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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, Re= 2003 (Hoyas et al., 2006)

• DNS Computations of turbulent separated flows

– 151 M grid points, Re = 395 (Marquillie et al., 2008)

– DNS at lower Reynolds number (Experiment at Re= 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

xLES

zLES

yLES

ySS

xSS zSS

x

y

z

Large-Scale Grid

Small-Scale Grid

Resolution requirements

Number of LES control volumes: NLES3 NLES <NSS

Grid points for TLS-LES: NLS3 + 3NLS

2NSS 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 LExact Field is split into LS and SS fields:

Continuous large scale field is defined by adopted LS grid:

( , ) ( , )

( , )

( , )

L

Lk

u x t L u x t

F L u x t

F u x t

( , ) ( , ) ( , )L Su x t u x t u x t

SS field is defined based on LS field from decomposition:

( , ) ( , ) ( , )S Lu x t u x t u x t

Sampling at LS grid nodes:L

Interpolation to the SS nodes:F

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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.

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 Yk

y Y

1 21

2 2

11 tanh tanh

1 22

c y d ck 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 Li i iu x t R u x t ku x t k u x t

1

1

R F Lij ij ij

F LF F R R L L R Ri j i j i j i j

k k

k u u u u k u u u u

• 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,Ri

i

u

x

Continuity:

Momentum:2

2

RR RRR ijR Ri i

i ji i j i

u upu u

t x x x x

( ) ( )R R R Rij i j i ju u u u The unresolved term in the momentum equation

1/ 2 22

3 F

ij t sgs ij ij sgsC k S k

R R RL R S S R S S

ij i j i j i ju u u u u uTLS:

LES:

1 1F LR F L F F R R L L R R

ij ij ij i j i j i j i jk k k u u u u k u u u u

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,Si

i

u

x

2

2

S SSR S R S Ri ii i j j i

j i j

u upu 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= 1200

Re= 590

3-D energy spectra

Red line : Instantaneous energy spectraBlue line : Volume average spectraBlack line: k-5/3 slope

Re= 2400

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

– 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 Re = 395

– DNS(~151 M) at Re = 395 by Marquillie et al., J. of Turb., Vol. 9, 2008

– Experiment at Re = 6500 by Bernard et al., AIAA J., Vol. 41, 2003

• Spatial resolution (%75 coarser than DNS)

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

– TLS-LES-SS (8 SS points/LS): x+SS = 9.6, z+SS = 2.4, y+SS = 0.68

– DNS (1536 x 257 x 384) : x+ = 3, z+ = 3, y+|max = 4.8

• Inflow turbulence from a separate LES channel study at Re = 395

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 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

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 = 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 = 500

Inclination angle: 100

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• Spatial resolution

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

– TLS-LES-SS (8 SS points/LS) : x+SS = 6.7, z+SS = 6.2, y+SS = 0.72

– LES (278 x 80 x 80) : x+ = 25, z+ = 25, 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 Yk

y Y

1 R F Lij ij ijk k

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

• Cp 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 costNext 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