The State of the Art in Turbulence Modelling in Brazil

RBCM-J

.

of theBraz.Soc.Mechanical Sciences

Vol

.

XX

-

No.1

-

1998

_-

pp.1-38 ISSN_Printed0100_in

-

_Brazil7386

The

State

of

the

Art

in

Turbulence

_{Modelling in}

Brazil

Atila Pantaleão

Silva

Freire

MilaR

.

Avelinoand

Luiz Claudio C

.

Santos

Universidade Federaldo Riode Janeiro

COPPE

-

Programade EngenhariaMecânica C.P. 68503

21945

-

970Rio de Janeiro,RJBrasil Email: atila@_{serv.com.ufri}

.

_br

Abstract

The present work discussesat lengththe currentstatus ofturbulent research in Brazil, Aftereight introductory

sectionson the subject,where some generalaspects of the problemare presented, and abrief reviewof some

scientific and engineering approachesis given,the paperstrolls over four specific sections,analyzing all work

carried out inBrazilin_{the past twenty five years on turbulence}

_.

_Infact,thepresent compilationis restrictedto the

mainevents sponsored bythe BrazilianSocietyof Mechanical Sciences.Thepresentreviewquotes284references, presents6tablesand ¡6figures.Thepapercontentsis:PaperOutline,SomeInsights,TheTraditional Approaches,

SomeBasicWorking Rules, TurbulenceModels,One-PointTurbulence Closure Models,Some OtherAproachesto

TurbulenceModelling,Some Major Achievements,A Bitof History,Statistics, A Personal View, Gallery, Final

Remarks, CitedBibliography and Compiled Bibliography.

Keywords

.

Turbulence Flow, TurbulenceModelling

Paper Outline

Thepresent reviewwascommittioned by the Brazilian Society of MechanicalSciences,ABCM,in order togive a clear picture of the presentstatusofturbulence modelling in Brazil. As a guidingline,

the

paper

was supposed toconcentrate on work that appeared in the last five yearsin the two major

Conferences sponsored by the Association

:

the Brazilian Congress of Mechanical Engineering

-COBEM,and the Brazilian National Meetingon Thermal Sciences, ENCIT.Thereview,if possible,

should not be a mere collection of annotated bibliographies, but offer

a

critical evaluation of the

published literature

.

AH these aspects were in the mindsof the present writers when preparing this manuscript. The

broad nature ofthesubject,however, togetherwith its importance andgeneral interest for the public,

made the writing a difficult task

.

Big issues,such as the inclusion of indexed publications, and of

publications appearing in vehicles other thanthose commonlylooked upbythe mechanical engineering community,hadtobetackledina very positive andquick way

.

Intheend,it

was

decidedtoadhere quite stringentlytothe main guidelines

.

The only exception hadtodo with the period of time covered by this review

.

Inordertogivea good historicperspective of the subject,it wasdecidednot toimpose

anybound on thenumber,and date,of works consideredforreviewhere.Inadopting this procedurewe

particularlyregretthe exclusion ofworkspublishedby the physical and mathematicalsocieties

.

Some

Insights

Since we are goingtodiscuss atquitesomelength thephenomenonofturbulence,perhapsit would

beappropriate at this stagetodefine turbulence

.

It can besaid,ina general manner,that a turbulent flow is aflow _{which is disordered in time and space.Of course, thisdefinition is vague,lacking a}

precise mathematical formulation. This, however, is the kind of definition we commonly find in

ireatises that deal with the subjectof turbulence.Theflows that arepresent in nature and technology

and which are termed turbulent are very complex, exhibiting fairly different dynamics from

one

occurrencetoanother. In applications,theflowsmay bethree-dimensional,orquasi-two

-

dimensional,

or mayeven havesome sort of large scale organized structures.All thesefeatures, coupled with the randombehaviour of the measured properties of the flow, seem todeem the flow mathematically

undefinable

.

Theimportantpropertywhich is required of them is thattheyshould be capable ofmixing all transportable quantities much fasterthanif only molecularprocesses where involved.Tocutshort an endlessdiscussion,and after Lesieur(1990), we proposehere thefollowing simplified definition of turbulence:

J. of the Braz. Soc.Mechanical Sciences-Vol.20,March1998 2

•

Aturbulentflowmustbeunpredictable,in the sensethata small uncertaintyastoitsknowledge

ata given initial time willamplify so astorenderimpossiblea precise deterministicprediction

of

its evolution;

•

Ithastosatisfytheincreasedmixingproperty definedabove;

•

Itmustinvolveawiderange ofspatialwavelengths.

Despite,the already mentioned vaguenessofthe above definition,forfluid dynamicists the

word

turbulencehas aprecise meaning.Mathematically,in studying turbulent flows we are tryingto

solve a

problem which

is

based on the assumption that the instantaneous flow

variables

satisfy the Navier

-Stokes equations.These area deterministicset ofequations whichapparently yield solutions witha

random nature. Thus, the philosophical issues raised on how

statistical

theories could be used to

describe

a

phenomenon which is deterministic

could

never

be squarely faced until recently.

New

ideas

andmathematical toolsadvancedbyRuelleandTakens(1971)and by Feigenbaum(1979) haveshown that

there

arenon-linear dynamicalsystemswhosesolutions,

because of

sensitive dependence

on

initial

conditions,exhibit a weakcausalityand henceapparentlyrandomnature. Eventhediscrete algebraic

equationZ_**

,

=

A Z„(1 -

ZJ

demonstrates random solutions whose long time solutions have a critical

dependenceonvanishingly small changes in the valueofparameter A. Becauseofthe dependence of

thelongtermsolution onthe vanishingly small changesinA,the solutionsare non-deterministic.The

conclusion is that fluid turbulence is

a

deterministic problem that evolveswith timeand

space

in avery complex fashionduetonon-linearinteractions.

Althoughturbulence is a random phenomenon,

it

isaccepted

that

theflow is completely governed

by the Navier-Stokes equations. Considering that a host

of

numerical

schemes

exist to deal with

systems of non-linear partial differential equations, one should have, in principle, no problem in computing turbulent flows. Actually, this isnot thecase. The richness

of

scales that characterizes

a

turbulentflowimplythat,iftheflowistobecorrectlynumericallysimulated,all lengths from the

large-scaleenergy containingeddies, tothe dissipative

scales

under which the motion is damped through viscosity,mustbe resolved.

The dissipation of mechanical energy into heat, however, may be shown through dimensional

argumentstooccuratvery small scales.The dissipative scales are, infact,givenby r|

=

ord(

v7

c

)l /

\

where eis theoverall viscousdissipationperunitmass,andvisthekinematic viscosity.The energy

dissipation rateisgivenbye = ord(u

7

l) whereuand1 arerespectively the characteristic velocityand length scales of the largest eddies of

a

turbulentflow,being,therefore,determined by thegeometry

of

theflow domain.Theminimum number ofpointsrequired in athree dimensional simulationof theflow

shouldthenbe N

the large scale eddies. Here, Nleng,h represents the number of grid points that the computation or

measurement

would have

to have inorder to describe the entire range of scales. In nature and

in

technological applications,theReynoldsnumber assumes typically values between 106 and10*,which

willresult inapproximately10'4

to10'*requiredgrid points.

The minimumtemporalresolutionisfoundby dividing thesmallest spatial scalebythe convective velocityofthelargesteddies.Likewise,theratiobetweenthelargestscalesand theconvective velocity

gives the large scale temporal resolution. The result is that

the

number of required time steps is

Nun

*

=

R|V4

-The number of points required to resolve the spectral properties in turbulence is then Ntoti

.

1=(R|

9/4

)(

RtV

4₎

=

R|\ Thisexpression show'sthat anorderof magnitude increase in the Reynolds

number of the flow will result in a three order of magnitude increase in the number of required computationalgridpoints.

These numbers preclude any direct simulation of flows of practical interest. In fact, even if powerful enough computers were available, the performance

of

direct simulation of flows at high Reynolds number would be an unfeasible exercise. We realize that firstly by recognizing that the

specification of properinitial and boundary conditions on the levelofdetail requiredby thesmallest dissipative scales is simply not possible. This, unfortunately, in dealing with the Navier-Stokes equations,isavery seriousdrawback.Thenon-linearcharacter of theadvectiontermsmay giverise to

spatial and temporal instabilitiesintheflowthatwillexcite and amplify thesmall scales inthemotion.

Thelackofuniqueness of thesolution,togetherwiththe amplificationofthesmallperturbationson the boundariesoftheflow,result ina motion which isapparently random innature.

The

Traditional

Approaches

Thedecomposition of flowfield variablesinto meanandfluctuating componentshasledto_w'hat is

now knownas the Reynolds

-

averaged equations. In many engineeringand geophysical problems, the

=

ord(I7q3₎

=

ord(R ,v/4₎

,whereR,represents theReynoldsnumberassociatedw ith

A

.

P

.

S.Freire etal.:The StateoftheArtinTurbulenceModelling...

fine details oftheflowarereallynotrequired,sothatoften itsufficestohave dataconveyedbylong

-time averages

.

Although the

averages were first defined

interms

of

time averages,

further

developments have introduced the concepts

of

space averages and of ensemble averages

.

The averaged Reynolds

equations can simplybederived from the Navier Stokes equation to which they are verysimilar,but

with additional terms involving the turbulent fluctuations

.

These additional terms result from the advection terms and

may

be

seen

,from a statistical point of view,

as

second order correlations

or

moments

.

Sincethepractical interestlies usuallyinthemean propertiesof theturbulentflowfield,the

statistical theoriestry to establish functional relationships betweenthe

mean

variablesof the flow and

thefluctuations

.

This is normally referred toas the

''

closure" problemin turbulence

.

The attempt at

establishing these functional relationships has been one of the dominant themes of research in

turbulence.

Exactequations for theterms involvingfluctuationscanthederiveddirectlyfromtheNavier-Stokes

equations.

However

,thedetermination of transportequationsfor high order correlations always

results

in theappearance of higher order correlations or moments. We are then faced with the problem of always having moreunknowns thanequationsavailable for their solution.One wayout would beto

neglect correlations ofsomeorder. Unfortunately,this provestobe unsatisfactory

.

The reason forthisis

that despiteits apparentrandomnature,turbulentflowsexhibitsomedegree of"order

"

,whichmustbe

captured by the mathematical

model

.

In

order torender thesystemof transportequations closed,we

need,therefore,some"closurehypotheses"_{independent of the physicalconservationlaws}.

Perhaps,

we

should then nowto consider the conditions,orsomegeneral principles,towhich the closuremodelsshould obey.

3

Some

Basic Working

Rules

Thesheer complexity of theturbulence phenomenon has prevented researchers from achieving

a

rational closureof the Reynolds

-

averagedequations.This hasmotivatedthe inventionofa numberof

mathematical models based on different principles and rules,and the evolution of certain heuristics

which are supposed tohelp us toconjecture thebroadfeaturesof turbulent flows without having to

appealtoanyof those modelsof doubtful validity

.

Of course,as theseheuristics

are

deduced from common knowledgeandexperience,they willalways be opentocriticism.However,they must be seen

as useful rules that have been subjected to a large scrutiny, and that, for this reason, constitute a serviceable body ofknowledge fortheengineer.

Some years ago, Narasimha(1989)suggested that the accumulatedexperience with the various types ofheuristicrules could besummarized in the formofthefiveworking rules listedin Table1

.

Table 1 Thefive basic workingrules

As the Reynoldsnumberof any turbulent flowtendstoInfinity,the fraction of energy contained in the length andtime scalesdirectlyaffectedbytheviscosityof the fluid becomes vanishingly small;sodo the scalesthemselves,compared to thoseaccounting for theenergy

.

Anyturbulentflowsubjected to constant boundaryconditions evolves asymptotically toa stateindependent of all detailsofits generationsave those demanded byoverall mass,momentumandenergy conservation.

If the equationsandboundaryconditionsgoverninga turbulent flowadmit a self preserving (or equilibrium) solution,theflowasymptotically tendstowards that solution.

1

2.

3.

4. Aturbulentflow may,beforereachingequilibrium,attainamature stateinwhich the differentenergetic parameterscharacterizingthe flowobeyinternal relationships,irrespectiveofthedetailedinitial conditionsasinRule 2

.

Betweentheviscousand the energetic scales inanyturbulent flow existsan overlap domainover whichthesolutionscharacterizing the flow inthetwo corresponding limitsmustmatchas theReynoldsnumber tends toinfinity. 5.

«

These rules have been used by engineers for over

a

long period without any special

acknowledgmentof them.Rule4,inparticular,is the basis of all handbook charts and correlations,such

J

.

of theBraz.Soc.Mechanical Sciences-Vol.20,March1998

4

comesnaturallytoworkersinturbulence whenevertheyencountera totally unfamiliar flow

’

1

_.

_Therules

establishsomepositionsuponwhichany turbulence model should conform

.

Atthis pointwejust remind the reader that the working rules havenotbeen deducedfrom the basic

laws of fluid motion and that for this reason they will always remain open to doubt

.

As a final

assessment

,

we quote Narasimha's very own words when the rules were first presented:"They (the rules)

are

obviouslyvery useful,being close toreality:but they cannot still be elevated to the status of

scientific laws,because the small departures noted from them cannot bedismissed as experimental error, and seem to indicate that the principles are strictly valid only undercertain as-yet unstated conditionswhichwould notalways be easily obtained

.

"

The most frequently challenged working rules

concern

the postulated independence from initial

conditions

.

Rules 2 and 4.

Turbulence Models

Turbulence modellingofacertaintypehas become a widespread subject

in

academiaand industry. The pressing demand for more efficient models,capableof dealingwithevermorecomplexflows,has

resulted in the proliferation of different schools of thought,many of them remotely resembling each

other

.

The implication isa continuousgrowthinthe number,complexity and sophistication of models.

The presumptionthatany monograph on thesubject shouldbeableto

cover

allaspectsof turbulence

modelling is,therefore,notsharedbythe present authors

.

Here,wewilltry toshow thefailures and the

successesof some of themostpopular approaches

.

The turbulencemodels canbe ensembled,in general,in four classes(Narasiniha(

1989

))

.

The Impressionistic Models, thatstrive togain insight into the structureand the solution of the

problem without any claim to quantitative accuracy in prediction. Examples are the Burgers equation for turbulence,and theLorenz equation for weather forecast

.

The Physical Models, that aim at predicting quantities of interest based on assumptions not inconsistent withthe observedorunderstood physics of the problem,appealing to experimental data for model parameters when necessary

.

The models of Emmons for boundary-layer transition, of Kutta-Joukowskifor thelift of an aerofoil,andof Kolmogorovforthespectrumof turbulence,all

fall within thiscategory.

The Rational Models,that investigatethenatureofproblemand solutionsthrough simplermodels derived from more complete systems by some limiting process. The Burgers model for weak

shocks, the Newtonian model for hypersonic flows and the rapid distortion of turbulence areall

rational models

.

The Ad Hoc Models, that provide estimates of quantities of interest without insistence that all

assumptions be physically or mathematically justified in detail

.

The Boussinesq eddy viscosity model,the mixing lengthmodel,theK

-

e

differentialmodel,theseare allAdHoc models

.

Themodels thatserveindustry

are

all AdHoc Models

.

Inparticular,gradient transport models have become very popularoverthe years. The appeal of an attractive blend of simplicity and acceptable

predictive ability hasstimulated a continued development of these models. In the next section, we discuss these models indetail.

One

-

Point

Turbulence

Closure

Models

One

_-

point closure models were developed in order to model inhomogeneous flows in practical

applications. Starting from the Reynolds

_-

averaged equations, they aim at establishing functional

relationships between the turbulent fluctuations,through their second

-

order moments at thesamepoint

in space, and the mean flow variables

.

This can be made in several ways

.

The simplest of the possibilities istointroduceanalgebraic relationship.

Using the notion thatthe collectiveinteractionof eddies can be represented by an increase in the coefficientof viscosity,Boussinesq postulated the second

-

order momentstobe proportional to the local

mean

rate of strain through an "eddy viscosity",

v

,.

This notion is

a

direct translation of Newton's viscosity lawtoturbulent flows. Since turbulence is apropertyoftheflow,the eddy viscosity shouldbe

a

function of the

mean

flow parameters

.

For three-dimensional flows, it should even be a vectorial quantity

.

Despite the boldnessof thisassumption,clear even totheearliest workersin turbulence,the eddy

viscosity hypothesishas receivedcritical attention over the last 120years.Turbulence models have

A.P.S.Freireetai

.

:TheStateofthe Artin Turbulence Modelling...

However,as pointedoutbyBradbury(1997),theideathat turbulent stressesarestrongly dependent on

local strain rates (whatever model for

v

( is used) implies that the length

scale

of the

turbulence

is

smaller than the scale of the velocity localgradient

-

this scaleis typically of the order of theflow

width.

However

,even casual observation of flow visualization showsthat the main turbulent structures

areofthesameorder asthe fluid width

.

It is,therefore,amazingthatproportionalityrelationsinvolving

only stresses and strain rateworksowell

.

Of course, the conceptofeddy

viscosity

is phenomenological,having no mathematical basis

.

The

resultis that a constitutive relationship forv,still needs to be constructed

.

The eddy viscositycanbe expressed algebraically intermsof quantitiesderivedfrom algebraic or differential equations

.

Depending on the natureand on the number ofequations used,a classification for the eddy viscosity models follows

.

Zero

-equationor algebraic models.Thesemodelsuseanalgebraicexpressionfor the definitionof V

,.

Typicalexamples are the mixing-length concept of Prantdl,the Cebeci Model andtheBaldwin

-Lomax Model

.

One-equation models.Thesemodelsemploy

a

single transport differential equationwhichhasto

besolvedforthefluctuatingfield

.

Anextra algebraicexpression isnormallyalso required for the

completespecification of theflow.Examplesofone-equationmodelsarethe models of Bradshaw,

of Nee andKovasnay,ofJohnson and King and ofGoldstein.

Two

-

equation

models

. These models use two transport equations for the description of the

fluctuating field

.

Examplesare the K/Emodel,the K/CO model,the K/Qmodel,theK/T model,and

manyothers

.

So far, thesimplest prescription forthesecond-moments has been achievedthroughthealgebraic models

.

Thesemodelsare built

on

an

analogy between the randomizingeffectsof the turbulent eddies onthemeanflow,andthe correspondingrandommolecular motion in a gas.Theimplication is that the

’’turbulent viscosity” can be seen as formed from a product of the density of the fluid, of a local

characteristiclength,_f,andof alocalcharacteristic velocity,

uc

. The analogy withthekinetic theory of gasesisclear; however, inaturbulentflow, both the characteristicparametersmust beexpectedtovary locally with the flow.

The mixing length theory assumes the fluctuations in both directionsto have thesame order of

magnitude, and these to be proportional to the distance from which discrete "lumps of fluid” are

fetched, times the local mean velocity gradient

.

This hypothesisreduces

our

closure problem tothe determinationof thecharacteristic lengthof theflow, thatis,of the mixing length

.

For simpleboundary

layerflows,

_lt

can

beassumedtobe proportionaltothe distancefromthewall

.

Forfree turbulentflows,

k

can be assumed tobe proportional tothe width of the turbulent region

.

The list is not exhaustive.

Here,it sufficestosay thateveryclassof flow has its particular formulationofthemixing length.

The mixing length concept works quite well for a class of problems, but suffers

from

all the

problems inherent to the eddy viscosity hypothesis

.

In particular, in complex flows, it may be

impossibletospecifyanyformforthemixing-_length

.

_Ingeneral,the model has showntobe appropriate for two-dimensional flows with mild pressure gradients and mild curvature. No flow separation or

rotation effects are allowed in these models. In addition, the algebraic models are useful for two

-dimensional shock-separated boundary layers with weakshocks,and in computing three dimensional

boundary layerswith small cross flow'.Infact,several authors havesought extensions ofthe algebraic

eddyviscosity modelto threedimensionalflows(

see

,for example,Rotta(1979)).The

success

,however,

hasbeenvery limited.

We draw ourcurtains on the mixing length model by pointingout to the reader that this model

assumes the flow to be isotropic, and takes no account of processes of convection or diffusion of turbulence

.

In the one

-

equation models,a transport equation is used to compute the second-

order

moments. This canbemade intw

_'

oways

.

Throughthe specification of a singlecharacteristicparameterthatcanbe

usedtorepresent thefluctuating field,or,alternatively,relating thecharacteristic velocityin the eddy

viscosity modelto a turbulent property

.

In the latterapproach, thecharacteristiclengthstill hastobe determined throughanalgebraic equation

.

Almost allone-equation modelsusethe turbulentkineticenergy, K,as the reference parameter.An

exactequationcan be derivedfor Kthrough somesimple algebraicmanipulationof theNavier-Stokes

equations

.

Thisequation, however,presentssome turbulent correlations which havetomodelled. The

diffusion and the production tenus

are

modelled

as

gradient transport terms. The dissipation term,

obtainedthrough a local isotropy assumption,is definedin termsofan algebraic equation thatinvolves

thefluiddensity,the turbulent kineticenergyand thelength scale.

J.ofthe Braz. Soc. Mechanical Sciences

-

Vol. 20,March 1998 6

In

practice,very littleadvance of this type ofmodels

over

thezero-equationmodelsexists

.

The fact

thatthe characteristic length scale still hastobefixedbyempirical argumentsimportstothesemodels many of the drawbacks of the algebraic models. The material found in literature,however, clearly

supports the ideathat

Kin

isa bettervelocityscalethanthoseemployed inthealgebraicmodels

.

A recent attemptat revivingtheone equation models has been made by Johnson and King.They

begin with a beforehand assumededdy viscosity algebraic behaviourwhichhas asitsvelocityscalethe maximumReynolds stress

.

Next,themaximum turbulentkineticenergy, tcm,is supposedto varyvery

little withthetransversal directionsothatan ordinarydifferential equation

can

beconstructed toits

prediction. Theclosure isconcluded consideringthat the

ratio

of the local kinetic energy tothe shear

stress isconstant. This model is aimed at a very limited class of flows,namely, at two

-

dimensional

separated flows without curvatureorrotation

.

The generalresultsof the model are verygood.

The next level of sophistication is to introduce a second transport equation, from which the

characteristiclengthcan becalculated.This givesrise to

the

two-equation models

.

These

models are

consideredtoembodymore physicsthan theprevious ones.Alist ofthevariousavailabletwo-equation

models can befoundin Launder andSpalding(1972)and morerecentlyinanumber of review'articles

(see,

e

.

g

., Lakshminarayana(1986),Wilcox(1988))

.

Thevariablesadoptedbyresearchers to determinethelength scalevary from oneworktoanother

.

Onemay chooseasthesecond variable a parameterassociated with themeanfrequencyof the most

energetic motions

.

That wasexactly theoriginalproposalofKolmogorov(1942)

.

Theturbulent

energy

dissipation rate per unit mass,

e

, has been preferred by

a

number of authors

.

Another very popular choice has been the specific dissipation rate,

co

. A transport equation for the length scale itself was derivedby Rotta(1951).

Thestandard two-equation models failto capture manyof thefeatures associated with complex

flows. The factthattheyare stillimprisionedto theeddy viscosity concept makesthemveryvulnerable

when dealing with flows that present curvature,separation, rotation, and three-dimensionally among other effects

.

The advantages and simplicity of the two

-

equation models, however, should not be

overlooked. These models are much superior to the algebraic and one-equation models in mildly

complexflows.Typicalsuccessful application of thesemodelsaretheflows in jets,channels,diffusers, andannulus wall boundarylayers withoutseparation.

Theweaknessesand shortcomings of thesemodelshavebeen listed by Hanjalic(1994);theyare:

•

Linearstress

-

strain relationship through theeddyviscosityhypothesis;

•

Scalarcharacter ofeddy viscosity;

•

Scalar characterofturbulencecharacteristicscales

-

insensitivitytoeddyanisotropy;

•

Limitationstodefineonlyonetime-or length scale ofturbulenceforcharacterizingallturbulent

interactions;

•

Failure to account for all viscous processes governing the behaviour of

e

or other

scale-determiningquantitiesby virtueof the simplistic form ofthebasicequationforthat variable;

•

Inadequate incorporation of viscosity dampingeffects on turbulencestructure (low Reynolds

numbermodels);

•

Inability to mimic the preferentially oriented and geometry-dependent effects of

pressure

reflectionand the eddy-flatteningand squeezing mechanisms duetotheproximity ofsolid-

or

interfacesurface;

•

Frequently inadequatetreatmentofboundary conditions,in particularatthe solid wall

.

A logical wayofderivingmoresophisticated models,of acertain universality,consistsinderiving full transport equations for each of the second-order momentsdirectly from the Reynolds-averaged equations

.

Thisprocedure,inevitably,results thatevery derivedequation involvesvarious correlations amongfluctuatingquantities w'hichare notexactlydetermined. These,ofcourse,mustbe modelledin

termsof the meanflow variables,the second

-

momentsthemselves,andat least onecharacteristic time

scale

.

Modelsof thisnatureare usually termed Reynolds

-

stresstransport models

.

Accordingtosome

authors, these models represent the highest degree of complexity that has been exercised in

conventional turbulence modeling

.

Additionally, to a more universal description of the turbulence, these models are considered to

adheremorerigorouslytosomemodellingprinciples whichare judgedbysometobe crucial

.

Thefirst

of the principles has to do with the mathematical formalism of the closure problem

.

Consistency

conditions such as dimensional coherence of all terms, tensorial consistency, coordinate-frame and

material frameindependence,satisfactionof realizabilityconditions,limitingproperties of turbulence,

all thesearesatisfied by the Reynolds-stressmodels.Thesecond principleisrelated tothephysicsof

the turbulence. For example, the decreasing influence of higher-order moments upon the mean flow

A.P.S

.

Freire et al.: The State oftheArt inTurbulenceModelling... 7

The Reynolds stressmodels

are

reported(Hanjalic(1994) )toprovide

a

superior representation of

two

-

dimensional non-_equilibrium

_flows

_.

_{They are also supposed to}

_perform

_{better in unsteady and}

periodic

flows

and

in

all situations

where

the turbulent field exhibits

hysteresis

as

compared with the

mean flow field. The models account for the effects of anisotropy, curvature, rotation, three -dimensionality, and hence normally

performs better

than

the

two

-

equation models under

these conditions. A shortcomingfeature of themodelsis the needfor a scale-determining equation,which

grows in importance in pressure dominated flows, or in flows with unproperly treated boundary conditions

.

Forthreedimensional flows,the Reynolds-stressmodels will resultin6transportequations which

mustbesimultaneouslysolved

.

together with theequationsof motion.This resultsin10

-

12equationsto

be solved for the mean flow and turbulence quantities

.

This is certainly a Herculean task even for

present daymodemcomputers.Hence,theorder istosimplifythe Reynoldsstresstransportequations

.

One tryistoreducetheseequationsto algebraicequations

.

Such models aretermed algebraicReynolds

stressmodels.

The

essential idea

of

these models is that only the convection and diffusion terms contain stress

gradients

.

Thus,if through somealternativeprocedure, theseprocessescould be modelled by equations

notcontaining stress gradients,thenthe differential transportequationsfor the Reynoldsstresseswould bereduced toalgebraic equations. Theneedfor solving turbulenttransportequations is noteliminated

here; however,the calculated flow variables arenow scalars

.

A still simplerapproachconsiders the stresstransport processes tobecompletely negligible.This

local-equilibrium approximations are used mainly for understanding the physical nature of complex

flows.In maycases,theyoftenleadtouseful overallmodelsof thestresses.

Infact,many of theideasthatwe have beenconsideringinthelastparagraphs wereadvancedquite

sometimeago

.

As earlyas 1940,Chou introduced,inanabsolutelyoriginalpaper,the equations for the

secondandthirdmomentsoftheturbulent fluctuations

.

Atthetime,heproposedto’

’

close

”

the equation for thetriplevelocity correlationbyassumingthefourth-ordermomentstobeproportional tothesum

of the three products of two double velocity correlations

.

In

addition, further hypotheses on the correlations involving turbulent pressure gradients and velocity fluctuations were needed

.

These correlationswerederivedbyChou fromaPoisson equation.Theequationsforvorticity decay were also

foundand the termsof energydecay

were

improved.

Essentially, Chou argued that because the Navier-stokes equations are the basic dynamical equations offluidmotion,itdoesnotsufficetoconsider only the mean turbulent motion

.

The turbulent fluctuations should be as important as the mean motion and,for this reason, theequations for them shouldalsobe considered.

What Chou really achieved was a systematic _{way of building up differential equations for the}

velocity correlations for each successive order from the equations of turbulent fluctuations

.

The

problem now lies on the difficulties to be found in solving simultaneously the Reynolds-averaged equationsandthe equationsforthehighorder moments. The threemajordifficulties foundat thetime were:

•

Thenon

_-

linear characterof theequations;

•

The _{correlation functions are slowly varying functions of space and time, whereas the}

fluctuationsare all rapidly varying functionsofthem;

•

Tomathematicallysolvethe setofnon-linear differentialequations,anextraphysical condition

isneeded(similarity hypothesis)

.

These difficulties were finally overcome by Chou (1987) himself by developing the method of successive substitution,tosolvethe

mean

motion and velocity fluctuation equations simultaneously

.

Other Approaches to Turbulence Modelling

Another method of describingtheturbulent fluctuationsisthe spectral method,firstintroduced by

Taylor(1935,1938)

.

The introductionofFourier analysisintothe problem leadstosomebenefits.The differential operators are converted into multipliers, the physics of turbulence is given a relatively simplepicture,and the degreesofliberty of theturbulent system arc better defined. The use ofFourier

analysis is particularly useful when the turbulence is homogeneous, that is, when it is statistically

invariantundertranslation

.

Unfortunately,thespectral approachdoes notsolvetheproblem of closure,

it just appearsinadifferentform.

Usingthespectralapproach,several theorieswere builtforthedescriptionofturbulent flows.Some

J.oftheBraz.Soc.MechanicalSciences-Vol. 20,March1998 8

•

Thedirect

-

interaction approximation;

•

The

quasi

-normal theory.

The first approach is analytical and has only given useful results for isotropic, homogeneous, turbulence. The quasi-normal theories, whichassume all cumulants abovesome given order greater

than two to vanish, lead toa negative energy spectra. Attempts at fixing this problem have led to

considerations that require an extra assumption about time scales that is questionable in theenergy

containing range. These methods are of narrow application, make a variety of hypotheses whose validity is difficulttoassessinphysicalterms,and

.

forthisreason,willnotbeconsidered further here

.

Othermethodsthat heavily relyon numericalsimulations have been developedmorerecently

.

One

of

them,directsimulation,hasreceived a lot

of

attention.

The remarkable recent advances in computing power have opened an era of simulation. The

application of paralleledcodes

on

computerswithefficient networking and large memory capacity has

becomea verypowerful way of computing turbulence. Ideally,wewouldliketoperformcomputations

by solving the completeNavier-Stokesequations

.

However,dueto theresolutionrestrictionsobserved

in section 2,thecalculations are presently restricted toflows with

a

Reynolds numberof theorder of

104

.

Despitetheseverelimitationsimposed by the resolution requirements,directsimulation has proved

to be a powerful tool for understanding the physics of turbulence and furnishing data for the

development and the improvement of turbulence models

.

In fact, it is capable of providing data on turbulence that is virtually unobtainable from experimentssuchaspressure-velocitycorrelations.

So far, our models for the simulation of complex turbulent flows, have relied on the use of

statistical averages forthe variables of interest.Another possibility is the useof filterson thesame

variables

.

This approach,normally termed largeeddy simulation, presents the advantage of naturally

introducing the scales of the resolvedvariables.The resultis that,i)inprinciple,it is simpletoprepare

theinput data for numericalmodelsthatare consistentwith thosescales;ii)theinteraction thatoccurs

between numerics and physicsinthe solution of theequationsof motion is very strong,and the useof this approach makes it easier to have a better understanding of this process.The term large eddy simulation is frequently associated in literature with a space filtering operation

.

The use of time

filtering,however,alsohas been tested;this results in the elimination ofhighly fluctuatingcomponents

in time,allowingtheuseof large timesteps in the numericalintegrationof the motion equations.

Let usnowlookin more detail atthephilosophy of largeeddysimulation.The methodrequires that

a filteringoperation isapplied tothe Navier-Stokes equations.Next,the velocityfield isdecomposed

into a large scale velocity and a sub-grid velocity scale. This produces a new problem

.

The

non-linearity of the advective term will now result in four different ternis

.

Indeed, for a general space

filtering operation, the classical averaging rules do not apply. Only one of these resulting terms is analogoustothe Reynoldsstresses. The other terms arise from the fact that, as we havejustsaid, the

filtering operation in not idempotent. The subgrid scale modelling problem can then be defined

as

findingexpressionsfor the subgrid terms asa functionof the largescalevariables. Actually,the subgrid

scale modelling problemisnot a well posed problem. In thephysical problem,the propagation tothe

largescalesof theuncertaintiescontained in the subgrid scales willcontaminate theformer yielding a

flowwithan unpredictablenature.Now,consider

we

haveconstructed

a

subgrid

scale

model,and

that

,

therefore, we haveatour disposal a closedsetof motion equations whereeverythingisexpressed in

terms ofthe large scales.Then thesimulationsconductedwiththesesetof equationswill notbe ableto

propagate any disturbances which would generatedifferent flow fields.Thisimpliesthatalargeeddy simulation of turbulence will not faithfully reproduce the large scale evolution from a deterministic pointofview,atleastfortimes greater thanthepredictability time

.

In largeeddy simulation,thecalculated flow must then be interpreted asadifferentrealization of

the actual flow

.

What one would hope to happen is that realization to have the same statistical

propertiesof the real flowand thesame spatially organizedstructures(though, atadifferent position

fromthereality).

Basedontheseconcepts,Lesieur(₁₉₉₀)defineswhatagoodlargeeddysimulation of turbulence

should be.

Low gradedefinitionthesimulationmust predictcorrectlythestatisticalproperties of turbulence

(spectral

distributions

,turbulent exchange coefficients,et cetera).

High gradedefinition moreover,thesimulationmustbe abletopredicttheshape and topology(but

notthe phase)of the organizedvortexstructures existingintheflowat thescalesofthesimulation. Animportant question tobeasked now is; _Howsmall thescaleof theresolved motion hastobe?

Webeginto

answer

by reminding the reader that in high Reynolds numberturbulentflow,thesmaller

the scaleof the motion,the more isotropic it becomes,and that,in fact, an "inertial"subrangeexists

.

A.P.S. Freireetal.:TheStateofthe Art in TurbulenceModelling...

scales could then beassessed by invoking theirnearisotropic properties. The implication isthat the

proper

master_{scale to be used in large eddy simulation of turbulence}is

precisely

the grid

spacing for

isotropic meshes

.

For

anisotropic meshes,

a

product

average

ora Euclidean norm can be used.

Some Major

Achievements

The_{classical theories of turbulence have achieved}

_some

_{resultsinthepast whichclearly have}

_{had a}

definitive influenceinour perceptionof the problem.Next,we shalldiscuss someofthesesuccesses

.

The second

-

order two-point correlations play a leading part in turbulence theory

.

We have

discussed in some detail in the previous sections the engineering approaches for the one-point

correlations.However,ifourinterest isthe underlyingstructureof theturbulence,weshould consider the velocitycorrelationsat twoor more_{points. These fundamental concepts wereadvanced byTaylor}

(1935) in

a

paper where he also introduced the concepts of statistical homogeneity and isotropy

.

Subsequently, Taylor(1938) introduced thethree-dimensionalenergy spectrum in wavenumber (i.e

.

,

the Fourier transform of

the

two-pointcorrelation inspace),

an

entity whosecalculationhas become one of the fundamental objectives of turbulence theory

.

This function

measures

how much energy is

contained between the

wave

numbersKandK+die.

Theclosureproblemforisotropic turbulence

can

be formulated in

wave

-number space.In thisway, when we consider the transport of turbulent energy, this will be in wavenumber rather than configuration space

.

Large scale structures are associated with small values

of

K, whereas small

structuresareassociatedwith high valuesof K.Thus,thetransfer ofenergywill

occur

fromone range

of eddy scales to another

.

This process is known as the"cascadeofenergy".

The_{energy balance equation is obtained fromtheequation forthesingle-timecorrelations.Thisis}

justthemeanmotion equation,after some manipulation,specializationtohomogeneousturbulence and

Fourier transformed. A further specialization tothe isotropic case,reduces the spectral tensor to its isotropic form

.

The traceof the tensorthengivestheenergyspectrum,E.Theresulting equationfor E,

involves then a production term, a dissipation term, and a transport term

.

The transport term

correspondstothe triple velocity correlations coming from non-linear interactionsofthe Navier-Stokes equations;thistermjustredistributes energyinwavenumber space

.

The usual interpretationofthe energy balance equationisthatthe energy in the flowstoredatsmall

K(that is,at largescales) _{is transferred by the}

_non

-linear transport term tolarge _{K (}that _{is, tosmall}

scales), where it isdissipated throughheat by theaction of viscosity

.

The non-linear term describes

conservativeprocesses,namelyinertial transfer_{of energy fromonewave numberto aneighboringone}

_.

Now, from ample experimental evidence, we know that the energy is determined by the lowest

wavenumber,thatthedissipationrateis determined bythehighest wavenumber,and that thetwo ranges

do notoverlapeven forverylowvaluesofthe Reynoldsnumber

.

Itfollowsthat thenon-linear transport

tenucanbemade todominateoveranaslargeas welike portionof the wavenumber space,bysimply

increasing theReynoldsnumber.

Theabove ideas weretorinalizedbyKolmogorov(1941)_{in two}famousassumptions.

Kolmogorov’sfirsthypothesisof similarity. At very high,but notinfinite Reynolds numbers,all

the small-scalestatistical properties are uniquely and _{universally determined by the scale}, I, the meanenergy dissipationrate,e,and the kinematic viscosity,

v

.

Kolmogorov’ssecondhypothesisof similarity.Inthe limit of infinite Reynolds number_{,all small}

-scale statistical properties are uniquely and universally determined by the scale, 1, and the mean

energydissipationrate,e.

By a simple dimensionalargument , the firsthypothesis implies that the energy spectrum can be

written as

:

E(K )

=

v

5/4

_e

! /4_f(

id),wheref isauniversal function.

Thesecondhypothesis implies that,inthelimitas Reynolds numbertends toinfinity,E(K)should

become independent of the viscosity. This amounts to the energy spectrum assuming the form

E(K)OC

e

2

'

3Kv

\

thefamousKolmogorov’s law.This law is remarkablywellverified experimentally,

althoughthefine-scale motiondoes notnecessarily havethedesired degreeofisotropy postulated by

the author,_{and the proportionality constantcannotbededuced convincinglybytheory}

_.

A second major success of turbulence theory is the theory of Taylor (1921) for the turbulent diffusion of fluid particles. A random processwhere at any instant the future state of the process is

entirelydetermined byits stateatthat particular instant and independentofitsprehistory is called a

Markov process.Alternatively,wesaywehave a Markov processwhenthefutureis independent of the

pastfor a known present.

For

theturbulent

diffusion

of fluid particles,thisis certainlynotthecase

.

In

the turbulent motion of a fluid, the _{motion of the particles is continuous - so is the exchange of} transferable quantities- and there is a correlation in _{time between properties of a fluid particle at}

J.oftheBraz.Soc.Mechanical Sciences

-

Vol.20,March1998

10

subsequent times

.

This memory behaviouroí turbulent diffusion implies that this processcannot be considered

a

Markov

process

. Taylor extended these notions to the turbulent flow diffusion by

considering the pathofa marked Huid particle during its motion through the flow field.

Considering the displacement

of

marked fluid particles, Taylor introduced

the

Lagrangian

autocorrelation and the integral time scale.Then, further considering the flowtobe homogeneous in

space and time, an exact result for the mean square distance traveled by a diffusing marked fluid

particle was derived in terms of the Lagrangian correlation coefficient, R|..The difficulty is that a

theoretical solution for

RL

cannot befound.Also,experimental measurementsof Lagrangian quantities in turbulent flow are difficult to make, so that information about

RL

is scarce

.

Nevertheless, two

important results concerning the limitingcases oft

—

» 0andoft

—

>_{°° were enunciated}

.

Short diffusion times:

RL

₌

LThe distancetraveled by the marked particle isequal toits velocity multipliedbythetime elapsed.Aclassical result from Newtonian mechanics.

In

otherwords,one

can

say that,at shortdiffusion times,themotion of the markedparticle ispredictable, provided we

knowtheinitial conditions: thatis,themotionisdeterministic.

Long diffusion times. The root mean square particledisplacement is proportional tothe square

rootof thetime elapsed.Thisis thesame resultone would find fortheclassical random walk of discontinuous movements, where the particle variance is proportional to the square root

of

the

numberofsteps

.

Thelatterresult characterizes the analogy betweenthegaskinetic theoryand thediffusive motion

ofeddies.Thus,the r.m.s

.

distancetraveled bya fluid particlecan be expressed in termsofa diffusion coefficient defined as a function of single-timemeansanda Lagrangian integral time-scale.Theabove

results,although simple,gave us a deep insight intothe nature of turbulent diffusion

.

Nothing was

really solved by the results,

as

the closure problem still persisted, but some connections and

’'transforms'1

ofthe problem were established

.

Another great achievement of the traditional approaches is the universal log-law for the mean velocity distribution inthe nearwallregion. Usingthenotionthatthe nearwallturbulentflowcould be

divided into distinct regions, with distinct dominant physical effects, scaling velocities and lengths,

Millikan, (1939) resorted to a "matchability" argument to work out a functional relationshipfor the velocity profile. The resulting logarithmic expression has become one of the great paradigms of

turbulencetheory. Validated by a huge number ofexperimentaldata,theso-called law of thewallhas

becomeabenchmarkresulttowhichall simulationsofwallflow havetoconformto

.

Evidenceinfavor

of the law is

.

therefore,strong.Some authors,however, have placedthe law under a severe scrutiny,

claimingthevelocity profileto bepower-like,andnotlogarithmic-like!

Infact,thederivation of both laws isequally consistent and rigorous.However,theyare based on

entirely different assumptions

.

Both derivationsstart from similarity and asymptotic considerations

.

The log-law is obtained from the assumption that for sufficiently large local Reynolds number and sufficiently large flow Reynolds number,thedependence of the velocity gradient on the molecular

viscositydisappears completely. In thealternative approach that leadsto the power-law

.

the velocity gradient is assumed to possess a power

-

type asymptotic behaviour, where the exponent and the

multiplying parameter are supposed todepend somehow on the flow Reynolds number. Thus, the

velocity gradient dependence onmolecular viscosity does notdisappearhoweverlargethe local and the flow Reynolds numbers may be. The form of the power-like scaling law' yields a family of curves

whose parameter is the Reynods number

.

Theresultingenvelop ofcurves isshowntobe very closeto

theuniversal log

-

law

.

The original result,_however, has remained unshaken. The fact that the log

-

law can be obtained fromformal asymptotic methodsand that ithas beenshown toapply toa variety of flows, has firmly silenced its critics.Infact,thelargebodyofusefulservices covered bythe lawhasmadeittotranscend

inimportance. The universallaws offriction forsmoothand rough pipesarejust oneofthe pungent

examplesofapplication ofthelog

-

law.

Typical examples of application of the law include caseswithwalltranspiration,roughness,transfer of heat,compressibility, three-dimensionality,and

even

separation of flow'.

Of

course,for allof these

situations,some modificationsneedtobemade inthe original formulationtocomply with specificflow

requirements. However,theessenceof the law and its generalformarc the same for all cases.

For

boundary layer flows, a second universal law was added to our collection of great

achievements

.

In theouter region ofthe boundary layer the log-law may nolongerapply,sincethe

conditionsin which itisbased are nolongervalid

.

However,experimental evidence has shown that,if

weconsider as ourreference quantities the skin

-

friction velocity and the boundary layer thickness, a

A

.

P

.

S.Freireetal

.

:The Stateofthe Art inTurbulence Modelling

...

part

of

the

boundary

layer

.

This

result

was

suggested

by

Von

Karman

to

be considered a

postulate,

the

velocity-defectlaw.

Impressed bythesimilar behaviour between the flowinthe outer region ofaboundary layer and the flow in a wake,for both exhibit large scale mixing processescontrolled primarily by inertiaeffects

rather thanviscous effects,Coles, (1956)_{proposed a} correctionfunction totheclassicallog

-

law

.

The

resulting expression, built purely on experimental grounds, includes in its validity domain both the

turbulent partof the wall region and theouter region.The equation for the correction function was called by Coles the law ofthe wake

.

The actualfunction maybeapproximatedbyanantisymmetrical

sine function multiplied by

a

profile parameter which does not depend on the transversal flow

coordinate

.

The above listofsuccessesis,of course,notexhaustive. Some othergreatachievements such as the stability theory for the flow between two rotating cylinders developed by Taylor, (1923), or the

boundary layer approximations duetoPrandtl, (

1904

),could easily have been included here

.

Themain

feature of the chosen achievements is that they were advanced by exploiting plausible physical

argumentsthrough dimensional and similarity analyses.

The reader will be

keen in

identifying some

of

the above results with some of

the

basic working

rules advanced inthe sectionSome BasicWorkingRules.

11

A

Bit

of

History

The

progress

in theartof turbulence modelling in Brazil must,insome way or another,be related

tothedegreeof development of itsgraduatecoursesinphysics, applied mathematics and engineering.

Thisisequivalent tosay that_{turbulence is a contemporary subject in Brazil},still in its infancy and,

therefore, in averyincipientstate

.

Infact,the very first graduate engineering courseinBrazilwasonly inaugurated attheFederal University of Rio de Janeiro(UFRJ)in thesixties. In a historical deed,the Chemical EngineeringDivisionof the Institute of Chemistryfoundedin March of

1963

the Chemical Engineering Graduate Program.Thisactwasa direct result from a previous official visittothe United States of America byProf. Alberto LuizCoimbra tostudy in detail all theaspectsof the engineering

graduate

courses

taught at theUniversitiesof Houston,Rice, California(Los Angeles and Berkeley),

Stanford, Cal.Tech

.

, Minnesota, M.I.T. andMichigan

.

The visit, which took place in December of 1960,was sponsoredby theOrganization of American States, (OAS),and by the Congregationofthe ChemicalSchoolof the Federal Universityof Rio de Janeiro.Asa follow uptothisvisit, inAugustof 1961 the Deans of the EngineeringSchoolsof the UniversitiesofHoustonandofTexas cametoRiode

Janeiro.

Together with their Brazilian pears, the visiting deans established a preliminary plan for the

implementationofagraduate engineering courseatthe FederalUniversity of Rio de Janeiro that was

finally presented in October,1961,tothe braziliancoordinator of theAlliance forProgress. This plan

was madeaccessibletothecommunityin Decemberof1961ata seminar on universityreformandon

the teachingofengineeringorganized bytheEngineering ClubofRiode Janeiro.Asan exercisetoa

complete implementation of thegraduate program,it was decidedthat shortand intensive courses on

the subjects of Boundary LayerTheory and Turbulence, Flow in a Porous Medium and Computer

Programmingwouldbeoffered in the monthsof July and Augustof1962

.

Thecourseswouldbejointly

sponsored by the OAS, the Brazilian National Research Council (

CNPq

), the Chemical Institute

of

UFRJand theUniversityof Houston(UH),andwould begiven by the UH lecturersDrs

.

Abraham E

.

Dukler andElliot I.Organick.

Thisinitiative,in fact,

was

determinantin thedefinition of theareasofexcellence that the graduate

courses tobe established were this have throughout the sixties

.

The special care dedicated to fluid

mechanics

was

clear,andthatiswherethisreviewwantstoconcentrate

on

.

The formal inauguration of the Chemical Engineering Program occurred in March, 1963

.

Still benefitingfrom financial helpfromtheOAS, andwithfurtheraidfromtheFulbright Commission and the Rockfeller Foundation,thecoordinators of thegraduateprogram invited four Americanlecturers to, together with six other Brazilian colleagues,provide the initially required teaching and supervision. The fourU

.

S

.

lectures were Profs

.

Donald Katz,LouisBrand, Cornelius John PingsJr.and FrankM

.

Tiller. Only two years later, and under the guidance of the Chemical Engineering Program, the Mechanical Engineering Programstartedfunctioning.In the yearsthatfollowed,nine othergraduate

programswerecreatedgiving origintothePost-GraduateEngineeringSchoolof theFederal University of Riode Janeiro(COPPE/UFRJ)

.

Theroleofthosethreeinitialcourses onthedevelopmentof the subjects of fluidmechanicsandof

J

.

ofthe Braz. Soc.MechanicalSciences*Vol. 20,March 1998

1 2

chemicaland mechanicalgraduateprograms becamean importantcenterfor fluid

mechanics

and heat transfer studies

.

Indeed, all twenty four M.Sc.(Magister

Sciential

theses that were presented at COPPE/UFRJ from 1964 to 1966 had todo in some way or another withfluid mechanics or heat transferprocesses.Onlyin 1968,and only with theentranceofthegraduate programsinMetallurgical

.

Electrical,Civil

.

Nuclear,Naval and Manufacturing Engineering,thetideofpowerstartedtochange.

Inabroader

sense

,as

we

shall

see

,the abovementionedfacts unfolded in a national

level

.Thecast

of great scientists temporarily associated to COPPE/UFRJ was to become a key factor in the

development of the sciences of fluid mechanics and of heat transfer in the wholeof Brazil

.

Several

examples immediately leaptothemind.The closerelation,almost intimate,betweenCOPPE/UFRJand

HoustonUniversitythroughProfs

.

Coimbra

and Tillerprompted severalstudents togo abroadtostudy

with Prof. A.E. Duklerandother

Houston

lecturers. Thesestudents were latetocome

back

toBrazil

and,on moving todifferent

institutions

,they were ultimately theresponsiblefor theestablishment of

many other graduate fluid

mechanics

programs

in

Brazil. Another notable example

was

the

passage

of

Prof. Ephraim M

.

Sparrow, currently an Emeritus Professor at Minnesota University, through COPPE/UFRJ

.

Afterspending twoyearsin Brazil,hewenton tosupervise a large numberof Brazilian studentsoveraspanof more than twenty years

.

His impression

on

the development of fluidmechanics science in Brazil is therefore unquestionable

.

Indeed, when the Mechanical Engineering Program started,its staff consisted only of four lecturers: Profs.Sparrow and JacquesLouis Mercier,andthe

assistantlecturersThéoF

.

C. Silva and Franscisco N

.

de Farias

.

As Prof. Mercier left afteraspanof

one year,the roleof turning the graduate program in mechanicalengineering intoasuccess wasleft

almost entirely toProf

.

Sparrow.

Inthe next sections we willtry toestablish afirm connectionbetween allthe abovefactsandthe

presentstateof turbulenceresearch in Brazil. Here,justfor the sake of future reference,wewillquotea

numberofselectedtheses

.

Theyare:

•

GilenoA.Barreto

.

"Hot

-

wireAnemometer-Constructionand Calibration",M.Sc.Thesis,1964

.

•

Ralf Gielow,"The Graetz-NusseltProblemforaTurbulent Flow",M.Sc. Thesis,1965.

•

Joã

o

S

.

DAvila, "Solutions of the Boundary Layer Equations for a Turbulent Flow", M.Sc

.

Thesis

,1968.

•

AmoBollmann,"Studyof aTurbulent Jetin aHighly Turbulent Environment",M

.

Sc

.

Thesis,

1969

.

•

JoséAugusto F. Gouvêa,"Heat Transfer inTurbulentFlows",M.Sc. Thesis,1969.

•

CarlosE. Lopes,"Turbulencein theVicinityofFluidInterfaces",M.Sc. Thesis,1970.

Theseexampleswerechosentoillustratethestate ofturbulenceresearchin Brazil by the end of the

sixties. They, by no means, _represent the totality of work done in turbulence at that time. Note,

however,that anyof the above titlescould be perfectly valid titlesfor thesestobe presented in thepast

few years.

The growth in the engineeringsciencesin Brazil provided by theconsolidationofseveralgraduate courses quickly started pressing for the creation of an appropriateforumwhere discussionsofscientific

interestweretotakeplace.As aresult,severalsymposia andconferences weretohave birthin the late

sixties,early seventies

.

For the mechanical engineering community, the landmark was the creation of the Brazilian CongressofMechanical Engineering.Aimingatdiscussing theproblems-in a broadsense

-

related to

mechanicalengineering in Brazil,asmall groupof twelveresearchersassembled in Florianópolis,Santa

Catarina,for theFirstBrazilian Symposium on Mechanical Engineering.TheSymposium washeldin

1971andproducedtwelve technical

papers

.Since then,and ateverytwoyears,theBrazilianCongress ofMechanical Engineering(COBEM)hasbeen stagedon a regular basis;it is,unquestionably,themost

importantconferencein Brazilon MechanicalEngineering.Inits latest edition, theastonishingnumber of 631 papers were presented.

By 1986thearea of thennosciences hadachievedsucha degree of development in Brazil that the creationofanevent uniquely devoted toit was amust.Anewconference wasthen organized which

wastobe termed the BrazilianNationalMeeting on ThermalSciences(_ENCIT).Theconference,which

is held every two years,became an immediatesuccess

.

Inthelast event, heldin Florianópolis,Santa

Catarina,331

papers were

presented

.

Statistics

Beforewemake anycriticalcommenton theexisting

literature

on turbulentflow,let us firstwork out thestatisticsof thepastCOBEM's and ENCIT's.On compiling all relevant worktothis reviewwe

A.P.S.Freire et ai

.

: TheState ofthe Art in TurbulenceModelling... 13

havetried

as

much

as

possibletoinclude

all

material ofinterest

.

Hence

,the orientation

was

toconsider

every possiblecontributionto thesubject

.

As such, all

papers

thatdealtwithturbulence in

one

way or anotherwere considered

.

The keytoabbreviations isshown inTable2.Table3shows thedetailsofall theeventscoveredby Table2 Keytoabbreviations

Aero. Comb

.

TurbMach. Int.Flow Heat Exch

.

E.F

.

M

.

Ind.Proc. Instr

.

M.-Phase Hot-W. Visual. Aeronautics Combustion Turbomachines Internal Flow HeatExchangers

EnvironmentalFluid Mechanics Industrial Processes

Instrumentation Multi

-

PhaseFlow Hot

-

WireAnemometry FlowVisualization

=

=

the present review. A comparison between the total number of published work and thenumber of

papers

on

turbulence can thedrawn from this tableand Table 4.

Note

that the papers

on

turbulence

accountfor roughly10% of the total contributiontotheENCIT's

.

Concerning the COBEM

_'

s,about 5%

Table 3 ABCM conferences

Conference Date,OrganizingCommittee No.Papers

COBEM

-

1971 COBEM-1973 COBEM

_-

1975

19

-

24/Nov,Prof

.

C.E.Stemmer/$C 5

-

7/Nov,_COPPE/UFRJ

12 85 106 9-11/Dec,COPPE/UFRJ

12-14/Dec

.

CT/UFSC

12-15/Dec

.

UNICAMP

15-18/Dec, PUC/Rio 13-16/Dec, UFU/MG 10

-

13/Dec,ITA/SP

10-12/Dec

.

PUC/Rio 7-11/Dec,DEM/UFSC 6

-

8/Dec,INPE,CTA 5

-

8/Dec,PEM/COPPE/UFRJ 10-12/Dec

.

DEM/UFSC

11-13/Dec,IPT

.

UNESP,UNICAMP,USP I-4/Dec,PUC/Rio

7-10/Dec,DEM/UnB/DF

7-9/Dec,EPUSP

.

IPT/SP

12

-

15/Dec,UFMG

.

PUC/MG

.

CEFET/MG II-14/Nov,DEM/UFSC COBEM-1977 COBEM

-

1979 COBEM

_-

1981 COBEM-1983 COBEM-₁₉₈₅ ENCIT-1986 COBEM

-

1987 ENCIT-1988 COBEM-1989 ENCIT-1990 COBEM-1991 ENCIT-1992 COBEM-1993 ENCIT

_-

1994 COBEM-1995 ENCIT-1996 133 169 162 197 239 61 443 94 350 216 401 189 521 137 631 331 of thecontributed

papers

w

ere

relatedtoturbulence.

Table 4 classifies all gathered literature accordingtothearea of application. A classificationofall theoretical work according to the type of adopted turbulence model or problem methodology is

presentedin Table 5

.

The experimentalworksareclassifiedin Table6

.

Table4 Workmotivation

Aero. Comb

.

TurbMach

.

Int.Flow HeatExch

.

E.F.M.

1973 ₂ 1975 1 1977 1979 1 1 1 2 1981 2 1983 1 1985 ₂ 1986 _{3 1} 1987 3 2 2

J.oftheBraz.Soc.Mechanical Sciences

_-

Vol.20,March 1996 14 (continued) 1 1988 2 1 2 1989 4 1 3 1 1990 5 1 1 2 1 4 6 3 4 1991 2 2 1992 5 1 1 3 3 2 1 1993 1 2 1994 2 2 1 2 1995 1 1 1 8 6 5 2 4 1996 2

Fundamental Ind

.

Proc

.

Inatr

.

M

.

-Phaae Total

1 4 1973 1 5 1975 4 7 10 1977 1 7 1979 4 3 1 1981 3 7 1983 3 5 7 1985 9 1986 5 3 10 1987 8 2 2 1988 9 1 5 22 1989 21 1990 6 2 18 2 1 1 1991 2 6 18 1992 1993 8 1 1 20 14 1994 5 1 16 10 1995 1 5 44 1996 12

Table5 TurbulenceModela

Algébrele Strem» Model»

2.Eq. Cíatele RNG Generic

Eddy Vlecoelty Medeie

OEq

.

₁

.

Eg

.

1973 1975 1 1 3 1 1977 1979 1 1 1 1981 1 1 1983 1 1 1985 1 1986 1 3 1987 1 1988 1 1 4 1989 1 6 1990 1 5 4 1991 3 2 2 1992 5 1993 2 1 6 1994 1 4 1995 1 2 14 1 1996 1

Transition Asyrnp

.

Tech. VortexMath. Other Total

1 1 1973 3 1975 1 6 2 1977 5 1979 2 2 1981 1 1 1983

A

.

P.S.Freireetal

.

: The State oftheArtin Turbulence Modelling

...

₁₅ (continued) 1985 1 2 5 1986 1 1 4 1987 1988 4 4 1 1 1989 4 3 11 1990 3 7 ₁₈ 1991 1 6 16 1992 1993 1 2 2 12 4 4 1 16 1994 1 1 2 12 1995 6 _{2 13} 1996 3 6 1 4 32

Table6 Experimental works

Hot-W- Optical Prtot Tube Visual Other Total

1973 1 1 1 3 1975 1 1 2 1977 3 1 4 1979 1 1 2 1 1981 ₁ 2 1983 1985 1986 1987 1988 1 3 1 7 2 2 4 1 2 7 1 4 2 7 1 2 1 4 2 1989 4 4 1 11 1990 2 1 3 1991 1 1 2 4 1992 1993 1994 1995 1996 2 3 1 6 1 1 2 4 2 2 1 1 1 3 3 4 1 5 13

A

Personal View

Letus nowmake acritical evaluationof allthe worklistedintheprevioussection

.

Thenumber of produced worksthat touch

on

turbulent

flow

is,asfullydemonstratedbyTable2,

notnegligible

.

Infact,wearesure that those numbers will surprisemostpeople.Inparticular,inthelast ENCIT forty fivepapers

on

turbulence were presented. This figure would justify in itsown right the realization of

a

conference only devoted to turbulence. Here, we remind the reader that a large, worldwide,_{conference on turbulence would attract about}

_one

_{hundred papers.Of course,}

_one

_may

arguethat the number corresponding to ENCIT96cannotbetakenasageneraltrend for the number of published works

.

However, even the average number of published works over the lasteight years,

twentytwo,isnotbad atall

.

The distribution of the worksaccordingtotheir area of interest,Table 3, showsthat Aeronautics and Fundamental Studies are still the leading scorers

.

Examples of works on Internal Flows, Heat Exchangers, Combustion or Turbomachinery, are very few

.

The

areas

of Instrumentation and Multiphase Flows are almost barely touched

.

Thetruepictureoftheturbulenceresearch carriedoutin Brazil by the mechanical engineers and

scientists,however, is provided byTable 4

.

Thestrikingfeaturethere is the overwhelming majority of

works on the two-equation modellingof turbulence and on similarity andasymptotic methods

.

The

numberof works in any of thesetwo classes iseven larger than the number of works that employ

algebraicturbulence models

.

Nowork

was

found

on

thedirectnumerical simulationof flows.Onlyone workwasfoundon large eddysimulation,andtwoon Reynoldsstressmodelling.