Transcript BRIEFING TITLE - ALL CAPS 30 Jan 01

Computational Combustion Lab
Aerospace Engineering
Multi-scale Simulation of Wall-bounded Flows
Ayse G. Gungor and Suresh Menon
Georgia Institute of Technology
Atlanta, GA, USA
Supported by Office of Naval Research
WALL BOUNDED SHEAR FLOWS: TRANSITION AND TURBULENCE
Isaac Newton Institute for Mathematical Sciences
Cambridge, UK
September 11th, 2008
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Motivation
• Flows of engineering relevance is at high Re
– Wall bounded flows, wake and shear flows
• The cost of simulations that resolve all the scales of motion is of the
order of Re3
• Almost 90% of this cost is a result of attempting to explicitly resolve
near-wall boundary layers
– Near-wall turbulence contains many small, energy containing,
anisotropic scales that should be resolved
• DNS Computations of channel flows
– 18 B grid points, Ret = 2003 (Hoyas et al., 2006)
• DNS Computations of turbulent separated flows
– 151 M grid points, Ret = 395 (Marquillie et al., 2008)
– DNS at lower Reynolds number (Experiment at Ret= 6500)
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Motivation
• Conventional LES requires very high near-wall resolution
– Near-wall Models
• Use algebraic relationships to compute wall stresses
• Resolution requirement reduced significantly
• Additional source of errors due to the modeling the dynamics in
the near-wall region
– Zonal Approaches
• Two Layer Approach – Solves boundary layer equations and/or
employ local grid refinement
• RANS-LES Approach – Uses RANS near the wall and LES in
the core region
• Most of the cost-effective approaches do not properly resolve the
turbulent velocity fluctuations near the wall
• Here, a two-scale approach for high-Re flows is discussed
– attempts to resolve near-wall fluctuations
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Multi-Scale Simulation Approaches
•
Multi-scale approaches:
 Dynamic multilevel method (Dubois, Temam et al.)
 Rapid Distortion Theory SS model (Laval, Dubrulle et al.)
 Variational multiscale method (Hughes et al.)
 Two-level simulation (TLS*) (Kemenov & Menon),
extended for compressible flows (Gungor & Menon)
•
Simulate both LS and SS fields explicitly
– Computed SS field provides closure for LS motion
•
All use simplified forms of SS equations
–
•
Some invoke eddy viscosity concept for SS motions
TLS simulates the SS explicitly inside the LS domain
*Kemenov and Menon, J. Comp. Phys., Vol. 220 (2006), Vol. 222 (2007)
Gungor and Menon, AIAA-2006-3538
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Two-Level Simulation: Key Features
•
Simulate both large- and small-scale fields simultaneously
– large-scales (LS) evolve on the 3D grid
– small-scales (SS) evolve on 1D lines embedded in 3D domain
– 3D SS equations collapsed to 3x1D equations with closure
•
Scale Separation approach employed
– No grid or test filtering invoked
– No eddy viscosity assumption invoked
•
High-Re flows simulated using a “relatively” coarse grid
– Efficient parallel implementation needed
– Cost becomes acceptable for very high-Re flow
– Potential application to complex flows
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Two Level Grid in the TLS-LES Approach
DyLES
DzSS
DxSS
Large-Scale Grid
y
Small-Scale Grid
z
x
DySS
DzLES
DxLES
Small scale equations are solved on three 1D lines embedded in the 3D domain
Resolution requirements



Number of LES control volumes:
NLES3
NLES <NSS
Grid points for TLS-LES:
NLS3 + 3NLS2NSS
NLS <NLES, NLS <<NSS
Grid points for DNS:
NSS3
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TLS v/s LES
•
Two degrees of freedom in Conventional LES
–
•
Filter Width and Filter Type
Two degrees of freedom in TLS:
– Sampling/Averaging Operator (SS <=> LS)
– Interpolation Operator (LS <=> SS)
•
•
TLS does not require commutativity to derive LS Eqns.
Full TLS approach described earlier
– isotropic turbulence, free shear and wall-bounded flows*
•
Here, a new hybrid TLS-LES approach demonstrated**
–
Application to wall bounded flows with separation
* Kemenov and Menon, J. Comp. Phys. (2006, 2007)
** Gungor et al., Advances in Turbulence XI (2007)
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TLS – Scale Separation Operator L
Exact Field is split into LS and SS fields:
u( x, t )  uL ( x, t )  uS ( x, t )
Continuous large scale field is defined
by adopted LS grid:
LD : Sampling at LS grid nodes
u L ( x , t )  L  u( x , t )
 F D LD  u( x , t )
 F D  u L ( xk , t ) 
SS field is defined based on LS
field from decomposition:
uS ( x, t )  u( x, t )  uL ( x, t )
F D : Interpolation to the SS nodes
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A priori analysis of scale separation operators (LES and TLS)
Fully resolved signal (black) from a 1283 DNS of
isotropic turbulence study.
The resolved field is represented with a 16 grid point.
The top hat filtered LES field (red) obtained by taking a
moving average of the fully resolved field over 8 points.
The TLS LS field (green) truncated from the fully
resolved signal.
The TLS SS field (blue) obtained by subtracting the LS
field from the fully resolved field.
TLS has higher
spectral support
The longitudinal energy spectra of a fully resolved signal (black) and (a) LES
energy spectra (red), (b) The TLS LS (green) and SS (blue) energy spectra.
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Hybrid TLS-LES Wall Model
• The TLS equations are used in the near-wall region
• The LES equations are used in the outer flow
• All zonal approaches (Hybrid RANS-LES) use some form of
domain decomposition
•
Hybrid TLS-LES uses functional decomposition
– No need for interface boundary conditions
– Need to determine the transition region dynamically
LES
LES
RANS
prescribed
y interface
Hybrid RANS-LES Strategy
TLS
prescribed y interface
for wall-normal lines
TLS-LES Strategy
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Hybrid TLS-LES Formulation – Scale Separation
• Hybrid TLS-LES scale separating operator R defined as an additive
operator that blends the LES operator F with the TLS operator L
R=kF
+ (1 - k) L
– LES operator F is the standard filtering operator
– k is a transition function relating TLS and LES domains
Step Function
1
k
0
if
y  Y TLS
y  Y TLS
Tanh Function

 c1  y d  c2  
1
k  1  tanh 
 tanh c1 
2 

  1  2c2  y d  c2 
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Hybrid TLS-LES Equations
• Application of additive scale separation operator
– Velocity components
uiR ( x, t )  R u( x, t )  kuiF ( x, t )  1  k  uiL ( x, t )
– Turbulent stress
t ijR  kt ijF  1  k t ijL
Hybrid Terms
 k  u u  u u   1  k   u u  u u 
F
i
F
j
R
i
R
j
F
L
i
L
j
R
i
• Hybrid terms also in RANS-LES formulation (Germano, 2004)
• Combination of time and space operation
• Here, the hybrid terms appear due to LES and TLS combination
• both are space operators !!
R
j
L
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Hybrid TLS-LES Equations
•
Resolved / Large Scale Equations
Continuity:
Momentum:
uiR
 0,
x i
R
2 R
R

t
R
uiR

u


p
ij
i
 uiR u Rj   



t
x i
x i
x 2j
x i
t ijR  (ui uj )R  (uiRuRj )
The unresolved term in the momentum equation
t  kt  1  k t  k  u u  u u   1  k   u u  u u 
2
F
1/ 2
LES: t ij  2C t Dk sgs S ij   ij k sgs any SGS model
3
Specific closures
R
R
R
for each model
TLS: t L   u R u S    u S u R    u S u S 
ij
i
j
i
j
i
j
R
ij
F
ij
L
ij
F
i
F
j
R
i
R
j
F
L
i
L
j
R
i
R
j
L
The scale interaction terms are closed if the small scale field is known
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Hybrid TLS-LES Equations
•
•
•
Small Scale Equations
Continuity:
uiS
 0,
x i
Momentum:
2 S
S
uiS

ui


p
R
S
R
S
R

u

u
u

u





F


i
j
j
i
t x j i
xi
x 2j


Represents the smallest scales of motion
“Hybrid TLS-LES SS domain”
–
–
–
discrete set of points along 3- 1D lines
3D evolution of small-scales in each line
Full 3D SS equations “collapsed” on to these 1D lines
– Cross-derivatives modeled based on a priori DNS analysis
– Channel and forced isotropic turbulence (Kemenov & Menon, 2006, 2007)
•
Explicit forcing by the large scales on these 1D equations
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Numerical Implementation of SS Equations
1) Approximate LS field on each 1D SS line by linear interpolation
2) Evolve SS field from zero initial condition until the SS energy
matches with the LS energy near the cut off
3) Calculate the unclosed terms in the LS equation
Time evolution of the SS velocity and SS spectral energy
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Hybrid TLS-LES of Channel Flow
•
Mean velocity profiles demonstrates the
capability of the model
•
Wall skin friction coefficient provides
–
–
good agreement with DNS
well comparison with Dean’s correlation
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Hybrid TLS-LES of Channel Flow
3-D energy spectra
Ret = 590
Ret = 2400
Hybrid TLS-LES approach recovers both
LS and SS spectra near the wall
Ret = 1200
Red line : Instantaneous energy spectra
Blue line : Volume average spectra
Black line: k-5/3 slope
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Numerical Solver
• Incompressible Multi-domain Parallel Solver
– 4th order accurate kinetic energy conservative form (used here)
– 5th order accurate upwind-biasing for convective terms
– 4th order accurate central differencing for the viscous terms
– Pseudo-compressibility with five-stage Runge – Kutta time stepping
– Implicit time stepping in physical time with dual time stepping
– DNS, LES (LDKM), TLS-LES
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Turbulent Channel Flow
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Turbulent Channel Flow
•
Coarse DNS – 192 x 151 x 128
–
Well prediction of the mean velocity,
turbulent velocity fluctuations and
turbulent kinetic energy budget.
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Hybrid TLS-LES of Separated Channel Flow
•
Hybrid TLS-LES(~0.18M) and LES(~1.6M) at Ret = 395
–
–
•
Experiment at Ret = 6500 by Bernard et al., AIAA J., Vol. 41, 2003
Spatial resolution (%75 coarser than DNS)
–
–
–
•
DNS(~151 M) at Ret = 395 by Marquillie et al., J. of Turb., Vol. 9, 2008
TLS-LES-LS (64 x 46 x 64)
: Dx+LS = 77.4, Dz+LS = 19.2, Dy+LS
= 5.4
TLS-LES-SS (8 SS points/LS): Dx+SS = 9.6, Dz+SS = 2.4, Dy+SS = 0.68
DNS (1536 x 257 x 384)
: Dx+
= 3,
Dz+
= 3,
Dy+|max = 4.8
Inflow turbulence from a separate LES channel study at Ret = 395
Total vorticity on a spanwise plane (LES)
Streamwise vorticity on a horizontal plane (LES)
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Time evolution of the SS velocity
Spanwise line in the
separation region
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SS evolution effect on the instantaneous flow
SS iterations: 20
SS iterations: 100
• simulations on each line
• optimal parallel approach
SS iterations: 300
SS vorticity magnitude isosurfaces colored with SS streamwise velocity
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Hybrid TLS-LES of Separated Channel Flow
• Hybrid TLS-LES grid is chosen very coarse deliberately
• Hybrid TLS-LES Cp shows good agreement with DNS
– ~%30 off from experiments (higher Re) for all studies
• Hybrid TLS-LES Cf in reasonable agreement with DNS and LES
– Separation is not properly predicted due to coarse LS resolution
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Hybrid TLS-LES of Separated Channel Flow
Streamwise velocity fluctuation
Wall-normal velocity fluctuation
DNS-151M (circles and shaded contours), TLSLES (red), LES (green)
The authors would like to thank Dr. J.-P. Laval for providing the DNS data
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Hybrid TLS-LES of Asymmetric Diffuser Flow
•
Hybrid TLS-LES(~0.25M) and LES(~1.8M) at Ret = 500
–
–
•
Experiment by Buice and Eaton, J. of Fluids Eng., Vol. 122, 2000
The main features of this flow
–
–
–
•
LES(~6.5 M) by Kaltenbach et al., J. of Fluid Mech., Vol. 390, 1999
A large unsteady separation due to the APG
A sharp variation in streamwise pressure gradient
A slow developing internal layer
Inflow turbulence from a separate LES channel study at Ret = 500
Inclination angle: 100
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Hybrid TLS-LES of Asymmetric Diffuser Flow
•
Spatial resolution
–
–
–
–
•
TLS-LES-LS (110 x 56 x 40)
= 5.72
TLS-LES-SS (8 SS points/LS) : Dx+SS = 6.7, Dz+SS = 6.2, Dy+SS = 0.72
: Dx+ = 25, Dz+ = 25, Dy+ = 0.98
LES (278 x 80 x 80)
LES by Kaltenbach et al., 1999 (590 x 100 x 110)
Step function (
–
: Dx+LS = 54, Dz+LS = 50, Dy+LS
t ijR  kt ijF  1 k t ijL, F: LES, L: TLS operator)
pre-defined interface, Y+TLS = 152
1
k 
0
if
y  Y TLS
y  Y TLS
TLS
LES
TLS
Isosurfaces of the second invariant of the velocity gradient
tensor colored with local streamwise velocity predicted
with LES model
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Hybrid TLS-LES of Asymmetric Diffuser Flow
•
•
•
Cp along the lower and upper wall predicted reasonably well
•
Overall, TLS-LES shows ability to predict separation regions without any model
changes
Hybrid TLS-LES shows reasonable agreement with the experiment
Skin friction coefficient over the upper flat wall displays a strong drop and a long
plateau starting near the separation region in the bottom wall, and a more
gradual decrease downstream
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Hybrid TLS-LES of Asymmetric Diffuser Flow
•
The total pressure decreases 30% in the streamwise direction due
to frictional losses.
•
Mean velocity predicted reasonably with the hybrid TLS-LES
model
– Separation location agrees well
– But reattachment is observed further downstream
Exp. (symbols), TLS-LES (dashed lines)
LES (solid lines)
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Conclusion and Future Plans
• A generalized hybrid formulation developed to couple TLS-LES
– New hybrid terms identified but they still need closure
• TLS as a “near-wall” model for high-Re flows used in a TLS-LES
approach without the hybrid terms
– Reasonable accuracy using “relatively” coarse LS grid
– Potential application to complex flows with separation
•
Efficient parallel implementation can reduce overall cost
Next Step
– Analyze the hybrid terms in the TLS-LES equations and develop
models for hybrid terms
– A priori analysis of SS derivatives for arbitrarily positioned SS lines