velocity and temperature boundary- layer modeling using

IUVSTA 2011| Leinsweiler, May 16th 2011

R. Groll | 1Center of Applied Space Technology and Microgravity

Computational Thermofluid Dynamics Division

Velocity and Temperature Boundary-Layer Modeling Using Averaged

Molecule cluster Transport Equations

R. Groll

Center of Applied Space Technology and MicrogravityUniversity of Bremen, Am Fallturm, D-28359 Bremen




LCharacteristic

layer thickness

Micro- und macroscopic approach

Micro channel flow | continuum approch

Molecular number density inside transsonic nozzle




Modification of parabola near the wall

u Gas

-uW

all

Ls

Slip Length

Slip Velocity

x2

x1

Poiseuille flow of high Knudsen numbers

Micro channel flow | slip velocity




0 00

limV

NnV

Number densityMolecule numberControl volume 0

NV

Mean velocity ( , , )i i Bn f t x d

Molecule velocity i

1i i iu n

n Number-density weighted

''i i i Velocity deviation

x

Molecule number density and statistical filters

Statistical modeling| Filtering




Continuity of molecular diffusion

'

'

jj

i j i jj i

nn Dx

n D n nx x

Velocity correlation

32

Sc=1

23

k kB

i i

k kk k

m

mmn Tk

u mnD

n mp mE d

Analogy of macroscopic values

0 0

'

j

j jj j

j jj j

nDx

n nn nt x t x

n n nt x x

Statistical modeling | Velocity correlations




jj

E n nE n n

0 0j jj j

n nn nt x t x

Molecule velocity ξ

Number-density weighted averaging

Momentum transport equation

2

0

2

1

i i j i i j i jj j j

ji kij

j j i k

ijj i j k k

nn n n nt x t x x

Dnx x x x

D n n n nx n x x x x

UnsteadyConvectionSourceDiffusionConversion

Statistical modeling | Momentum transport




Turbulence analogy

12 k k

2

2

2 1 2

ij

j i ki j ij ij ij

i j k k i j k

D n n n n Dn x x x x x x x

Energy transport equation

Terms of kinetic dissipation and other energetic production

Depending on velocity and number density gradients

Energy Diffusion without density gradients works with Pr=κ/(κ +2) = 1/γ < 1 .

compressionconvection dissipationdiffusion

2

2 20

2 1

k ij j ij

j j j j k j

i

j i i i j

n n n n Dn n nt x t x x x x x

D n n D nDnx x n x x n x

conversion

Statistical modeling | Energy transport




Thermodynamic consistence of transport equations

Constitutive equation

Variation energy / intrinsic energy

compressionconvection

1 kj

j k

ue eu Ts et x x

Compression Energy increase Expansion Entropy increase

Only forγ>1

2

de Tds pdve nTs n n p v

Entropy increase

32

2

dissipation conversion diffusion

1i iij

j j i i j i j j j

D n n n nnTs n D Dnx x n x x x x x x x

Statistical modeling | thermodynamic consistance




**

2/3 2/30 0

111 /

w w ww

ww

LKnL

Boundary conditions

Well-knownwall shear stress

23w

w wL L

Boundary Conditions | Wall shear stress

* *

2* * *00

20 0 0 0

2123 3 3

w

w w w ww

w wKnc

L L p Kn cL p L

2/33

11 /

w ww ww w w

L LL L L

Modeled influence of large mean-free-paths on wall shear stress




Kn=0.05 Kn=1.00

*

0*

dp Kndx

* *0

0 0

*0

0 0

ii

u pu pHp p

xKn xL L

Pressure gradient condition:

Non-dimensional values:

Boundary Conditions | Velocity profiles

Computational resuts




Conclusions

Algebraic model describes scalar diffusion coefficient

(→ diffusion modelling)

For high Knudsen numbers : Diffusion isdescribed by formulation of dilute gases

(→ slip modelling)

Continuity, momentum and ernergy are calculated by transport equations ofnumber density n and the first and second statistical moments of themolecular velocity. (→ statistical modelling)

Conclusions | Scale-transitioning Analogy

*2/3

11 wL Kn

Analogy of macroscopic values

32

23

k k

m

mE d

jj

E n nuE n n

( , , )i i Bn f t x d

1

2 k ke




Many thanks for your attention!

velocity and temperature boundary- layer modeling using

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