Turbulence Modelling: Large Eddy Simulation

Turbulence Modeling: Large Eddy Simulation and Hybrid RANS/LES

• Introduction

• LES Sub-grid Models

• Numerical aspect and Mesh

• Boundary conditions

• Hybrid approaches

• Sample Results

Turbulence Structures

Introduction: LES / other Prediction Methods

• Different approaches to make turbulence computationally tractable:

– DNS: Direct Simulation.

– RANS: Reynolds average (or time or ensemble)

– LES: Spatially average (or filter)

DNS

3D, unsteadyRANS

Steady / unsteady

LES 3D, unsteady

RANS vs LES

RANS model1

RANS model 2

LES

Why LES?

• Some applications need explicit computation of accurate unsteady fields.

– Bluff body aerodynamics– Aerodynamically generated noise (sound)– Fluid-structure interaction– Mixing– Combustion – …

LES : difficulties

• Cpu expensive for atmospheric modelling:

– Unsteady simulation

– Turn around time is weeks/months (rans is hours or days)

– Cannot afford grid independence testing

• Still open research issues

– combustion, acoustics, high Prandtl, shmidt mixing problem, uns.

– Wall bounded flows

• Need expertise to reduce cpu, human cost

– Mesh – models – strategy

– Analysis of the instantaneous flow

– Knowledge about turbulent instabilities, turbulent structures

Energy Spectrum

• Large eddies: responsible for the transports of

momentum, energy, and other scalars.

anisotropic, subjected to history

effects, are strongly dependent on boundary conditions, which makes their modeling difficult.

k,f

Small eddies tend to be more isotropic and less flow-dependent (universal), mainly dissipative scales, which makes their modeling easier.

Eu,ET

LES: Filtering - Decomposition

E

Energy spectrum against the length scale

scalesubgrid scaleresolved

ttt ,,, xuxuxu

t,xu

f2

t,xu

AnisotropicFlow dependent

IsotropicHomogeneousUniversal

Filtering

• The large or resolved scale field is a local average of the complete field.

– e.g., in 1-D,

– Where G(x,x’) is the filter kernel.

– Exemple: “box filter” G(x,x’) = 1/D if |x-x’|<D/2, 0 otherwise

( ) ( , ') ( ') 'i iu x G x x u x dx

Filtered Navier-Stokes EquationsFiltering the original Navier-Stokes equations gives filtered Navier-Stokes equations

that are the governing equations in LES.

j

i

jij

jii

x

u

xx

p

x

uu

t

u 1

Filter

j

ij

j

i

jij

jii

xx

u

xx

p

x

uu

t

u

1

jijiij uuuu Needs modeling

Sub-grid scale (SGS) stress

N-S equation

Filtered N-S equation

Available SGS model

• Subgrid stress : turbulent viscosity

• Smagorinsky model (Smagorinsky, 1963)

– Need ad-hoc near wall damping

• Dynamic model (Germano et al., 1991)– Local adaptation of the

Smagorinsky constant

• Dynamic subgrid kinetic energy

transport model (Kim & Menon 2001)– Robust constant calculation

procedure– Physical limitation of backscatter

2

t sv C S

2/1sgskt kCv

j

sgs

k

sgs

j

sgs

j

iij

j

sgsjsgs

x

k

x

kC

x

u

x

ku

t

k

2/3

12

3ij kk ij t ijv S

2

t Dv C S

Smagorinsky’s Model

• Hypothesis: local equilibrium of sub-grid scales• Simple algebraic (0-equation) model (similar to Prandtle Mixing

length model in RANS)

• Cs= 0.065 ~ 0.25• The major shortcoming is that there is no Cs universally applicable

to different types of flow.• Difficulty with transitional (laminar) flows.• An ad hoc damping is needed in near-wall region.• Turbulent viscosity is always positive so no possibility of

backscatter.

ijijSSS 2,3/1 with

2

t sv C S

Dynamic Smagorinsky’s Model

• Based on the similarity concept and Germano’s identity (Germano et al., 1991; Lilly, 1992)

• Introduce a second filter, called the test filter with scale larger than the grid filter scale

• The model parameter (Cs ) is automatically adjusted using the resolved velocity field.

• Overcomes the shortcomings of the Smagorinsky’s model.

– Can handle transitional flows

– The near-wall (damping) effects are accounted for.

• Potential Instability of the constant

• Basic Idea : consider the same smagorinsky model at two different scales, and adjust the constant accordingly

• Constant value = error minimization using least square method and Germano’s Identity

Tij

ij

Lij

Error minimization

Test Filter Grid Filter

ij i j i jL u u u u

ijij ijE L T

Dynamic Smagorinsky’s Model

Dynamic Subgrid KE Transport Model

• Kim and Menon (1997)

• One-equation (for SGS kinetic energy) model

• Like the dynamic Smagorinsky’s model, the model constants (Ck, Ce) are automatically adjusted on-the-fly using the resolved velocity field.

• Backscatter better accounted for

ijsgskijkkij SkC 2/123

1

j

sgs

k

sgs

j

sgs

j

iij

j

sgsjsgs

x

k

x

kC

x

u

x

ku

t

k

2/3

Mesh

• Grid resolution :

– Constraint Based on LES hypothesis:• Explicit Resolution of production mechanism (whereas production

is modeled with RANS)• Resolution of anisotropic and energetic large scales

– Cell size must be included inside the inertial range, in between the integral scale (L) and the Taylor micro-scale (l).

– Integral scale L• Energy peaks at the integral scale. These scales must be resolved

(with several grid points).• Crude Estimation of L :

– Use correlation (mixing layer, jet) for L– Perform RANS calculation and compute L = k3/2 / e

Energie E(k)Dissipation D(k)

1/1/L 1/

Mesh

• Grid resolution:

– Taylor micro-scale l:• Dissipation rate peaks at l.• Not necessary to resolve l but useful to define a lower bound for the cell

size.• Estimation of l ~ L ReL-1/2

– ( for an homogeneous and isotropic turbulence = 151/2 L ReL-1/2)

– Temporal resolution: resolve characteristic time scale associated to the cell size (ie CFL=U Dt / Dx <1).

• As for the numerical scheme (minimization of numerical errors) use of hexa and high quality of mesh (very small deformations) is recommended

Numerics: time step

– Dt must be of the order (or even less for acoustic purpose) of the characteristic time scale t corresponding to the smallest resolved scales.

– As t~Dx/U , it correspond to approx CFL = 1 (Courant Dreidrich Levy Number) (where U is the velocity scale of the flow)

Numerics: discretization scheme

• Discretization scheme in space should minimize numerical dissipation– LES is much more sensitive to numerical diffusion than RANS– 2nd Order Central Difference Scheme (CD or BCD) perform much better than high order

upwind scheme for momentum• A commonly used remedy is to blend CD and FOU.

– With a fixed weight (G = 0.8), this blending scheme has been found to still introduce considerable numerical diffusion

– Bad idea! • Most ideally, we need a smart, solution-adaptive scheme that detects the wiggles on-the-fly

and suppress them selectively.

Boundary Conditions (LES)

• Near-wall resolving

• Near-wall modelling

• Inlet Boundary Conditions

Inlet Boundary Conditions Inlet Boundary conditions :

– Laminar case:

• Random noise is sufficient for transition

– Turbulent case:

mean velcocity field turbulent fluctuations

, ,i i iu t U u t x x x

Precursor domain

Realistic inlet turbulence

Cpu cost – not universal

Vortex MethodCoherent structures – preserving turbulenceNeed to use realistic profiles (U, k, ) – otherwise risk to force the flow

Spectral synthesizer No spatial coherenceFlexibility for inlet (profiles of full reynolds stress or k constant values, correlation)

Boundary Conditions

• Near Wall treatment

• 1/ Near Wall Resolving

– All the near–wall turbulent structures are explicitly computed down to the viscous sub-layer.

• 2/ Near Wall Modeling

– All the near-wall turbulent structures are explicitly computed down to a given y+ >1

• 3/ DES:

– No turbulent structures are computed at all inside the entire boundary layer (all B.L. is modeled with RANS).

Boundary Conditions

• 1/ Near wall resolving : explicit resolution of the boundary layer.

– Motivation: separated flows, complex physics (turbulence control)

– High resolution requirement due to the presence of (anisotropic) wall turbulent structures: so called streaks.

– Necessary to resolved correctly these near wall production mechanisms

–

• Boundary layer grid resolution :

• y+<2

• Dx+ ~ 50-150, Dz+ ~ 15-40

•

• Not only wall normal constraints, but also Span-wise and stream-wise constraints due to the streaky structures

Boundary Conditions

• Near wall modeling :

• Wall function:

– Schumann/Grozbach:

• Instantaneous wall shear stress at walls and instantaneous tangential velocity in the wall adjacent cells are assumed to be in phase.

• Log law apply for mean velocity (necessary to perform acquisition)

– Werner & Wengle (6.2):

• Instantaneous wall shear stress at walls and instantaneous tangential velocity in the wall adjacent cells are assumed to be in phase.

• Filtered Log law (power 1/7) applied to instantaneous quantities

Hybrid LES-URANS

• Near Walls: URANS 1-equation model

• Core region: LES 1-equation SGS model

Hybrid LES-URANS

• Navier Stokes time averaged in the near wall and filtered in the core region reads:

LES-URANS hybrid

• Use 1-equation model in both LES and URANS regions

LES region

RANS region

LES-URANS hybrid

• Problems:– LES region is supplied with bad BC from the URANS

regions – The flow going from URANS to LES region has no

proper time or length scale of turbulence

• Solution:– Add synthesized isotropic fluctuations as the source

term of the momentum equations at the LES-URANS interface.

Inlet BC and forcing

35 M 65 M

Side Box (max) 8 mm 6 mm

Rear Box (max) 8 mm 6 mm

Nb Prism layer 5 5

Side box Rear box

Volume mesh: Gambit & « Sizing Functions » to control both growth rate and cell size in specified box

LES of a realistic Car model exposed to Crosswind

Results:

Model Exp SST k-w

LES WALE (35 M cells)

LES WALE (65 M cells)

RSM v2-f

Drag (SCx)

0,70 0,66 0,69 0,68 0,71 0,73

Side (SCy)

2,22 2,00 2,19 2,18 2,30 2,10

For

ces

Lift (SCz)

1,40 1,66 1,27 1,30 1,82 1,77

Yawing (SCn)

-0,64 -0,60 -0,57 -0,59 -0,47 -0,47

Rolling (SCl)

-0,42 -0,36 -0,49 -0,49 -0,46 -0,41

Mom

ents

Pitching (SCm)

0,12 0,10 0,21 0,23 0,03 0,07

Simulation of flow over a 3D mountain

Comparision between RANS and LES-RANS hybrid model

• RANS using SST model

• Hybrid RANS-LES model

Conclusion

• RANS/URANS is not always reliable.• LES is closer to reality than RANS/URANS.• LES is computationally very expensive.• In the absence of enough experimental data one is left

with no choice but to use LES wherever feasible.