June 6-10, 2011 FVCA 6 1

Unsteady Numerical Simulation of the Turbulent

Flow around an Exhaust Valve

6th International Symposium on Finite Volumes for Complex Applications

CTU in Prague,

Czech Republic

6th – 10th June 2011

FVCA 6 2June 6-10, 2011

Unsteady Numerical Simulation of the Turbulent Flow around an Exhaust Valve

Milan ŽALOUDEK

prof. Jaroslav FOŘT, Czech Technical University, Prague, Czech Republic

prof. Herman DECONINCK, von Kármán Institute for Fluid Dynamics, Rhode-St-Genése, Belgium

FVCA 6 3June 6-10, 2011

Outline

Motivation Physics Solved Numerics Used Results Conclusions

FVCA 6 4June 6-10, 2011

Motivation• exhaustion = complicated issue

• physical – determined by many factors (unsteady, 3D, turbulent, chemistry, ...)

• numerical – see later

• difficult comparisons of results

• one of the least explored engine domains

• hardly no experimental data

• doubtful results from comercial CFD codes

• exhaust valve and exhaust pipe

• sudden area widenings and restrictions

• sharp corners causing flow separation

• wide velocity range

• goal: insight of the flow structure

FVCA 6 5June 6-10, 2011

Physics Solved

FVCA 6 6June 6-10, 2011

Flow characteristics• during real operating cycle

• fully 3D flow

• exhaust valve opens and closes very quickly

• turbulent flow, possibly involving some chemistry

• problems of numerical solution

• moving boundaries

• unsteady flow conditions

• wide velocity range

• formulation of outlet

boundary condition

(recirculation zone leaving

and re-entering the domain)

ilustrating solution, one half of the valve contour

velocity streamlinesMach

number

FVCA 6 7June 6-10, 2011

Governing Equations• Reynolds-Averaged Navier-Stokes equations Q

x

F

x

F

t

W

i

Vi

i

Ci

T

jjt

T

it

itk

tiijijii

Vi

Tiii

Ci

T

xx

kFPkPQ

xx

kqqwF

kpepwpwwF

kewwW

21

2*

21

2211

21

2)1(,,0,0,0,0

,,,,,0

,,,,,

,,,,,

• constitutive relations closing the set of equations

conservative variables

convective fluxes

viscous fluxes

t

ti

ti

ii qq

x

Tq

Pr

Pr

Pr

k

kij

i

j

j

itij x

w

x

w

x

w 3

2

2

1

ρ density, (w1,w2) velocity components, p pressure, e internal energy, T temperature,

k turbulent kinetic energy, ω specific dissipation rate

source term

stress tensor

heat fluxes

equation of state, Sutherland’s law, Fourier’s law

FVCA 6 8June 6-10, 2011

Turbulence Modelling• main variables decomposed to mean part and fluctuation part• density weighted averaging, suitable for compressible flows• Boussinesq hypothesis – analogy between molecular and turbulent

transport of momentum• Reynold’s stress tensor• turbulent viscosity extracted from turbulence model

kkijk

kij

i

j

j

itji ww

x

w

x

w

x

wuu ''

3

1

3

2

2

1

t

Models implemented and used

• Menter’s baseline model (BSL)

• Wilcox k-ω model, rev. 2008

• Explicit Algebraic Reynolds Stress Model (EARSM)

FVCA 6 9June 6-10, 2011

• turbulent viscosity

• model constants combined via

Menter’s BSL model

k

t*

*21

*11

*111

* ,41.0,2

1,

2

1,075.0,09.0

*22

*22

*222

* ,41.0,0.1,85.0,0828.0,09.0

jjj

tjj

i

t

ij

jt

jj

iij

xx

kF

xxx

U

Dt

D

x

k

xk

x

U

Dt

Dk

21

2

**

21

~

~

41231

202-k*

132

2221

tanh,,maxmin

10,2

maxCD 4

500

F

xx

k

y

k

CDy

k

y jjk

• blending between k-ω model near walls and k-ε model in freestream

• transport equations of ε added through blending function F1

• transport eq. for turbulent kinetic energy k and specific dissipation ω

(1994)

k-ω :

k-ε :

2111 1 FF

FVCA 6 10June 6-10, 2011

Wilcox’s k-ω model

*lim

2,max~

ijijSS

C

~k

t

ijk

kijij x

wSS

3

1

• limiting magnitude of turbulent viscosity

• transport eq. for turbulent kinetic energy k and specific dissipation ω

(2008)

• cross-diffusion term

0

00

jjdO

jjd

xx

kif

xx

kif

8

7,

8

1,

5

3,

2

1,

100

9,0708.0,

25

13lim

** CdO

jjd

jjj

iij

jjj

iij

xx

k

x

k

xx

U

kDt

D

x

kk

xk

x

U

Dt

Dk

2

**

~

~

• model constants

FVCA 6 11June 6-10, 2011

)(

3

22 ex

ijijijttij kakS

• transport eq. for k and ω in a form of Kok’s TNT k-ω model

• coefficients

• model constants

Wallin’s EARSM

kt 12

1

****4

)(kjikkjik

exij SSa

• turbulent viscosity

• anisotropy term

• explicit algebraic Reynolds stress model

• non-linear relation for turbulent stresses with respect to Sij

(2000)

• normed tensors

• turbulent time-scale

k

kij

i

j

j

iij

i

j

j

iij x

w

x

w

x

wS

x

w

x

w 3

2

2

1,

2

1 **

8.1,6,09.0 '1

* CC

k

C**

,1

max

'144'111 ,,,,,,,, CSkCS

FVCA 6 12June 6-10, 2011

• W fulfils equation

• W fulfils initial condition and boundary conditions

Mathematical Formulation• searching for a function W(xi, t), on a domain Ω such that

0)0( WtW

02

1

Qx

F

x

F

t

W

i

Vi

i

Ci

t

t

FVCA 6 13June 6-10, 2011

• Boundary conditions• Inlet: total pressure, total temperature, incidence angle, turb. variables

• Outlet: pressure, velocity, temperature, turbulent variables

• Wall: no-slip condition, turb. variables according to literature (F.R. Menter)

• Symmetry: non-permeability condition

ref

inlet

inlett

inletinletTT L

wkTp ,

100,,,

Mathematical Formulation

0,

nn

k

n

T

n

wpp ioutlet

0nw

201

60 , 0 ,0

ykw wallwall

• Computational domain

FVCA 6 14June 6-10, 2011

Numerics Used

FVCA 6 15June 6-10, 2011

Numerical Solution• in-house CFD code COOLFluiD

• developed by group of engineers, based at VKI

• based on finite volume method (FVM)• cell-centered approach, variables stored in centroids of each cell• continuous problem discretized with the Gauss theorem• explicit or implicit time integration

• spatial accuracy improved by a linear reconstruction (obtained by a least squares interpolation method) and the Barth limiter

• the arbitrary Lagrangian-Eulerian (ALE) formulation used for unsteady flow simulations

• time accurate computations using Crank-Nicholson scheme and/or backward differentiation formula (BDF2)

Computational Object Oriented

Library for Fluid Dynamics

FVCA 6 16June 6-10, 2011

• steady flow computation• original equation transforms to

J Jacobian matrix, ΔW difference in vector of unknowns, R right hand side (time dependent terms, numerical fluxes, source terms)

• linear system solved by GMRES iterative solver (provided by PETSc)

• Jacobian matrix computed numerically

Implicit Time Integration

nn WRWWJ

i

faces

kkk

nVkk

nC

i

ni

ni QnWFnWF

t

WW

#

1

111 ~~1

• unsteady flow computation• dual time stepping

• outer t.s. (Δt) – real time accurate step, 1st step C-N, further on BDF2

• inner t.s. (Δτ) – solving the system at each real time step

• linear system solved by GMRES

faces

kkk

nnnVkk

nnnC

i

ni

ni

ni

ni

ni nWWWFnWWWF

t

WWWWW #

1

1,1,11,1,111,1,11,1

,,~

,,~1

2

43

FVCA 6 17June 6-10, 2011

Convective Fluxes• AUSM+up, Advection Upstream Splitting Method, modified for all speeds• based on solution of Riemann problem – flux over 1D discontinuous step

between two states WL, WR

otherwise 162

1 if 1

otherwise 161

1 if

)2()2(

)1()5(

)2()2(

)1()4(

MM

MP

MM

MM

MM

MM

M

• MLR, pLR computed using splitting polynoms, ΦLR upwinded

corrRRLLLR

corrRLLR

pMpMpp

MMMM

,, )5()5(

)4()4(

PP

MM 0 if

0 if

LRR

LRLLR

MW

MW

LRLRLR pMF

pa

a

uMp

ape

ua

a

a

u

upe

pu

u

F

~

, ,

0

0~ 2

• Mcorr, pcorr corrections to ensure better convergence at low speeds

2)2(

)1(

14

12

1

M

MM

M

M

FVCA 6 18June 6-10, 2011

• Mcorr, pcorr corrections to ensure better convergence at low speed

LR

RL

a

a

uu

M

MMM

MMf

2

,max,1min

2

22

0

00

LRaRLucorr

LR

a

pcorr

uuafKp

a

ppM

f

KM

PP

2

20 ,1 max

Convective Fluxes

inviscid flow in a channel, solved with numerical scheme without (left), with (right) corrections

sequence for different freestream velocities M=0.020, M=0.200, M=0.675 (top to bottom)

• M∞ freestream Mach number, affect correction terms through fa

FVCA 6 19June 6-10, 2011

• computed by a central approximation• using diamond dual cells• derivatives computed by the Gauss theorem

Viscous Fluxes

• evaluated cell-wise

Source Terms

Computational Grids• structured triangular grids

(splitted quadrilaterals)

• due to large grid displacements, unsteady flow simulations employ set of grids for different valve lifts

• solution between grids interpolated by the Shepard method

FVCA 6 20June 6-10, 2011

Results

FVCA 6 21June 6-10, 2011

Flow Structure (1/3)

contours of Mach number

• steady flow computation• 2D, turbulent flow model (BSL)• valve opening 4 mm• temperature 500 K• pressure ratio 0.4

100

400

kPa

kPa

p

p

outlet

inlet

FVCA 6 22June 6-10, 2011

Flow Structure (2/3)detail of valve seat detail of outlet boundary

supersonic expansion around corner

overallmax M = 2.85

outletmaximal M = 1.12average M = 0.72

no backflow

shock waves deflecting flow

causing separations on both sides

artificial channel throatdetermined by recirculationsallowing further expansion

FVCA 6 23June 6-10, 2011

Flow Structure (3/3)detail of expansion detail bottom corner

separation behind valve seat

separation along the exhaust valve

separation ona straight wall

separation behind the pipe corner

separation at corner, where

valve meets its casing

FVCA 6 24June 6-10, 2011

Influence of Turbulence Model

• 2D model• steady flow computation• identical boundary conditions

• valve opening 4 mm• pressure ratio pin/pout = 2.5

(exhaust to atmosphere)• temperature 500 K

• qualitative agreement of all models• BSL and Wilcox model very close• EARSM predicts different flow

topology

contours of Mach number

(1/3)

FVCA 6 25June 6-10, 2011

Influence of Turbulence Model(2/3)Comparison of pressure and Mach number throughout pipe

• data extracted along (main) streamline passing through the middle of the channel throat

• BSL, Wilcox models almost identical• EARSM holds trend, but predicts

milder peaks and higher outlet velocity

FVCA 6 26June 6-10, 2011

Influence of Turbulence ModelPositions of separation zones• zero coordinates at channel

throat (aerodynamically choked)• distances in milimeters

(3/3)

lower wall [mm] upper wall [mm]

start end length start end length

BSL 2.8231 21.237 18.414 2.8284 35.730 32.901

Wilcox 2.1243 22.024 19.899 2.8284 35.774 32.945

EARSM 0.3643 18.091 17.727 0.0000 34.873 34.873

FVCA 6 27June 6-10, 2011

Unsteady Flow Simulation (1/5)• 2D, turbulent flow model (BSL)• movement corresponds to RPM = 3.500, one valve loop ≈ 1.5 10-2 s• boundary conditions set according to the literature (J. Heywood)

• various inlet pressure evolution - spark ignition / compression ignition• same outlet pressure evolution

• max. valve lift 11 mm, treshold 0.5 mm• time step Δt = 10-6 s• computational grids for different lifts: 0.5 – 2.5 – 7.0 mm• remeshing + interpolation points

FVCA 6 28June 6-10, 2011

Unsteady Flow Simulation (2/5)Spark ignition (SI) engine• valve loop ≈ 1.5 10-2 s, time step Δt = 10-6 s

• step solution displayed Δt = 10-4 s

• see full movie

FVCA 6 29June 6-10, 2011

Unsteady Flow Simulation (3/5)Compression ignition (CI) engine• valve loop ≈ 1.5 10-2 s, time step Δt = 10-6 s

• step solution displayed Δt = 10-4 s

• see full movie

FVCA 6 30June 6-10, 2011

Unsteady Flow Simulation (4/5)Comparison of un-steady approach

• valve lift 7.0 mm

• inlet BC corresponds to CI engine (closing phase)

• Mach number contours

• pressure evolution in exhaust pipeunsteady solution

steady solution

FVCA 6 31June 6-10, 2011

Unsteady Flow Simulation (5/5)Mass flow comparison

• CI engine detects higher ṁ, due to higher operating pressure ratio

• ṁ coincides in early stages for both engines, due to aerodynamical choking

• ṁ at very low lifts negligible

• differences against steady solutions approx. ≈ 10%

Lift 7 mm

open

11 mm 7 mm

close

SI 2.523 3.775 2.472

CI 3.162 4.707 2.532

steady 3.469 - 2.220

diff [%] +9.7 - -12.3

mass flow rate [kg/s]

FVCA 6 32June 6-10, 2011

Conclusions• reasonable results of gas exhaustion acquired with in-house

developed CFD code (respecting physical assumptions)

• gas exhaustion phenomena

• even small geometrical difference cause dramatic flow changes

• careful capturing of separation zones required

• exhaustion unsteadiness can not be neglected

• final target: insight of the flow topology

Future Prospects• more turbulence models for unsteady flow computations

• extension to 3D unsteady computations

• optimization of the exhaust valve shape

• complete simulation of a 4-stroke engine (cylinder domain)

FVCA 6 33June 6-10, 2011

Acknowledgements

This work has originated

thanks to

• team of patient co-workers

• developers of CFD package COOLFluiD

• grant GAČR No. P101/10/1329

• Josef Božek Research Center 1M6840770002