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CFD TURBULENCE MODELLING: RANS, LES &

DNS

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STRUCTURE

The formidable modelling task Overview some RANS models &

limitations Show for LES model not that important Outline the things for LES that matter

and the order of importance Discuss mixing LES & RANS models RANS model defects and management

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KEY ROLE OFTURBULENCE

Drag generation Heat transfer Particle dispersion Scalar mixing Sound generation

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TURBULENCE

da Vinci - describes the“clouds as scattered and torn” Van Gogh

Transition

l

y+

5

FORMIDABLE TASK

“I am an old man now, and when I die and go to heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am rather optimistic” Sir Horace Lamb

FRS (1849-1934) 2nd Wrangler Trinity College

“Turbulence is the last great unsolved problem in classical physics”Richard Feynman (Nobel Prize in Physics - quantum

electrodynamics )

Do not even know the Karman constant (l = κy – 0.38 < κ < 0.45) or if it is a constant!!! – Spalart (2006) 2% decrease in κ gives 1%

decrease in predicted aircraft drag

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MODEL BASIS

Phenomenological – but we do not fully understand the phenomena!!!

Spalart & Allmaras (1994) – La Recherche Aerospatiale, No 1, 5-21

Abstract – A transport equation for turbulent viscosity is assembled based

on empiricism and arguments of dimensional analysis ……

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DICTIONARYDEFINITION

Empiricism- Philosophy. the doctrine that all

knowledge is derived from sense experience.

- Undue reliance upon experience, as in medicine; quackery.

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SA MODEL BASIS

4 nested models: (I) Free shear flows; (II) log-outer layer; (III) buffer and viscous sublayer & (IV) laminar and trip region

Model (I)

],C[TermDiffusionSCDt

Dt

t σµµ

21 +=

Shearing for production

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CALIBRATION

2D mixing layer

Wake

Calibration suggests 0.6<σ<1; 0.1375< C1<0.1275 & 0.6< C2<0.7

Pick:2/3, 0.1355, 0.622. Acknowledge plane jet spreading rate 38% too high

( )2010= U∆.maxτ

( )2060= U∆.maxτ

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RANS µt BASEDMODELS

Boussinesq (1877)

ijtji Suu µρ =′′ + QUADRATIC TERMS + CUBIC TERMS

Gatski & Speziale (1993) Craft, Launder & Suga (1996)

∂U/∂y

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BASIC RANS MODELS

Classified by the number of differential equations

Realizability - whether the model is constrained so that it does not break basic physical principles k > 0

Hundreds of basic RANS models - indicative of quest for something better.

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ZERO EQUATIONMODELS

Dimensional grounds µt α Length x Velocity

Incomplete model - l eqn takes different forms for different flow zones.

» Log layer l = κy » viscous sublayer l = κ y D» Outer BL l = C δ» Mixing layer l ≈ 0.07 x the layer width;» Round jet l ≈ 0.075 x the jet half width;» Plane jet l ≈ 0.09 x jet half width and» Wake l ≈ 0.16 x the wake half width

Numerous l choices and these, in themselves, can be subjected to further corrections – see later.

Zero equation models require relatively modest grids - slow iterative convergence. Seldom be expected to provide high accuracy unlikely to massively err.

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ONE EQUATION TURBULENCE MODELS

Zero eqn. assume turbulence equilibrium. One-eqn allow turbulence transport

SA model, Secundov, Baldwin-Barth µt α φ; k-lmodel φ = k

Grid forgiving – frequently used just near walls

φφφ

SΓDt

D+∇= 2

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TWO EQUATION TURBULENCE MODELS

Vast number - most are based on k & ε

Secundov et al. µt and l; Kim and Chung k-µt; Wilcox k-ω; Warner et al. k-kl

k-ω - sensitive to the specified free stream turbulence intensity level - zonal Menter (1993) model.

ερµ µ

2

=k

Ct

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RSM MODEL

Around 12 massive differential equations to solve Most exact model Contains substantial empiricism Computational cost is extreme Numerous gradient terms → grid demands Separation or buoyancy → large scale unsteady structures →

failure Uncertain near wall performance: modelling dissipation,

pressure-strain, ad hoc modifications Wide range of options suggests a weakness and unresolved

issues Simpler eddy viscosity models used near walls

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NON-LINEAR EDDY VISCOSITY MODELS

• ‘Half-way-house’ between costly RSM and simple linear EVM

•Extended forms of the Boussinesq approximation: quadratic or cubic

•Quadratic - anisotropy modelling; cubic - streamline curvature

•Marketed as a means of getting RSM type performance at linear eddy viscosity model cost

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QUALITATIVE PREDICTIVE ACCURACY

•Round jet/plane jet anomaly - dδ/dx opposite traits•Ma > 1 dδ/dx decreases x2 but RANS insensitive•Axis switching µt based models can’t handle

0.086-0.0960.101Round jet

0.1-0.110.091Plane jet

Measuredk-εFlow

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WALL JETS

RANS 30%over-predictspreading rate

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URANS

Linear models

Non-linearmodels

OK - has spectralgap - unusual

Liu and Tucker (2007)IJNME

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URANS

T [K]

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The Resolved Solution inDifferent Approaches

•

By Strelets group

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WHAT IS THE SOLUTION? &WHAT IS LES?

LES = Resolve all large eddiesRANS = Resolve time average of flow

x

y

∆

Resolved/solved forModelled < 2∆

l

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L. F. Richardson’s (1922) Rhyme& Kolmogorov (1941)

•Big whorls have little whorls,which feed on their velocity,and little whorls have lesser whorls,and so on to viscosity (in the molecular sense).

•Kolmogorov (1941), smaller whorls or eddies isotropic Energy α k-5/3

Big whorls

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LESHIERARCHY

WALL MODELLING

GRID/SOLVERCOMPATIBLITY

PROBLEM DEFINITION

MODEL

BCs, Soln uniqueness

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ULTIMATE NUMERICAL INFLUENCES FOR LES

NUMEROUS FORMS OFNS EQNS – Chow & Moin (JCP) 2003

NUMEROUSDISCRETIZATIONS

dissipation

Staggered grids, cell centered/vertex,

codes with smoothers, Rhie and Chow, axis treatment one legged and two

legged

+

GRID TOPOLOGY dissipation

+

+ turbulence model validity + wall modelling + problem definition + solnuniqueness + ……= Do not get too hung up on turbulence model

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HEAT TRANSFER

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GRID INFLUENCE

θ

Denton code 10-20 x fasterthan HYDRA

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LES

Boussinesq (1877)

ijsgsji Suu µρ =′′ + Non-linear terms

Comte-Bellot and Corsin

Clark et al. (1979) [+Leonard]Kosovic (1997), Leray, LANS α [Geurts & Holm (2005)]Cubic [Lund & Novikov (1992)]

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LES MODELS TESTED

ui,j = ∂ui/ ∂xj

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LES MODEL MINIMAL INFLUENCE – GOOD

RESULTS WITH NO LES MODEL!!

Re = 4 x 103

Circa 1 million cells

x/D = 15

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KEY LES PROBLEM

Hinze (1975)

•Resolving streaks

•Pope (2004), Boeing 777 at cruise108 streaks

•LES Cost α Re2.5*

•Hybrid LES-RANS Cost α Re0.5

(Based on outer layer modelling)

y+=90

*Piomelli, AIAA-2008-396

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WING-FLAP [Re = 23 x 106, 3.3 million cells]

ZONAL ILES-RANS vorticity contours

Model CL

% Error

RANS +24

Zonal LES-RANS +10

Zonal ILES-RANS -5

ILES -16

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CHEVRON ILES-RANS

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JET WITH CO-FLOW

Re = 300,000

x/D=1

x/D=2

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COMPRESSOR/TURBINELES

s

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DES of F-15 Post-Stall

By Forsythe, Wurtzler, Squires, Cobalt

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Work of L. Hedges, NASA Funded

URANS

DES

VorticityMagnitude

SRANS,Partly Converged

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Generic Heavy Truck in Cross-Wind

By Wurtzler, Forsythe, Cobalt

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Electronics

Heat transferzone

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LES SIMULATIONS

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WASHING OUT OF EDDY VISCOSITY

Surface orthogonalflow direction

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PART 2: RANS MODEL DEFECTS AND MANAGEMENT

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SIMPLER MODELPROBLEMS

» Erroneous predicted turbulence for curved shear layers and adverse pressure gradients;

» Separation suppression on curved surfaces» Failure at separation» Excessive turbulence in stagnation zones» Wrong behaviour for rotating flows» Insensitivity to density gradients» Excessive heat transfer at reattachments points » Insensitivity to system rotation

Cf. Boussinesq - u’u’=f(∂U/∂y), RSM - Du’u’/Dt=……

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STAGNATIONPROBLEM

Turbulence massively over predicted Suppression of leading edge separation;

excessive predicted heat transfer Ad hoc corrections:

» Kato and Launder (1993);» Yap (1987) &

» Cµ modifications

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CURVATURE PROBLEM

Turbulence can be virtually eliminated around convex surfaces

Basic Richardson correction (various forms, can be used in various places in same model)

Many other curvature corrections available e.g. modification of the k or ε equation or Cµ based on contractions of more advanced models

r

U

R

URi ∂

∂= θθ

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STREAMLINE CURVATURE

B. E. Launder Int. J. Heat and Fluid Flow,1989

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BODY FORCE/SWIRLPROBLEM

Turbomachinery flows often involve rotation/swirl & local swirl

Stable

Rossby number (different forms)

Many other correction forms- SARC

0>)rdr/()rU(d θ

y/U

ΩRo ∂∂=

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BUOYANCY

•∂T/∂r > 0 unstable, Rayleigh-Bernard instability

•Large scale unsteady structures

•Way beyond even RSM

•Again a range of different corrections e.g. gradient based hypothesis

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CONVEX FEATURESPROBLEM

Eddy viscosity contours

| ∇φ|=1

| ∇φ|=1+ Γ∇2φ Γ = ε φ

wall distances

Flow direction

Tucker, Rumsey and Spalart … AIAA J. (2005)

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CONVEX FEATURESPROBLEM

Hamilton-Jacobi equation – Tucker, Rumsey, Spalart ….(AIAA J.)

RANS still vital for design

| ∇φ|=1

| ∇φ|=1+ Γ∇2φ Γ = ε φ 2.77Normal calculation

2.90Palliative dfunction

3.10Measured

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TRANSITION MODELLING

Transition modelling is of some importance for LPT, engine inlets

Influenced by roughness, acoustics and external disturbances, pressure gradient, freestream velocity change, surface curvature, temperature gradients and rotation

Generally associated with the growth of Tollmien-Schlichting (T-S) waves.

Bypass transition, linear T-S wave process is bypassed, free stream intensities greater than 0.5%

Some turbulence models can naturally predict the bypass transition process

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TRANSITION MODELLING

k Convection = k Diffusion + P - D k convects and diffuses into BL Increases µt, increases Pk = µt (dU/dy)2

Substantial kave growth until transition

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TRANSITION MODELLING

Standard transition procedure, estimate the transition point, exp. correlations (Abu-Ghannan and Shaw (1980), Arnal (1992))

Trip model Lots of models, active research area With fine enough grids LES will capture

transition

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REAL GEOMETRY, CAA AND DESIGN OPTIMISATION

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REAL GEOMETRY

Blisters

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DESIGN OPTIMISATION

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SILENT AIRCRAFT

Shadow region

Acoustic waves reflect from upper surface

Sound reflected from underside of wing

Reflection of jet noise for a conventional podded engine installation

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NOISE SHIELDING

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Low Noise Design: High-Lift System

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Combustion Noise

URANS + LES + High-Fidelity Models

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CONCLUSIONS

Vast number of RANS models, choice can have substantial impact - CFD use a specialist activity

CFD predict correct delta’s

To predict exact levels extreme insights into turbulence model and many other CFD aspects + calibration data

Rationalism is desirable with every effort being made to place the CFD solution on a solid rational basis

Many practical flows are highly three dimensional in which inviscid pressure driven structures occur and then turbulence stresses become less important

However, if the 3D structures are unsteady in nature, other challenges arise

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CONCLUSIONS

URANS can help Zonal RANS-LES & LES with take over but when? Depends on HPC/GPU developments Zonal RANS-LES & LES still need physical insight by analyst

Problem definition, solution uniqueness, transition, separation + Near wall LES modelling + Grid structure/quality > LES Model

LES Hierarchy?: