropic mesh refinement in boundary layers. The layer meshes are supposed tobe of tensor-product type. In particular, we prove a modified inf-sup condition with a constantindependent of the viscosity and of critical parameters of the mesh. Full proofs of the mainresults can be found in [1].

Some numerical tests for a laminar and a turbulent channel flow problem confirm the theoret-ical results. In Fig. 1, we show some results for the three-dimensional channel flow at Reτ = 395using an improved statististical turbulence model based on the unsteady Reynolds averagedNavier-Stokes model (URANS), the v2

− f model by P. Durbin, see also [3]. The results are ingood agreement with the DNS data in [2] and partly better than those reported in [3].

Achnowledgement: This paper is based on joint work with Th. Apel (Neubiberg), T.Knopp (DLR Gottingen) and R. Gritzki (TU Dresden).

References

[1] Apel, Th., Knopp, T., Lube, G.: Stabilized finite element methods with anisotropic mesh

refinement for the Oseen problem, submitted to Appl. Num. Math. 2006

[2] Kim, J., Moin, P., Moser, R.: Turbulence statistics in fully developed channel flow at low

Reynolds number, J. Fluid Mech. 177 (1987) 133–166.

[3] Laurence, L.R., Uribe, J.C., Utyuzhnikov, S.V.: A robust formulation of the v2− f model,J. Flow, Turbulence and Combustion 23 (2004) 169–185.

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G. LUBE: A stabilized finite element method with anisotropic mesh refinement for theOseen equations

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%Speaker: LUBE, G. 109 BAIL 2006

xxxx

xx

xx

xx

xx

xx

xx

xx

xxx

xxxxxx

y+

U+

10-1 100 101 102 1030

5

10

15

20

25

30 ParallelNSLaurenceDNS

x

x

x

x

x

x

x

x

x

xx

xxxx

xx

xx

xx

xx

xx

xx

xx x x x x

y+

k+

0 100 200 300 4000

1

2

3

4

5

6 ParallelNSLaurenceDNS

x

x

xxxxxxx

xx x x x

xx

xx

xx

xx

xx

xx

xx x x x x x

y+

ε+

0 20 40 60 80 1000

0.05

0.1

0.15

0.2

0.25

0.3

0.35 ParallelNSLaurenceDNS

x

xxxx

xx

x

x

xx

xx

x x x x x xx

xx x x x x x x x x x x x

y+

v2

+

0 100 200 300 4000

0.5

1

1.5

2 ParallelNSLaurenceDNS

x

Figure 1: Plot of u+, k+, ǫ+, (v2)+ vs. y+ := yuτ

νcompared to DNS-data in [2] and results in [3]

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G. LUBE: A stabilized finite element method with anisotropic mesh refinement for theOseen equations

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Detached Eddy Simulation of Supersonic Shear Layer Wake Flows

Heinrich Ludeke∗ DLR Institute of Aerodynamics

and Flow TechnologyLilienthalplatz 7, D-38108 Braunschweig

e-mail: heinrich.luedeke@dlr.deTel. +49 (531) 295-3315

Key words: detached-eddy simulation, axisymmetric base flow, compressible wake, turbulentseparation

One challenge of numerical investigations of unsteady super- and hypersonic flow fields is thestudy of the turbulent wake at complex vehicle configurations. A recent promising approach is thetechnique of detached-eddy simulation (DES) proposed by Spalart et.al. Detached-eddy simulationis a hybrid approach for the modelling of turbulent flow fields at complex geometries. The ideais to combine the best features of both, the Reynolds-averaged Navier-Stokes (RANS) and thelarge eddy simulation (LES) approach to predict massively separated unsteady flow fields at highReynolds-numbers especially in the wake of Re-entry vehicles during descent. In this study theSpalart Allmaras one equation turbulence model is used as an accurate and efficient base for DES.

The intention of the current work is an investigation of the shear flow in the wake of a bluntcylinder at M = 2.4 at high Reynolds numbers. This is used as a basic configuration for re-entryvehicles with a blunt base. The typical time averaged flow field for this configuration is shownin fig. 1 with pressure contours and streamlines. The large turning angle behind the base causesseparation and a region of reverse flow as visible in the wake. The point at the axis of symmetriewhere the streamwise velocity is zero is considered to be the shear layer reattachment point. Inthis region the flow is forced to turn along the axis of symmetry causing a reattachment shockto be formed. The detailed experimental data base, provided by Herrin and Dutton [1], is usedfor comparison of numerically predicted and measured data. The practical applicability of theapproach is demonstrated considering as example an axisymmetric re-entry capsule.

All simulations were carried out by the hybrid structured-unstructured DLR Tau code whichis extensively validated for sub- trans- and hypersonic cases [2]. For the axisymmetric cylinder-configuration structured and unstructured grids have been generated at different resolution of theturbulent wake. They are designed with a similar resolution used by Forsythe and Sqires in [3].Especially the near wake of the cylinder is refined extensively. The results have shown good gridconvergence for the cylinder cases for all averaged quantities, namely the velocity components andthe pressure, as well as for the diagonal elements of the Reynolds stress tensor and the resolvedturbulent kinetic energy in the free shear layer.

In the wake the turbulent kinetic energy and turbulent intensities in radial and streamwisedirection, computed by extracting the standard deviation of the velocitys over the time, was suc-cessffully compared with the measurements. Furthermore the pressure level along the base wascompared with the experimental data of Herrin and Dutton [1]

Finally simulations of the SARA capsule, a facility conceived to be a recovery orbital platformto perform orbital flights, are carried out to give a practical demonstratin of DES modelling forhypersonic flight [4]. To ensure turbulent wake flow the trajectory point at M = 5.1, Re = 20 · 106

was chosen. The influence of unsteady loads, generated by the turbulent wake flow was investigatedfor this vehicle. A preliminary snapshot of the vorticity contours in the wake is shown in fig. 2.

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H. LUDEKE: Detached Eddy Simulation of Supersonic Shear Layer Wake Flows

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References

[1] J.L. Herrin, J.C. Dutton: Supersonic Base flow Experiments in the Near Wake of aa CylindricalAfterbody. AIAA Journal, Vol 32, No. 1, January 1994.

[2] A. Mack, V. Hannemann: Validation of the unstructured DLR-TAU-Code for HypersonicFlows, AIAA 2002-3111, 2002.

[3] J.R. Forsythe, K.A. Hoffmann, K.D. Squires: Detached-Eddy Simulation with compressibilityCorrections Applied to a Supersonic Axisymmetric Base Flow. AIAA 02-0586, 2002.

[4] Alessandro La Neve, Flavio de Azevedo Correa Junior: SARA Experiment Module Project:Using Skills to Enlarge Experiences. International conference on Engineering Education,Manchester, U.K., August 18-21, 2002

Reattachment Shock

Expansion Waves

Separation Bubble

cp: -0.18 -0.14 -0.10 -0.07 -0.03 0.01 0.05

Figure 1: Flow topology and Cp contours ofthe axisymmetric cylinder wake at M∞ = 2.4,Re = 1.4 · 106.

vorticity: 1600 2106 2773 3650 4804 6325 8326 10960 14427 18991 25000

Figure 2: Instantanious vorticity contours in theSARA wake at M∞ = 5.1, Re = 20 · 106.

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H. LUDEKE: Detached Eddy Simulation of Supersonic Shear Layer Wake Flows

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%Speaker: LUDEKE, H. 112 BAIL 2006

Boundary Layer Solution For laminar flow through a Loosely curved Pipe by Using Stokes Expansion

By Kamyar Mansour

Department of Aerospace Engineering and New Technologies Research Center Amir Kabir University of Technology

Tehran, Iran, 15875-4413 mansour@aut.ac.ir

And Flow Research and Engineering

P.O. Box # 20543 Palo Alto, CA, 94309

Keywords: Curved pipe, high Dean Number

We consider fully developed steady laminar flow through a toroidal pipe of small curvature ratios .The solution is expanded up to 40 terms by computer in powers of Dean Number. The major conclusion of this investigation is that the friction ratio in a loosely coiled pipe grows asymptotically as the 1/4 power of the similarity parameter and not as the 1/2 power as previously deduced from boundary-layer analysis. This work confirmed the results obtained by [1]. The goal of this analysis is to provide as complete a description as possible of the flow. The analysis yields a solution for all values of Reynolds number from zero to infinity in a continuous fashion The paradox concerns the discrepancy between the solution obtained using the extended Stokes series method [1] and that obtained using boundary layer techniques [2],[3],[4],[5],[6]and experimental work of [7],[8],[9],[10]and numerical work of [11]for the ratio of the friction factors in coiled tubes to that in straight one in steady, fully developed laminar flow. . In the ensuing debates several papers of[12],[13].[14],[15],[16].[17],[18].[19],[20],[21],[22],[23] various explanations and new evidence have been given; however, the paradox , the resolution of which is important still remain as a open problem The author in [20] raise the possibility of cause of this paradox is the use of only 24 terms to estimate the asymptotic limit obtained by [1]. In this paper we extend the Stokes series from 24 terms used by [1] to 40 terms. We confirm the major result of the friction ratio in a loosely coiled pipe grows asymptotically as the 1/4 power of the similarity parameter and not as the 1/2 power and that confirm the result of [1]. What is particularly important in problems of this type is the presence of analyticity. Not every stokes expansion, for examples that of the flow past sphere as described by the full Navier-Stokes equation are analytic in Reynolds number. In this case, method of matched asymptotic expansions is required and can be automated

An alternative explanation of the departure of the experiments from our curve might be that for more tightly coiled pipes the steady laminar flow is succeeded not by turbulent flow, but by an intermediate regime of unsteady laminar motion, with higher friction. Taylor [25] expressed the possibility that there may exist an intermediate flow regime between the laminar and turbulent ranges. He observed a transition from a steady laminar flow to a laminar vibrating flow as the speed increased. The onset of turbulence accrued only at a significantly higher speed.

The effect of diameter ratios a/L on the relation between the friction factor and the Dean number has been investigated numerically by [16] and is found to be negligibly small. They concluded the friction factor ratio is relatively insensitive to the diameter ratios at a given Dean number. However in 1988 [18] for corresponding curved pipe problem tried to make experiment to match the series extension, would have to satisfy that is the flow be fully developed at high Dean number namely bigger than 500 and laminar as well as a/L less than .03. But his result adds another mystery to this paradox and they obtained data, which lay closer to the present method. In

K. MANSOUR: Boundary Layer Solution For laminar flow through a Loosely curvedPipe by Using Stokes Expansion

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%Speaker: MANSOUR, K. 113 BAIL 2006

this paper we have shown that what proposed by [20] as the cause of the discrepancy lies in the use of the only first 12 terms for corresponding curved problem is not correct. In this work we increase number of terms from 12 in [1] to 20 and still the major conclusion of the discrepancy between the solution obtained using the extended stoke series method and that obtained using other method persist as it was the case for the rotating pipe[26]. REFERENCES [1] Van Dyke, M. 1978 Extended stokes series: laminar flow through a loosely coiled pipe. J.Fluid Mech.,86, 129-145 [2] Adler,M.”Flow in curved tubes” Z.Angew.Math.1934 Mech.,14,257 [3]Barua,S.N. 1963 On secondary flow in stationary curved pipes. Quart. J. Mech. Appl. Maths. 26. 61-77 [4] Mori, Y. & Nakayama, W. 1965 study on forced convective heat transfer in curved pipes (1st report, laminar region). Intl. J. of Heat Mass Transfer 8, 67-82 [5] Ito, H., 1969 Laminar Flow in curved pipes Z. agnew. Math. Mech. 11. 653-663 [6] Smith, F. T. 1976 Steady motion within a curved pipe. Proc. Roy. Soc. A 347, 345-370 [7] White, C. M. 1929 Streamline flow through curved pipes. Proc. Roy. Soc. A 123, 845-663. [8] Hasson,D. 1955 “Streamline flow resistence in coils. Research 8(1) supplement p.S1 [9] Trefethen, L. 1957b Flow in rotating radial ducts: report R55GL 350 on laminar flow in rotating, heated horizontal, and bent tubes, extended into transition and turbulent regions. Gen. Elec. Co. Rep. 55GL350-A. [10] Ito, H.,1959 Friction factors for turbulent flow in curved pipes.Trans.ASMEJ.BasicEngng.81,123-134 [11] Collins, W. M. &Dennis, S. C. R. 1975 The steady motion of a viscous fluid in a curved tube. Quart. J. Mech. Appl. Maths. 28, 133-156. [12] Dennis, S.C.R. 1980 “Calculation of the steady flow through a curved tube using a new finite difference method. J.Fluid. Mech.,99, 449-467 [13] Dennis, S.C.R. &Ng,M.C. 1982 “Dual solution for steady laminar flow through a curved tube.Q.J. Mech. Appl. Maths 35,305-324 [14] Nandakumar, K. & Masliyah, J. H. 1982 Bifurcation in steady laminar flow through curved tubes.J. Fluid Mech. 119,475-490 [15] Mansour, K. 1985 Laminar flow through a slowly rotating straight pipe. J. Fluid Mech. 150, 1-24. [16] Soh,W.Y.&Berger,S.A. 1987 “fully developed flow in a curved pipe of arbitrary curvature ratio.Intl J. Numer.Mech. Fluids 7,733 [17] Winters,K.H. 1987 A bifurcation study of laminar flow in a curved tube of rectangular cross-section. J. Fluid Mech. 180, 343-369 [18] Ramshankar, R. & Sreenvasan, K. R. 1988 A paradox concerning the extended Stokes series solution for the pressure drop in coiled pipes. Phys. Fluids, 1339-1347. [19] Van Dyke, M. 1989 some paradoxes in viscous flow theory. “Some unanswered question in fluid mechanics “ (ed. L. M. Trefethan & R. L. Panton).Appl. Mech. Rev. 43, 153-170 [20] S. Jayanti & G.F. Hewitt. "On the paradox concerning friction factor ratio in laminar flow in coils", Proc. R. Soc. Lond. A (1991) 432, 291-299 [21] Mansour, K. 1993 Using Stokes Expansion For Natural Convection inside a two-dimensional cavity. Fluid Dynamics Research, 1-33 [22 ] Dennis,S.C.R. and Riley,N. 1993 ”On the fully developed flow in a curved pipe at large Dean number” Proc. Roy. Soc. A.434, 473-478 [23] Ishigaki, H. "Analogy between laminar flows in curved pipes and orthogonal rotating pipes". J. Fluid Mech. (1994), vol.268, pp. 133-145 [24] Dean, W.R. 1927 Note on the motion of fluid in a curved pipe. Phil.Mag. (7) 4,208-223 [25] Taylor, G.I. 1929 The criterion for turbulence in curved piped. Proc. R. Soc. Lond. A 124, 243-249. [26] Mansour, K. 2002 Boundary layer solution using Stokes expansion for Laminar flow through a slow- rotating pipe, Proceedings of BAIL 2002 ,An international conference on boundary and interior layers computational & Asymptotic methods, Perth, Western Australia.

K. MANSOUR: Boundary Layer Solution For laminar flow through a Loosely curvedPipe by Using Stokes Expansion

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%Speaker: MANSOUR, K. 114 BAIL 2006

Mass conservation of finite element methods for coupled flow-transport

problems

Gunar Matthies1 and Lutz Tobiska2

1 Fakultat fur Mathematik, Ruhr-Universitat Bochum2 Institut fur Analysis und Numerik, Otto-von-Guericke-Universitat Magdeburg

We consider a coupled flow-transport problem in a bounded domain Ω ⊂ Rd, d = 2, 3. The

system is described by the instationary, incompressible Navier–Stokes equations

ut − ν4u + (u · ∇)u +∇p = f in Ω× (0, T ],

div u = 0 in Ω× (0, T ],

u = ub on ∂Ω× (0, T ],

u(0) = u0 in Ω,

(1)

and the time-dependent transport equation

ct − ε4c + u · ∇c = g in Ω× (0, T ],

(cu− ε∇c) · n = cI u · n on Γ−× (0, T ],

ε∇c · n = 0 on Γ+ × (0, T ],

c(0) = c0 in Ω.

(2)

Here, u and p denote the velocity and the pressure of the fluid, respectively, ν and ε are smallpositive numbers, c is the concentration, cI the concentration at the inflow boundary Γ

−:=

x ∈ ∂Ω : u · n < 0, and Γ+ := ∂Ω \ Γ−. We assume that ub is the restriction of a divergence

free function onto the boundary ∂Ω. Note that the velocity u from the Navier–Stokes equationsenters the transport equation as a convection field.

Due to incompressibility constraint, the weak solution c of (2) satisfies the global massconservation property

d

dt

∫Ω

c dx +

∫Γ−

cIu · ndγ +

∫Γ+

cu · ndγ =

∫Ω

g dx. (3)

It is well-known [1], that the finite element solutions satisfy in general the incompressibilityconstraint and the global mass conservation property (3) only approximately.

Different discretisation methods [2, 3, 4] for the instationary, incompressible Navier–Stokesequations and stabilised schemes for the transport problem like SDFEM will be studied numer-ically.

References

[1] C. Dawson, S. Sun, M. F. Wheeler, Compatible algorithms for coupled flow and transport.Comput. Methods Appl. Mech. Engrg., 193, 2565–2580 (2004).

[2] G. Matthies, L. Tobiska, The inf-sup condition for the mapped Qk/Pdisc

k−1element in arbi-

trary space dimensions. Computing, 69, 119–139 (2002).

[3] G. Matthies, L. Tobiska, Inf-sup stable non-conforming finite elements of arbitrary orderon triangles. Numer. Math., 102, 293–309 (2005).

[4] G. Matthies, Inf-sup stable nonconforming finite elements of higher order on quadrilater-als and hexahedra. Bericht Nr. 373, Fakultat fur Mathematik, Ruhr-Universitat Bochum(2005).

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G. MATTHIES, L. TOBISKA: Mass conservation of finite element methods for coupledflow-transport problems

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%Speaker: MATTHIES, G. 115 BAIL 2006

Global Interactive Boundary Layer (GIBL) for a Channel

J. Mauss♦ and J. Cousteix♣

♦ IMFT and UPS, 118 route de Narbonne, 31062 Toulouse Cedex - FrancePhone: 05 61 55 67 94 - mauss@cict.fr

♣ ONERA and SUPAERO, 2 av. E. Belin, 31055 Toulouse Cedex - FrancePhone: 05 62 25 25 80 - Jean.Cousteix@onecert.fr

We consider a laminar, steady, two-dimensional flow of an incompressible Newtonian fluidin a channel at high Reynolds number. When the walls are slightly deformed, adverse pressuregradients are generated and separation can occur. The analysis of the flow structure has beendone essentially by Smith [4]. Later, a systematic asymptotic analysis has been performed bySaintlos and Mauss [3]. With the Successive Complementary Expansion Method, SCEM, weassume a uniformly valid approximation (UVA) based on generalised expansions. This method,developed by Cousteix and Mauss [1], has been used by Dechaume et al. [2].

Navier-Stokes dimensionless equations can be written

div−→V = 0 , (grad

−→V ) ·−→V = −gradΠ +

1Re

−→V . (1)

The basic plane Poiseuille flow is

v(x) = u0 = y − y2 , v(y) = 0 , Π = Π0 = − 2x

Re+ p0 . (2)

The flow is perturbed, for instance, by indentations of the lower and upper walls such as

yl = εF (x, ε) , yu = 1− εG(x, ε) , (3)

where ε is a small parameter. If we seek a solution in the form

v(x) = u0(y) + εu(x, y, ε) , v(y) = εv(x, y, ε) , Π− p0 = − 2x

Re+ εp(x, y, ε) , (4)

the Navier-Stokes equations become

∂u

∂x+

∂v

∂y= 0 , (5a)

ε

(u

∂u

∂x+ v

∂u

∂y

)+ u0

∂u

∂x+ v

du0

dy= −∂p

∂x+

1Re

(∂2u

∂x2+

∂2u

∂y2

), (5b)

ε

(u

∂v

∂x+ v

∂v

∂y

)+ u0

∂v

∂x= −∂p

∂y+

1Re

(∂2v

∂x2+

∂2v

∂y2

). (5c)

It is clear that, for high Reynolds numbers, the reduced equations are of first order leadingto a singular perturbation. In the core flow, we are looking for approximations coming fromasymptotic generalised expansions such as

u = u1(x, y, ε) + · · · , v = v1(x, y, ε) + · · · , p = p1(x, y, ε) + · · · . (6)

Formally, neglecting terms of order O(ε, 1/Re) , for the core flow, we obtain

∂u1

∂x+

∂v1

∂y= 0 , (7a)

u0∂u1

∂x+ v1

du0

dy= −∂p1

∂x, (7b)

u0∂v1

∂x= −∂p1

∂y. (7c)

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J. MAUSS, J. COUSTEIX: Global Interactive Boundary Layer (GIBL) for a Channel

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%Speaker: MAUSS, J. 116 BAIL 2006

It is very interesting to observe the singular behaviour of the solution of (7a–7c) as weapproach the boundaries. For instance, when y → 0, we have

u1 = −2p10 ln y + c10 + · · · ,

where p10 and c10 are functions of x and ε.The generalised asymptotic expansions for the velocity are given by

v(x) = u0(y) + εu(x, y, ε) + · · · , v(y) = εv(x, y, ε) + · · · . (8)

Using the SCEM, the problem consists of solving the continuity equation together with themomentum equation

∂u

∂x+

∂v

∂y= 0 , (9a)

ε

(u

∂u

∂x+ v

∂u

∂y

)+ u0

∂u

∂x+ v

du0

dy= −∂p1

∂x+ ε3 ∂2u

∂y2. (9b)

But, now, we have to solve simultaneously (9a–9b) and the core equations (7a–7c). The sameform as Prandtl’s equations is recovered if we let

U = u0 + εu , V = εv ,∂Π∂x

= −2ε3 + ε∂p1

∂x, (10)

leading to

U∂U

∂x+ V

∂U

∂y= −∂Π

∂x+ ε3 ∂2U

∂y2. (11)

The boundary conditions are now U = V = 0 on the walls. As four conditions must be satisfied,it is clear that the pressure gradient must be adjusted in order to ensure the global mass flowconservation in the channel. Calculations have been performed with a simplified model comingfrom the triple deck theory. An example is given in Fig. 1 which gives the evolution of the skin-friction coefficient along the lower and upper walls of a channel whose lower wall is deformed.

-5 -4 -3 -2 -1 0 1 2 3 4 5

0,00.2

0.4

0.6

0.8

1.0

1.2

Cf

2Re

x/L

upperwall

lowerwall

R = 103

Figure 1: Application of GIBL in a channel whose lower wall is deformed

References

[1] J. Cousteix and J. Mauss. Approximations of the Navier-Stokes equations for high Reynoldsnumber flows past a solid wall. Jour. Comp. and Appl. Math., 166(1):101–122, 2004.

[2] A. Dechaume, J. Cousteix, and J. Mauss. An interactive boundary layer model compared tothe triple deck theory. Eur. J. of Mechanics B/Fluids, 24:439–447, 2005.

[3] S. Saintlos and J. Mauss. Asymptotic modelling for separating boundary layers in a channel.Int. J. Engng. Sci., 34(2):201–211, 1996.

[4] F.T. Smith. On the high Reynolds number theory of laminar flows. IMA J. Appl. Math.,28(3):207–281, 1982.

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%Speaker: MAUSS, J. 117 BAIL 2006

On the implementation of turbulence models in incompressible

flow solvers based on a finite element discretization

O. Mierka and D. Kuzmin

Institute of Applied Mathematics (LS III), University of DortmundVogelpothsweg 87, D-44227, Dortmund, Germany

omierka@math.uni-dortmund.de

kuzmin@math.uni-dortmund.de

1. Introduction

The numerical implementation of turbulence models involves many algorithmic componentsall of which may have a decisive influence on the quality of simulation results. In particu-lar, a positivity-preserving discretization of the troublesome convective terms and nonlinearsources/sinks is an important prerequisite for the robustness of the numerical algorithm. Thispaper presents a detailed numerical study of several eddy viscosity models implemented in theopen-source software package FeatFlow (http://www.featflow.de) using algebraic flux cor-rection to enforce the positivity constraint. The underlying finite element discretization anditerative solution techniques are presented and relevant algorithmic details are revealed.

2. Algebraic flux correction schemes

The design of high-resolution finite element schemes for numerical simulation of turbulent incom-pressible flows on the basis of eddy viscosity models is addressed. A robust positivity-preservingalgorithm is developed building on the algebraic flux correction paradigm for scalar transportproblems [1]. It is explained how to get rid of nonphysical oscillations in the vicinity of steepgradients and to remove excessive artificial diffusion in regions where the solution is sufficientlysmooth. To this end, the discrete operators resulting from a standard Galerkin discretizationof convective terms are modified so as to enforce the desired matrix properies without violatingmass conservation.

3. Implementation of eddy viscosity models

The developed algebraic flux correction techniques are applied to high Reynolds number flowsthat can be described by the incompressible Navier-Stokes equations coupled with an eddyviscosity model of turbulence. A global Multilevel Pressure Schur Complement (discrete pro-jection) method is employed to enforce the incompressibility constraint at the discrete level [2].The turbulent eddy viscosity is introduced in several different ways using

• the RANS approach as represented by various modifications of the k − ε model;

• Large Eddy Simulation (LES) with explicit and implicit subgrid scale modelling.

In particular, the advantages of Monotonically Integrated LES algorithms as compared to explicitsubgrid scale models of Smagorinsky type are explored. The implementation of the standard k−ε

model is based on a block-iterative algorithm featuring a positivity-preserving representation ofsink terms [2]. The main highlight of the present paper is a unified solution strategy for stronglycoupled PDE systems which result from 2D/3D finite element discretizations of RANS andLES models on unstructured meshes. Special emphasis is laid on the near wall treatment and

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O. MIERKA, D. KUZMIN: On the implementation of turbulence models in incom-pressible flow solvers based on a finite element discretization

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%Speaker: MIERKA, O. 118 BAIL 2006

implementation of initial/boundary conditions. A set of representative benchmark problems isemployed to evaluate the performance of the turbulence models under consideration as appliedto incompressible flows at high Reynolds numbers.

4. Numerical examples

The first example deals with a 3D simulation of the turbulent incompressible flow past a back-ward facing step (Re = 44, 000) using the standard k− ε model with logarithmic wall functions.The numerical solutions displayed in Figure 1 are in a good agreement with those published inthe literature [3].

Figure 1: Backward facing step simulation results (Re = 44, 000). Contour lines of a),b) -turbulent kinetic energy; c),d) - turbulent eddy viscosity. a),c) - reference solution [3].

A preliminary code validation for Chien’s low-Reynolds number modification was performedfor a simple channel flow at Re = 13, 750 and compared with Kim’s [4] DNS data in Figure 2.Numerical experiments using both versions of the k− ε model as well as Large Eddy Simulationwith explicit and implicit subgrid scale modelling are currently under way. The results of anin-depth comparative study will be reported at the Conference.

Figure 2: Channel flow simulation results (Re = 13, 750) compared with reference data [4].

References

[1] D. Kuzmin and M. Moller, Algebraic flux correction I. Scalar conservation laws. In: D.Kuzmin, R. Lohner and S. Turek (eds.) Flux-Corrected Transport: Principles, Algorithms,and Applications. Springer, 2005, 155-206.

[2] S. Turek and D. Kuzmin, Algebraic flux correction III. Incompressible flow problems. In: D.Kuzmin, R. Lohner and S. Turek (eds.) Flux-Corrected Transport: Principles, Algorithms,and Applications. Springer, 2005, 251-296.

[3] F. Ilinca, J.-F. Hetu and D. Pelletier, A Unified Finite Element Algorithm for Two-EquationModels of Turbulence. Comp. & Fluids 27-3 (1998) 291–310.

[4] J. Kim, P. Moin and R. D. Moser, Turbulence statistics in fully developed channel flow atlow Reynolds number. J. Fluid Mech. 177 (1987) 133–166.

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O. MIERKA, D. KUZMIN: On the implementation of turbulence models in incom-pressible flow solvers based on a finite element discretization

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%Speaker: MIERKA, O. 119 BAIL 2006

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K. MORINISHI: Rarefied Gas Boundary Layer Predicted with Continuum and KineticApproaches

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%Speaker: MORINISHI, K. 120 BAIL 2006

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K. MORINISHI: Rarefied Gas Boundary Layer Predicted with Continuum and KineticApproaches

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%Speaker: MORINISHI, K. 121 BAIL 2006

QUALITATIVE ANALYSIS OF NAVIER-STOKES SOLUTIONS

IN VICINITY OF THEIR CRITICAL LINES

by Adriana Nastase

Aerodynamik des Fluges, RWTH-Aachen, Germany

Fax: 0049/241/809-2173, E-Mail: nastase@lafaero.rwth-aachen.de

Own developed, zonal, spectral solutions for the compressible stationaryNavier-Stokes layer (NSL) over flattened flying configurations (FCs), are hereproposed. A new spectral coordinate η was introduced in the NSL :

η =x3 − Z(x1, x2)

δ(x1, x2)(0 ≤ η ≤ 1) (1)

and the dimensionless axial, lateral and vertical velocities uδ, vδ and wδ, thedensity function R = lnρ and the absolute temperature T on the upper NSL(which is here only considered) are expressed in the following spectral forms,as in [1], [2], namely :

uδ = ue

N∑i=1

ui ηi , vδ = ve

N∑i=1

vi ηi , wδ = we

N∑i=1

wi ηi ,

R = Rw + (Re −Rw)N∑

i=1

ri ηi , T = Tw + (Te − Tw)

N∑i=1

ti ηi . (2a-e)

Hereby ue, ve, we, Re and Te are the edge values, which can be obtainedfrom the outer potential flow, Rw and Tw are the given values of R and T onthe wall of the FC and ui, vi, wi, ri and ti are the unknown spectral coefficientsof the velocity’s components, which are used to fulfill the partial differentialequations (PDEs) of the NSL. The first and the second derivatives of the veloc-ity’s components uδ, vδ, wδ are linear functions versus the spectral coefficientsui, vi, wi of the velocity’s components.

The impulse equations of the NSL, which are PDEs of second order, arenow considered. If the spectral forms of the velocity’s components uδ, vδ andwδ are used, the impulse equations are reduced to three equivalent quadraticalalgebraic equations (QAEs) with slightly variable coefficients, versus the spec-tral coefficients ui, vi and wi, as in [1], [2]. In these equations the free termsare proportional to the gradients of the pressure p in the direction of the axisof coordinates O xi and, therefore, these terms have greater variations thanthe other coefficients of these equations. The influence of the variation of freeterms and of one of coefficients of the linear term of the QAE over the existenceof real values of the spectral coefficients and the performing of the qualitativeanalysis of the asymptotical behaviours of the three-dimensional PDEs of thecompressible NSLs in the vicinity of their singular points and lines, are treatedhere by using the qualitative analysis of the equivalent QAEs.

The visualizations of asymptotical behaviours of these equivalent QAEs aremade in a here introduced ”Euclidian M-orthogonal space of the NSL’s freespectral coefficients”, which are here treated as variables.

Further the assumption is made that all QAEs of the same type (elliptical

A. NASTASE: Qualitative Analysis of the Navier-Stokes Solutions in Vicinity of theirCritical Lines

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%Speaker: NASTASE, A. 122 BAIL 2006

or hyperbolical), of the same size (same space dimension M), with the samenumber of positive eigenvalues, for which the free terms a are varied from−∞ to +∞ and all the other coefficients are maintained constant, have, in thevicinity of their singular points, qualitatively, similar asymptotical behaviours.

The visualizations of the asymptotical behaviours of elliptical and hyperbol-ical QAEs with variable coefficients, are made versus their principal coordi-nates. Their critical points and lines are obtained by cancelling of their greatdeterminants ∆i.

The visualization of elliptical QAEs with variable free coefficients is repre-sented, in two-dimensional cuts, in form of coaxial ellipses, which collapse, ifthe free term a is equal to the critical value ac , which is located in the commoncenter C of the coaxial ellipses. If a < ac , the visualization of the ellipticalQAEs in two-dimensional cuts are coaxial ellipses, which all approach the crit-ical point a = ac , when the free term increases. If a = ac , the elliptical QAEscollapse in this critical point (black point). If a > ac , the elliptical QAEs haveno more real solutions and the spectral velocity’s components are partially ortotally imaginary for each value of M .

The visualization of the hyperbolical QAEs in the vicinity of their criti-cal point (b = bc) is totally different. If their free coefficients b < bc , thetwo-dimensional cuts are coaxial hyperbolas with two branches (for M = 2),which approach their common asymptotes, if b increases. Up M ≥ 3 breakand rebreak of hyperhyperboloids occur. The two-dimensional cuts in hy-perboloids (M = 3) or hyperhyperboloids (M > 3) are coaxial hyperbolaswith two sheets or coaxial ellipses, because the hyperboloids and hyperhyper-boloids can be with one or two sheets. If b = bc , the two-dimensional cuts inthe hyperhyperboloids degenerate in their asymptotical lines. If b > bc , thetwo-dimensional cuts are coaxial hyperbolas with two sheets, which are jump-ing in the opposite double angles of their asymptotical lines or coaxial ellipses,which are approaching their critical point, located in their common center.

The asymptotical behaviours of the elliptical and hyperbolical QAEs withvariable values of the coefficients of their free terms and of one of their linearterms in the vicinity of their critical parabolas are also very different.

The elliptical QAEs (for M = 2) are visualized in form of coaxial ellipses,which collapse in each point of their critical parabola and, inside this parabola,there are no more real solutions. A black parabolic hole occurs.

The hyperbolic QAEs (for M = 2) degenerate in their asymptotes, in eachpoint of their critical parabola. By crossing of this parabola, the hyperbolasapproach their asymptotes in one of their double angles and, after jumping,they are going away from these asymptotes.

The collapse points and lines of the elliptical QAE are useful for the determi-nation of the position of detachment lines and for the beginning of transition.

The saddle points and lines of the hyperbolical QAE are useful for the de-termination of the position of characteristic and shock surfaces and for thebifurcation.

REFERENCES1. NASTASE, A., New Zonal, Spectral Solutions for the Navier-Stokes Partial Dif-

ferential Equations, Proc. of the International Conference on Boundary andInterior Layers, ONERA Toulouse, 2004, France.

2. NASTASE, A., Zonal, Spectral Solutions for Navier-Stokes Layer and Applica-tions, 4th European Congress on Computational Methods in Applied Sciencesand Engineering (ECCOMAS), Jyvskyl, 2004, Finland.

A. NASTASE: Qualitative Analysis of the Navier-Stokes Solutions in Vicinity of theirCritical Lines

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%Speaker: NASTASE, A. 123 BAIL 2006

Application of the Smith Factorization to DomainDecomposition Methods for the Stokes Equations

FREDERIC NATAF, GERD RAPIN

Laboratoire J. L. Lions, Universite Pierre et Marie Curie, 75252 Paris Cedex 05,France; nataf@ann.jussieu.fr

Math. Dep., NAM, University of Gottingen, D-37083, Germany;grapin@math.uni-goettingen.de

In this talk the Smith factorization is used systematically to derive a new domain de-composition method for the Stokes problem. In two dimensions the key idea is thetransformation of the Stokes problem into a scalar bi-harmonic problem. We show,how a proposed domain decomposition method for the bi-harmonic problem leads toa domain decomposition method for the Stokes equations which inherits the conver-gence behavior of the scalar problem. Thus, it is sufficient to study the convergence ofthe scalar algorithm. The same procedure can also be applied to the three dimensionalStokes problem.

As transmission conditions for the resulting domain decomposition method of theStokes problem we obtain natural boundary conditions. Therefore it can be imple-mented easily.

A Fourier analysis and some numerical experiments show very fast convergence ofthe proposed algorithm. Our algorithm shows a more robust behavior than Neumann-Neumann or FETI type methods.

The last decade has shown, that Neumann-Neumann type algorithms, FETI, andBDDC methods are very efficient domain decomposition methods. Most of the earlytheoretical and numerical work has been carried out for scalar symmetric positive def-inite second order problems, see for example [8, 5]. Then, the method was extendedto different other problems, like the advection-diffusion equations [1], plate and shellproblems [12] or the Stokes equations [10, 11].

In the literature one can also find other preconditioners for the Schur complementof the Stokes equations (cf. [11, 2]). A more complete list of domain decompositionmethods for the Stokes equations can be found in [10, 13]. Also FETI [6] and BDDCmethods [7] have been applied to the Stokes problem with success.

Our work is motivated by the fact that in some sense the domain decompositionmethods for Stokes are less optimal than the domain decompoition methods for scalarproblems. Indeed, in the case of two subdomains consisting of the two half planes itis well known, that the Neumann-Neumann preconditioner is an exact preconditionerfor the Schur complement equation for scalar equations like the Laplace problem (cf.[8]). A preconditioner is called exact, if the preconditioned operator simplifies to theidentity. Unfortunately, this does not hold in the vector case. It is shown in [9] that thestandard Neumann-Neumann preconditioner for the Stokes equations does not possessthis property.

Our aim in this talk is the construction of a method, which preserves this property.Thus, one can expect a very fast convergence for such an algorithm. And indeed, thenumerical results clearly support our approach. In this talk the ideas of [4] are extended

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F. NATAF, G. RAPIN: Application of the Smith Factorization to Domain Decomposi-tion Methods for the Stokes equations

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in several directions. We also give some hints how this approach can be applied to theOseen equations. For an application to the compressible Euler equations see [3].

References[1] Y. Achdou, P. Le Tallec, F. Nataf, and M. Vidrascu. A domain decomposition

preconditioner for an advection-diffusion problem. Comput. Methods Appl. Mech.Engrg, 184:145–170, 2000.

[2] M. Ainsworth and S. Sherwin. Domain decomposition preconditioners for p andhp finite element approximations of Stokes equations. Comput. Methods Appl.Mech. Engrg., 175:243–266, 1999.

[3] V. Dolean and F. Nataf. A New Domain Decomposition Method for the Com-pressible Euler Equations. M2AN , 2005. accepted.

[4] V. Dolean, F. Nataf, and G. Rapin. New constructions of domain decompositionmethods for systems of PDEs. C.R. Acad. Sci. Paris, Ser I, 340:693–696, 2005.

[5] Ch. Farhat and F.-X. Roux. A Method of Finite Element Tearing and Intercon-necting and its Parallel Solution Algorithm. Internat. J. Numer. Methods Engrg.,32:1205–1227, 1991.

[6] J. Li. A Dual-Primal FETI method for incompressible Stokes equations. Numer.Math., 102:257–275, 2005.

[7] J. Li and O. Widlund. BDDC algorithms for incompressible Stokes equations,2006. submitted.

[8] J. Mandel. Balancing domain decomposition. Comm. on Applied NumericalMethods, 9:233–241, 1992.

[9] F. Nataf and G. Rapin. Construction of a New Domain Decomposition Methodfor the Stokes Equations, 2005. Submitted to the Proceedings of DD16.

[10] L.F. Pavarino and O.B. Widlund. Balancing neumann-neumann methods for in-compressible stokes equations. Comm. Pure Appl. Math., 55:302–335, 2002.

[11] P . Le Tallec and A. Patra. Non-overlapping domain decomposition methods foradaptive hp approximations of the Stokes problem with discontinuous pressurefields. Comput. Methods Appl. Mech. Engrg., 145:361–379, 1997.

[12] P. Le Tallec, J. Mandel, and M. Vidrascu. A Neumann-Neumann Domain De-composition Algorithm for Solving Plate and Shell Problems. SIAM J. Numer.Anal., 35:836–867, 1998.

[13] A. Toselli and O. Widlund. Domain Decomposition Methods - Algorithms andTheory. Springer, Berlin-Heidelberg, 2005.

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Numerical approximation of boundary layersfor rough boundaries

N. Neuss

April 5, 2006

In physical problems, interesting phenomena often occur at boundaries orinterfaces between different media. Often these phenomena are complicateddue to the nature of the process or due to the intricate geometry of theinterface. Therefore, they are usually described by effective boundary orinterface laws.

In this talk we will discuss some instructive cases in a quasi-periodicsetting, where the constants in those effective conditions can be calculatedfrom the microscopic setting.

First, we consider the Poisson problem

−∆uε = f, x ∈ Ωε

uε = 0, x ∈ ∂Ωε

where Ωε is a domain with a boundary ∂Ωε featuring a micro-structure of sizeε. A good approximation to uε is given by the solution ueff to the problem

−∆ueff = f, x ∈ Ω

ueff = c(x)∂ueff

∂n, x ∈ ∂Ω

where Ω is an approximating domain with smooth boundary ∂Ω and thefunction c : ∂Ω→ R can be computed solving auxiliary problems taking intoaccount the fine-scale structure of ∂Ωε, see [?].

Similarly, solving the Navier-Stokes equation in such a domain Ωε can bereplaced by solving it on Ω with a Navier boundary condition. Related to thisis the Beavers-Joseph problem, where one can derive an effective interfacelaw for a rough interface between free flow and porous medium flow, see [?].

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Finally, the same idea can be used to obtain effective boundary conditionsfor numerical approximations Ωh of arbitrary domains Ω. This allows us toobtain good O(h2) approximation errors although the domain Ωh only needsto approximate Ω up to order O(h).

References

[1] W. Jager, A. Mikelic, N. Neuss: Asymptotic Analysis of the LaminarViscous Flow Over a Porous Bed. SIAM J. Sci. Comp. 22,6, pp. 2006-2028 (2001).

[2] N. Neuss, M. Neuss-Radu, A. Mikelic: Effective Laws for the PoissonEquation on Domains with Curved Oscillating Boundaries. To appear inApplicable Analysis.

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%Speaker: NEUSS, N. 127 BAIL 2006

An Augmented Lagrangian based solver for the low-viscosity

incompressible flows

Maxim A. Olshanskii∗

We describe an effective solver for the finite element discretization of the Oseen problem

−ν∆u + (w · ∇)u +∇p = f in Ω (1)div u = 0 in Ω (2)

u = g on ∂Ω (3)

with a known, divergence-free vector function w. Discretization of (1)–(3) using LBB-stablefinite elements is based on the following stabilized finite element formulation

L(uh, ph;vh, qh) +∑

τ∈Th

στ (−ν∆uh + w·∇uh +∇ph − f ,w·∇vh)τ

= (f ,vh) ∀vh ∈ Vh, qh ∈ Qh (4)

withL(u, p;v, q) = ν(∇u,∇v) + ((w · ∇)u,v)− (p,div v) + (q,div u)

and a suitable choice of the stabilization parameters στ . With certain assumptions finding FEsolution results in solving the linear system of the form

(A BT

B O

)(up

)=

(f0

). (5)

Linear systems of the form (5) are often referred to as generalized saddle point systems. Inrecent years, a great deal of effort has been invested in solving systems of this form. Most ofthe work has been aimed at developing effective preconditioning techniques; see [1, 3] for anextensive survey. In spite of these efforts, there is still considerable interest in preconditioningtechniques that are truly robust, i.e., techniques which result in convergence rates that arelargely independent of problem parameters such as mesh size and viscosity. In this paper wedescribe a promising approach based on an augmented Lagrangian formulation [4]:

(A + γBT W−1B BT

B O

)(up

)=

(f0

)(6)

with a positive-definite matrix W and parameter γ > 0. The system (6) has precisely the samesolution as the original one (5). Rather than treating (5), a block-triangular preconditioner isconstructed for solving (6) with a Krylov subspace iterative method.

The success of this method crucially depends on the availability of a robust multigrid solverfor the (1,1) block (submatrix) in (6); we develop such a method by building on previous workby Schoberl [5], together with appropriate smoothers for convection-dominated flows. Thismultigrid iteration will be used to define a block preconditioner for the outer iteration on the

∗Department of Mechanics and Mathematics, Moscow State M. V. Lomonosov University, Moscow 119899,Russia (Maxim.Olshanskii@mtu-net.ru). The work was supported in part by the Russian Foundation for BasicResearch and the Netherlands Organization for Scientific Research grants NWO-RFBR 047.016.008

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coupled saddle point system. We will show that this approach is especially appropriate fordiscretizations based on discontinuous pressure approximations, but can be used to constructpreconditioners for other discretizations using continuous pressures. As an example of finiteelement (FE) method with discontinuous pressures we will use the isoP2-P0 pair. In this case,our numerical experiments demonstrate a robust behavior of the solver with respect to h andν for some typical wind vector functions w in (1). Further, the isoP2-P1 finite element pairis used for the continuous pressure-based approximation. For this case numerical experimentsshow an h-independent convergence rates with mild dependence on ν, when the viscosity becomesvery small. We note that this approach does not require a sophisticated preconditioner for thepressure Schur complement of (5) or (6).

Some spectral estimates for the preconditioned problem are shown; basic components forbuilding appropriate multigrid method are discussed. The role of parameter γ in (6) is addressed.We also present a comparison with one of the best available preconditioning techniques andcoupled multigrid methods (Vanka multigrid), showing that our method is quite competitive interms of convergence rates, robustness, and efficiency. Finally we discuss that using SUPG typestabilization in finite element formulation (4) is vital not only for capturing effects of unresolvedsubscales in solution to (1)–(3), thus improving accuracy of the discrete solution, but also fordeveloping robust and effective iterative methods.

This presentation is based on the collaborative research with M.Benzi [2].

References

[1] M. Benzi, G. H. Golub and J. Liesen, Numerical solution of saddle point problems, ActaNumerica, 14 (2005), pp. 1–137.

[2] M. Benzi, M.A. Olshanskii, An augmented lagrangian-based approach to the Oseen prob-lem, (submitted), available at www.mathcs.emory.edu/~molshan

[3] H. C. Elman, D. J. Silvester and A. J. Wathen, Finite Elements and Fast Iterative Solvers:with Applications in Incompressible Fluid Dynamics, Numerical Mathematics and ScientificComputation, Oxford University Press, Oxford, UK, 2005.

[4] M. Fortin and R. Glowinski, Augmented Lagrangian Methods: Applications to the Numer-ical Solution of Boundary-Value Problems, Studies in Mathematics and its Applications,Vol. 15, North-Holland, Amsterdam/New York/Oxford, 1983.

[5] J. Schoberl, Multigrid methods for a parameter dependent problem in primal variables,Numer. Math., 84 (1999), pp. 97–119.

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Numerical and Asymptotic Analysis of Singularly Perturbed PDEs ofKinetic Theory

N. Parumasur, J. Banasiak and J.M. Kozakiewicz

University of KwaZulu-Natal, Durban, 4041, South Africa parumasurn1@ukzn.ac.za

We consider the numerical solution of various equations occurring in kinetic theory. Thenumerical algorithm is applied in conjuction with a modified asymptotic procedure. Variousmathematical derivations and numerical algorithms are provided for the following singularlyperturbed models of kinetic theory

∂tu +Au + Su +1εCu = 0 (1)

where u is the particle distribution, ∂t is the time derivative and the operators A ,S , and Cdescribe attenuation, streaming and collisions of particles, respectively. We begin by outliningthe features of a modified asymptotic method [2, 1] which is quite useful when solving (1).

Let P be a bounded operator in the Banach space X having zero as its simple isolatedeigenvalue and the corresponding eigenspace V . Then X can be expressed as a direct sumX = V ⊕W, where both V and W are invariant subspaces of the operator C and C is one-to-onefrom W onto itself. Let P be the spectral projection associated with the eigenvalue λ = 0 sothat

V = PX, W = QX,

where Q = I −P is the complementary projection. We use a projection method [2] to write (1)as a system of evolution equations in subspaces V and W . Applying the projections P and Qon both sides of (1), successively, we obtain

∂tv = P(A+ S)Pv + P(A+ S)Qw

ε∂tw = εQ(A+ S)Qw + εQ(S +A)Pv +QCQw, (2)

with the initial conditionsv(0) =

ov, w(0) =

ow,

whereov = P o

u,ow = Q o

u. Taking into account that the projected operators PSP, PAQ andQAP vanish for most types of linear equations we obtain the following form of (2)

∂tv = PAPv + PSQw

ε∂tw = εQSPv + εQSQw + εQAQw +QCQw (3)

v(0) =ov, w(0) =

ow,

Next we apply the modified asymptotic approach to (3). We represent the solution of (3) as asum of the bulk and the initial layer parts:

v(t) = v(t) + v(τ), w(t) = w(t) + w(τ), (4)

where the variable τ in the initial layer part is given by τ = t/ε. The bulk solution will beconsidered as a function of ρ of order zero and the function w(N) will be assumed to be of theform

w(N)(t) =N∑

n=0

εnWnρ(t), (5)

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N. PARUMASUR, J. BANASIAK, J.M. KOZAKIEWICZ: Numerical and AsymptoticAnalysis of Singularly Perturbed PDEs of Kinetic Theory

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%Speaker: PARUMASUR, N. 130 BAIL 2006

where the superscript N indicates the order of the approximation and W are time-independentbounded linear operators from V to W . Substituting this expansion into the first equation in(3) yields

∂tρ = PAPρ +N∑

n=0

εnPSQ(Wnρ). (6)

Expressing the time derivative ∂tρ in (6) in powers of ε and comparing terms of the same powerin ε yields at first order

W0 = 0, W1 = −(QCQ)−1QSP. (7)

The operator W1 can be evaluated since QCQ is invertible on the subspace W. Using (7) in (6)gives the equation

∂tρ = PAPρ− εPSQ(QCQ)−1QSPρ. (8)

A similar procedure yields the initial layer terms

v0(τ) ≡ 0, v1(τ) = PSQ(QCQ)−1eτQCQ ow,

and the initial condition for (8)

v(0) =ov −εPSQ(QCQ)−1 o

w . (9)

We apply the procedure to a wide range of problems of kinetic theory.

References

[1] J. Banasiak, J. Kozakiewicz, N. Parumasur, Diffusion Approximation of Linear KineticEquations with Non-equilibrium Data – Computational Experiments, Transport TheoryStatist. Phys. (accepted).

[2] J. R. Mika, New asymptotic expansion algorithm for singularly perturbed evolution equa-tions, Math. Methods Appl. Sci. 3 (1981) 172-188.

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B. RASUO: On Boundary Layer Control in Two-Dimensional Transonic Wind Tunnels

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B. RASUO: On Boundary Layer Control in Two-Dimensional Transonic Wind Tunnels

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%Speaker: RASUO, B. 133 BAIL 2006

A Comparison of Stabilization Methods for Convection-Diffusion-ReactionProblems on Layer-Adapted Meshes

Hans-G. Roos, TU Dresden

The use of layer adapted meshes allows to prove robust convergence results of Galerkin finite element

methods for convection-diffusion-reaction problems. However, the robust solution of the generated

discrete problems which are in general nonsymmetric is a nontrivial task.

The situation improves if one uses some stabilization. But the question is: which of all the existing

stabilization techniques is optimal?

H.-G. ROOS: A Comparison of Stabilization Methods for Convection-Diffusion-Reaction Problems on Layer-Adapted Meshes

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Discontinuous Galerkin stabilization for convection–diffusionproblems

Hans–Gorg RoosInstitut fur Numerische Mathematik, Technische Universitat Dresden,

Germany

Helena ZarinDepartment of Mathematics and Informatics, University of Novi Sad,

Serbia and Montenegro

Abstract. A convection–diffusion problem with Dirichlet boundary conditionsposed on a unit square is considered. The problem is discretized using a com-bination of standard Galerkin FEM and h–version of the nonsymmetric discon-tinuous Galerkin FEM with interior penalties on a layer–adapted mesh. Withspecially chosen penalty parameters for edges from the coarse part of the mesh,we prove uniform convergence (in the perturbation parameter) in an associatednorm. Numerical tests support our theoretical results.

References

[1] Roos, H.–G., Zarin, H., The discontinuous Galerkin method for singularlyperturbed problems. Numerical Mathematics and Advanced Applications(eds. M. Feistauer et al.), Proceedings of the ENUMATH Conference 2003(Prague, August 18-22, 2003), Springer Verlag, 2004, pp. 736–745

[2] Zarin, H., Roos, H.–G., Interior penalty discontinuous approximations ofconvection–diffusion problems with parabolic layers. Numerische Mathe-matik, 100(4), 2005, pp. 735–759

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H.-G. ROOS, H. ZARIN: Discontinuous Galerkin stabilization for convection-diffusion problems

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%Speaker: ZARIN, H. 135 BAIL 2006

Abstract submitted to the Local Organization Committee of the BAIL 2006 Conference 1

On Turbulent Marginal Separation: How the Logarithmic Law of the Wall is

Superseded by the Half-Power Law ∗

B. Scheichl and A. Kluwick

Institute of Fluid Mechanics and Heat TransferVienna University of Technology

Resselgasse 3/E322, A-1040 Vienna, Austriabernhard.scheichl@tuwien.ac.at

1. The Asymptotic Theory of Marginally Separating Turbulent Boundary Layers

A novel rational theory of the incompressible nominally steady and two-dimensional turbulentboundary layer (TBL) along a smooth and impermeable surface and exposed to an adversepressure gradient, which is impressed by the prescribed external potential free-stream flow, hasbeen developed recently by the authors. This asymptotic flow description exploits the Reynolds-averaged Navier–Stokes equations by taking the limit Re →∞ where Re denotes a Reynoldsnumber formed by using a global length and velocity scale characteristic for the external bulkflow. In the following all quantities are non-dimensional with that global reference scales.

The so-called classical theory, see for instance [1], is capable of describing a strictly attachedTBL only as it employs the assumption of an asymptotically small streamwise velocity defectwith respect to the external flow in the fully turbulent main part of the TBL. The new theory,however, is an extension of the classical approach insofar as it is essentially based only on thehypothesis that the turbulent time-mean motion is governed locally by a single velocity scale. Asan important consequence, by taking a streamwise velocity deficit of O(1), which is a necessarycharacteristic of flows that may even undergo marginal separation, the boundary layer thicknessis measured by a small parameter denoted by α which is seen to be independent of Re as Re →∞.It then can be shown that in the primary limit α→ 0, Re−1 = 0 the TBL is represented by atwo-tiered wake-type flow; remarkably, it thus closely resembles a turbulent free shear layer. Thedescription of the outer main layer is addressed in [2]. There it is demonstrated analytically andnumerically by adopting a local viscous/inviscid interaction strategy that in the primary limitconsidered marginal separation is associated with the occurrence of closed reverse-flow regionswhere the surface slip velocity Us, which is a quantity of O(1) in general, assumes negativevalues along a streamwise distance of O(α3/5). Here we add that the overall slip velocity at thebase of the whole wake region comprising both layers is written as us = Us +O(α3/4), wherethe perturbations reflect the effect of the inner wake having a thickness of O(α3/2).

2. The Near-Wall Flow Regime for Finite Values of Re

It is the main purpose of our contribution to rigorously elucidate how high but finite values ofRe affect the near-wall flow regime in order to provide a rational basis for the calculation of thewall shear stress, in particular immediately up- and downstream of separation and reattachment.We note that the investigation of the latter flow situation is not only a challenge to settle an,for the time being, unsolved problem in turbulent boundary layer theory, which has attractedmany researchers in the past, but also of basic engineering relevance, as none of the presentlyadopted turbulence models are applicable when the wall shear stress changes sign.

In Figure 1 the resulting four-layer TBL structure is depicted where the case of the oncomingfirmly attached flow chosen. We start the analysis by focusing on the viscous wall layer adjacent

∗This research is granted by the Austrian Science Fund (FWF) under project no. P 16555-N12.

B. SCHEICHL, A. KLUWICK: On Turbulent Marginal Separation: How the Logarith-mic Law of the Wall is Superseded by the Half-Power Law

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Abstract submitted to the Local Organization Committee of the BAIL 2006 Conference 2

PSfrag replacements

yy

u

OW

IW

IL VWL

y ∼ δ = O(α)

u ∼ Us

u ∼ us

α3/2 u = O(√y)

u/uτ ∼ κ−1 ln y

Figure 1: Asymptotic splitting and streamwise velocitycomponent u of the initially attached turbulent bound-ary layer with thickness δ, which evolves along the sur-face given by y = 0. The notations OW, IW, IL, andVWL mark the outer and the inner wake, the inter-mediate and the viscous wall layer, respectively. Theasymptotic relationships are explained in the text.

to the surface where convective terms are negligibly small and Reynolds stresses have the samemagnitude as the molecular shear stress. Let y and τw denote, respectively, the coordinateperpendicular to the surface and the wall shear stress. As far as low-order results and theattached case τw > 0 are concerned, the asymptotic description of the wall layer turns out to befully analogous to that in the classical theory. That is, sufficiently far from the positions whereτw vanishes the streamwise velocity component u is expanded in the usual manner according to

u/uτ ∼ u+0 (y

+)+· · · , sgn(τw)uτ = |τw|1/2 = (|∂u/∂y|y=0/Re)1/2, y+ = y |uτ |Re = O(1). (1)

Most important, the hypothesis stated above is seen to require the celebrated logarithmicmatch with the fully turbulent flow regime on top of the viscous wall layer,

u+0 ∼ A±

−1 ln y+ +B±, A± > 0, y+ →∞. (2)

Here the subscripts + and − denote the cases τw > 0 and τw < 0, respectively. Therefore, A+

equals the v. Karman constant, and the values of B± shall refer to a perfectly smooth surface.Note that for separated flows (τw < 0) an asymptotic behavior akin to (2) was already proposedin [3] on semi-empirical grounds. Moreover, the velocity components u in the wall layer and theintermediate layer of thickness τw, see Figure 1, match provided that the skin friction law

uτ/us ∼ A±ε+O(ε2), ε = 1/ ln |u3τRe| → 0, Re →∞, (3)

holds. This relationship between uτ and the aforementioned slip velocity us represents a gener-alization of the well-known classical result which, in principle, is included in (3).

Let η = y/u2τ = O(1) be the coordinate characteristic for the intermediate layer. The match

of u with the inner wake reveals a half-power law, u/uτ = O(√η) for η →∞. Such a behavior

is believed to hold also on top of the viscous wall layer of a separating TBL. On the otherhand, the viscous wall and the intermediate layer collapse and, in turn, (3) ceases to be validwhen us/uτ = O(1), that is, for us = O(Re−1/3). As a highlight of the asymptotic analysis, it ispointed out how that merge of the near-wall layers generates a new strongly viscosity-affectedflow regime near the locations where τw = 0. There (1) is replaced by the appropriate expansion

u/up ∼ u×0 (p×, y×) + · · · , p× = (up/uτ )

3, up = (px/Re)1/3, y× = y upRe = O(1). (4)

Herein the local value px = O(1) of the leading-order streamwise pressure gradient enters thevelocity scale up. Finally, then the generalized logarithmic law (2) is gradually transformed into

u×0 ∼ C(p×)√

y× +D(p×), C(p×) > 0, D(p×) ∼ us/up = O(1), y× →∞. (5)

References

[1] G.L. Mellor, “The Large Reynolds Number, Asymptotic Theory of Turbulent Boundary Layers” J. Engn

Sci., 10, 851–873 (1972).

[2] B. Scheichl and A. Kluwick, “Turbulent Marginal Separation and the Turbulent Goldstein Problem”, AIAA

paper 2005-4936 (2005), also AIAA J. (submitted in extended form).

[3] R.L. Simpson, “A Model for the Backflow Mean Velocity Profile”, Technical Note, AIAA J., 21 (1), 142–143(1983).

B. SCHEICHL, A. KLUWICK: On Turbulent Marginal Separation: How the Logarith-mic Law of the Wall is Superseded by the Half-Power Law

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%Speaker: SCHEICHL, B. 137 BAIL 2006

Boundary and Interior Layers in Turbulent Thermal Convection ∗

O. Shishkina & C. Wagner

DLR - Institute for Aerodynamics and Flow Technology,Bunsenstrasse 10, 37073 Gottingen, Germany

Olga.Shishkina@dlr.de, Claus.Wagner@dlr.de

1. Introduction

Numerous scientifical problems and industrial applications require solutions of the Rayleigh-Benard problem, i.e. turbulent convection of fluids heated from below and cooled from above,with the Rayleigh number (Ra) from 105 up to 1020. For a review on this classical problem andfor the references to earlier literature we refer to the paper Ahlers, Grossmann and Lohse [1].

Since the diffusion coefficient in the Navier-Stokes equation, which is inversely proportional tothe square root of Ra, is very small, the solution - both the temperature and the velocity fields -have very thin boundary layers near the horizontal walls. For moderate Rayleigh numbers (from105 up to 108) interior layers, i.e. thermal plumes, are also observed. The boundary layers, thethermal plumes and the turbulent background are indicated, respectively, by high, moderate andsmall values of the temperature gradient norm and, hence, by large, moderate and small valuesof the thermal dissipation rate. By means of direct numerical simulations (DNS) we invesigateboundary and interior layers which take place in turbulent Rayleigh-Benard convection.

2. Governing equations and the numerical method

The governing dimensionless equations for the Rayleigh-Benard problem in Boussinesq approx-imation can be written in cylindrical coordinates (z, r, ϕ) as follows

ut + u · ∇u +∇p = Γ−3/2Ra−1/2Pr1/2∇2u + Tz, (1)Tt + u · ∇T = Γ−3/2Ra−1/2Pr−1/2∇2T, (2)

∇ · u = 0, (3)

where u is the velocity vector, T the temperature, ut and Tt their time derivatives, p thepressure. Here Ra = αgH3∆T/(κν) denotes the Rayleigh number, Pr = ν/κ the Prandtlnumber, Γ = D/H the aspect ratio with H the height and D the diameter of the cylindricalcontainer. Further, α is the thermal expansion coefficient, g the gravitational acceleration, ∆Tthe temperature difference between the bottom and the top plates, ν the kinematic viscosity andκ the thermal diffusivity. The dimensionless temperature varies between +0.5 at the bottomplate to −0.5 at the top plate. An adiabatic lateral wall is prescribed by ∂T/∂r = 0. Finally, onthe solid walls the velocity field vanishes according to the impermeability and no-slip conditions.

To investigate turbulent Rayleigh-Benard convection in cylindrical containers of the aspectratios Γ = 10 and Γ = 5 for Ra from 105 to 107 and Pr = 0.7 we conducted DNS. The simulationswere performed with the fourth order accurate finite volume method developed for solving (1)– (3) in cylindrical coordinates on staggered non-equidistant grids. For the numerical methodused in the DNS we refer to [2].∗The work is supported by Deutsche Forschungsgemeinschaft (DFG) under the contract WA 1510-1

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Ra = 105 Ra = 106 Ra = 107

Figure 1: Snapshots of the temperature, −0.37 ≤ T ≤ 0.37, for Γ = 10 and z = H/(2Nu).Colour scale spreads from blue (negative values) through white (zero) to red (positive values).

3. Mesh generation

In [3] it was proven that the ratio of the area averaged (over the top or the bottom plates) tothe volume averaged thermal dissipation rate εθ = Γ−3/2Ra−1/2Pr−1/2(∇T )2 is greater thanor equal to the Nusselt number (Nu = Γ1/2Ra1/2Pr1/2 〈uzT 〉t,S − Γ−1

⟨∂T∂z

⟩t,S≥ 1) for all Ra,

Pr and Γ. It means that the largest values of εθ take place in the boundary layers near thehorizontal walls. To resolve these boundary layers special meshes are required.

The grid equidistribution ansatz (see for example [4]) enables to detect the boundary layersand construct appropriate meshes for their resolution. In our solution-adapted mesh genera-tion algorithm we used grid equidistribution of the arc-length of the mean temperature profilescomputed on equidistant meshes. The algorithm produces meshes that lead to principally moreaccurate solutions in comparison with the equidistant meshes.

4. Numerical experiments

Using the solution-adapted meshes with 110, 192 and 512 nodes in z-, r- and ϕ-directions,respectively, we conducted the DNS of three-dimensional Rayleigh–Benard convection in widecylindrical containers of the aspect ratios Γ = 10 and Γ = 5 and for moderate Rayleigh numbersfrom 105 to 107. The snapshots of the temperature field on the borders between the lowerthermal boundary layers and the bulk are presented in Fig. 1 as they were obtained in the DNSfor Γ = 10. The coherent flow patterns reveal the horizontal extension of hot and cold plumes.Analysing spatial distribution of εθ for different Ra we conclude that the role of the thermalplumes in thermal convection decreases with growing Ra. Some other physical results obtainedin the DNS of turbulent Rayleigh–Benard convection in wide containers are discussed in [3].

References

[1] G. Ahlers, S. Grossmann & D. Lohse, “Hochprazision im Kochtopf: Neues zur turbulentenWarmekonvektion”, Physik Journal, 1, 31–37 (2001).

[2] O. Shishkina & C. Wagner, “A fourth order accurate finite volume scheme for numericalsimulations of turbulent Rayleigh-Benard convection”, C. R. Mecanique, 333, 17–28 (2005).

[3] O. Shishkina & C. Wagner, “Analysis of thermal dissipation rates in turbulent Rayleigh-Benard convection”, J. Fluid Mech., 546, 51–60 (2006).

[4] N. Kopteva & M. Stynes, “A robust adaptive method for a quasilinear one-dimensionalconvection-diffusion problem”, SIAM J. Numer. Anal., 39, 1446–1467 (2001).

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O. SHISHKINA, C. WAGNER: Boundary and Interior Layers in Turbulent ThermalConvection

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Using rectangular Qp elements in the SDFEM for a convection-diffusion

problem with a boundary layer

Martin Stynes1 and Lutz Tobiska2

1 National University of Ireland, Cork, Ireland.m.stynes@ucc.ie

2 Otto-von-Guericke Universitat Magdeburg, Postfach 4120, D-39016 Magdeburg, Germanytobiska@mathematik.uni-magdeburg.de

We consider the convection-diffusion problem

−ε∆u + b · ∇u + cu = f on Ω = (0, 1)2, u = 0 on ∂Ω,

where the parameter ε lies in the interval (0, 1], the function b(x, y) = (b1(x, y), b2(x, y)) withb1(x, y) > β1 > 0 and b2(x, y) > β2 > 0, c(x, y) ≥ 0 on Ω, and c(x, y) − (div b(x, y))/2 ≥ c0 > 0on Ω. We assume that the functions b, c, and f are sufficiently smooth. Layer-adapted meshesare usually used for solving the singularly perturbed boundary value problem since its solutiontypically has boundary layers at the sides x = 1 and y = 1 of Ω.

The convergence properties of the streamline diffusion finite element method (SDFEM; themethod is also known as SUPG) on a rectangular Shishkin mesh are analyzed. The trial functionsin the SDFEM are piecewise polynomials that lie in the space Qp, i.e., are tensor productsof polynomials of degree p in one variable, where p > 1. In [1, 2], for sufficiently small ε(ε ≤ N−1/2 ln2 N), the error bound

‖uI − uN‖SD ≤ C N−2 ln2 N

has been proven in the case p = 1, where uI is the nodal interpolant of the solution u, uN isthe SDFEM solution, and ‖ · ‖SD is the streamline-diffusion norm. This error bound is based onanisotropic interpolation estimates, superconvergence of the piecewise linear nodal interpolation,and a detailed study of the behaviour of the solution in the different parts of the domain.The main objective is to extent the error analysis to the case p > 1 by using a nonstandardinterpolation. A detailed study of the approximation and superconvergence properties leads tothe estimate

‖uI − uN‖SD ≤ C N−(p+1/2), p > 1.

Comparisons are made between this result and the corresponding result for the case p = 1, whichturns out to be exceptional [3]. Moreover, we discuss the possibilities to derive similar estimatesfor ‖u − PuN‖SD and ‖u − uN‖d,SD, where PuN denote a suitable postprocessed numericalsolution and ‖ · ‖d,SD a discrete version of the streamline-diffusion norm, respectively.

References

[1] M. Stynes and L. Tobiska, “Analysis of the streamine-diffusion finite element method fora convection-diffusion problem with exponential layers”, East-West J. Numer. Math., 9,59–76 (2001).

[2] M. Stynes and L. Tobiska, “The SDFEM for a convection-diffusion problem with a boundarylayer: Optimal error analysis and enhancement of accurary”, SIAM J. Numer. Anal., 41,1620–1642 (2003).

[3] L. Tobiska, “Analysis of a new stabilized higher order finite element method for advection-diffusion equations”, Tech. Rep. 05-36, Department of Mathematics, Otto-von-GuerickeUniversity (2005).

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M. STYNES, L. TOBISKA: Using rectangular Qp elements in the SDFEM for aconvection-diusion problem with a boundary layer

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%Speaker: TOBISKA, L. 140 BAIL 2006

Numerical Approximation of Flow

Induced Airfoil Vibrations

Petr SvacekDepartment of Technical Mathematics, Karlovo namestı 13,121 35 Praha 2, CTU, Faculty of Mechanical Engineering.

The strong interaction between the aerodynamic field and the aero-elastic fieldplays an important role in the design of aerospace vehicles. The fluid-structureinteraction is usually a complex nonlinear problem. The aero-elastic stabilityof aerospace vehicles has a great impact on their design. In this paper we areinterested in the interaction of two dimensional incompressible viscous laminarflow and a solid airfoil. The airfoil can rotate and oscillate in vertical direc-tion. The numerical simulation of such a problem is very challenging topic - itconsists of discretization and stabilization of the Navier-Stokes equations for ahigh Reynolds number. Also the nonlinear discrete problem has to be treatedcarefully. Moreover, the computational domain is time dependent and one hasto recompute it for each time step together with the used grid. In order to takethe grid motion into account the Arbitrary Lagrangian-Eulerian formulation ofthe Navier-Stokes equation is used.The most difficult part of the fluid-structure interaction is a fluid flow simulation.We consider viscous laminar incompressible two dimensional flow which is fullytime dependent

DAuDt

− ν4u + ((u−wg) · ∇)u +∇p = 0, in Ωt,

∇ · u = 0, in Ωt

equipped with appropriate boundary conditions.The above problem is discretized by the finite element method(FEM). Never-theless, the Galerkin FEM leads to unphysical solutions if the grid is not fineenough in regions of strong gradients (e.g. boundary layer). In order to obtainphysically admissible correct solutions it is necessary to apply suitable meshrefinement combined with a stabilization technique giving stable and accurateschemes. In our paper we present a special version of the GLS stabilizationmethod for Navier-Stokes equations, see [1], [2].Further, the computation of the force acting on the airfoil requires correct eval-uation of boundary integral of the stress tensor. A straightforward evaluationof the stress tensor integral may lead to inaccurate results. This obstacle isavoided with the aid of a weak formulation of the force acting on the profile.

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The fluid forces acting on the profile causes its motion or deformation, whichdepends on the airfoil properties. We simulate the situation, when the airfoil isa solid body with two degrees of freedom. Its motion is obtained as the solutionof two ordinary differential equations coupled with the Navier-Stokes system.We discuss the GLS stabilization of the FEM, evaluation of forces acting onthe profile, time discretization and the solution of the discrete problem. Theobtained results are compared with available data.

References

[1] Svacek, P., Feistauer, M. Application of a stabilized FEM to problemsof aeroelasticity. In: Feistauer, M., Dolejsı, V., K., N. (Eds.),NumericalMathematics and Advanced Applications, ENUMATH2003. Springer, Hei-delberg, pp. 796–805, 2004

[2] Svacek, P. , Feistauer, M., Horacek, J., Numerical simulation of flowinduced airfoil vibrations with large amplitudes. Journal of Fluids andStructures(submitted), 2004.

P. SVACEK: Numerical Approximation of Flow Induced Airfoil Vibrations

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%Speaker: SVACEK, P. 142 BAIL 2006

Full asymptotic analysis of the Navier-Stokes equations

in the problems of gas flows over bodies

with large Reynolds number

N.V.Tarasova

Baltic State Technical University1, 1st Krasnoarmeiskaya ul., 190005 St Petersburg, Russia

E-mail: natali_t3@professor.ru, tsrknv@bstu.spb.su

Investigation of the gas flows over bodies with large Reynolds number in most cases can besignificantly simplified when the flow area is divided into two parts: the external inviscid oneand the narrow area near the body surface well-known as viscous boundary layer. As this takesplace the Navier-Stokes equations describing such flows are splitted into the Eulier equations forthe external flow area and the Prandtl boundary layer equations. Such splitting of the problemis obtained on the base of the matched asymptotic expansion method proposed by Van Dike forhypersonic flow [1]. All gas parameters are written in the form of power series in the standardsmall parameter ε = 1/

√Re∞ (Re∞ is the Reynolds number in the free stream flow) and the

system of the Navier-Stokes equations reduces to the sequences of partial differential equationsystems in the external flow area and internal flow area. As a result, in a first approximation wederive the Euler equations (in external flow area) and the Prandtl boundary layer equations.

Traditionally the model of incompressible gas flow is used both in the external flow areaand inside the boundary layer if the Mach number in the free stream M∞ is less than 0.3 andthe characteristic temperature drop is small enough, otherwise the model for compressible gasflow is used. At the same time in practice in many problems of gas flows over bodies, such ascalculation of induced convection in heat exchanges, we have hyposonic gas flows in which thegas temperature across the boundary layer on the body surface can be changed significantly thatleads to considerable changes of the gas density. As a result, the gas flow outside the boundarylayer can be considered as incompressible but inside the layer as essentially compressible one.In the described case the researchers usually use the common model of the compressible gasboth inside the boundary layer and in the external inviscid flow area. However, such approachimplies the necessity to solve the compressible Euler equations in the external area that leads tosignificant computational difficulties if the Mach number in the free stream flow is small enough(in the case of hyposonic flows).

The main aim of this paper is to carry out the strict asymptotic analysis based on the methodof matched asymptotic expansions of the complete system of the Navier-Stokes equations in allpossible situations:

1) flows with the Mach number greater than 0.3 and the large temperature drop across theboundary layer,

2) flows with the Mach number greater than 0.3 and the small temperature drop across theboundary layer,

3) flows with the Mach number less than 0.3 and the large temperature drop across theboundary layer,

4) flows with the Mach number less than 0.3 and the small temperature drop across theboundary layer,

and as a result to suggest the setting of the problem for the Euler equations and the boundarylayer equations including the procedure of sewing together the appropriate gas parameters.

Cases 1) and 4) are seen to give us the well-known models for the compressible and incompres-sible gas flow in both areas respectively. Because of this, they have not been the major subjectof investigation in the present study. At the same time cases 2) and 3) give us some intermediatesituations which are of great interest. This problem is not trivial for example for case 3) because

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N.V. TARASOVA: Full asymptotic analysis of the Navier-Stokes equations in theproblems of gas flows over bodies with large Reynolds number

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the application of the Euler equations for incompressible flow outside the boundary layer and theclassical equations of the compressible boundary layer near the body surface makes impossiblethe agreement between both equation systems.

To consider the proposed problems from common positions another special dimensionlessvariables for gas parameters in the Navier-Stokes equations are introduced. These new variablesare varied inside the studied flow area usually from 0 to 1 (in other words, they can vary to thevalue of the order of unit). The main idea to use such special dimensionless variables is that inthis case all governing parameters that can be both small and not small are indicated explicitlyas the coefficients in the equation system. Among these parameters are M 2

∞and ∆T/T0 (∆T is

the temperature drop and T0 is the characteristic value of the temperature in the area).It should be noticed that in the hyposonic flows the gravity force can affect significantly the

gas flow (in studies of free and induced convection). In term of asymptotic analysis it meansthat it is necessary to take account of one more parameter, for example 1/Fr (Fr is the Frudnumber) that can be both small and not small.

The full asymptotic analysis is carried out for all mentioned situations on the base ofcomparison of all parameters mentioned above (M 2

∞, ∆T/T0, 1/Fr) with the standard small

parameter ε in order of magnitude. As a result, the model for gas flow in both areas is formulatedand an attempt to construct the procedure of the agreement of the solutions in both areas ismade. It should be stressed that the equation systems derived under the assumptions 1) - 4)differ from each other.

This model constructed for the cases 2) and 3) possibly can occupy the intermediate placebetween two classical approaches when a gas is considered as incompressible or compressible oneover the whole flow field.

The equations describing hyposonic flows (M → 0) with the arbitrary values of the Reynoldsnumber (Re), ∆T/T0 and 1/Fr were derived in [2] from the Navier-Stokes equations on the baseof the asymptotic analysis with the Mach number (M) as a small parameter. In this work anattempt to compare the model constructed for cases 2) and 3) with the results obtained in [2]when considered flow with the large Reynolds number was made.

References

1. M. Van Dyke. 1962 In: Hypersonic Flow Research (Ed. F.R.Riddell). Academic Press.2. A.I. Zhmakin, Yu.N. Makarov. 1985 Numerical modelling of hyposonic flows of viscous gas,

Dokl. AN SSSR, Mekh. Zh. i Gaza 280 (4), 827–830. [in Russian]

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N.V. TARASOVA: Full asymptotic analysis of the Navier-Stokes equations in theproblems of gas flows over bodies with large Reynolds number

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%Speaker: TARASOVA, N.V. 144 BAIL 2006

International Conference: BAIL 2006 Boundary and Interior Layers

- Computational & Asymptotic Methods - Effects of golf ball dimple configuration on aerodynamics, trajectory, and acoustics

*Chang-Hsien Tai ＋ Chih-Yeh Chao＋＋ Jik-Chang Leong＋

Qing-Shan Hong＋

*Corresponding author Department of Vehicle engineering, National Pingtung University of Science and Technology＋

Department of Mechanical engineering, National Pingtung University of Science and Technology＋＋

1, Hseuh Fu Road, Neipu Hsiang, Pingtung Taiwan, R.O.C. Fax: 886-8-7740398 E-mail: chtai@mail.npust.edu.tw

Abstract

In many reports about golf ball, including the history of its development, have introduced the standards on golf ball specification. However, there is not a single well-documented solid requirement found for the design of golf ball surface. Not only have a lot of reports discussed the material and structure of a golf ball, but also most of the golf ball manufacturers improve their products by modifying the number of layers beneath the golf ball surface and their materials. Even so, there are relatively very few papers focused on the influence of different concave surface configurations on the aerodynamic characteristics of the golf ball. Furthermore, the noise a golf ball generates in a tournament is very likely to affect the emotion and hence the performance of the golf ball player. For these reasons, this study investigates the performance of a golf ball based on the CFD method with the validation using a wind tunnel. In 1938, Goldstein [1] had proposed an important parameter – the spin ratio. In corporation with different Reynolds numbers, this parameter makes the study of life and drag effects feasible for whirling smooth bodies. In his book, Jorgensen [2] especially emphasized that the main objective of concaved surfaces on a golf ball is to generate small scale turbulence. When flying, this turbulence postpones air separation, reduces the low pressure region trailing the golf ball, and eventually lowers the air drag. Warring [3] used numerical approach to perform a series of studies related to golf ball using Excel spreadsheets. The goal of his paper was to provide guidance for golf ball players and manufacturers so that their golf ball was capable of flying for a longer distance. In the study of acoustics, Singer, et al. [4] calculated

the noise level from a source using a hybrid grid system with the help of Lighthill’s acoustics analytic approach. On the other hand, Montavon, et al. [5] combined CFD method and Computational Aeroacoustics Approach (CAA) to simulate noise generation from a cylinder. Using CFX-5 with LES (Large Eddy Simulation) as their turbulence model and Ffowcs-Williams Hawkings formulation, they had successfully shown that their predicted sound levels agreed very well with theoretical ones for Reynolds numbers about 1.4×105. Figure 1 shows the flow field around a typical golf ball (Case 1). In Case 2, additional dimples are added onto the golf ball considered in Case 1. The orientation of these additional dimples is depicted in Figure 3. It is found, based on Figure 2, that the flow field associated to Case 2 is no longer symmetrical because of the presence of the additional dimples. Figure 3 demonstrates the distribution of lift and drag coefficients of Cases 1 and 2. Clearly, the addition of small dimples increases the drag. This implies that the golf ball in Case 2 suffers more serious drag effect at low trajectory speeds. The lift the golf ball in Case 2 experiences at moderate Reynolds numbers increases so greatly that it becomes greater than that for Case 1. The life force in overall is therefore greater for Case 2 than Case 1. Although the drag imposed on the golf ball is always smaller for Case 1 than for Case 2, the drag in Case 1 is only about 38.5% less than that in Case 2. However, the lift in Case 2 is 103% greater than that in Case 1. This somewhat indicates the lift effect is 2.68 times of the drag effect. The overall performance of the golf ball for Case 2 is much greater than that for Case 1. Therefore, the former golf ball is capable of traveling further, as shown in Figure 4.

C.H. TAI, C.-Y. CHAO, J.-C. LEONG, Q.-S. HONG: Effects of golf ball dimpleconfiguration on aerodynamics, trajectory, and acoustics

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%Speaker: HONG, Q.-S. 145 BAIL 2006

International Conference: BAIL 2006 Boundary and Interior Layers

- Computational & Asymptotic Methods - This thesis used structured and non-structured grids to come out with the most appropriate grid systems for the current golf balls simulations. Then, numerical simulations were carried out to estimate the aerodynamics parameters and noise levels for various kinds of golf balls having different dimple configurations. With the obtained aerodynamics parameters, the flying distance and trajectory for a golf ball were determined and visualized. The results showed that structured grids produced more accurate results. In terms of dimple layout, the lift coefficient of the golf ball increased if small dimples were added between the original large dimples. When launched at small angles, golf balls with deep dimples were found to have greater lift effect than drag effect. Therefore, the golf balls would fly further until a critical depth of 0.25 mm. As far as noise generation was concerned, deep dimples produced lower noise levels.

Keywords: golf ball, CFD, dimple, flying trajectory, noise

Fig.1 The velocity vector contour on rotation of

Case1(Re=1×105)

Fig.2 The velocity vector contour of

Case2(Re=1×105)

Fig.3 Life, Drag coefficient of Case1 and Case2

Distance (m)

Height(m)

50 100 150 200 2500

50

100

150

200

250Case1 = 215.2mCase2 = 262.1m

Fig.4 Flying trajectory of Case1 and Case2

Reference

[1] Goldstein, S., “Modern Developments in Fluid Dynamics,” Vols. I and II. Oxford: Clarendon Press, 1938.

[2] Jorgensen, T. P., “The Physics of Golf, 2nd edition,” New York: Springer-Verlag, pp. 71-72, 1999.

[3] Warring, K. E., “The Aerodynamics of Golf Ball Flight,” St. Mary’s College of Maryland, pp. 1-37, 2003.

[4] Singer, B. A., Lockard, D. P. and Lilley, G. M., “Hybrid Acoustic Predictions,” Computers and Mathematics with Application 46, pp. 647-669, 2003.

[5] Montavon, C., Jones, I. P., Szepessy, S., Henriksson, R., el-Hachemi, Z., Dequand, S., Piccirillo, M., Tournour, M. and Tremblay, F., “Noise propagation from a cylinder in a cross flow: comparison of SPL from measurements and from a CAA method based on a generalized acoustic analogy,” IMA Conference on Computational Aeroacoustics, pp. 1-14, 2002.

Reynolds number

Dragcoefficient

Liftcoefficient

103 104 1050

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16Case1-Drag coefficientCase2-Drag coefficientCase1-Lift coefficientCase2-Lift coefficient

C.H. TAI, C.-Y. CHAO, J.-C. LEONG, Q.-S. HONG: Effects of golf ball dimpleconfiguration on aerodynamics, trajectory, and acoustics

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Uniformly Convergent Numerical Methods for Singularly Perturbed DelayDifferential Equations∗

Hongjiong Tian

Department of Mathematics, Shanghai Normal University,100 Guilin Road, Shanghai 200234, P. R. China

hjtian@shnu.edu.cn

1 Introduction

Singularly perturbed problems form a special class of problems containing a small parameterwhich may tend to zero. Singularly perturbed delay differential equations (DDEs) has arose inmany fields, such as in the study of an “optically bistable device” [1] and in a variety of modelsfor physiological processes or diseases [2]. Such problems include a subclass of what we frequentlythought of as “stiff” equations. We will concentrate on uniformly convergent numerical methodsfor linear and nonlinear singularly perturbed delay differential equations with a fixed lag.

2 Uniformly convergent schemes

Linear problemsConsider linear singularly perturbed delay differential equations

Ly(t) ≡ εy′(t) + a(t)y(t) = b(t)y(t− 1) + g(t), 0 ≤ t ≤ T,y(t) = φ(t), −1 ≤ t ≤ 0,

(1)

where ε > 0 is a small parameter, a(t) ≥ α > 0, b(t), φ(t) and g(t) are smooth functions. Weconcentrate on the following difference schemes

Lhyi ≡ εσi(ρ)D+yi + a(ti)yi+1 = b(ti)yi−m + g(ti), i ≥ 0,y−j = φ(t−j), j = 0, 1, 2, . . . , m,

(2)

andLhyi ≡ εσi(ρ)D+yi + a(ti+1)yi+1 = b(ti+1)yi+1−m + g(ti+1), i ≥ 0,y−j = φ(t−j), j = 0, 1, 2, . . . , m,

(3)

where the step length h satisfies the constraint 1 = mh with a positive integer m, ti = ih, ρ = hε

and

σi(ρ) =

ρa(0)

1−exp(−ρa(0)) exp(−ρa(0)), i = 0, 1, · · · ,m− 1,ρa(j)

1−exp(−ρa(j)) exp(−ρa(j)), i = jm, jm + 1, · · · , (j + 1)m− 1, j = 1, 2, · · · .(4)

Nonlinear problemsConsider nonlinear problems of the form

εy′(t) = f(t, y(t), y(t− 1)), 0 ≤ t ≤ T,y(t) = φ(t), −1 ≤ t ≤ 0,

(5)

∗The work of this author is supported in part by E-Institutes of Shanghai Municipal Education Commission(No. E03004), Shanghai Municipal Science and Technology Commission (No.03QA14036), Science and TechnologyFoundation of Shanghai Higher Education (No.03DZ21), and The Special Funds for Major Specialties of ShanghaiEducation Committee.

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H. TIAN: Uniformly Convergent Numerical Methods for Singularly Perturbed DelayDifferential Equations

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where we assume that −fy(t, y, z) ≥ α > 0 for all t ≥ 0 and all real y, z. We now linearize (5)and introduce the Newton sequence ys(t)∞s=0 for the initial guess y0(t) satisfying the initialcondition y0(t) = φ(t), t ∈ [−1, 0]. This is done by defining ys+1(t), for all s ≥ 0, to be thesolution of the linear problem

Lsys+1(t) ≡ εdys+1(t)dt − fy(t, ys(t), ys(t− 1))ys+1(t)

−fz(t, ys(t), ys(t− 1))ys+1(t− 1)= f(t, ys(t), ys(t− 1))− fy(t, ys(t), ys(t− 1))ys(t)

−fz(t, ys(t), ys(t− 1))ys(t− 1), t > 0,ys+1(t) = φ(t), −1 ≤ t ≤ 0.

(6)

We may show that not only the convergence of this sequence is quadratic, but also its propor-tionality constant is independent of s and ε. If the initial guess y0(t) is sufficiently close to y(t),then the Newton sequence ys(t)∞s=0 converges to y(t).

3 Numerical experiments

Considerεy′(t) = −y(t) + y2(t− 1), t ≥ 0,

y(t) = 2, t ∈ [−1, 0].

We take initial guess as

y0(t) =

4− 2e−

tε , t ∈ [0, 1),

16− (8 + 16 t−1ε )e−

t−1ε − 4e−2 t−1

ε , t ∈ [1, 2).

The true solution and the numerical solution using the optimal scheme after two iterations areplotted in Figure 1. This figure indicates that the uniformly convergent scheme works well alsofor nonlinear problem.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 22

4

6

8

10

12

14

16

18

t

y

Numerical and analytic solutions:ε y’(t)=−y(t)+y2(t−1),ε=10−2

Numerical solutionTrue solution

Figure 1: Comparison between numerical and analytical solutions.

References

[1] M. W. Derstine, H. M. Gibbs, F. A. Hopf, D. L. Kaplan, Bifurcation gap in a hybrid opticalsystem, Phys. Rev., A, 26(1982)3720–3722.

[2] M. C. Mackey, L. Glass, Oscillation and chaos in physiological control systems, Science,197(1977)287–289.

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H. TIAN: Uniformly Convergent Numerical Methods for Singularly Perturbed DelayDifferential Equations

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Highly accurate 9th-order schemes and their applications to

DNS of thin shear layer instability

A.I.Tolstykh , M.V.Lipavskii, E.N.Chigerev

Computing Center of Russian Academy of Sciences, Vavilova str.40, 119991 Moscow GSP-1,Russia e-mail tol@ccas.ru

ABSTRACT

The principle of constructing arbitrary-order approximations and schemes is outlined. Itsessence is forming linear combinations of basis operators from certain types of one-parametricoperators families by fixing distinct values of the parameter. The details of the procedure firstproposed in [1] can be found in [2],[3] where the linear combinations were referred to as mul-tioperators. The multioperators were designed for parallel machines providing approximationorders which are linear functions of numbers of processors involved in calculations.

In the present talk, extremely accurate ninth-order multioperators-based schemes for fluiddynamics equations are presented, the basis operators being Compact Upwind Differencing(CUD) ones from [4]. The schemes preserve upwinding and conservation properties of CUDschemes; they are characterized by very small phase & amplitude errors for physically relevantwave numbers supported by grids and damping spurious oscillations. They are capable toresolve properly small scale phenomena using reasonable meshes and allow to perform high-accuracy unsteady calculations for large time intervals. Their properties make them very useful,in particular, for thin layers, DNS and LES computations.

Illustrative examples followed by direct numerical simulations of thin incompressible 2D shearlayers instability are presented. The Navier-Stokes calculations were carried out for various highReynolds number flows with complete resolution of turbulent scales as well as for zero molecularviscosity. In the latter case, ninth-order dissipative mechanism was responsible for generatingsmall-scale vorticity. The results obtained for large time intervals show the full history of theflow development with rolling-up,pairing,generation and decaying of turbulence.The resultingenergy and enstrophy spectra are discussed.

References

[1] A. I. Tolstykh, Multioperator high-order compact upwind methods for CFD parallel calcula-

tions, in Parallel Computational Fluid Dynamics, Elsevier, Amsterdam, 1998, pp. 383-390.

[2] A. I. Tolstykh, On ultioperators principle for constructing arbitrary-order difference

schemes, Applied Numerical Mathematics 46 (2003),pp.411-423

[3] A. I. Tolstykh, Centered prescribed-order approximations with structured grids and result-

ing finite-volume schemes, Applied Numerical Mathematics, 49(2004),pp.431-440.

[4] A. I. Tolstykh, High accuracy non-centered compact difference schemes for fluid dynamics

applications, World Scientific, Singapore, 1994.

A.I. TOLSTYKH, M.V. LIPAVSKII, E.N. CHIGEREV: Highly accurate 9th-orderschemes and their applications to DNS of thin shear layer instability

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%Speaker: TOLSTYKH, A.I. 149 BAIL 2006

ABOUT UNSTEADY BOUNDARY LAYER ON A DIHEDRAL ANGLEM.M. VasilievKeldysh Institute of Applied Mathematics, Miusskaja sq. 4, Moscow 125047, Russiavasiliev@keldysh.ruThe unsteady ow of the viscous incompressible uid is considered. This ow is causedby the sudden motion of the dihedral angle with the constant velocity U in the uidbeing at rest. Let denote the linear angle of . It is assumed, that the angle moves inthe direction of the edge (0z) and only one velocity component of uid w in this directionis dierent from zero. Such ows are called by layered[1].The unsteady layered ow caused by the sudden motion of an innite at plate isinvestigated rst by Stokes[2]. Steady boundary layer on the right dihedral angle wasconsidered by Loytsjansky[3].Let us introduce the cylindrical coordinates (r; #; z).We connect this coordinates withthe angle . Since is innite in the direction of axis 0z and the ow is caused only bythe wall motion, we shall assume that all hydrodynamic functions are independent fromcoordinate z. According to made assumptions, the motion equations reduce to followingone equation: @w@t @2w@s2 + 1s @w@s + 1s2 @2w@#2 ! = 0: (1)where t is the time-coordinate, s = r=p, kinematic viscous coecient.The considered ow simulates roughly a boundary layer in the neighborhood of theintersection the wing and the fuselage of an aircraft at enough distances from the leadingand the trailing edges of the wing.In this work the power geometry methods[4] are used for obtaining of self-similarsolutions of boundary-value problems. These methods have simple algorithms. Theywere applied succesifulle both to linear and nonlinear problems in works [5] [9] andothers.For the self-similar variables 1 = s=pt; 2 = # the equation (1) is12 @2@12 + 1 1 + 1212 @@1 + @2@22 = 0: (2)where w = U(1; 2). Solution of the equation (2) by corresponding boundary conditionswas obtained in the form w = U 1(1) cos (#); (3)where = = and function 1 is determined as a result of the solution of the boundary-value problem for the equation12100 + 1 1 + 121210 21 = 0; (4)The analytical solution of the considered problem in Cartesian coordinates (x; y; z) wasobtained only in case = =2. This solution isw = U erf erf ; (5)where = x2pt; = y2pt; erf ' = 2p 'Z0 el2 dl:

M. VASILIEV: About unsteady Boundary Layer on a dihedral angle

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%Speaker: VASIELIEV, M. 150 BAIL 2006

It is checked that asymptotics of the solutions (3) and (5) coinside both by r ! 0 andr ! +1. References[1] Schlichting, H., Grenzschicht-Theorie. Verlag G. Braun. Karlsruhe. 1951.[2] Stokes, G.G., On the eect of internal friction of uid on the motion of pendumlums.Trans. Cambr. Phil.. IX, 8. 1851.[3] Loytsjansky, L.G., Laminar Boundary Layer. Fizmatgiz, Moscow, 1962 (Russian).[4] Bruno, A.D., Power geometry in algebraic and dierential equations. Fizmatlit,Moscow, 1998 (Russian). = Elsevier Science, Amsterdam, 2000.[5] Bruno, A.D., Algorithmic analysis of singular perturbations and boundary layersby power geometry. Proc. of the International Conference on Boundary and InteriorLayers (BAIL 2002), Perth, Western Australia, 2002, pp. 251-256.[6] Vasiliev, M.M., Asymptotics of some viscous, heat conducting gas ows //Proc.of the International Conference on Boundary and Interior Layers (BAIL 2002), Perth,Western Australia, 2002, pp. 251-256.[7] Vasiliev, M.M., About the obtaining self-similar solutions of the Navier-Stokesequations by methods of power geometry//Proc of ISAAC-2001, World Scientic, Singapore,2003, pp. 93-101.[8] Bruno, A.D., Shadrina, T.V., The axially symmetric boundary layer on a needle//Proc. of the International Conference on Boundary and Interior Layers(BAIL 2004),Toulouse, France, 2004, 9 pp.[9]Vasiliev, M.M., On the self-similar solution of some magnetohydrodynamic problems//Proc. of the International Conference on Boundary and Interior Layers(BAIL 2004),Toulouse, France, 2004, 6 pp.

M. VASILIEV: About unsteady Boundary Layer on a dihedral angle

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%Speaker: VASIELIEV, M. 151 BAIL 2006

High-order symmetry-preserving discretization on strongly stretched grids

Arthur E.P. Veldman

Institute of Mathematics and Computing Science, University of Groningen

P.O. Box 800, 9700 AV, Groningen, The Netherlands

Many physical phenomena feature thin boundary layers, in which the solution varies much morerapidly than elsewhere in the domain of interest. A ‘natural’ approach is to adapt a computational gridto the variations of the solution. In this way one obtains grids with a large diversity in mesh size. Alsothe size of adjacent mesh cells can be quite different, i.e. the grid shows strong stretching.

‘Traditional’ discretization methods (based on Lagrangian interpolation) focus on minimizing localtruncation error, but experience has shown that these methods prefer low grid stretching rates (e.g. [2]and the refernces therein). The problems arise because this approach does not take into account theproperties of the discrete system matrices that arise after discretization. An alternative approach is todevelop discretization schemes with the properties of the system matrix in mind - a general name forthis philosophy is mimetic discretization. Here certain properties of the analytic operator are mimicedin their discrete counterpart.

At RuG we have chosen to retain the symmetry properties of the operator, in our application acombination of convection and diffusion. In particular, we discretize convection such that the discreteversion remains skew-symmetric. An almost immediate consequence is that the system matrix remainsdiffusively stable (hence never can become singular) on any grid. In formula: Let the system under studybe given by

dφ

dt+ Lφ = 0,

then the evolution of the kinetic energy ‖ φ ‖H= φ∗Hφ, where H represents the local mesh size, is givenby

d

dt‖ φ ‖= −φ∗((HL)∗ + HL)φ.

With skew-symmetric convection, the symmetric part (HL)∗ + (HL) of the system matrix comes onlyfrom diffusion, and the above assertion follows.

Another consequence is that the convective discretization does not produce unphysical numerical dif-fusion, which unavoidably will interfere with the physical diffusion. This strategy has been applied e.g.in direct numerical simulation of turbulent flow, where the small-scale balance between convection anddiffusion is quite delicate [3].

In the presentation we would like to illustrate the performance of various symmetry-preserving finite-volume methods: second- and fourth-order central schemes (that we apply in DNS), but also higher-orderupwind schemes. It is less known that on contracting and expanding grids the traditional upwind methodproduces negative diffusion [4]; a symmetry-preserving upwind version (a suitable combination of skew-symmetric odd derivatives and symmetric even derivatives) will repair this.

For instance, the fourth-order symmetry-preserving discretization of a first-order derivative ∂φ/∂xbecomes

∂φ

∂x≈−φi+2 + 8φi+1 − 8φi−1 + φi−2

−xi+2 + 8xi+1 − 8xi−1 + xi−2

.

After muliplication with the denominator (as is usual in a finite-volume setting), the convective contri-bution to the coefficient matrix is clearly seen to become skew symmetric.

As a main example we consider a one-dimensional convection-diffusion equation on the unit interval ata diffusion coefficient k = 0.001. Our ‘favorite’ grid is the Shishkin grid, consisting of piecewise-uniformgrid regions with an abrupt (very large) change in mesh size (e.g. [1]). Figure 1 shows a comparison be-tween the individual solutions of second- and fourth-order traditional methods and symmetry-preservingmethods on an abrupt Shishkin grid and on a (smoothly stretched) exponential grid. Both grids contain

1

A.E.P. VELDMANN: High-order symmetry-preserving discretization on stronglystretched grids

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%Speaker: VELDMANN, A.E.P. 152 BAIL 2006

28 grid points, with half of the grid points located in [1 − 10k, 1]). On both grids the ‘traditional’fourth-order method suffers from an almost singular system matrix.

0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

error=3.5e−02

2nd Lagrange

0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

error=7.0e−04

2nd symm−pres

0 0.2 0.4 0.6 0.8 1−150

−100

−50

0

50

error=7.4e+01

4th Lag (+ exact bc)

0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

error=2.2e−04

4th symm−pres (+ 2nd bc)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

error=3.7e−03

2nd Lagrange

0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

error=2.5e−04

2nd symm−pres

0 0.2 0.4 0.6 0.8 1−800

−600

−400

−200

0

200

error=4.6e+02

4th Lag (+ exact bc)

0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

error=1.6e−05

4th symm−pres (+ 2nd bc)

Figure 1: Discrete solutions for k = 0.001 at an abrupt grid (left) and an exponential grid (right). The ’traditional’

discretization, especially the fourth-order version, has problems with the stretching of the grid.

Figure 2 shows a grid-refinement study of the second- and fourth-order traditional and symmetry-preserving methods on both typs of grids (abrupt and exponential). The much more regular and forgivingcharacter of the symmetry-preserving discretization is evident.

10−2

10−1

10−5

100

Abrupt: d=10k

mean mesh size

glob

al e

rror

10−2

10−1

10−5

100

mean mesh size

Exponential: d=10k

2L2SP4L4SP

Figure 2: The global error as a function of the mean mesh size. Half of the grid points is located in the bound-

ary layer of thickness d. Four methods are shown: 2L (second-order Lagrangian), 2S (second-order symmetry-

preserving), 4L (fourth-order Lagrangian with exact boundary conditions) and 4S (fourth-order symmetry-preserving

with second-order boundary treatment).

References

[1] P.A. Farrell, J.J.H. Miller, E. O’Riordan and G. I. Shishkin: A uniformly convergent finite differencescheme for a singularly perturbed semilinear equation. SIAM J. Num. Anal. 33 (1996) 1135–1149.

[2] A.E.P. Veldman and K. Rinzema: Playing with nonuniform grids. J. Eng. Math. 26 (1992) 119–130.

[3] R.W.C.P. Verstappen and A.E.P. Veldman: Symmetry-preserving discretisation of turbulent flow. J.

Comput. Phys. 187 (2003) 343–368.

[4] G. Golub, D. Silvester and A. Wathen: Diagonal dominance and positive definiteness of upwind ap-proximations for advection diffusion problems. In: D.F. Griffiths and G.A. Watson (eds.) Numerical

Analysis: A R. Mitchell 75th Birthday Volume, World Scientific, Singapore (1996) 125–132.

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A.E.P. VELDMANN: High-order symmetry-preserving discretization on stronglystretched grids

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%Speaker: VELDMANN, A.E.P. 153 BAIL 2006

Application of Bifurcation Method to Computing Numerical

Solutions of Lane-Emden Equation ∗

Zhong-hua Yang, Ye-zhong Li, Ying ZhuDepartment of Mathematics,Shanghai Normal University

Shanghai,200234,China

AbstractIn this paper, the Lane-Emden equations of index p

∆u + up = 0, (x, y) ∈ Ωu|∂Ω = 0, (x, y) ∈ ∂Ω

(0.1)

where Ω is a bounded open domain in R2, p > 0, are concerned. Equation (0.1) describes thebehavior of the density of a gas sphere in hydrostatic equilibrium in appropriate units. Theindex p, which is called the polytropic index in astrophysics, is larger than 1

2 . It means that nopolytropic stellar system can be homogeneous in galactic dynamics([3],[7]).Using the Liapunov-Schmidt method and symmetry-breaking bifurcation theory, we compute and visualize multiplesolutions of Lane-Emden equation on a bounded domain of R2 with a homogeneous Dirichletboundary condition, which plays an important role in stellar structure and evolution theory.The domains we consider here include the unit square and the square cut by small square.

The critical point theory was applied to prove the existence and multiplicity of solutionsunder various assumptions([2], [8]). But what distribution and structure they have and how tocompute them have attracted the attention of many mathematicians, physicists and engineers.Due to the multiplicity, degeneracy and instability of the critical points with higher Morse index,the computation of multiple solutions encounters essential difficulties and is truly challenging.Since 90’s of last century, numerical works to compute numerical solutions of (0.1) appearedin the literature. The mountain-pass algorithm, the scaling iterative algorithm ,the monotoneiteration, the direct iteration algorithm and the research extension method([4],[5],[6]) are usedto compute the solutions of (0.1). But in these algorithms “ good guess of solution” of (0.1),which is a difficult task, is needed. Therefore only few solutions of (0.1) are computed yet.

In this paper, we try to use the bifurcation method to overcome this difficulty. Our mainidea is to embed (0.1) into nonlinear elliptic BVP with parameter λ of the form

F (u, λ) = ∆u + λu + up = 0, (x, y) ∈ Ω,u|∂Ω = 0, (x, y) ∈ ∂Ω.

(0.2)

According to the bifurcation theory (0.2) has nontrivial solution branches which bifurcate fromits bifurcation points, so we can compute the solutions of (0.1) by using continuation method([1])along these nontrivial solution branches of (0.2) until λ = 0. Many new solutions of (0.1) withdifferent symmetry are computed by the bifurcation method.

∗Supported by Shanghai Development Foundation for Science and Technology (No.03QA14036), ScienceFoundation of Shanghai Municipal Education Commission(05DZ07), Shanghai Leading Academic DisciplineProject(T0401).

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Z.-H. YANG, Y.-Z. LI, Y. ZHU: Application of Bifurcation Method to ComputingNumerical Solutions of Lane-Emden Equation

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%Speaker: YANG, Z.-H. 154 BAIL 2006

On the other hand, Lyapunov-Schmidt reduction is used to get the bifurcation equation of(0.2), from which we can easily obtain the initial guesses for computing directly the differentsolutions of (0.1) with different symmetry. Therefore the bifurcation method also offers a moreeffective way to computing the multiple solutions of Lane-Emden equation and other nonlinearelliptic boundary value problem.

References

[1] E.L.,Allogower, K.Georg, Numerical continuation Methods, An Introduction, Springer Se-ries in Computational Mathematics(Springer, Berlin), 1990.

[2] A.Ambrosetti, P.H. Rabinowitz, Dual variational mehtods in critical point theory and ap-plication, J.Funct. Anal., Vol.14(1973), 327-381.

[3] S.Chandrasekhar, An Introduction to the stellar structure, Dover Publication, Inc., NY,1967.

[4] Chuanmiao Chen, Ziqing Xie, Structure of multiple solutions for nonlinear differential equa-tions. Science in China Ser.A Mathematics Vol.47 Supp.(2004), 172-180.

[5] Goong Chen,Jianxin Zhou, Wei-ming Ni, Algorithms and visualization for solutions of non-linear elliptic equations.International Journal of Bifurcation and Chaos,Vol.10,No.7(2000)1565-1612.

[6] Y. Li, J.X. Zhou, A minimax method for finding multiple critial points and its applicationsto semilinear PDEs, SIAM J.Sci. Comput., Vol.23(2002), 840-865.

[7] R. Kippenhaln, A. Weigert, Stellar Structure and Evolution, Springer-Verlag, New York,Berlin, 1990.

[8] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differ-ential Equations, CBMS Regional Conf. Series in Math.65, Amer.Math.Soc., Providence,1986.

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Z.-H. YANG, Y.-Z. LI, Y. ZHU: Application of Bifurcation Method to ComputingNumerical Solutions of Lane-Emden Equation

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%Speaker: YANG, Z.-H. 155 BAIL 2006

Numerical simulation of turbulent boundary for stagnation-flow in the spray-painting process

Q. Ye

Institut für Industrielle Fertigung und Fabrikbetrieb Universität Stuttgart Nobelstr. 12, 70569 Stuttgart, Germany

(Abstract for submission to the BAIL 2006)

In order to optimise the painting process, which amounts to a high percentage of fixed and flexible costs in automotive production, numerical simulations of spray painting for the automotive industry, especially using high-speed rotary bell and electrostatically supported methods, have been performed [1]. Previous numerical studies were mainly concerned with the calculation of the two-phase turbulent flow of the spray jet, the modelling of the electrostatic field including space charge and the prediction of the film thickness distribution on the coated work piece. Less attention was paid to the near-wall turbulent flow in the painting process, which is, however, quite important for particle deposition. The current numerical investigation is aimed at the turbulent boundary of the stagnation-airflow. A CFD code (Fluent) based on Reynolds-averaged Navier-Stokes equations (RANS) has to be used because of the complicated turbulent flow in the spray-painting process (Fig. 1). A two-dimensional turbulent channel flow first is calculated using different turbulent models with near-wall functions. The simulated results are compared with the DNS data [2] (Fig.2), in order to obtain a suitable model for the investigation of the real near-wall flow. The complicated three-dimensional turbulent flow is then calculated using a real geometry of the atomizer, high-speed rotary bell, and simple target geometry, e.g., a flat plate. The influences of the near-wall mesh resolution and the near-wall functions on the velocity distributions (Fig. 3) and the turbulent magnitudes close to the target are analysed. The computational results provide useful information for the further study of the particle deposition in spray-painting processes.

Figure 1: Contours of the air velocity magnitude (m/s)

up

down

up

down

Q. YE: Numerical simulation of turbulent boundary for stagnation-flow in the spray-painting process

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%Speaker: YE, Q. 156 BAIL 2006

k-yplus

00.5

11.5

22.5

33.5

44.5

5

1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03

yplus

norm

aliz

ed k

standard-wallDNSenhanced-wallReynolds-stressk-omega

velocity magnitude

0

0.5

1

1.5

2

2.5

3

3.5

-0.5 -0.3 -0.1 0.1 0.3 0.5

z(m)

velo

city

(m/s

)

enhanced -w all

non-equilibrium

[1] Q. Ye, J. Domnick, A. Scheibe, K. Pulli: Numerical Simulation of the Electrostatic Spray-

painting Process in the Automotive Industry. High-Performance Computing in Science and Engineering’04, Springer-Verlag Berlin, Heidelberg, , 2004, pp. 261-275.

[2] J. Kim, P. Moin, R. Moser: Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, pp.133-166. 1987.

Figure 2: Comparison of non-dimensional turbulent kinetic energy in the fully developed turbulent channel flow between different turbulent models

Figure 3: Comparison of velocity magnitudes along the painting target at a distance of 0.1 mm to the wall for two different near-wall functions

Q. YE: Numerical simulation of turbulent boundary for stagnation-flow in the spray-painting process

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%Speaker: YE, Q. 157 BAIL 2006

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163 BAIL 2006

Authors

Alizard, F., 67Alrutz, Th, 69Anthonissen, M., 57Apel, Th., 71

Banasiak, J., 130Bause, M., 72Bleier, N., 37Boguslawski, L., 73Branley, D., 44Buschmann, M.H., 32

Cangiani, A., 75Chao, C.-Y., 145Chigerev, E.N., 149Chou, Y.-C., 106Clavero, C., 77Cousteix, J., 113

Das, D., 35Dunne, R.K., 14

Eisfeld, B., 79

Firooz, A., 81

Gad-el-Hak, M., 32Gadami, M., 81Gaponov, S.A., 85Georgoulis, E.H., 75Gersten, Kl., 3Gracia, J.L., 77Gravemeier, V., 8

Holling, M., 91Hammouch, Z., 87Hamouda, M., 88Hartmann, R., 22

Hegarty, A.F., 44, 56Hemavathi, S., 53Herwig, H., 91Heuveline, V., 24Hong, Q.-S., 145Houston, P., 4Huerre, P., 5Hussong, J., 37

Il’in, A.M., 93Islam, W.S., 96

Jang, J.-Y., 106Jensen, M., 75Jimack, P., 28

Kachuma, D., 98Kameswara Rao, A., 39Kaushik, A., 100Kluwick, A., 136Knobloch, P., 102Knopp, T., 69, 104Kozakiewicz, J.M., 130Kuzmin, D., 118

Ludeke, H., 111Lenz, S., 8Lesshafft, L., 5Leu, J.-S., 106Li, S., 59Li, Y.-Z., 154Likhanova, Y.V., 108Linß, T., 47Lipavskii, M.V., 149Lisbona, F., 77Liseykin, V.D., 108Lube, G., 109

Mackenzie, J.A., 25Madden, N., 47Mansour, K., 113Matthies, G., 71, 115Maubach, J., 57, 58Mauss, J., 116Mierka, O., 118Morinishi, K., 120

Nastase, A., 122Nataf, F., 124Nayak, A., 35Neuss, N., 126Nicola, A., 25

O’Riordan, E., 14Olshanskii, M.A., 128

Parumasur, N., 130Patrakhin, D.V., 108Perotta, S., 27Petrov, G.V., 85Purtill, H., 44

Raghavan, V.R., 96Ram, V.V., 37Rapin, G., 124Rasuo, B., 132Robinet, J.-Ch., 67Roos, H.-G., 134, 135

Sagaut, P., 5Savic, L., 41Scheichl, B., 136Schneider, R., 28Sedykh, I., 57Sengupta, T.K., 39Sharma, K.K., 100Shishkin, G.I., 16, 44, 48, 49, 51, 56, 59Shishkina, L.P., 49, 59Shishkina, O., 138Sikwila, S., 56Smorodsky, B.V., 85Sobey, I., 98Stanculescu, I., 18

Steinruck, H., 41Stynes, M., 6Stynes, M., 140Svacek, P., 141

Tai, C.H., 145Tarasova, N.V., 143Temam, R., 88Terracol, M., 5Tian, H., 147Tobiska, L., 117, 140Tolstykh, A.I., 149Tselishcheva, I.V., 51Turner, M.M., 14

Valarmathi, S., 53Vaseva, I.A., 108Vasieliev, M., 150Veldmann, A.E.P., 152

Wagner, C., 138Wall, W.A., 8Wang, H., 19

Yang, Z.-H., 154Ye, Q., 156

Zadorin, A.I., 61Zarin, H., 135Zegeling, P., 63Zhu, Y., 154