PUBLICAÇÃO DA ABCM • ASSOCIAÇÃO BRASILEIRA DE CIÊNCIAS MECÂNICAS 

V O L. XX • No.1 • MAR CH 1998 ISSN 0100-7386 



)OURNAL OF THE BRAZILIAN ~OCIETY OF MECHANICAL ~CIENCE~ 

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RBCM - J . of lhe Braz. Soe. Mechanlcal Sciences 
Vol. XX· No. 1 • 1998 · pp. 1·38 

ISSN 0100-7386 
Prin1ed on Brazil 

The State of the Art in Turbulence Modelling in 
Brazil 
Atlla Pantaleão Silva Freire 
Mlla R. Avelino and 
Lulz Claudio C. Santos 
Universidade Federal do Rio de Janeiro 
COPPE ·Programa de Engenharia Mecânica C.P. 68503 
21945-970 Rio de Janeiro. RJ Brasil 
Emaíl: ahlaCserv.com.ufrj.b~ 

Abstract 
Th~ preunt work discusse.f ai ltmgrh rhe current sratus of rurbulent resean:h ;, Brm;il. After eight introductory 
secrions on lhe subject, where some general aspecls of lhe problem are prttst:nted, and a brief rf'view of somt! 
scit!lltific Olld t!llgineering appmachts is given, tht' paptr stmUs over four sptcific srctions. analyt.i11g ali work 
carritd our 111 Bro:j/ in riu~ pa.ft twenty ftve years on turbulell<·e. ln fact, rht prumr campiúuion is restricted to the 
main ~·e111s sporuurnl by rhe Brazi/ian Society nf Meclumical Sciences. The present revit"w quotes 284 rt!ftrences, 
prtsents 6 tables and 16 ftgurn The pttpa COflltiiiS is: Paper Ourline, Some lnsiglus. 111e Traditiunal Approache.<, 
Som~ Ba.<ic Worli.ing Rufes. Turbulence Models, One-Poilll Turbulence Cln.wre Models, Some Olher Aproaches to 
Turbulence Modelling. Somt> Major A('hievemcnts. A Bil of History. Stmistk.f. A Personal View, Gallery. Final 
Remarks, Ciled Bibliograph-y and Compiled Biblio~raphy. 
Keywords:Turbu/ence Flow, Turbulence Modelling 

Paper Outline 
Thc present review was comm.ittioned by the BraziJi.an Society of Mechani.cal Sciences, ABCM, i.n 

order to give a clear picture of the present s tatus of turbulence modelling in Brazi l. As a guidíog li.ne, 
lhe papel' was supposed to l:oncentrale on work lhat appeared in the lasl five years in lhe two major 
Conferences sponsored by lhe Association: the Brazitian. Congress of Mechanical Engineering -
COBEM, and the Braz.ilian National Meeting on TI1ermal Sciences, ENCIT. The review, if possible, 
should not be a mere collection of annotated bibliographi.es, but o ffer a criticai evaluation of lhe 
published literature. 

Ali lhese aspects were in the mineis of t:he present writers when preparing thi.s manuscript. The 
bwad nature of tbe subject, bowever. together with its importance and general interest for t:he public, 
made Lhe writi.ng a difficult task. Big issues, such as Lhe inclusi.on of indexed publications. and of 
publications appearing in vehicles other than tho~e l:OmmonJy looked up by lhe mechani.cal engineering 
corruuuJúty. bad to be tackled in a very positive and quick way. Ln lhe end. it was decided to adhere 
qu.itc stringently to the main guidc I ines. The only exception had to do with the period of time covered 
by this review. ln order to give a good historie perspective of the subject. it was decided nor to impose 
any bound on tbe number. an<.l date, of works considered for review hcre. ln adopting thü procedure we 
panicularly regrei the exdusion of works published by the physical and mathematical socicties. 

Some lnsights 

Since we are going to discuss at quite some length lhe phenomenon of turbuJence, perhaps ir would 
be appropriate ar this stagc to define turbulencc. 11 can be sai.d, in a general manner. that a turbulent 
flow is a tlow which is disordered in time and space. Of coursc, this delinition is vague, lacking a 
precise matbematical fonnulation. This, however, is thc k.i.nd o f definition we commonly find in 
treatises thar deal with rhe subject of lurbulence. Thc tlows that are present in nature and technology 
and which are termed turbulent are very complex. exhibiting fairly different dynarnics from one 
occurrence to another. ln applicaúons. the tlows may be three-dimensi<mal, or quasi-two-dimensional, 
or may even have some son of large scale organized structures. Ali Lhese features , coupled with the 
random behaviour of lhe measured properties of the flow. seem lO deem the tlow malhematically 
undelinable. The imponant property which is required of them is lhat they should be capable o f mixing 
ali transportable quantities much faster than if only molecular processes whcre invo lved. To cut short an 
endlesS' dlscussion, and after Lesieur (1990), we propose here lhe fo llowing simplified del'inítion of 
rurbulence: 
lnvlt&d Arrie/e. Tectmieal Edrtor: Leonardo Goldstein Jr. 



2 J. of lhe Bra.z. Soe. Mechanlcal Sciences- Vol. 20. March 1998 

• A rurbulent flow must be unpredictable, in the sense lhat a small uncertainty as to ill; knowledge 
at a given initial time will amplify so as to render impossible a precise deterministic prediction 
of its evolution: 

• h has to satisfy thc increased mixing propeny defined above; 
• lt must involve a wide range of spatial wave lengths. 
Despite. the already mentioned vagueness of the above definilion. for tluid dynamicists the word 

turbulence has a precise mcaning. Mathemalically. in studying turbulent flows we are trying to solve a 
problem which is based on the assumption lhat the instantaneous flow vaciablcs satisfy lhe Navier­
Stokes equations. These are a deterministic set of equations which apparenlly yield solutions with a 
random nature. Thus, lhe philosopbical issues raised on how statistical lheories could be used to 
describe a phenomenon which is deterministic could oever be squarely faced until recent.Jy. New ideas 
and malhematical tnols advanced by Ruelle and Takens (1971) and by Feigenbaum ( 1979) h ave shown 
Li1at there are non-linear dynamical systems whose solutions, becausc of sensitive dependence on initial 
conditions. ex.hibit a weak causality and hence apparently random nature. Even the discrete algebraic 
equation z .. , = A z. (I - Z,) demonstrates random solutions whose long time solutions h ave a criticai 
depcndence on vanishingly small changes in the value of pararneter A. Because of the uependence of 
the long term solution on lhe vanisbingly small changes in A, lhe solutions are non-deterministic. The 
cooclusion is that fluid turbulence is a deterministic problem that evolves wilh time and !oipace in a very 
complcx fashion due to non-lineac interactions. 

Ahhough turbulence is a random phenomeoon, it is accepted that the flow is complctely govemed 
by lhe Navier-Stokes eyuations. Considering that a host of numerical schemes exist Lo deal wit.ll 
systems of non-boear partia! differential equations. one should have. in principie. no problem in 
computing turbulent flows. Actually, this is not the case. The richncss of scales that characterizes a 
turbulent flow imply Li1at. ii' the t1ow is to be correctly numerically simulatcd. alllengths from thc largc­
scule energy containing eddies. to t.lle dissipative scales under which the ml.llion is darnped through 
viscosity. must be resolvcd. 

The dissipation of mechanical energy into heat, bowever. may be !.hown through dimensional 
arguments to occur at very small scales. The dissipative scales are. in fact, given by 11 = ord (vl/€.)1-''. 
wbere E is the overall viscous dissipation per unit mass. and v is Lllc kincmatic visco~ity. The energy 
dissipation rate is given by E: = ord (u1/l) where u <1nd I are respective ly the characteristic velocity and 
length scales of the largest eddies of a turbulent llow, bcing, therefore, determincd by lhe geometry of 
the flow domain. The rninimum numbcr of.l-:lint~ required in a three dimensional simulation of the llow 
should thcn be N,"'QQ"' = ord (Jlfl)l )= ord (R1 •). wbere R, represents the Reynolds numbcr associmed with 
the large scale eddies. Here. N,..,,b represcnts the number of grid points that lhe computation or 
measurement would have to have in order to describe tbe entire range of scales. ln nature and in 
technological applications. the Reynolds number asswnes typically values between 100 and 10". wbich 
will result in approxirnately 10" to 1011 required grid points. 

The minimum temporal resolution is found by dividing lhe smallest spatial scate by the convective 
velcx:ity of the largest eddics. Likewise, Llle ratio between thc largest scales and the convective velocity 
gives the large scale temporal resolution. The rcsult is that the number of required time stcps is 
N, .... = R,)". 

Thc numher of points required to resolve lhe spectral propertics in turbulence is then 
N,..., = (Rt4)(R,Y') = R,J. This expression shows that an arder of magnitude incrcase io tbe Reynold~ 
number of the flow will resuh in a Lllree arder of magnitude increase in the number of rcquired 
compULational grid poinL<;. 

These numbers preclude any direct simulation of flows of practical interest. ln fact, evcn if 
powerful enough computers were available, the performance of direct ~imulation of tlows at higb 
Reynolds number would be an unfeasible exercisc. Wc realize that firstly by recognizing that the 
specilication of proper initial and boundary conditions on Li1e levei of detail required by the smallest 
dissipative scales is simply not possible. This, unfortunately, in dealing with the Navi.er-Stokes 
cquations. is a very serious drawback. The non-linear character of the advection terms muy give rise to 
spatial and temporal instabilities in lhe flow that will excite and arnplify lhe small scales in the morion. 
The lack of uniqueness of lhe solution, together with the amplification of thc small petturbatil)ns on Lhe 
boundarics of lhe flow. result in a motion wb.icb is apparently random in nature. 

The Traditional Approaches 

The decomposition of tlow field variables into mean and fluctuatillg componentli has lcd to what is 
now known as lhe Reynolds-averaged equations. ln many engineering and geophysical problems, the 



AP.S. Freire et ai.: The State of lhe Art in Turbulence Modelling ... 3 

fine details o f the now are really not required. so thal oft.en il suffices 10 have data conveyed by !ong­
time averagcs. Although thc avcrages were firsl defined in lerrns of time averages. further development~ 
have mtroduced the concepts of space averages and of ensemble aventges. The averaged Reynolds 
equarions can simply be derived from lhe Navier·Stokes equaúon to which lhey are very similar, but 
with additional lerrns involving the turbulent fluctuations. These addilional terms result from the 
advection terrns and may be seen. from a slalislical point of view. as second order correlations or 
momenrs. Since lhe practical interesl lies us ual ly in lhe mean properlies of thc lurbulent flow field. the 
staristical theories try to establish functional relalionships between lhe mean variables of the flow and 
the t1uc tuations. This is norrnaJiy referred to as the "closure" problem in ntrbulence. The att.empl at 
eslablishing lhese functional relalionships has becn one of tbc dominam themes of research in 
turhulence. 

Exact equations for the terms involving nuctuations can Lhe dcrivcd dircctJy from the Nuvier-Stokes 
. equalions. However. the determ.ination of transport equations for higb order correlalions aJways results 
in the appearance of higher arder correlations or momenL~. We are then faced with the problem of 
always having more unknowns than equat.ions available for tbeir solution. One way ou1 would be to 
neglecl correlalions of some order. Unfortu nately. this proves lo be unsa1isfac1ory. The reason for this is 
thal despite its apparenl random narure. turbulent flows exh.ibi t some degree of "order'', whicb musl be 
captured by lhe mathematical model. ln order to 'render tbe system of transport equalions closed, we 
need, lherefore, some "closure bypotheses'' indcpendent of the physic:rl conservation laws. 

Perhaps, wc sbould tbcn now to consider the conditions. or some generul principies, Lo whicb the 
closure modcls should obey. 

Some Basic Working Rules 

The sheer complexity of the rurbulence phenomenon ha.<; prevented researchers from achieving a 
ralional closure of the Reynolds-averaged equations. Tbis h~ motivated lhe invention of a number of 
mathemalical models based on different principies and rules , and tlle evoluúon of certain heuristics 
which are supposed to help us to conjecture lhe broad fearures of turbulcnt flows without having to 
appeal 10 any uf those models of douhtful validity. Of course. as these heuristics are deduced from 
common knowledge and experience. they will always be open to criticism. However, they musl be seen 
as useful n•le~ that have been subjected to a large scrutiny, and that. for this reason, constitule a 
serviceahle body of knowledge for the cnginecr. 

Some years ago. Narasimha ( 1989) suggested that the acc umulated expericnce with the various 
types of heuristic rules could be sununari:ted in the forrn of the five working rules listed in Table I. 

Table 1 The flve baalc working rulee 

1. As lhe Reynolds number of any turt>ulenl flow tends to lnfinity, lhe fraction of 
energy contained in lhe length and time scales directly affected by lhe viscosrty of 
the fluld becomes vanishlngly smalf: so do lhe scales lhemselves, compared to 
lhose accounting for lhe energy. 

2 Any lurt>ulent flow subjected to constant boundary conditions evoiVes 
asymptotically lo a stale independent of ali detalls of its genera!Jon save lhose 
demanded by overall mass. momentum and energy conservation 

3. lf the equations and boundary conditlons goveming a turt>ulent llow admit a salf 
preserving (or equillbrium) solulio~. lhe flow asymptotically tends towards lhat 
solulion. 

4. A turbulent flow may, before reachlng equlllbrium. attaln a mature stata ln whlch 
the d1lferent energetic parameters characterizing the flow obey Internal 
relationships, irrespec1ive of lhe detaíled initial condltions as in Rute 2. 

5 Between lhe viscous and the energelle scales in any turbulent flow exists an 
overiap domain over whicll the sotutions characterizing lhe flow ln lhe two 
corresponding limits must match as the Reynolds number tends to infinlty. 

These rules have been use~ by engineers for over a long period without any special 
acknowledgmenl of them. RuJe 4 . in particular. is thc basis of ali handbook charts ru1d correlations. such 
as. e.g .. Lhe Moody diagram for pipe losscs. They are, thus, according to Narasimha ( 1989). "what 



4 J. ol lhe Braz. Soe. Mechanical Sciences · Vol. 20. Marcn 1998 

comes naturally to workcrs in turhulem:e whenever lhcy encoumer a totally unfamiliar flow". The rules 
establish some positions upon which any turbuJence model should conform. 

At lhis poinl we just remind the reader that the working rules h ave not been dcduced &um the basic 
laws of 11uid motion and lhat for thil. n:ason they will always remain open to doubt. As a fmal 
assessmem, we I.JUOte Narasimba's veiy own words when lhe rules were first presented: "Thcy (tlle 
rulcs) are obviously very uscful, being close to reality; but lhey cannot stiJJ be elevated to lhe status of 
scicntific laws. because the ~mail departures noted from thcm cannot he disnussed as experimental 
error. and seem to indicate that lhe principies are strictly valid only under certuin as-yet unstated 
conditions which would not always be easily obtaincd." 

The most frequently challenged working rules concem lhe postulated independence from initial 
conditions. Rule:. 2 and 4. 

Turbulence Models 
Turbulen~.:e modeUing of a certain type ha.~ become a widesprcad ~ubject in acadenlia and industry. 

The pressing demand for more efficient models, capuble of dealing with evermore complex 11ows, has 
resulled in tlle proliferaüon of differcnt schools of thought, many of them remotely resembling cach 
olher . The implicaticm is a cominuous growth in lhe number. complcxity and sophistication of modcls. 
The presumption that any monograph on lhe subject should be able to covcr ali aspecL5 of turbulence 
mudelling is, thercfore, not ~hared by the present authors. Here. we will try to show lhe failures and the 
successes of some of the most popular approaches. 

The turbulencc models can be ensembled. in general, in four classes (NaJasimha ( 1989)). 
The lmpressionistic Models, lhat s tri ve to gain insight into the structure and the solution of the 
prohlem withoul any clairn to quantitative accuracy in predit:tion: Examples are the Burgers 
equation for turbulence, and the Lorenz equation for weather forecast. 
The Physical Models, that aim at predicting quantities of interest based on assumptions not 
inconsistent with lhe obscrved or understood physics of the problem, appealing to experimental data 
for model pansmeters when nccessat'y. The models of Emmons for boundary-layer transition, of 
Kutta-Joukowsld for lhe lift of an acrofoil, and of Kolmogorov for the spectrum nf turbulence, ali 
faJJ wilhiu this category. 
The Rational Models. that investigatc lhe nature nf problem and solutions through .5impler models 
derived from more wmplete systems by some limiting process. Tbe Burgers modcl for weak 
shocks. lhe Newtonian m<)del for hypersonic flows ru1d thc rapid djstorlion of turbulence are ali 
rational models. 
The Ad Hoc Models. lhat providc estimates of quantities uf interest without insistence that ali 
a<;sumptions be physically or malhematically justified in detail. The Boussinesq eddy viscosity 
model, the mixing length model. the K' - e. differential modcl. these are ali Ad Hoc models . 
The moucls thnt serve industry are ali Ad Hoc Models. ln particular, grad:ient transport modcls have 

become very popular over lhe years. The appeal of nn attracti vc blend of simplicity and acccptable 
predictive ability has stimulated a cootinued developrnent of lhese models. ln the next section, wc 
discuss these models in dewil. 

One-Point Turbulence Closure Models 
One-point closure model~ were developed in order to mo<lel inhomogeneous flows in practícal 

applit:aüons. Starting from the Reynolds-averaged cquations, Lhey aim at es1ablishing functional 
re lationships belween the turbulentlluctuntions, through their secünd-ordcr moments at thc same point 
in space. and the mean flow variables. This can be made in severa! ways. The simplest of the 
possibilitie ~ i~ tl) introduce an algebraic relationship. 

Using lhe notion lhat the collective interaction of eddies can bc represented by an increase in thc 
coefficient of viscosity. Boussinesq postulated the second-order momcnts to be pmportional to the local 
mean rate of strain lhrough an "eddy viscosity". v,. This notion is a direct u·anslation of Newton's 
viscosity law 10 turbulent tlows. Since turbulence is a property of the flow, thc cddy viscosity s hould be 
a function of Lhe mean tlow paran1eters. For three-dirnensional tlows, it should eveo be a vectorial 
quantity. 

Despite the boldness of lhis assumption, clear even to the earliest workers in turbulence. the eddy 
viscosity hypothesis has received criticai attention over the last 120 years. Turbule.uce models have 
become more and more sophisticateu but lhe usefulness of lhe eddy viscosity concept is still large. 

j 



A.P.S. Freire et ai. : The State of lhe Art in Turbulence Modelling . 5 

However, as pointed out by Bradbury ( 1997), lhe idea that turbulent stresses are strongly dependent on 
local strain rates (whatever model for v, is used) implies that lhe length scale of the turbulence is 
smaller lhan lhe scale of the velocity local gradient - this scale is typically of the order of the flow 
width. However. even casual observation of flow visualization shows that the main turbuJent structures 
are of tbe sarne order as the fluid width. lt is, therefore, amazing that proportionality relations involving 
on.Jy stresses and strain rate work so well. 

Of course, the concept of eddy viscosity is phenomenological. having no mathematical hasis. The 
result is that a constitutive relationship for v, still needs to be constructed. 

The eddy viscosity can be expressed algebraically in terms of quantities derived from algebraic or 
differential equations. Depending on the nature and on the number of equations used, a classification 
for the eddy viscosity models foUows. 

Zero-equation or algebraic models. These models use an algebraic expression for the definition of 
v,. Typical examples are the mixing-length concept of Prantdl, the Ccbeci Model and the Baldwin­
Lomax Model. 
One-equation models. These models employ a single transport differential equation which has to 
be solved for the fluctuating field. An extra algebraic expression is normally also required for the 
complete specification of the tlow. Examples of one-equation models are the rnodels of Bradshaw. 
of Nee and Kovasnay. of Johnson and King and of Goldstein. 
Two-equation models. These models use two transpor! equations for t11e description of the 
floctuat.ing field. Examples are lhe KIE model, the Klúl model, the KIQ model, the Klr. model, and 
many others. 
So far. the simplest prescription for lhe second-moments bas been achieved through the algebraic 

models. These models are built on an aoalogy between the randomiz.ing effects of the turbulent eddies 
on the mean tlow, and the corresponding random molecular motion in a gas. The implication is that tbe 
"rurbulent viscosity" can be seen as formed from a product of the density of the fluid, of a local 
characteristic length. 1.. and of a local characteristic velocity, u<. The analogy with the k:inetic theory of 
gases is clear; however, in a turbulent tlow, both the characteristic pararneters must be expected t.o vary 
locaUy with the flow. 

The mixing length theory assumes the fluctuations in both directions to have the sarne order of 
magnitude, and these to be proportionai to the distance from whicb discrete "lurnps of fluid" are 
fetched, times the local mean velocity gradient. This hypolhesis reduces our closure problem to the 
detem1ination of the characterisric length of lhe flow, that is. of lhe mix.ing leogtb. For simple boundary 
layer flows, I. can be assumed to be proportional to the distance from the wall. For free turbuJent tlows. 
I. can be assumed to be proportional to the width of the turbuJent regiou. The list is not exhaustive. 
Here. it suftices to say that every class of tlow bas its particular formulation of thc mix.ing length. 

The mixing length concept works quite well for a dass of problems, but suffers from ali the 
problems inherent to the eddy viscosity hypoU1esis. ln particular, in complex tlows, it may be 
impossible to specify any form for the mix.ing-le-ngtJ1. ln general, lhe model has shown to be appropriate 
for two-dimensional tlows with mild pressure gradients and mild curvature. No flow separation or 
rotation cffects are allowed in these models. ln addition. the aJgebnúc models are useful for two­
dimensional shock-separated boundary layers with weak shocks, and in computing U1ree dimensional 
boundary layers with small cross flow. ln fact. severa! authors have sought extensions of the algehraic 
eddy viscosíty model to three dimensional tlows (see. for example. Rmta( 1979)). The success, however, 
has been very limited. 

We draw our curtains on the mixing length model by pointing out to thc reader lhat this model 
assumes the t1ow to he isotropic, and takes no account of processes of convectíon or diffusion of 
turbulence. 

ln the one-equation models. a transport equation is used to compute the second-order moments. 
This can be made in rwo ways. Th.rough U1e spc~ification of a single. characteristic parameter that can be 
used to represem the fluctuat.ing field. or, altcmatively, relating the characteristic velocity in the eddy 
viscosiry model to a turbulent property. ln lhe latter approach, the characteristic length still has to be 
detennined through an algebraic equation. 

Almost ali one-equation models use the turbulent k.inetic energy, K, as lhe reference parameter. An 
e.xact equation can be derived for K through some simple algebraic rnanípulation of the Navier-Stokes 
equations. This equation. however, presents some turbulent correlatioos which have to modelled. The 
diffusion and the production tenns are modelled as gradient transport terms. The dissipation term, 
obtained through a local isotropy assumption. is defined in terms of an algebraic equation that involves 
the tluid density, the turbulent kinetic energy a.nd t11c length se ale. 



6 J. of the Braz. Soe. Mechanocal Sciences - Vol. 20. Man:h 1998 

ln practice. v~ry ünJe advancc of this Lypc of models o ver the z.ero-equation models exists. The fact 
that the char.tcteristic length scaJe still has to be fued by empirical arguments imports to these models 
many of the drawbacks of t:he aJgebraic models. The material fo1and in Literature. however. dearly 
suppor~ the ide a that K'r. is a beuer velocity scale than those employed in the àlgebraic models. 

A recent auempt at reviving the one equation models ha~ been made by Johnson and King. They 
hegin with a beforehand assumed eddy viscosity algebraic behaviour which has as it~ velocity scaJe the 
maximurn Reynolds stress. Nex.t. the maximum turbulent kinetic energy. K ... is supposed to vary vcry 
littlc wilh lhe transversal direction so that an on.Linary differential cqualion can be constructed to its 
prcdiction. The closure is concluded considering that Lhe ratio of thc local kinelic energy to the shear 
stress is constam. This model is aimed at a very linúteu cla~s of flows. namely. at two-cümensionaJ 
separatcd flows witholll curvature or rotation . The general resuhs of tlte modelare very good. 

The next levei of soph.istication is tu introduce a second u·anspon t.>quation, from wbicb tlle 
characteristic lengtl1 can bc caJculated. This g ives rise to the t.wo-cquation models_ These models are 
cunsidered LO emlxldy more physics than the previous ones. A Iist of the vari~JUS available two-equation 
model~ ca.r1 be found in Laundcr and SpaJding ( 1972) and more recently in a number of review articles 
(see. e.g_, Laksbtlúnarayana ( 1986), Wilcox. (1988)}. 

The variablcs adopted by researchers to determine th~ length scale vary from one work to another. 
One may choose as the second variable a parameter associated wi th the rnean frequency of the most 
energctic motion~. That was exactly the originul proposal of Kolmogorov ( 1942). The turbulent energy 
dissipation rate per unit mass. e. has been preferred by a number of authors. Another very popular 
choice has been the specific dissipution rate. w. A transport equation for the length scale itself was 
derived by Rotta (I 951 ). 

The standard two-equation mode ls fuil LO capture ma.ny of thc features associated with complex 
flows. The fact that lhey are still imprisioned to the eddy viscosity concepl makes them very vulnerable 
when dealing with flows that present curvaturc, scparation, rotation, and three-dimensionally arnong 
other effecL~. The advant.ages and s tmpücity of lhe Lwo-equalion models. however. sbould not be 
overlooked. Thcse models are rnuch superior to the a lgebraic and one-equation models in mildly 
complex t1ow~. Typical successful application of the.~e models are lhe flows in jeL~. channels. diffusers, 
and annulus waJI boundary Jayers withoul separation. 

The wcakltesscs and sbortcomingli of these models h ave been li~ted by HanjaJic (I 994}: tbey are: 
• Linear strcss-strain relationship through lhe eddy viscosity hypothesis: 
• ScaJar charac1er of cddy viscos ity; 
• ScaJa.r character of turbule.nce chumctcristic ~cales- insensitivity to eddy auisotropy: 
• Limitations to define only one time- or length scale of turbule nce for characteri7jng ali turbulent 

interactions: 
• Failure to account for ali v i ~cou.~ processes goveruing thc bchaviour of E or other scaJe­

dctermining quantities hy virtue of tbe simplistic form of tbe basic equation for that variable: 
• fnadequatc incorporation of visco~i1y damping effects on turbulence structllre Oow Reynolds 

number models): 
• lnahilily to mimic t11c prcfcrc ntially oriented and geometry·dependent effects of pressure 

refleclion and t11e cddy-flattening and squeez.:ing mechanism1- due to Lh~ prox.im.ity of solid- or 
interface surface: 

• Frequeutly inadcquate treatment of bottndnry condition:s. in particular at Lhe M)}id walL 
A logical way of deriving more sophisticatcd motlels. of a certain universality. consists in deriving 

full transport equations for each of the second·onler mome rm di.rcctly from lhe Reynolds-averaged 
equations. T his procedure. inevitably. results that every derived equation invoJves various correJations 
among fluctuating quantities which are not exactly determiued. Thcsc, of COltrse. must be modellcd in 
terms of lhe mean flow variables. the second- moments thcmselves, and at least one characteristic time 
scaJe. Models of this nature ur~ u_osually termed Reynolds-stress transport models. According to some 
authors. thcsc m<.><lels represent Lhe h.ighcst degree of complexity that has been exercised in 
convcntion<tl turbulence modeling. 

AddiLwnally. lo a more univer~aJ de~cription of lhe turbulence. these motleb are considered lo 
adhere mure rigoronsJy lo some modelling principies which are judgcd by some to be crucial. The firsl 
of Lhe principies has to do wilh thc malhematical formalism of lhe closure problem. Consisteocy 
conditions such ru. dimensional coherence of ali terms. tensorial consistency. coord.inate-frame and 
matcrral frame independence, satisfaction ol r~ali7.abilily condilions, limiung properties of rurbuJence. 
ali thcse are ~atisfted by tbe Reynold:s-s1re~1> moJels. The second princ1plc is related to the physics of 
lhe turbulence. For example. lhe de~:re<osi ng innuence of lúgber-ordcr momenL~ upon the mean flow 
properties is a feature that must he ohserved in turbuJem models. 



A.P.S. Freore et ai · The State ol lhe Art in Turtlulence Modelhng ... 7 

The Reynollb stress models are reponcd (Hanjalic (I 994)) to provi de a superior representatioo of 
t\vo-dimensional non-equilibrium nows. Thcy are also s upposed to perform be"er in unsteady and 
periodic nowli and in ali situations wherc the turbulent field e:dlibits hystere~is a~ compared wilh the 
mean llow field. Thc models accoum for the effect~ of anisotropy. curvarurc. rotation, lhree­
dimensiooality. and hence normally performs beuer lhan lhe two-equation models under these 
conditions. A shortconling feature of lhe models is the need fm a scale-determining equation. whicb 
b'TOWS in i111po11ancc in pressure dominated fiows, or in t1owS with unproperly trcatcd boundary 
conditions. 

For thrce dimensional nows. the Rcyn()Jds-su·ess models will resull in 6 transport equations which 
must be simultancously solve.d .wgelher with the equations of motion. Til is rcsults in I 0-12 equations to 
be solved for the mean now and turbulence quantities. This is certainly a Herculean task even for 
IJl'~~eut day modero computcrs. 1-lcncc, the order is to simplify lhe Rcynold!! ~trcss transpor! equutions. 
One try is to reduce dlese cquations to algebraic equations. Such models are termed algcbraic Reynolds 
stress mode Is. 

The essential idea of Lhese models is that onlv the convection and diffusion tcrms contain stress 
gradients. Thus. if lhrough some altemative proçe(Íuré, Lhese processes could bc modelled by equations 
not containing strcss gradients, Lhen Lhe differentiul Lran~port equations for tbe Reynolds stresses would 
be reduced to algcbraic equations. The need for S(>lving turbulent transport cquations is not eliminated 
hcrc; howcver, Lhe calculated now variable~ are now scalars. 

A st.i ll simpler approuch considers Lhe stress transport processes to be completely negligible. This 
local-equilibrium approximations are used rnainly for understanding the physical nature of complex 
flows. ln may cases, lhey often lead to use fui overall modcls of the stresses. 

ln fact, many of lhe idea~ Lhat we have becn considering in lhe last paragraphs wcrc advanced quite 
some time ago. As early as I 940. Chou introduced. in an absolutely original paper. Lhe equations for lhe 
second and third mnments of Lhe turbulent fluctualions. At lhe time, he proposed to "dose" lhe equation 
for the triple velocity correlation by assuming Lhe fourth-order moments te) be proponional to thc sum 
of the three pruducts of two double velocity correlations. ln additinn, furlher hypotbeses on the 
correlaLions involving turbulent pressure gradients and veloci ty nuctuations wcrc needed. These 
correlations werc dcrived by Chou from a Poisson equatiun. The equations for vonicity decay were also 
found and thc tcrms of energy decay were improved. 

Essentially, Chou argued that because the Navier-stokes equaúons are the basic dynamical 
equations of nuid motion, it does not suffi<:e to consider only tbe mean turbulent motion. The turbulent 
nuctuations should be as important a~ the mean motion and. for this reason, the equations for them 
should also be considered. 

What Chou rcally achieved was a systcmatic way of building up differcntial equations for the 
velocity correlations for each successivc ordcr from the equations of turbulenL fluctuations. The 
problem now lies on the difficuJties to bc found in solving simultaneously Lhe Reynolds-averaged 
equations and the equations for the higb ordcr moments. The three majtlr uifficulties found at tbe time 
were: 

• TI1e non-linear character of Lhe equations: 
• Thc correlation fuoctions are slowly varying funct.ions of spacc and úme. wbereas Lhe 

fluctuations are all rapidly varying functions of them; 
• To malhematically solve lhe set of non-linear differenúal equations. an extra pbysicaJ condition 

is needed (similarity hypothesis). 
These difficulties were finally oven:ome hy Chou ( 1987) himself by developing the method of 

successive substitution. to solve the mean rnotion and velocity fluctuation equations simultaneously. 

Other Approaches to Turbulence ModeiUng 

Another rncthod of describing lhe rurbulent fluctuations is tbe spe.ctTal melhod. lirst introduced by 
Taylor (1935. 1938). The introduction of Fourier analysis inro lhe problem leads to some benefits. The 
differential opemtors are convened imo mulúplicrs, lhe physics of turbulence is given a relatively 
simple picture, and the degrees of libeJ1Y of lhe turbulent system are better defined. The use of Fourier 
analysis is panicularly useful when Lhe turbulcnce is homogeneous, that is. whcn it is statistically 
iovariant under translation. Unfortunately. Lhe ~pectral approach does not solve Lhe prohlem of closure. 
it just appears in a d.iffcrcnt fom1. 

Using the spectral approach. se.veral theories were built for the description of turbulent flows. Sorne 
are: 



8 J o! lhe Braz. Soe. Mechanical Sciences • Vol . 20. March 1998 

• The direct-interacúon appro.ltimaúon: 
• The qua.si-nonnal theory. 
The first approach is analytical and has only given useful resultJo. for isotropic. homogeneous, 

turbulcncc. Thc quasi-normal theories. which as.sume aJl cumulants above sorne given ortler greater 
than two l(l vunish. lead to a negative energy spectra. Anempts at fuing this problem have led to 
considerutions that require an extra assumption about time scales that i ~ questionable in the energy 
wntaining runge. TI1ese melhods are of narrow applicaúon. make a variety of hypotheses whose 
vaüdity i:, difficulllll assess in physicalterms. and. for this reason, will not be considcred furtl1er here. 

Other melhods lhat heavíly rely on numerical simulations have been devcloped more recently. One 
of lhem. dírect :;imulation. has received a lot of attention. 

The remarkahle recent advances in computing power have opened an era of simulation. The 
applicution of paralleled codc!! on n1mputcr~ with cfficiem network.ing and targc mcu10I y ~;Upa~.:ity h ali 
tx:come a very powerful way of computing turbulcncc. ldcaJJy, we would like to perform computations 
by solving the complete Navier-Stokes cquations. However, due to lhe re!.olution restrictions observed 
in section 2. the calculations are presently rcstricted to flows with a Reynolds number of the order of 
10". Déspite the severe lirnitations imposed by the resolution requiremcntl>. direct simulation ha~ provcd 
to be a pnwerful too! for understanding the physics of turbulcncc and furnishing data for the 
development and the improvement of turbulence models. ln fact. it is capable of providing data on 
turbulence lhat is virtually unobtainablc frorn experiments such as pressure-vclocity correlations. 

So far. our models for lhe simulation tlf complex turbulent tlows, have relied on the use of 
statislical avcrages for the varíables of interest. Another possibility is the use of fílters on the sarne 
variablcs. This approach. normally termed !urge eddy simulation, presents the advantage of naturally 
introducing lhe scales of the resolved variables. Thc result is that. i) in principie, it is simple to prepare 
the input data for numerical models lhat are cnnsistent with lhose scalcs: ii) the interaction that occurs 
between numerics and physics io the solution of lhe equations of motion is very srrong, and the use of 
this approach makes it easier to have a better understanding of this process. The term large eddy 
simulation is frequently associatcd in literature with a space fiiLering open1tion. The use of time 
filtering. however. al~o has been tested; this rcsults in thc eliminaúon of highly nuctuating components 
in lime. allowing the use of large time ~tep~ in lhe numerical integration of the motion equations. 

Let u~ nnw look ín more detail at lhe philosophy of large cddy simulation. The melhod requires that 
a !iltering operatíon is upplied to the Navier-Stokes equations. Nex.t, thc velocity field is decomposed 
into a large ~cule velocity and a sub-grid velocity scale. This produces a new problem. The non­
lincaríty of the atlvective term will now resull in four different terms. lndecd, for a general space 
fillcring operation, the classi~.:al averuging rules do not apply. Only one of Lhese resulti.ng terms is 
analogous to thc Reynolds stresses. The othcr tem1s arise from lhe fact that, as we have just said, the 
fíltering opcralion in not idempotent. The suhgrid scaJe modelling problem can then be defined as 
finding exprcssions for the subgrid terms as a function of lhe large scale varinbles. Actually. the subgrid 
scale modclling problcm is not a well posed problem. ln the physical problem. the propagation to the 
large scales of the uncenaintics contained in t11e subgrid scales will contaminate the fom1er yielding a 
llow with an unpredictable nature. Now, considcr wc have constructed a subgrid scale model. and that. 
therefore. we have at our disposal a closed .set of motion equaúons where everylhing is expressed in 
terms of thc large scales. Then the simulation~ conductcd with these set of equations will not be able to 
propagate any disturbances which would generate different flow ficlds. TI1is implies that a large eddy 
simulation of turhulence will not faithfully reproduce lhe large scale evolution from a deterrninistic 
point of view. at lea~t for limes greater lhan the predictability time. 

ln large eddy simulation, the cakulated now must then be interpreted as a different realization of 
the actual flow. What one would hope to huppen is that realization to have the sarne statistical 
propertics of Lhe real t1ow and t11e 11ame spalially nrganized slructures (lhough. at a different position 
from the rcality). 

Based on Lhcse concepts, Lcsicur ( 1990) defines what a good large eddy simulation of turbulence 
should be. 

I.A.Jw gr adr definition lhe simulation must prcdict correctly the ~tatisucal propenies of turbulence 
tspectral diçtributions. turbulent exchange cocfficients, et cetera). 
High gr.tde definiti on moreover, the simulation must be ablc to predict the shape and topology (but 
not lhe pha.~e) of the organized vonex structures existing in the flow at thc scaJes of the simulat..ion. 
An imponant question to be asked now is: How small lhe scale of thc rcsolved motion has to be? 

We begin to answer hy rcrninding the render that in high Reynolds numbcr turbulent flow, the smaller 
thc scale of Lhe motion. lhe more isotropic it becomes. and that. in facl, an "incrtial" subrange exists. 
Thus. if tl1e scale of the large ~cale motitm lies on the inertial subrange. lhe behaviour of thc unresolved 



A.P.S. Freire et ai.: Tha State of the Art in Turbulence Modelllng ... 9 

scales could then be asscssed by involcing their near i~utropic properties. Thc implication is Lhat Lhe 
proper ma.~ter ~cale to be use{! in large eddy simulation of turbulence is precisely the grid spacing for 
isotmpic me..~he~. For anisotropic mesbcs. a product average ora Euclidean nonn can be used. 

Some Major Achievements 

The cla.ssical the11ries of rurbulence havc achieved some results in the past which clearly have had a 
defini tive innuence in our perception of the problem. Nellt. we shall diswss some of Lhese successes. 

The second-order two-point correlations play a leading part in turbulence theory. We have 
discussed in some detail in lhe previous sections the engineering approaches for Lhe one-point 
correlacions. However. if ow· intcrcsc is Lhe underlying structure of the turbulence. we sbould consider 
the velocity corrclations at two or more points. These fundamental coocepts wcrc advanced by Taylor 
(1935) in u paper where he also introduccd the concepts of statisticaJ homogeneity and isotropy. 
Subsequently. Taylor ( 1938) introduced Lhe three-climensional energy spectrum in wavenumber (i.e., 
the Fourier transforrn of Lhe two-point correlation in space), an entity whose calculation has become one 
of Lhe fundamental objectives of rurbulence Lheory. This function meal.ures how mucb energy is 
contained between the wave nwnbers K and K + dK. 

The clnl>ure prohlem for isotropic turbulence can be formulated in wave-number space. ln this way, 
when we con~ider the transport of turbule nt energy. this will be in waveoumbcr rather than 
configuration space. Large scale strucn.res are associated wilh small valucs of K, whereas small 
structurcs are associated wilh high values of K. Thus. the transfer of energy will occur from one range 
of eddy scalcs to another. Thjs process is known as Lhe "ca:;cade of energy". 

Thc cncrgy balance equation is obtajned from Lhe equatioo for lhe singlc-time correlations. This is 
just the mean motion equation. after some munipulation, specialization to homogeneous turbulence and 
Fourier trans fonned. A further specializ.utinn to Lhe isouopic case, reduces the spectrul tensor to its 
isotropic fonn. The trace of tbe tensor then gives tbe energy spectrum, E. The resulting equation for E. 
involves then a production terrn. a ili~sipation term. and a rranspor1 tenn. The Lmnsport cerm 
corresponds to the triple velocity correlations coming from non-linear interaction:. of Lhe Navier-Stokes 
equations; this terrn jul>l redistributes energy in wave number space. 

The us ual interpretation of Lhe energy balance equation is that the energy in Lhe flow storcd at small 
K (Lhat i~. ai large scales) is transferred by lhe non-linear trJllsport term to largc K (that is. to small 
scales), where it is dissipated Lbrough heat by the action of viscosity. The non-lioear terrn describes 
cOJL~ervative processes. namely inertial rransfer of energy from one wave nw11ber to a ncighboring one. 
Now, from ::unple experimental evidence, we know that Lhe energy is dctcrmioed by lhe lowest 
waveuumber. thatthe dissipation rate is determined by the highest wavenumbcr, and that the two ranges 
do not ovcrlap even for very low values of the Reynolds numbcr. lt follow:. Lhat the non-Linear traosport 
tcrm can be made to dominate over an as largc a~ we like p01tion of lhe wavenumber space. by simply 
i11creasing Lhe Reynolds number. 

The above ideas were fonnabz.ed hy Kolmogorov ( 1941) in two famous assumptions. 
Kolmogorov's first hypothesis of similarity . At very hjgh, but not intinile Reynolds numbers, ali 
the small·scale stati~tical properties are uniquely and universally detem1ined by thc scale, I, Lhe 
mean cnergy di!-isipation rate, E, aud thc k.inematic viscosity, v. 
Kolmogorov's second bypothesis or similarity. ln the limit of infinite Reynolds uumbcr, ali small­
scale s tatistical propcrties are uniquely and univerl><llly determined by Lhe scale. I, and Lhe mean 
energy di sl> ipation rate, E. 
By a simple dimensional argument, the tirst hypothesis implies that Lhe energy spectmm can be 

written as: E (K) =v'" Eu• f(Jd), where f is a univcr~al function. 
The second hypothesis implies Lhat. in Lhe lirnit as Reynolds number tends to intinily, B (K) should 

become independent of lhe viscosity. This amoums to lhe energy s pectrum assuming the form 
E(K) "" 1::~3 K ' 1

) . the famous Kolmogomv's K':.t• law. This law is remarkably weiJ vcrilicd experimentally, 
allhough the fine-scale motion does not necessarily have the desired degree of isotn)py postuJated by 
the author. and lhe pruportionality cooslaJll cannot be deduced convincingly hy theory. 

A second major success of turbulcnce lheory is Lhe theory of Tay lor ( 1921) for the turbuJent 
diffu..;ion uf nuid particles. A random process where at any instant the future state of t:he process is 
enúrely determioed by its state at that panicular ins tant and independent of llS prehistory is called a 
Markov proccss. Alternatively, we say we have u Markov process wheo tbc future is independent of the 
past for a known present. For the turbulent diffu~ion of tluid particles. tbis is certai nly not tbe case. ln 
the turbuJcnt motion of a tluid. the motion vf Lhe particles is continuous - so is the e1tchange of 
transferablc quantities - and there is a corrdation in time bctween properties of a fluid panicle at 



10 J. ol the Braz. Soe Mechanical Scoences- Vol. 20. March 1998 

subsequ~nl limes. This memory behaviour of IUrbulcot diffusion implic~> that lhis process cannot be 
con~idert"d ~ Markov process. Taylor cxtended lhese notions to the turhulent tlvw diffusion by 
t:Onl.tdering lhe path Of 8 marlted tlujd parucJe during ÍIS ffiOUOO through the tlow field. 

Cun!>idering the displacement of marked tluid paniclcs. Taylor itltroduced lhe Lagra.ngian 
autocorrelation and the integr.1l time scale. Then. further consideriJ'Ig the Oow to be homogeneous in 
space and time. an exact result for lhe mean square distancc u·aveled by a diffusing marked fluid 
particle was derived in terms of the Lagrangian correlation cocfficient. RL· The difficulty is that a 
LheoreticaJ solution for R, t:annot be found. Also, experimental measurements of Lagrangian quantities 
in turbulenl flow are difficult 10 make. so that information about R, is scarce. Nevertheless, lwo 
impon.ant rcsults conceming the Jimjting cases of t -tO and of t -t.,.., werc cnunciated. 

Short diffusion times: RL = I. The distance traveled by the marked particle is cqual to its velocity 
muhiplicd hy rhe rime elap~ed A t:las~ical result from Ne~'tonian mechnnics. ln other words. one 
can ~ay th<tl, at short diffusion times, the motion of tbc marked particle is predictable, provided we 
know the initial conditions; lhat is, lhe motion is determ.inistic. 
Long dif"f'u_~ion times. The root mean !>quare panicle displacement is proportional to Lhe square 

root of thc time elap~d. This is the sarne result one would find for lhe classical random walk of 
d.iscontinuou~ movements, where lhe panicle variance js proportional to thc square root of the 
nurnber of steps. 
1l1e laucr result charat:teri7.es lhe unalogy hetween lhe ga~ kinelic theory and the diffusive motion 

of eddies. Thus, thc r.m.s. distance traveled hy a fluid particle can be expressed in terms of a diffusion 
cocfficient dcfincd as a function of ~inglc-timc means anda Lagrangian integral time-scale. The abovc 
results, allhough simple, gave us a dcep in~ight into the nature of turbulenl diffusion. Nothing was 
rc;~lly solved by Lhe rcsults, as lhe dosure prohlcm still pers.isted, but some connections and 
"lr;~nsforms" of the problem wcrc established. 

Another great achievement of lhe traditional approaches is Lhe universal log-law for the mean 
velot:ity distribution in lhe near wall rcgion. Using the not.ion that Lhe near wallturbulent flow could be 
divided into distinct regions, with distinct dom.inant physical effecb, scaling velodties and lengths. 
Millikan. ( 1939) resoned to a •matchability" argument to work o ut a functional relalionsrup for the 
vclocity profilc:=. The resulting Iogarithmic C'<pression has become one of the great pamdigrns of 
lllrbulcnce theory. Validated by a huge number of experimental data, the so-called law of lhe wall has 
becomc a bench rnark result to which aiJ simulations of wall flow have to conform to. Evidence in favor 
of the law is. tllcrefore, strong. Some authors, however, have placed thc Jaw under a severe scrutiny, 
claiming t.hc vclociry profile lo he power-like, and not logarilhrnic-like! 

ln fact, t.hc dcrivation of bolh laws i:. equaiJ:r· consistem and rigorous. However, thcy are based on 
cntirely differcnt assuJllptions. 8oth derivations stnrt from sirnilarity and asymptotic considerations. 
Thc log-law is obtained from tbe assumption thut for sufficiently large local Reynolds nurnber and 
suffrciently large now Reynolds number. the dependence of thc velocity gradient on the molecular 
vist:osity disappears completely. ln the ahcrnative approach that le.ads to the power-law, the velocity 
graJicnt i~ as~umed to possess a powcr-type asymptotit: behaviour. where the exponent and t.he 
multiplying parametcr are supposed to depend l.omehow on the now Reynolds number. Thus. the 
,.elocity gradient tlependence o n molecular viscosity does not disappear however large the loeal and the 
fiow Reynolch numbers may be. The form of lhe powcr-like scaling law yields a family of curves 
who!;C paramcter i~ the Reynods number. The resuhing envelop of curves is shown to be very clnse to 
the univer!.allog-law. 

The original rcsult. however, has remained unshaken. The fact that the log-law can be obta.ined 
frorn fom1al asymptotic mcthods and that it has beco shown to apply to a variety of tlows, has firmly 
silenccd its c ritics. ln fact, thc large body of useful services covered by the law has made it 10 transccod 
in importance. The universal laws of fr.iction for smooth and rough pipes are just one of the pungent 
examplcs of application of thc log-law. 

Typical cxamples of application of the law include cases with walltranspirmion. roughness, transfer 
of heat. compressibility. three-dimen~ionality. and even separation uf llow. Of course, for ali of these 
situation~, !>Orne modifications need to be madc in the original formulation Ln comply with specific flow 
rcquirements. However. the essence of the law and its general formare the sarne for ali cases. 

For boundary layer flows. a second universal law was added to our collectinn nf great 
achievements. ln the outer region of the boundary layer the log-law may no longer apply. since the 
conditions in which it is ba~ed are no longcr valid. However, experimental cvidence has shown that, if 
we consider a~ our rcfcrence quanlities lhe ~kin-friction velocity and the boundary layer thickness, a 
velocity defect similarity relation can be constructed which tums out to be valid for t.he whole turbulent 



A.P.S. Fre1re et et.: The State of the Art in Turbulence Modellrng ... 11 

pan of lhe boundary layer. This resuh was suggcstcd by Von Kannan to be considere<! a posrulate, the 
velocity-defect law. 

lmpressed by the similar behaviour betwe.en the flow in the outer region of a bounda.ry layer and Lhe 
flow in a wa.ke. for bolh exlúbit large scale mixing processes contmlled prim:lrily by inertia effects 
r.tther than vis(OUS effects, Coles, ( 1956) proposed a correction function to lhe classical log-law. The 
resultjng expre~~ion, buih purely on experimental grounds. includes in its vulid.ily domain both thc 
n•rbulent part of lhe wall regioo and chc outer region. The equation for Lhe correclion function was 
called. by Coles the law of the wake. The actual function may be approximated by an anlisymmetrical 
sine function multiplied by a profile parametcr which does not depcnd on lhe transversal flow 
coordinate. 

The abovc list of succcsses is. of course. not exhaustive. Some olher great acrucvements such as the 
stabiliry thcory for thc flow bcrwccn two rotuting cylinders devoloped hy Tnylor, ( 1923), or the 
boundary layer uppro)(imations due to Prandtl, ( 1904), could easily havc been included here. The main 
fearure of the chosen achievements is that thcy were advaoced by exploiting plausible physical 
argument~ thmugh dimensional aod similarity analyses. 

The reader will be k.een in identifying some of the above resuJL' with sorne of the basic working 
rules advanced in the section Some Basíc Working Rules. 

A Bit of History 
The progress in the art of turbulence modciUng in Brazil must, in some way or another. bc related 

to the degrce of develt>pment of its graduatc courses in physics, applied mathematks and engincering. 
Th.is is C<Juivalent to say that turbulcncc is a contemporary subject in Bnuil. slil l in its infancy and, 
therefore. in a very incipient state. ln fact, Lhe very first graduare engíneering coursc in Braz.íl was only 
inauguraled at lhe Federal University of Rio de Janeiro (UFRJ) ín lhe sixties. ln a bistorical deed, the 
Chemical Engineering Divi.sion of lhe lnstitute of Cbemí~Lry founded in March of 1963 the Chemical 
Engineering Graduate Prograrn. This act w~ a direct result from a previous official visit to lhe United 
States of Amcrica by Prof. Alberto Luiz Coimbra to study in detail ali lhe aspects of the eogjneering 
graduatc cou~es taught at the Universitie.~ of Hou~ton, Rice, Califomia (Los Angeles and Berkeley), 
Stanford, Cal. Tech .. Minnesota. M.l.T. and Michigan. The visit. which took place in December of 
1960, was sponsored by lhe Organiz.ation of Americao States. (OAS). nnd by thc Congregation of the 
Cbemjcal School of the Federal University of Rio de Janeiro. As a follow up to thi~ visit. io August of 
1961 the Deans of the Engineering Schools of thc Universities of Houston and of Te)(as carne to Rio de 
Janeiro. 

Together with their Brazilian pea.rs, lhe visiting deans established a preliminary plan for the 
implemcntativn o f a graduate enginccriog course at lhe Federal University of Rio de Janeiro that was 
finally presentt:d in Ocrober, 1961, to the braziüar1 coordinat.or of the Alliancc for Progress. Th.is plan 
was made accessible to the community in December of 1961 ata seminar on unjversity refonn and on 
the teaching of eogi.neering organized by lhe Engineering Club of Rio de Janeiro. As an exercise to a 
complete implcmcntation of lhe graduare program, it was decided that short and intensive courses on 
the subjects of Boundary Layer Theory and Turbulence, Aow in a Porous Medium and Computer 
Programming would be offered in the monlhs of July and August of 1962. The courses would be jointly 
sponsored by lhe OAS. the Braziüan National Resean:h Council (CNPq), lhe Cbemical l.n.~titute of 
UFRJ and the University of Houston (UH), and would be given by the UH lecturers Drs. Abraham E. 
Dukler and Elliot I. Organick. 

This initialive, io fact, was detemúnanl ia thc definition of the areas of excellence Lhat lhe graduate 
courses to be estahlished were trus have throughout the sixties. The ~pecial care dedicated to fluid 
mechanics wa.~ dear, and that is whcre this review wants to concentrare on. 

The fonnal inauguraúon of the Chemical Engineering Program occurrcd in March, I 963. Still 
beneliting from finar1cial help from lhe OAS, and with further aid from thc Fulbright Commission and 
the Rockfeller Foundation, the coordinators of lhe graduate program invited four Americao lecturers to, 
togelher wíth six otber Brazilian colleagues, provide t11e inüially required teacbing and supervísion. 
The four U.S. lectures were Profs. Donald Katz, Lows Braod, Comelius John Piogs Jr. and Frank M. 
Tiller. Only two years later. and under Lhe guidance of the Chemical Engincering Program, the 
Mcchanical Engineering Program started funclioning. ln lhe years that followed. nine other graduate 
prograrns were created g.iving origin to lhe Post-Graduate Engineering School of lhe Federal Uoiversity 
of Rio de Janeiro (COPPFJUFRJ ). 

The role of those three initial courses on thc development of the subject~ Clf tluid mechanics and of 
heat tran~fer in Brazil was seminal. Levered by the presence of great scientists on those subjects, the 



12 J. of the Braz. Soe. Mechanlcal Sciences- Vol . 20, March 1998 

cbemical and mechanical graduate programs beca.me an important center for fluid mechanics and heat 
transfer ~tudies. lndeed. ali twcnty four M.St:.(Magister Scientia) lheses that were presented at 
COPPEIUFRJ from 1964 to 1966 had ro do in some way or anolher with fluid mechanics or heat 
transfcr processes. Only in J 968, and only with tJ1c entrance of lhe graduate programs in Metallurgical, 
Electrical, Civil, Nuclear. Naval and Manufacturing Engineering, the lide of power started to change. 

ln a broader sense. as we shall see. tbe above mentioned facts unfolded in a nationaJ levei. The cast 
of great scientist.s temporarily associated to COPPEIUFRJ was to become a key factor in lhe 
development of lhe sciences of tluid mechanics and of hcat transfer in the whole of Brazil. Severa! 
exarnples immediately leap to the mind. The close relation. almost intimate, between COPPEIUFRJ and 
Houston University through Profs. Coimbra and Tillcr prornpted severa! students to go abroad to study 
with Prof. A. E. Dukler and other Houston lecturers. Tbese students were late to come back to Brazil 
and, on moving to diffcrcnt inslitutions. lhcy wcrc ultimatcly thc respon~ible for lhe cstablishment of 
rnany other graduate fluid mechanics programs in Brazil. Anotber notable example was lhe passage of 
Prof. Ephraim M. Sparrow, currently an Erneritus Professor at Minnesota University, through 
COPPEIUFRJ. After spending two years in Brazil. he went on Lo supervise a large number of Brazilian 
students ovcr a span of more 1han twenty years. H is impression on the developmenl of fluid mechanics 
science in Brazil is lherefore unquestionable. Lndeed. when lhe M echanicaJ Engineering Program 
started. its staff consi),Led only of four lecturers: Profs. Sparrow am.l Jacques Louis Mercicr. and lhe 
assistam lecturers Théo F. C. Silva and Franscisco N. de Farias. As Prof. Mercier left after a span of 
o ne year. lhe role of turning Lhe gr.tduate program in mechanical engineering into a success was left 
almosl entirely 10 Prof. Sparrow. 

ln the next sections we will try to establish a firm connection between ali the above facts and lhe 
present state of turbulence research in Braz.il. Herc. just for the sal< e of future reference, we will quote a 
nurnber uf selected theses. Thcy are: 

• Gileno A. Barrelo, "Hot-wire AJlemomeLer - Construction and Calibration~. M.Sc. Thesis, 1964. 
• Ralf Gielow. "The GraetL-Nusseh P roblem for a Turbulenl Flow", M.Sc. Tbesis. I 965. 
• João S. D'Avila, "Solutions of lhe Boundary Layer Equations for a Turbulent Flow". M.Sc. 

Tbesis, 1968. 
• Arno Boll rnann. "Study of a Turbulent Jet in a Highly Turbulcnt Environment", M.Sc. Thesis, 

1969. 
• José Augusto F. Oouvea. "Heat Transfer in Turbulent Flows'', M.Sc. Thesis, 1969. 
• Carlos E. Lopes. "Turhulence in the Vicinity of Fluid Interface~". M.Sc. Thesis. 1970. 
These exarnplcs wcre chosen to illustrate lhe stale of turbulence research in Brazil by the end of the 

~ix tics. They. by no means. represem the totality of work done in turbuJence at thal time. Note. 
however. that any of lhe abovc ti tles could be perfectly valid titles for theses to be presented in the past 
few years. 

The groW1h in the enginecring sciences in Bruzil provided by t:he consolidation of .~everal graduate 
cour~es quickJy st.aned pres~ing for tbe creation nf un appropriate forum where discussions of scientific 
intcrest were to take place. As a resuJt, severa! symposia and conferences were to have birth in the late 
sixties. carlv seventies. 

For thé mechanical engineering community. the landmark was the creation of the Brazilian 
Congress of Mechanical F.ngineering. Aiming at discussing rhe problems - in a broad sense - re lated to 
mechanical engineering in Brazil, a srnall group of twelve researchers assembled in Florianópolis. Santa 
Cutarina, for lhe First Br:u.ilian Symposium on Mechanical Engineering. The Symposium was held in 
I lJ71 and produced twclve technical papers. Since Lhen. and at cvery two years. tbe Brazilian Congress 
of Mechanical Engineering {COSEM) has been staged on a regular basis: it is. unquestionably, the most 
irnpt)rtanl conference in Brazil on Mechanical F.ngineering . ln its latest cdition. the astonislúng number 
of 631 papcrs were presented. 

By J 986 Lhe arca of them1oscicuces had uchieved such a degree o f dcvelopment in Brazi l that the 
crcation of an event uniquely devoted to it was a must. A new conferencc was tben organized which 
was to be termed the Brazilian National Mecting on Thennal Sciences (ENCIT). Tbe conference. whicb 
is held cvcry rwo years. became an immediatc success. ln the last cvent, held in Florianópolis. Santa 
Catarina. 331 papers werc presented. 

Statistícs 

Before we make any critical commenL on lhe existing lileraturc 011 turbulent llow, Jet us first work 
out the statislics of the past COBEM's and ENC!T's. On compiling ali rclcvant work Lo lhis review we 



A.P.S. Freire et ai.: The Sta1e of the Art in Turbulence Modelling .. . 13 

have lried ru; much as possible to include aJI material of imerest. Hence, the orientation wa~ to consider 
every possíble contribution to the subject. As such. ali papers that deall with turbulence ín one way or 
another were cnnsidered. 

The key to abbreviation.s is shown in Table 2. Table 3 sbows lhe details of all the events covered by 

Table 2 Key to abbrevlatlons 

Aero. 
Comb. 
TurbMach. 
lnt.Aow 
Heat Exch. 
E.F.M. 
lnd.Proc. 
lnstr. 
M.-Phase 
Hot-W. 
Visual. 

Aeronautics 
Combustíon 
Turbomachines 
Internal Flow 
Heat Exchangers 
Environmental Fluíd Mechanics 
Industrial Processes 
lnstrumen1ation 
Multi-Phase Flow 
Hot-Wire Anemometry 
Flow Visuaiization 

the presem review. A comparison between lhe total number of published work and the number of 
papers on turbulence can the drawn from thls table aud Table 4. Note that the papers on turbulence 
account for rougWy I 0% of tbe total contribution to the ENCIT's. Conccrn.ing the COBEM's. about 5% 

Table 3 ABCM conferences 

Conference Date, Organizing Committee No. Pa[!!!rs 
COBEM-1971 19-24/Nov. Prof. C.EStemmer/SC 12 
COBEM-1973 5·7/Nov, COPPEIUFRJ 85 
COBEM-1975 9-11/Dec, COPPE!UFRJ 106 

COBEM-1977 12·14/Dec, CT/UFSC 133 
COBEM-1979 12·15/Dec, UNICAMP 169 
COBEM-1981 15·18/Dec. PUC/Rio 162 
COBEM-1983 13·16/Dec, UFU/MG 197 
COBEM-1985 10-13/Dec. ITNSP 239 
ENCIT-1986 10-12/Dec. PUC/Rio 61 
COBEM-1987 7-11/Dec. DEM/UFSC 443 
ENCIT·1988 6-8/Dec. INPE,CT A 94 
COBEM-1989 5-8/Dec. PEMICOPPEIUFRJ 350 
ENCIT-1990 10·12/Dec. DEM!UFSC 216 
COBEM-1991 11 -13/Dec. IPT,UNESP,UNICAMP,USP 401 
ENCIT·1992 1 4/Dec. PUC/Rio 189 
COBEM-1993 7-10/Dec. DEM/UnBIDF 521 
ENCIT·1994 7·9fDec. EPU$P,IPTf$P 137 
COBEM-1995 12·15/Dec, UFMG.PUC/MG,CEFET/MG 631 
ENCIT·1996 11-14/Nov DEM!UFSC 331 

of the contributed papers were related to turbulence. 
Table 4 classifies ali gathered literature according to the area of application. A classification of ali 

theoretical work according to the type of adopted turbulence model or problem methodology is 
presented in Table 5. The experimental works are classified in Table 6. 

Table4 Work motivation 

Aero. Comb. TurbMach. lnt.Fiow Heat Exch. E.F.M. 
1973 2 
1975 
1977 
1979 2 
1981 2 
1983 
1985 2 
1986 3 1 
1987 3 2 2 



14 J. of lhe Braz. Soe. Mechanlcal Sclences • Vol. 20, March 1998 

(contlnued) 

1988 2 1 
1989 4 2 
1990 5 1 2 3 1 
1991 6 3 4 4 
1992 5 I 2 2 
1993 3 3 2 1 
1994 2 2 1 1 2 
1995 1 1 2 
1996 8 6 2 5 2 4 

Fundamental lnd.Pro<:. lnstr. M.-Phaae Total 
1973 1 4 
1975 4 5 
1977 7 lO 
1979 4 7 
1981 3 
1983 3 3 7 
1985 5 7 
1986 5 9 
1987 3 10 
1988 2 2 8 
1989 9 5 22 
1990 6 2 21 
1991 2 1 18 
1992 6 2 18 
1993 8 20 
1994 5 14 
1995 10 16 
1996 5 12 44 

Teble 5 Turbulence Modela 

Edd~ Vlscoslty Modela Al!l!bralc Stress M~.!J.L. 
O Eq. 1. Eg. 2. Eq. Claaslc RNG Gene ri c 

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

Trensltlon As~m(!. Tech. Vortex Meth. Other Total 
1973 1 1 
1975 1 3 
1977 2 6 
1979 2 5 
1981 2 
1983 1 



A.P.S. Freire et ai. . The State of the Art ln Twtlulence Modelllng ... 15 

(continued) 

1985 2 5 
1986 4 
1987 4 
1988 1 4 
1989 4 3 ,, 
1990 3 7 18 
1991 1 6 16 
1992 1 2 2 12 
1993 4 4 1 16 
1994 2 12 
1995 6 2 13 
1996 3 6 4 32 

Table 6 Experimental works 

Hot-W. Optlcat Pltot~ube VIsual Other Total 
1973 1 1 3 
1975 1 2 
1977 3 4 
1979 2 
1981 1 1 
1983 2 3 7 
1985 2 2 
1986 4 2 7 
1987 4 2 7 
1988 1 2 4 
1989 2 4 4 11 
1990 2 1 3 
1991 1 2 4 
1992 2 3 1 6 
1993 2 4 
1994 2 2 
1995 1 1 3 

-1~~6 3 4 5 13 

A Personal View 
Let us now make a criticai evaluation of allthe work listed in the previous scction. 
The number of produced works lhat touch on turbulent Oow is, as fully demoostrated by Table 2, 

not negligible. ln fact. we are sure thatthose numbers will surprise most peoplc. ln particular, in the last 
ENCIT forty tive papers on turbulence were presentcd. This figure would justify io its own right the 
realiz.ation (lf a conference only devoted to turbulence. Here, we remind the reader that a large. 
worldwide, conference on turbulence would attract aboul one hundred papers. Of course, one may 
argue that the number correspondiog to ENCIT'96 cannot be taken as a general Lrend ll.x Lhe number of 
publíshed works. However, cveo lhe average number of published works over the last e ight years, 
twenty two, is not bad atai L 

Thc distribution of the works according to lheir are.a of i nteres~ Table 3, shows that Aeronautics 
and Fundamental Studies are stjU the leading scorers. Examples of works on Internal Flows, Heat 
Exchangers, Combustion or Turbomachinery, are very few. The areas of lnstrumentation and 
Multiphase Flows are almost barely touched. 

The true picture of lhe turbulence research carried out in Brazil by tbe mechanical engineers and 
scientists. however. i:; provided by Table 4. The stri.k.ing feature there is lhe overwhelrojog majority of 
works on the two-equation modelüng of turbuleoce and on similarity and asymptotic methods. The 
number of works in any of lhese two classes is even larger lhan the number of works lhat employ 
algebraic turbulence models. No work was fouod on the direct numerical sirnulation of flows. Only one 
work was found on large eddy sirnulation, and two on Reynolds stress modemng. 



16 J . of the Braz. Soe. Mechanical Sciences - Vot 20, March 1998 

ll1e compilalion of ali experimental work on turbulence is sbown in Table 5. Tbe numbers are 
extremely Líoúd. They show that no tradjúoo exists in Bra:úl for the performance of experimental 
studie~ on turbulcnt nows. ln particular, Lhe hot-wire anemometry technique, so important for the 
measurem,ent of turbulent Oow propenics, is shown to be vcry incipient in Brazil. 

The performance of good, reliablc and accurate experimcnts is however crucial for the develupment 
of Lhe ~cieoce of turbulence. As we saw before, the models lhat serve industry are alJ Ad Hoc models. 
We bavc also seen that moM of the work carried out in Braz.il makes use of two-equation differential 
modcls and that these models are vulnerable to a series of weaknesse:- and shortcomings. Unfonunately. 
lhesc difficulties must always be overcome on an individual basis through oew modelling procedurcs 
resulting from new experimental observations. 

ln fact, as observed by Prof. L. J. S. Bradbury in a personal communicaúon to the present aulhors, 
the buund betwccn the available data and lhe available turbulence models is very tight. lnd~t.l. a~ vaslly 
demonstrated by common experience, tbe progress in turbulencc modellíng is direcdy fastened to the 
progress in turbulence measurement tcchniques. Thus, accordiog to Bradbury: "Wbco only mean 
velociry rncasurements were availablc, only integr.1l lheories satisfying overoill momcntum conservaúon 
could be formulated. A small number of dispo~able coostants were needed but agreemcnt with 
experiments was poor outside lhe data set. Wilh turbulent mcasurements, differential models could be 
developcd which satisfied local momentum conservatioo. As better measurernents became available. 
modcls salisfying an enlarging hierarchy of conservation relationships were developed with an 
enJarging ruerarchy of disposable constaots. ln no case are the models rational rnodels - they are ali, to a 
grcatcr or lesser extent. empirical. and modelling is really a sophisticated way of "curve fitting" to lhe 
experimental data. The number of disposable constants in Reynolds stress models is large and thcy are 
adjusted to give the best fit to the largest set of experimental data". 

ln relation to lhe large number of lheorcticaJ work on two-equation differential modcls and, 
particulurly, on numcrical techniques, we ruust acknowledge this to be related to type of rraining that 
the Braz.ilian scientistS received in their uudergraduate and graduatc courses. ln fact, the increasing lack 
of training on experimental techniques and on lhe fundamental aspects of turbulence is a worldwide 
trend that is only being repeateJ in Braz.il. The pres.-,ing demand for publications, lhe absolute shortage 
of money for research. the computcr revolution. ali lhcse are factors that conspire against lhe 
establishment of experimental programs and time consuming fundamental srudies. Researchers these 
days have become more practical, choosing as t.heir research lines topics whicb can be developed at a 
certain pace and with the minimum of trouble: this will assurc lhem a prolitable carrier. 

Thc aggravating circumstance for the Brazilian case is lhat we have never had any tradition on lhese 
issues and, as a result, nothi.ng really was built in lhe past in those lines. The competence of Brazilian 
rescarchers to implement turbulence models for a varie.ty of flow situations is undoubt.edly high. ln the 
literature reviewing process severa! examples of vcry complex problem gcometries were found. Oespite 
a clear dominance of oumerical melhods that involve finite differenccs and finite volumes, quite a 
number of works were found on finite elements, vortex dynamics, integral methods, the generaUzed 
integrul transform tcchnique. integral metbods, and many others. 

The number of works which resortto perturbation tecluüques is also significative. These techniques 
have a long rustory of service to nuid mcchanics. dating back to Prandtl ( 1904) and the derivation of lhe 
boundary layer lheory. The law of thc waJI, is perhaps lhe soundest case of application of singular 
perturbalion methods to turbulent flow. ln recent ümes, in lhe past editions of COBEM's and ENCrrs. 
extensions of lhe law for more complex tlow situations have been sought wilh success. Suc.:h diverse 
effectS as now transpiration, flow compressibility, transfer of heat at lhe wall surface roughness, shock 
wave interaction, now separation, or a combination of ali these factors, have been incorporated to the 
classicnl formulation yielding a set of useful exprcssions for the local vclocity and temperature profiles, 
and for the skjn-friction cocfficient and the local Stanton number. 

The traditinn on theoretical nuid mechanics in Brazil has certai.nly ils roots on lhe early graduare 
courses set out in lhe sixties. At thattime, aod by vinue of the stroog malhematical character of the fust 
nuid dynanúcists, in particular Prof. Coimbra, the emphasis had always been on lhe development of lhe 
fundamental aspccts of problems. Experimental ihvestigations were always rare. It is, therefore, only 
natuml tu expect tl1is trend to he reflectcd in lhe past publications appearing in COBEM and in ENCIT. 
ln fact, since lhe early editions of COBEM some very interesting theorelical works on turbulent now 
were present. The years of 1975 and of L 977 were clearly vintage years for work.s on rurbulence. Ali 
five works presented in 1975 and ali ten works prescnted in 1977 were top quaJity. ln the years to 
follow some great works were continuously been addcd to ABCM's archive. 

Trus brings us to a paradox. Despite the superior qualiry of work that was being generated. few 
people managed to continue in the system producing quaJjry work. ln reality, many varushed after a few 



A.P.S. Freire et ai.: The State of the Art in Turbulence Modelllng ... 17 

years. ln more recent times, tbis has become a worrying panem. Researchers with a very good acadernic 
record. who perfom1ed very good research work in their Pb.D. courses, simply do not seem capable of 
following it up in f.beir carriers. The consequence is that, despitc a continued entrance of new people 
into the fluid dynamics comrnunity, lhe number of pubüshed works remains nearly constant in the Last 
eigbt years. The reasoos for tbis phenomenon are unknown to the present autbors. Maybe researchers 
are carrying out more appüed studies, which take them away from their original specialization field; 
maybe their academic loads become too heavy or the struggle for financial aid is tough and 
unrewarding. The number of feasible explanations is high and academic at lhis stage. The i.mportant 
point to notice is that a lot of well educated scientists are not surviving in lhe present system and tbat 
something must be done. 

Gallery 
This section was lhought as to provide the reader with a brief but faithful view of the type of studies 

carried out on turbulence in Brazil in the last twenty five years, illustrating lhe past achievements 
through picLures and graphs taken directJy frorn the original works. Tbc expectation is t:hat the selection 
will be representative enough to cover ali aspects discussed in previous sections. ln tbis sense, the main 
hody of work is concentrated on asymptotic techn.iques and on two-equation differential models. 

The first ABCM Conference with full procecdings was lhe 1973 COBEM. ln that occasion, only a 
single work was presented on tbe subject of turbulence. The work, due to Gaspareto and Giorgetti. was 
a very interesting attempt at developing hot-wire probes for the measurement of water flow properties. 
The wires were coated wilh epox.y, yielding an altemative method for the use of hot films. The 
description of tbc rna.in aspects of Lhe problem. including the whole probe production process and the 
calibration process was, unfortunately, very short and incomplete. what makes it virtually impossible to 
reproduce the author's results. ln taldng the measurements a DISA 55001 anenometer unit was used. 
Graphs for Lhe flow mean velocíty in the wake beh.ind a cylinder were presented for two d.ifferent 
Reynolds numbers. Typical measured velocity protiles are shown in Fig. I, which is here reproduced in 
its totality. In fact. in reproducing ali figures, we have decided to keep the original captions. 

.. 
rt0 

. 2 

. ' 
o 

li 

• o 

4 

''o 
l 

2 
! 

I 

O> ~ o 
o 

·I 

·2 

1----f--

o, v. 
0,2 o.• 0,6 0,11 1,0 

0/. 
(),2 0,4 0,6 0,6 1,0 o • 

P(I!FI~ OE llfLOCIOAOtS NA tSTtoP.O. Ro 0 • 102~ 

SONOAS: • fu..N-!1 o riO '!JM"t ~ O FIO ~~m 

fig. 1 Hot·wlre results of Gaspareto and Giorgetti. 

The next COBEM presented some high qualiry work. The return to Brnzil of seveml researcb 
students who had gone abroad for their Ph.D. degrees brought back some innovating ideas. All works 
presem in me 1975 proceedings were, in fact, to have far reaching intluence in me developtuent of 
turbulent research in Brazil. ln view of the statistics we have seeo before, we are obliged to mention 
here the work of Carajilescov. ln his work. an one-equation di.fferential model was used to describe the 
details of the flow through a triangular array of rods with differeot aspect ratios and Reynolds number. 



18 J. of the Bra.z. Soe. Mecl'lanical Sciences · Vol. 20, March 1998 

The calculations were in reasonable agreement with the experimental data of olher auú10rs, allowing the 
autbor to make a good assessmenl of any ex.isting secondary tlow. ResuJts for the shear stress 
distribution are shown in Fig. 2. 

1.01 

••• 

··" 

' / 
,• I 

.• · I 
/ 

i 
:/ 

r o 2!1 lO t 

o 

Fi&• ) . Wall &bear Strea• O!atribution 

• • • 
• 

" 

~· I.ZI7 ) 

At...-t.7X.IC 
0,. • 

-. " 

Pia. 4. Wa~l Shaar Stresa Oiatríbutíon 

Fig. 2 Results of carajileskov. 

The year of 1977 was also very good for works on turbulence. Therefore, the task of selecúng a 
representative reference of ali works published was really difficult. The works of Militzer. Pimenta. 
Alves, Pereira Filho, Alvim Filho and of Menon. ali re-sulted from their graduare dissertations. The 
works of Crabb and of Nickel were top qual.ity, but had no participation. direc:t or indirect of brazilians. 
From a historical perspective, it is the judgment of the present aulhors that perhaps the article with Lhe 
most relevam result was the work of Pimenta. His experiments were part of a great coUective effon of 
the Heat and Mass Transfer Group at Stanford University to assess the propcrties of transpired turbulent 
bouudary layers, and the new data be gathered for flow over a rough an porous wall became reference 
work. Some of his data for the local skiu-fricúon cocffteient and for the Stanton number are shown in 
Fig. 3. 



A.P.S. Freire et ai.: The State of the Art in Turbulence Modelllng . 

Ctl2 

Q002' 

QOJ1 

o 
o 

--correlação eq 19 

St 

4Xk' 

• 

F:O 
' 

QX)l 

F:Q0:)44 

" ;~u 

o 
ll tr 

Fig. 4 Kúmeros de Stanton com inje~ão 

------
· -----.... ·- -,- _a_ ---,_ --.. - I .... 

x/r 

20 

l . 
F 

00 

002 o. 

Flg.S C0eficient~ de ~trito rom injeção 

Fig. 3 The experimental resulta ol Pimenta. 

19 

The next COBEM did oot bring any great news. Compared with the previous conferences. the 
n:sults presented did not show any significative change. Seven works wbere published in the 
proceedings. A very interesting one was the experimental work of Ferreira on turbulent thermaJ 
convection in a horizontal layer between parallel plates. The work illustrates the various possiblc 
configurations for t11e thennal convectioo process, comparing the data of several authors, including his 
own. Fig. 4 shows t11e flow configurations tackJed in tllis work. Results for the higb order moments 
were presented. 

The year of I 98 I showed a sharp decline in the number of works on turbul~nce. Only three works 
were published that year. From these, the most interesting one was the work of Frota and Moffat. In this 
work, a rriple hot-wire system was developed to measure the insrantaneous values of the velocit.y 
components. The technique aJiowed tbe measurement of the turbulent properties of complex turbulenl 
three-dimensional flows with the use of a single probe. According to the authors. both the mean 
veiocily profiies and the turbulent shear stresses could be measured with an accuracy of 1.4% and 3% 
respectively. provided the probe axis had a maximum misalignment of 20" with the Oow d.irection. The 
paper is reasonably detaiied but the results are poorly presented. The figures and graphs bave a small 
size and for t.bis reason are difficult to reproduce. 

ln the next COBEM we had a singular fact: ali published works on turbulence wcrc experimental 
works. Besides. and more importanUy. five of the seven presented works were fully developed in 
Brazil, rwo of them dealing with the difficult subject of bydrodynamic stability. To iUustrate that 
vintage year for experimental work we show here graphs for the criticaJ Taylor number and the critica] 
Reynolds number obtained, respectively by Purquerio and by Santana et aJ., for two different tlow 
geometries: the tlow of a Newtonian tluid in the interior of two concentric rotating cylinders and the 
tlow of a non-Newtonian fluid in a capillar reometer. 



20 J, of the Braz. Soe. Mechanícal Sciences - Vol. 20. March 1998 

OUA·ORO I 

CONVECÇÃO t[AMICA TURBULfNTA ENTRE PLACAS PLANAS 

&ELO 

CONVECÇlO MO 

Pt:H!.TftA.TIWII 

co•vrcçlo 
P(MflltATIVA 

C STUOO E"'lCPERIMf H!AL 

t9-6 7 

Fig. 4 Posslble flow configurations for turbulent thermal convectlon ln a horizontal layer between parallel 
plates (after Ferreira). 

\. 
'• ~1 ~ _..,.,., •~oW•!• 

- ~ . .... " oj :· 

.. ,"" •t 
' ' ' 

~~ \ \, . _-::-::·.:::·: -···­
•, '\ ' : ::: ::.:~=-:~.~::: .. 
'; " ' ,..._ 
\, '.., ...... 

\ · ........... 

"',,, 

·. 
. . I 

· - --~ 
- .. 

•• t .. •• ~ • Vtfla ,i.O du •~• de 1a ytor {l.a u 4 •t t U«h(' l· 

••ntn 4a4 ltuut>JIJd•d,, L o.tl • r•. •t;t\1 t il 

o -~ ~-,...,· 
• ..-[ o o J' ~ • 

f .. -~ 
· ~ 1-----

A .. Htlll'• • 4% 
•.........,.. e l't 
O~ Cit.t.'i 

''"Lr· .. 
~ T .. , {fl.t11 

.•. '-:!:oo-o.---- -::.,..-----.. ,:::<1';-~--.J 

.r ............. ~ 
, ,..._. .... , M4tst C>0 Mf'IOOO Ol' 1'1.-NCS r ,...n 

Fig. 5 The transition results of Purquerio and of Santana et ai. 



A.P.S. Freire et ai.: Ttle State ol the Art ln Turbulence Modelling ... 21 

The following confcrence had the sarne number of published works. seven. From these. Lhe most 
imponant were Lhe rwo experimemaJ works of Leite et ai. on plane turbulenr jet.s. ln these works, the 
organized motion in the near fíeld of a rurbulei\1 plane jet under periodic controlled acoustic excitations 
were investigated visuaJJy and quantítalively througb smoke-wire and hot-wirc techniques. The 
StrouhaJ number was made to vary from 0.15 to 0.60 and tbe time-dcpcndent excitation frorn 0.5% to 
49%. The main conclusion was that the fundanJeotaJ cornponent attains a maximum value for StrouhaJ 
= 0.18 at x/H = 4.0 along the centerline. Flow visualization results obtaincd througb the smoke-wire 
technique are shown in Fig. 6. 

Fig. 6 Organlzed motlon ln the near fletd of a turbulent p lane jet. 

The year of J9S6 inaugurated lhe ENCITs. ln ali, nine papers on turbulence were presented in this 
COJú.erence. The work.s were about average, rel1ecting mainly the work carried out in Brazil :11 that time; 
the mccting had only oue forcign entry. ln fact, most works were jus! a small depanure of former 
work.s that were being studied for many years; some for more ú1an tcn years. Tbe work of Pimenta and 
Alvim Filho studicd the mixing flow provokcd by the interaction of two eonfined axisymmetric jets. 
Severa! graphs were presented with ilie axial distribution of static pressures and the transversal profiles 
of the averaged velocities and turbulent intensities. Figure 7 shows some of the data. 

r IR, 

1,00 

o 

1,00 

EI<SAIC ~ 

e X/O• 0,0 

0 K/0•0,14 

.t. X/0 •0,42 

t> 

ô K/0 • 1,38 

Q. X/O• 2,7~ 

)I )( I O • 4,78 . 
2,ao~~~------~~--~~------------~.2~0~--------~~~.~~~~x~. 

Figura 8 - Perfis de Intensidade de Turbulência 

Fig. 1 The experimental data of Pimenta and Alvim Filho 

The 1987 COBEM section on turbulent tlow had only three anicles. From those, the only one worth 
of a note wa~ the work of Leite and Scofano. By this time. Leite had a very comprehensive oeuvre on 



22 J. of the Braz. Soe. Mechanical Sciences · Vol. 20, March 1998 

plane turbulent jets, cenainly one of the most important in the history of events sponsored by ABCM. 
For this year, no figures will be presented. 

The 1988 ENCIT had no particular section oo turbulence. Even so, eight works oo the subject were 
presented that year. Embarking on a trend that was to dominate the studies on rurbulence in Brazjl in 
lhe coming years. the turbulent flow in a compressor was investigated by Deschamps, Ferreira and 
Prata through a rwo-equation differential model. The limitation in paper length resultcd in a very 
concise article where many aspects of the analysis could not be explained in detail. For example, only 
graphs of the pressure distrihution where presented. These graphs are shown in Fig. 8. ln the years to 
come t.hese authors were to publish many other interesting papers on the subject. 

N 40------~----~----~------r----, 
........ .. 
0.30'-::::::=~~. 
' !" 

11/d•O,~; Ro: 13325 
-o-E•~~erlm•ntol 1 a. 

20 -Num,roco 

10 

0~--

·lO 

·30 

fig . 6 - ~omparaçao entre result~dos numé rico 
e experimental; h/d•O,O~. Re•l3 .3Z5. 

N 40 ~--~---...,..------.----~--. 

........ 

'" a. 30 1'1/d•O,O!I; Ro<2321!1 --E--1 ...... .. 
20 - N ..... r.c:o 

10 

-10 

-~L---~----~----~--~~--~ 
O 0,3 o,8 0,9 1,2 R 1,5 

Fig.7 • Comp~raçãn entre Te$ulLadus numérico 
e expericnt!.utal; h/d•O,O.S, Re•23 .275. 

Fig. 8 The results of Oeachamps et alll. 

The tenth COBEM presented a vcry good selection of works on turbulence. The highlight here was 
the large number of works, nine, devoted to tbc fundamcutal aspects of tbe problem. Half of the twenty 
two articles were on experimental technique.s; five on two-equaúon differential models. The mosl 
important contribution to turbulence in this meeting was given by tbe twin works of Coelho oo the 
modelling of turbulent jets io cross flow. Some of his results are reproduced in Fig. 9. 



A.P.S. Freire et al .: The State ot lhe Art in Turbulence Modelling ... 

l'H,."UR! 'J.. 2 · Modol• for â jet in b croas-flow: (a.; ont.ra.in!.n,g aurfact~; 
(b) voreox patr. 

. .. 

c • c• 

; 

FICVRB 2. 3. N'omenc l at:.uxe for an ele:ment of t.he j o t . 

Fig. 9 The llow conliguratlon ot a turbulent tet ln cross llow. 

23 

The major conclusion reached by Coelho was that turbulent entrainment and lhe transpor! of the 
transversal component of voiticity have a strong influence on the dynamics of lhe mixing layer in lhe 
initial region of lhe jeL Fwther considerat.ious ou the format.ion of lhe wake behind the jet led to two 
main conclusions: I) The deflcction of lhe jet in the uear field of lhese tlows is mainly due to 
entrainment rather than to pressure drag. 2) The transversal component of vort.icity has a strong 
intluence on the formation of the pair of trailing vortices . inducing a rapid transference of transversal 
vorticity into the pai r of vortices which is being formed. 

ln 1990. the third ENCIT had two specific sections on turbulent tlow. The biggest contingent of 
papers lhis time wa~ on asymptotic techniques applied to turbulent tlow. The emphasis on the 
implementation aspects of numerh:al methods to turbulent flow wus also high. The sad news here was 
the small number of experimental works. The simulation of im:ompressible 20 and 3D turbulent tlows 
by a füúte element method performed by Brasil Jr. et ai. lhrough the K·f. model is presented in Fig. 10. 
Two cases were presemed in the paper: turbulent backward facing step and annular tttrbulent jel. Only 
the former case is shown here. 



24 J . of the Braz. Soe. Mechanical Scíences- Vol. 20, March 1998 

-
(t 1"\0f' .! ... 

Fig. 10 The results of Brasil Jr for the turbulent kinetic energy leveis. 

The 1991 COSEM section on boundary layer theory and turbulence had twelve papers. A very 
interesting work, however, was presented in the section on therrnal convection, vaporization and 
condensation. The work, due to Yanagihara and Torii, studied the inf1uence of an array of longitudinal 
vortices generated by half-delta wings on the heat transfer of laminar boundary layers. Hot-wire 
velociry measurements and beat transfer experiments were carried out to evaluate the mechanism of 
beat transfer increase. The main conclusion was that arrays of counter rotating longitudinal vortices 
presem better heat transfer characterisúcs rhan co-rotating arrays. Some of rhe resuJLS found by the 
autbors are sbowu in Fig. I I. 

,.I~'-

! " ::a·= .... , .... ... 
~. ~·· 

Fip,. 1 Ke.t IJ'ariJt=r raWu lftd vd«Jry <:Omc>en, for 
I pÚI wbb thc CXIm1MII fto., "P .a .r- 0 ... UI 

IIII., ~~ · 
,.. .. 1\ -.... ,fi ,...,, •• 
.,. • • 4 ..... 

.. 

F·~ 8 Stm~ w:&odty OOftiO\rn fM 10 am.y o! 
"'--"""lol ..,...,.. (o• IS' ... d 1• so ""') 

Fig. 11 Experimental results of Yanagihara and Torli. 



A.P.S. Freire et ai.: The State ol the Art in Turbulence Modelling ... 25 

The IV ENCIT wa~ oot a very good meet.ing for K-E modellists; only tbree work.s on t:wo-equation 
differential models were presented from a total of eighteen work.~. The selected reference for this 
compilation. however. deals wilh tbe K-E mode1 applied to a lhree-dimensional turbulent swirling flow 
io a recrangular duct of 1arge aspect ratio. The resuJts, obtained by Nogueira and Niecke1e, show the 
effects of the Reynolds number and of tbe swirl intensity in tbe tlow field, as sbown in Fig. 12. 

(&) Z/D• = 0,0, Re = 2 x 10', P/D ~ ~.4 

(h)'z;v, = 1,1, R• = z x to•, PfD = ~.~ 

. {c) ZjD, = 2,5, ll< = 2 x 10\ PfD = (,. 
F'1gura S. Des.,nvolvíment<> das V"locidad., Tr6.1U1veraaio 

Fig. 12 The numerical predictlons of Nogueira and Nleckele. 

The 1993 COBEM showed a relative balance among the severa! entries for lhe sections on turbulent 
flow. One of the most interesting works was surely lhe numerical prediclions of Kobayashi and Pereira 
for lhe flow over a two-dimensional h.ill covered wilh vegetation. The equations of motion were solved 
with the aid of an extended K-f turbulence model which included tenns due to the d.rag caused by the 
p1ant canopy. Typical results for the kinetic rurbulent energy are shown in Fig. 13. 

a. 

Figura S. Previsões de Perfis de energia cinttica turbulenl& lc com 
Co=0,8 e W. = 1,95 e resultados experimentais para as ats 

geometrias. 

Fig. 13 The oumerlcal predlctlons of Kobayashi and Pereira. 

The year of 1994 was vcry qujet. The works on turbulence were again dominated by numerically 
oricntcd stuclies. with a clear prevalence of two-equation models. To illustrate this fact. we quote here 
thc article of Vasconccllos and Maliska. Over lhe years Maliska realized an important work on the 



26 J . of lhe Braz. Soe. Mechanical Sciences - Vol. 20. March 1998 

de\'elopment of procedures for lhe computalion of lluid mecbanics. The next figure gi\'es lhe reader a 
g limpse of his work. The paper perform~ a numerical study of lhe turbulent tlow in a bifurcating 
ehannel using a multidoma.in procedure. 

Nombn St,..a.mfi•H I''(~ S\I'MI\tJina r~ tur 

I 1.743 lO"' -l.079 10" 

z - 7.276 JO"' -8.797 10" 1 

• 4.1164 10"' - 1.4K7 10"1 

I :t ~.11'l0 lO"' -~.310 10" I 

s - 2.SO'l •o-• 6.!>27 lO" ' 

6 ~-•~• •o-• l.814 .o-• 
1 - 8 .607 10"' 5.09"liO" ' 
g - I 035 to" 6.676 •o-• 

T&ble l : SIN'&mli• .. . l i &- s ud fis 9 

Fig. 14 The numerlcal predictions of Vasconcelfoa and Mallska. 

The Xlll COBEM hud 33 entries on turbulent tlow. Despite the lru·ge number of possible candidates 
for our rcpresentalin: article, the choice bere was obvious. ln fact, the only work on large eddy 
simulaúon in lhe history of COBEM's and ENClT's to date wus published in this evcnr. Pinho and 
Sil\'eira Neto perfonned the simulalion of a turbulent tlow in a rectangular cavity using lhe sub-grid 
Smagorin~ky isotropic model and lhe MacCormack compressible discretization melhod. The paper 
presents pictures of the calculated tlow. showing lhe temporal evolution of the vorticity field. The main 
re~ulls are shown in Fig. 15. 

Flg. 15 The large eddy slmulatlon of Pinho and Silveira Neto. 



A.P.S. Freire et ai.: The State of the Art ln Turbulence Modelllng ... 27 

ln 1996. forty five articles on turbulent flow were published in the ENCfT proceedings. From these, 
fourteen articles dealt with rwo-equatíon differential models: a obvious majority. The pick of a 
represemative paper became then. again, very· difficulr. One of the articles that really called the 
attention of tbe present authors was tbe article. of Queiroz et aJ. on the dispersion of contaminants 
released in an environment wíth a known dispersíve capacity. Different formulations for the diffusity 
tensor were tested which were numerically solved. The results were compared with an exact analytical 
solulion and with solutions provided by classicaJ integral methods. Results for tbe local concentration 
pro files are shown in Fig. 16 . 

.... 

.... 

.... 

.... 

.... 
o.oo ott e.eo O$ o..to o.bO o..to '0-"'0 •to .,. i.OO 

x,ll5 
xt!D 

Figura 6 - Cuo n, modelagem ga\J$SÍana 
com:spondente à classe B. figura 5 - Ca.'IO n, solu.çlo numérica. 

Figma 7 - Caso U, soluçllo ~ata 

Fig. 16 The experimental results of Queiroz. 

The feeling of the present authors is that tbe "turbulence" community has still much to mature. The 
high quality of the works presented in the fust editions of COBEM were ín part due lO the foreign 
supervision of most works. As the number of papers increased and lhe authors were left ro carry out 
their own research, the qualily started lO suffer. The number of people active in rurbulence is still very 
small and lhe nuclei of most work generated in Brazil easily idenúfied. lt is true that tbe recenl 
progresses have been remarkable; however, ruucb still remains to be dooe. 

Final remarks 
The purpose of this work was left clear at its outset: 10 give a picture of the present starus of 

turbulcnce modelling in Brazil. The strategy for doing this was also clear. We started with a tour on the 



28 J of the Braz. Soe. Machanical Sctences - Vol. 20. Marcll 1998 

subjecL. aiming at giving lhe reader a view over most approaches lo turbuJencc modeiUng. Next, aftcr a 
short bistorical recollection, we presented a detailed statistics of ali past COBEM's and ENCIT's in 
what coucerns ~urhulence. The criticai evaluution of this statistics was left to a separated section. We 
must emphatically point out lhttl every opinion expressed in this work is tbe respon!>ibility solely of its 
authors. The Braz.ilian Society of MecbanicaJ Sciences holds no responsability for any judgment or 
conclusioo upheld here. 

During lhe coUection of the relevant matcriaJ. lhe aulhors tried 10 be as careful a.~ possible so as to 
avoid lhe omission o f any relateo work. The task of reviewing ali proceeding&, however, was very 
difficuh and time consuming and this muy have caused some references not to be spotted. for whi~h we 
apologize in advancc. 

The present personal view on lhe subject of turbulcnce musl nol be taken herc as conclusive. The 
aulhors are surely biased by lheir own experiences on lhe field , so lhat further view~ on lhe subject must 
bc sought by the inlerested rcader. 

The general conclusion is tbat experimental and fundamental stuoies on turbulent tlow must be 
stimulated in lhe future. Also, thc mechanical engjneering community must seek closer links with lhe 
physics conununity. 

Bíbliography gen eration . Thcrc are many ways for fom1atting bíbUographies. The present work 
has made cxtensive use of the LATEX system ano lhe companion program BIBTEX written by Oren 
Pmashní.k. BasicaJiy, lhree .bib files were prcpared: dass.bib, encit.bib and cobem.bib. These can be 
obtained directly from lhe authors. 

Acknowlcdgments. ln writing this work tbc authors have been strongly influenced by many ideas 
of Profs. Leslie Bradbury ano Roddam Narasimha; fruiúul discussions on thc subject have challenged 
the authors to always re-think evcry fundamental aspect of lhe problem. forcing tbem to carefully 
consider every s ingle word laid here; ttús has left an indelibJe mark on lhe final format of lhe prcsent 
work. 

lu the refercnces' compilation process we bene fited from a valuable help from Mr. Eduardo Nunes. 
Tbe work wns financially supported hy the Brazilian National Research Council (CNPq} through 

thc Rcsearch Grant No 350183/93-7. 

Cited Literature 

The cited Lilcrature ~onstitutes only a small portion of lhe texts on lhe subject, relevant to lhe 
present review. Duc to Jack of space, we havc dccided to include here only tbose references essential to 
a wmplete understanding of Lhe text. They fonn a very short tour on lhe world of turlmlence, being a 
small representatíon of lhe main I ines of thought on the subjecl. 

References 
Bradbury. L.J .S .. 1997. Personal Cornmunication. 
Chou. P. Y ., 1940, "On an F.x rension of Reynolds' Mcthod of Finding Apparcnt Stress and the N<tture of Turbul~ncc'. 

Chine.sc J. Physics, pp. l -5.1. 
Chou. P.Y. LYI:P . 'On lhe Theory ofTurbulence for lncornpressibJe Fluid". Sei. Sinica. V. 4. pp.369-380. 
Cob. D, 1956. 'The Law of lhe Wlike in Turbulenl Boundary Layers'. J. Auid Mechamcs. V. I . pp.l91-226. 
l'cigenbaum. M .. 1979. "Qualitative Univen;ality for a Cla.o;.~ of Non-Unear Transfonnations·. J . Statist. Phys .. V. 21. 

pp.669-706. 
llanjalic, K .. I '194. 'Advam:ed Turbulenee Closure Models: a View of Current Status and Future Prospects", lnt. J. 

Heat and Fluid Fk1w. V. IS. Nu. 3. pp.l78-203. 
Kolmugorov, A.N .. 1941. "Thc Lo~:al Structure of Turbulence in lm:umpressible Viscous Fluid for Very Lnrge 

Reynolds Numbcr", DokL Aknd. Nauk. SSSR, V. 30. pp.9-n. 
Kolmogorov. A.N .. 1942. "The .Equauon of Tu