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
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