Multi-scale Simulation of Wall-bounded Flows

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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


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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