TURBULENCE MODEL ING FOR BEG INNERS

by

TONYSAAD

UNIVERSITYOFTENNESSEESPACEINSTITUTE

tsaad@utsi .edu http://jedi .knows. it

The purpose of this tiny guide is to summarize the basic concepts of turbulence

modeling and to a compile the fundamental turbulence models into one simple

framework.Intendedforthebeginner,noderivationsareincluded,unlessinsome

simple cases, as the focus is to present a balance between the physical

nderstandingandtheclosureequations.Ihopethismaterialwillbehelpful.u

TABLEOFCONTENTS

Introduction ........................................................................................................................ 3

First Order Models: ........................................................................................................... 6

Zero-Equation Models .............................................................................................................. 7

One-Equation Models: .............................................................................................................. 8

Two-Equation Models ( k ): .............................................................................................. 10Second Order Models ....................................................................................................... 12

The Standard Reynolds Stress Model (RSM) ....................................................................... 13

Turbulent Diffusion Modeling ............................................................................................................ 13

Pressure-Strain Correlation Modeling ................................................................................................. 14

Modeling of the Turbulent Dissipation Rate ....................................................................................... 16

The Algebraic Stress Model ................................................................................................... 18

A

cknowledgments............................................................................................................. 19

2

INTRODUCTION

Aturbulentflowfieldischaracterizedbyvelocityfluctuationsinalldirectionsand

hasaninfinitenumberofscales(degreesoffreedom).SolvingtheNSequationsfora

turbulentflowis impossiblebecausetheequationsareelliptic,nonlinear,coupled

(pressurevelocity, temperaturevelocity). The flow is three dimensional, chaotic,

diffusive, dissipative, and intermittent. The most important characteristic of a

turbulentflowistheinfinitenumberofscalessothatafullnumericalresolutionof

the flow requires the construction of a grid with a number of nodes that is

proportionaltoRe9/4.

nsforaNewtonianfluidareThegoverningequatio

ConservationofMass

0ii

ut x

+ =

(1)

Conservationofmomentum

+ = + +

3

j j j

Conservationofpassivescalars(givenascalarT )

j ii i i ui

i

u uu u p g sx x x x

(2)

+ = + p p j t

c T c u T Tk s j j jx x x

Sohowcanwesolvetheproblem?Oneofthesolutionsistoreducethenumberof

scales(frominfinityto1or2)byusingtheReynoldsdecomposition.Anyproperty

(whether a vector or a scalar) can be written as the sum of an average and a

fluctuation, i.e.

(3)

= + where the capital letter denotes the average and thelower case letter denotes the fluctuation of the property. Of course, this

decompositionwill yield a set of equations governing the average flow field. The

newequationswillbeexactforanaverageflowfieldnotfortheexactturbulentflow

field.By an average flow fieldwemean that anypropertybecomes constant over

time.TheresultofusingtheReynoldsdecompositionintheNSequationsiscalled

theRANSorReynoldsAveragedNavierStokesEquations.Uponsubstitutionofthe

Reynolds decomposition (for each variable, we substitute the corresponding

tainthefollowingRANSequations:decomposition)weob

ConservationofMass

0 + =

i

i

Ux

(4)

Conservationofmomentum

_____

+ = +

j ii i

i j uii

U UU U Pu u Sx

(5)j j jx x x

Conservationofpassivescalars(givenascalarT )

_____

+ = p p j

p j tj j j

c T c U T Tk c u tx x x

+S (6)

Note:aspecialpropertyoftheReynoldsdecomposition isthattheaverageofthefluctuating

componentisidenticallyzero,afactthatisusedinthederivationoftheaboveequations.

However,byusingtheReynoldsdecomposition,therearenewunknownsthatwere

introducedsuchas the turbulent stresses_____

i ju u and turbulent fluxes (where theoverbardenotesanaverage) and therefore, theRANSequationsdescribeanopen

setofequations.Theneedforadditionalequationstomodelthenewunknownsis

calledTurbulenceModeling.

Wenowhave9additionalunknowns(6Reynoldsstressesand3turbulentfluxes).

Intotal,forthesimplestturbulentflow(includingthetransportofascalarpassive

4

scalar,e.g.temperaturewhenheattransferisinvolved)there14unknowns!

A straight forwardmethod tomodel the additional unknowns is to develop new

PDEs for each termbyusing theoriginal set of theNS equations (multiplying the

momentumequations toproduce the turbulent stresses).However, theproblem

with this procedure is that it will introduce new correlations for the unknowns

(triplecorrelations)andsoon.Wethenmightthinkofdevelopingnewequationsfor

the triple correlations,nevertheless,wewill endupwithquadruplecorrelations

andsoonAnalternativeapproachistousethePDEsfortheturbulentstressesand

fluxes as a guide to modeling. The turbulent models are as follows, in order of

increasingcomplexity:

ls model)Algebraic(zeroequation)mode :mixinglength(firstorder

Oneequationmodels:kmodel,tmodel(firstordermodel) tordermodel)Twoequationmodels:k,kkl,k2,lowRek(firs Algebraicstressmodels:ASM(secondordermodel)

Reynoldsstressmodels:RSM(secondordermodel)

ZeroEquationModels

OneEquationModels

TwoEquationModels

AlgebraicStressModels

ReynoldsStressModels

FirstOrderModels

SecondOrderModels

5

FIRSTORDERMODELS

Firstordermodelsarebasedon theanalogybetween laminarand turbulent flow.

They are also called Eddy Viscosity Models (EVM). The idea is that the average

turbulent flow field is similar to the corresponding laminar flow. This analogy is

illustratedasfollows

ij23

Laminar FLow:

23

Turbulent Flow:

j jiij

j i j

ip i

jt iij i j t ij

j i

t ti i

6

p ic x

u uux x x

k Tqc x

UUu u kx x

k Tq u t

= + =

= = + = = ferredtoasthegeneralizedBoussinesqhypothesis.

(7)

whichisre

Notethat:

Tt urbulent ViscosityTurbulent Kinetic EnergykTurbulent Conduction Coefficienttk

===

Theturbulentviscosityandtheturbulentconductioncoefficientareflowproperties.

They are not properties of the fluid. They vary from one flow to another. So the

problemnowisthedevisemeansormodelstofindtheseunknowns,theturbulent

viscosity and the turbulent conduction, because the turbulent stresses and fluxes

illbeexpressedasfunctionofthesenewflowproperties.

w

ZEROEQUATIONMODELS

Inzeroequationmodels,asthenamedesignates,wehavenoPDEthatdescribesthe

transportoftheturbulentstressesandfluxes.Asimplealgebraicrelationisusedto

close the problem. Based on the mixing length theory, which is the length over

which there is high interaction of vortices in a turbulent flow field, dimensional

analysisisusedtoshowthat:

tt m mdUlu l l

7

dy

isdeterminedexperimentally.Forboundarylayers,wehave

= = (8)

ml

for

for m

m

l y yl y

=