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 11
th
, 2008
Computational Combustion Lab
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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 Re
3
•
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, Re
t
= 2003 (Hoyas
et al
., 2006)
•
DNS Computations of turbulent separated flows
–
151 M grid points, Re
t
= 395 (Marquillie
et al.
, 2008)
–
DNS at lower Reynolds number (Experiment at Re
t
= 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
Small scale equations are solved on three 1D lines embedded in the 3D domain
D
x
LES
D
z
LES
D
y
LES
D
y
SS
D
x
SS
D
z
SS
x
y
z
Large

Scale Grid
Small

Scale Grid
Resolution requirements
Number of LES control volumes:
N
LES
3
N
LES
<N
SS
Grid points for TLS

LES:
N
LS
3
+ 3N
LS
2
N
SS
N
LS
<N
LES
, N
LS
<<N
SS
Grid points for DNS:
N
SS
3
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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:
Continuous large scale field is defined
by adopted LS grid:
(,) (,)
(,)
(,)
L
L
k
u x t L u x t
F L u x t
F u x t
D D
D
(,) (,) (,)
L S
u x t u x t u x t
SS field is defined based on LS
field from decomposition:
(,) (,) (,)
S L
u x t u x t u x t
Sampling at LS grid nodes
:
L
D
Interpolation to the SS nodes
:
F
D
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Fully resolved signal (black) from a 128
3
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.
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.
TLS has higher
spectral support
A priori analysis of scale separation operators (LES and TLS)
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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
RANS
LES
prescribed
y interface
Hybrid RANS

LES Strategy
TLS
LES
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
= k
F
+ (1

k)
L
–
LES operator F is the standard filtering operator
–
k is a transition function relating TLS and LES domains
1
if
0
TLS
TLS
y Y
k
y Y
1 2
1
2 2
1
1 tanh tanh
1 2
2
c y d c
k c
c y d c
Step Function
Tanh
Function
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•
Application of additive scale separation operator
–
Velocity components
–
Turbulent stress
Hybrid TLS

LES Equations
Hybrid Terms
(,) (,) (,) 1 (,)
R F L
i i i
u x t R u x t ku x t k u x t
1
1
R F L
ij ij ij
F L
F F R R L L R R
i j i j i j i j
k k
k u u u u k u u u u
t t t
•
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 !!
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Hybrid TLS

LES Equations
•
Resolved / Large Scale Equations
0,
R
i
i
u
x
Continuity:
Momentum:
2
2
R
R R
R
R
ij
R R
i i
i j
i i j i
u u
p
u u
t x x x x
t
( ) ( )
R R R R
ij i j i j
uu u u
t
The unresolved term in the momentum equation
1/2
2
2
3
D
F
ij t sgs ij ij sgs
C k S k
t
R R R
L R S S R S S
ij i j i j i j
u u u u u u
t
TLS:
LES
:
1 1
F L
R F L F F R R L L R R
ij ij ij i j i j i j i j
k k k u u u u k u u u u
t t t
Specific closures
for each model
The scale interaction terms are closed if the small scale field is known
any SGS model
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Hybrid TLS

LES Equations
•
Small Scale Equations
•
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
0,
S
i
i
u
x
2
2
S S
S
R S R S R
i i
i i j j i
j i j
u u
p
u u u u F
t x x x
Continuity:
Momentum:
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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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Re
t
=‱㈰〠
Re
t
=‵90
3

D energy spectra
Red line : Instantaneous energy spectra
Blue line : Volume average spectra
Black line: k

5/3
slope
Re
t
=′400
Hybrid TLS

LES approach recovers both
LS and SS spectra near the wall
Hybrid TLS

LES of Channel Flow
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Numerical Solver
•
Incompressible Multi

domain Parallel Solver
–
4
th
order accurate kinetic energy conservative form (used here)
–
5
th
order accurate upwind

biasing for convective terms
–
4
th
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 Re
t
㴠=㤵
–
DNS(~151 M) at Re
t
= 395 by Marquillie
et al.
, J. of Turb., Vol. 9, 2008
–
Experiment at Re
t
= 6500 by Bernard
et al.
, AIAA J., Vol. 41, 2003
•
Spatial resolution (%75 coarser than DNS)
–
TLS

LES

LS (64 x 46 x 64) :
D
x
+LS
= 77.4,
D
z
+LS
= 19.2,
D
y
+LS
= 5.4
–
TLS

LES

SS (8 SS points/LS):
D
x
+SS
= 9.6,
D
z
+SS
= 2.4,
D
y
+SS
= 0.68
–
DNS (1536 x 257 x 384) :
D
x
+
= 3,
D
z
+
= 3,
D
y
+

max
= 4.8
•
Inflow turbulence from a separate LES channel study at Re
t
㴠=㤵
Total vorticity on a spanwise plane (LES)
Streamwise vorticity on a horizontal plane (LES)
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Spanwise line in the
separation region
Time evolution of the SS velocity
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SS iterations: 20
SS vorticity magnitude isosurfaces colored with SS streamwise velocity
SS iterations: 100
SS iterations: 300
SS evolution effect on the instantaneous flow
•
simulations on each line
•
optimal parallel approach
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Hybrid TLS

LES of Separated Channel Flow
•
Hybrid TLS

LES grid is chosen very coarse
deliberately
•
Hybrid TLS

LES C
p
shows good agreement with DNS
–
~%30 off from experiments (higher Re) for all studies
•
Hybrid TLS

LES C
f
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
The authors would like to thank Dr. J.

P. Laval for providing the DNS data
DNS

151M (circles and shaded contours), TLSLES (red), LES (green)
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Hybrid TLS

LES of Asymmetric Diffuser Flow
•
Hybrid TLS

LES(~0.25M) and LES(~1.8M) at Re
t
= 500
–
LES(~6.5 M) by Kaltenbach
et al.
, J. of Fluid Mech., Vol. 390, 1999
–
Experiment by Buice and Eaton, J. of Fluids Eng., Vol. 122, 2000
•
The main features of this flow
–
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 Re
t
= 500
Inclination angle: 10
0
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•
Spatial resolution
–
TLS

LES

LS (110 x 56 x 40) :
D
x
+LS
= 54,
D
z
+LS
= 50,
D
y
+LS
= 5.72
–
TLS

LES

SS (8 SS points/LS) :
D
x
+SS
= 6.7,
D
z
+SS
= 6.2,
D
y
+SS
= 0.72
–
LES (278 x 80 x 80) :
D
x
+
= 25,
D
z
+
= 25,
D
y
+
= 0.98
–
LES by Kaltenbach et al., 1999 (590 x 100 x 110)
•
Step function (
, F:
LES
, L:
TLS operator)
–
pre

defined interface,
Y+
TLS
= 152
1
if
0
TLS
TLS
y Y
k
y Y
1
R F L
ij ij ij
k k
t t t
Hybrid TLS

LES of Asymmetric Diffuser Flow
Isosurfaces of the second invariant of the velocity gradient
tensor colored with local streamwise velocity predicted
with LES model
TLS
TLS
LES
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Hybrid TLS

LES of Asymmetric Diffuser Flow
•
C
p
along the lower and upper wall predicted reasonably well
•
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
•
Overall, TLS

LES shows ability to predict separation regions without any model
changes
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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
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