The Understanding and Prediction of Turbulent Flow (P. BRADSHAW, BA) Bueno

8/13/2019 The Understanding and Prediction of Turbulent Flow (P. BRADSHAW, BA) Bueno

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he understanding and predictionofturbulent flowP BRADSHAW BA

Department of Aerona utics Imp erial Collegeof Science and Technology ion do n

I. INTRODUCTIONI t was easy for me to choose a subject for this lecture.First, turbulence is an important part of engineering fluiddynamics which has not so far been treated in the Reynolds-Prandtl lectures; secondly, it is the most obvious linkbetween the careers of our two heroes; and thirdly it is theonly scientific subject on which I am even remotely quali-fied to lecture before such a distinguished audience. Thefirst and second propositions are sufficiently demonstratedby the annual consumption of something of the order oflWOkg of kerosene to overcome the effects of Reynolds'stresses in Prandtl's boundary layers. Have any two humanbeings ever had such a spectacular memorial? Yes,Lanchester and Prandtl, to whose induced drag we sacri-lice another 10l0kgof fuel each year.)My title mentioned understanding and prediction, inthat order. Perhaps the most difficult part of any scienceis t~ assess how much understanding one must acquire tomake adequate predictions for engineering purposes. Inturbulence studies we are fortunate in having a completeset of equations, the Navier Stokes equations, whoseability to describe the motion of air at temperatures andpressures near atmospheric is not seriously in doubt (it1s easy to show that the smallest significant eddies aremany times larger than a molecular mean free path). Weare unfortunate because numerical solution of the fulltime-dependent equations for turbulent flow is not practic-able with present computers. Useful research results can beobtained from computer solution of intelligently simpli-lied versions of the time-dependent equations, but evento calculate the simple case of parallel duct flow, with arather coarse mesh, takes several hours on the fastest com-puters(ll. At present, engineering calculations are neces-sarily based on the time-averaged equations. At first sightthis is no shortcoming because engineers want time-averaged values: however, averaging eliminates some ofthe information contained in the Navier Stokes equationsand, by substituting apparent mean (Reynolds) stresses forthe actual transfer of momentum by the velocity fluctua-tions, increases the number ~f unknowns above the num-ber of equations.The Sixth Reyn olds-Pratidtl Lecture, given in Munich on 14thA p r i l 1972.

The problem hced by an engineer, then, is to supplythe information missing from the time-averaged equations(the Reynolds equations ) by formulating a model todescribe some or all of the six independent Reynoldstresses- 11,11,.* By weighting the Navier Stokes equationsbefore averaging we can obtain differential equations whichexpress conservation of each Reynolds stress, in the sameway that the Navier Stokes equations express conserva-tion of each component of momentum. We call thesethe Reynolds-stress transport equations. They are thesimplest exact equations for turbulence q ~n t i t i e s . heirterms represent the basic processes by which Reyaolds

stresses are created, transported and destroyed: and theyprovide a convenient, though not unique, framework for adiscussion of turbulent flow. Unfortunately, just 3s theReynolds equations for the mean velocity componentscontain the unknown Reynolds stresses, the Reynolds-stresstransport equations themselves contain further unknowns.Tfiese unknowns include pressure-fluctuation terms andterms involving eddy length scales, as well as higher-orderproducts of fluctuating velocities. An infinite number ofweighted equations would be needed to restore all the in-formation lost by time averaging: the alternative is to usea finite number of equations and supply the missing in-formation from experiments.Having established the framework of discussion. let USexamine the tasks facing the developer of a calculationmethod, the experimenter, and the user of calculationmkthods.1 I The devel perThe developer of a calculation method must decide howmany of the transport equations and other weighted equa-tions he can profitably use in his model of the Reynoldsstresses. The extra unknowns must be represented bydimensionally correct and physically plausible combina-tions of the mean velocity, Reynolds stresses and any otherquantities for which transport equations are available.These combinations will involve empirical functions or*When discussing generalities i t is helpful to use the suffix no-tation: i or j can be 1, 2 or 3 denoting components in thex y z directions respectively. If a suffix is repeated in a singleterm (e.g. u a u , / ~ l . ~ ,r a u , l a x , the term is summed over allalues of the suffix u t uiul=u: :+ u:.

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constants, whose values must be derived in some way fromexpzime~t , his problem of reducing the number ofunknowns to equal the number of equations is known asthe closure problem. It is closely related to the closureproblem inhomogeneous turbulence, and those who knowthe sad story of attempts to solve the latter problem maywince at the thought of a similar quest for the Philoso-pher's Stone in the much more difficult field of shear-flowturbulence. However, our ambition is limited to the pre-diction of Reynolds stresses rather than the completespectrum tensor, and we are willing to make full use ofexperimental data. On the other hand we require turbu-lence models which are potentially accurate to within a fewper cent; this disqualifies many interesting pieces of workwhich give only order-of-magnitude results.When calculations had to be done on a desk calcdatoror slide rule the most realistic models that could be used-and then only for painfully few cases-were the eddyviscosity and mixing length models, which relateReynolds stresses directly to the local mean velocity gradi-ent without necessarily considering the transport equa-tions at all. Computers have changed all that. However,we must remember that our descriptions of the physicalworld, including all our equations of motion, rest entirelyon observation. All a computer can do is to rearrange theinformation we give it, and the information does notacquire extra merit or accuracy by being subjected toa few million arithmetic operations. The value of thecomputer is that it can rearrange more complicated de-scriptions of the physical world than can be handled byanalytic methods or desk calculators. In particular itallows us to couple almost any turbulence model we canconceive with a suitable numerical procedure to make acomplete calculation method. Fig. shows the wholedevelopment process. Careful attention to the input (at thetop) reduces the number of iterations required. I shallnot say much about numerical procedures in this paperbecause, for every one person who knows enough aboutturbulence to produce a plausible set of differential equa-tions to describe it. there are tens or hundreds who know(or can learn) enough about numerical analysis to solvethose equations. This does not imply that numericalanalysis is a lower-grade subject but merely that we knowmore about it. In particular there are now several well-developed numerical procedures for the sort of equationsand the sort of boundary conditions found in turbulenceproblems (excepting, of course, the more complex flowsfor which even laminar solutions are difficult to obtain).

Turbulence umer catprocrdurr

Calculationmethod

f ~ g u r e Flow chart for development of calculation methods.

Although the computer has reduced the computationalconstraints on our choice of a turbulence model, we stillface the physical constraints of our lack of measurementsof turbulence quantities-the statistical properties of thefluctuation field. The question concerning how many equa-tions to use now becomes How many equations can wehandle before the benefits of increasing refinement areoutweighed by decreasing understanding of the turbulencequantities in the equations? . Donaldson@' has asked-and answered-much the same question. Clearly we shouldseek, not a perfect and all-embracing model to use for alltime, but the best model to use for the next few yearsuntil experimental work has improved our understandinglet us call this the optimum model. The views of differentresearch groups on the optimum model are nearer unan-imity today than in the past: most of the models beingdeveloped or coming into engineering use at the presenttime are based on the transport equations for the Reynoldsstresses themselves, with an associated equation for eddylength scale which may be a transport equation or, inrestricted cases, an algebraic formula. Closure of the Rey-nolds-stress transport equations requires third-order meanproducts of velocity fluctuations, rr v for instance, to beexpressed in terms of second-order mean products (theReynolds stresses themselves) and is known s second-order closure or Reynolds stress closure. First-orderclosure or mean-velocity-field closure expresses the Rey-nolds stresses as functions of the mean velocity field as inthe mixing length or eddy viscosity formulae: I shall callthese algebraic formulae to distinguish them from higher-order closures involving differential equations for turbu-lence quantities. All but the very simplest second-orderclosures require empirical information about turbulencequantities which have not been measured to the accuracyneeded in calculation methods-if they have ever beenmeasured at all. This is not an impossible situation: onesimply has to rely more on physical intuition and trial-and-error adjustment of constants thad on turbulencemeasurements (Fig. 1 . However unless more effort isdevoted to experimental work the trial-and-error part ofthe process will predominate, leading to the unhealthy situ-ation of too many computers chasing too few facts (Fig.2 .1 2 The experimenterFor some time, experimenters have used the Reynolds-stress transport equations, especially the turbulent energyequation, as a framework for basic research on turbulenceprocesses. Now that developers of calculation methodsare starting to do the same, we may hope for significantinteraction between theory and experiment, which is aprerequisite of true science. The experimenter's task cannow be defined, helpfully instead of just hopefully as inthe past, as the provision of qualitative inspiration andquantitative data for the developers of calculation methodsthey in turn can provide the experimenter with a point ofview. There seems little danger of an experimenter trust-ing a particular calculation method so completely as toperform experiments that are useful for that calculationmethod and no other, so that this choice of a point ofview can do nothing but good. The only danger is thatan interest in complicated turbulence quantities may deterexperimenters from making the accurate measurements ofmean flows and Reynolds stresses that are needed to testcalculation methods.Techniques for turbulence measurement have im-proved steadily with advances in electronics (Fig. 2 but thebasic sensor is still the hot wire anemometer. The laserdoppler anemometer will probably supplement rather than

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Figure 2. oo many computers chasing too few facts.supplant it. Perhaps the biggest advance in recent yearsis in the ability to extract complicated statistical proper-ties and conditional probabilities ("conditional sampling")by either analogue or digital processing. This ability isdoing for the experimenter what the ability to handle trans-port equations has done for the developer of calculationmethods-given him more degrees of freedom to use forgood or ill.Since one of the biggest difficulties in closing the Rey-nolds-stress transport equations is in dealing with thepressure-fluctuation terms, the recent appearance of plaus-ible measurements of pressure fluctuations within theatmospheric boundary layefl3),using a disc-shaped probe,is particularly timely. Probe shape appears critical and theconstruction of probes small enough for laboratory usemay be extremely difficult. Basic data on pressure fluctua-tions might be obtainable most economically by "computerexperimentsy'-numerical solutions of the time-dependentNavier Stokes equations.1 3 he userThe newer calculation methods, using second-order closure,are more realistic than the old, but, even when success-fully developed, require more knowledge of turbulence t oassess. A calculation method should be assessed by theuser, who will have to bear the consequences of inadequateprediction especially in aeronautics these consequencescan be very serious, and originators of calculation methodsfor turbulent flow sometimes make over-enthusiastic claimson the basis of insufficient comparisons with experiment(see Thompson's review('' for some examples). Assessmentby the user implies that individual users, or the communityof users in industry and research establishments, must keepin touch with the advances in physical understanding onwhich prediction methods are based, and I hope this reviewmay help. The prospective user of the newer calculationmethods should approach them with his eyes open: thismay seem an irritating demand to make of a busy engin-eer, but it is at least a more realistic demand than askinghim to keep his eyes shut, which is what he has sometimesbeen expected to do in the past Prospective users of thenewer methods may have misgivings about the amount ofcomputer programming or computer running time neededcertainly the days when an industrial user could pro-gram the method himself are over, and he must accepta program package from the originator, but the run-ning times involved are short by modern standards. About20 seconds on a machine like the IBM 360-65 or a minute

or so on an SM 7094, suffice for a typical boundary layercalculation by the most advanced "field" m ds -usingtwo or three transport equations for turbulence.quantitiesand various short cuts can be taken. It is not wise todistinguish-or choose-calculation methods on th6'basisof the numerical procedure employed, even though muchof the work in developing a calculation method may benumerical analysis and computer programming: a numeri-cal procedure without a turbulence model stands in thesame relation to a complete calculation method as an oxdoes to a bull.1 4 Complex turbulent flows 5)Experiments, calculation methods and reviews often con-centrate on the velocity field of the two-dimensional in-compressible boundary layer, ignoring the problems ofheat or pollutant transfer, three-dimensionality, compres-sibility and flows other than boundary layers. This can bepartly justified because1. the accuracy required in heat or pollutant transferproblems is either fairly low (so that Reynolds' analogyis acceptable in many cases) or impossibly high in

determining creep limits);2 turbulence is always three-dimensional, so that moderatethree-dimensionality of the mean flow should not aff~actthe turbulence structure and rational extensions of tur-bulence models for two-dimensional flow should giveunimpaired accuracy (but see next paragraph);3. compressibility should not directly affect the turbu-lence if the Mach number fluctuation is small (Mor-kovin's hypothesis("), although mean density gradients,or extra strain rates resulting from longitudinal pres-sure gradients, may affect the turbulence;4. viscosity and its dependence on temperature shouldnot directly affect the turbulence except in the uni-versal sublayer of a turbulent wall flow or in certainlow-Reynolds-number situations.However the extension of calculation methods to flowsother than boundary layers is a more serious matter. Evenother types of thin shear layer, such as jets or wakes,make greater demands on accuracy than boundary layers.More importantly, the presence of rates of strain otherthan a simple shear aU/ (e.g. strong deflections oraccelerations) may significantly affect turbulence structure,in a way that is not fully represented by including termsinvolving the extra rates of strain in the exact transportequation. To quote a case from my own experience, thepresence of a small value of dVld x (i.e. longitudinalcurvature) changes the Reynolds shear stress by a frac-tion which is an order of magnitude larger than(dl/ /dx)/(dU/dy) . I think it needs emphasising quitestrongly that thin shear layers alone are not likely to be asufficient test of a model intended for complex flows. Thisis not just an academic point-many users are concernedwith complex flows whether they like it or not.The present second-order-closure candidates for thetitle of "optimum model" include methods that aim atcomplete generality-applicability to any turbulent flow.There is no reason>o suppose that any general model in-dependent of, and simpler than, the Navier Stokes equa-tions can be found. Therefore engineering calculationmethods must be based on, or compatible with, empiricalsimplifications of the Navier Stokes equations. Thesesimplifications are virtually certain to have a smaller rangeof . validity than the Navier Stokes equations as well asless accuracy in a given case, so we must not expect toomuch of our optimum model.

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1.5. Scope of this papersad feature of most previous combined reviews of tur-bulence physics and calculation methods, a feature whichis no fault of the authors, is the almost complete lack ofconnection between the two parts of the review. This istrue even of the masterly survey by R~tta(~)--indeedhisexposure of the lack of physical relevance of the calcula-tion methods current in 1962 has inspired much later work.I hope that recent advances in calculation methods willhelp me to present a more unified view in this paper.Section 2 is a discussion of the physics of turbulence, re-lated to the behaviour of the Reynolds-stress transportequations: Ref 8 may be a helpful introduction for non-specialists, though not too successful as a unified viewSection 3 is a discussion of turbulent shear layers, especi-ally wall layers, ending with a list of the questions thatmust be answered in the development of any turbulencemodel. At the cost of some damage to my unified image Ihave given, in Section 4, an introductfon to the older tur-bulence models in current use, arranged as a historicalreview starting with the "eddy viscosity" model of Bous-sinesq and ending with the turbulence models presented atthe Stanford meeting in 1968. In my view none of themodels described in Section 4 is even potentially suitable

for flows other than thin shear layers. Section 5 returns tothe main theme with a discussion of the most recent modelsbased on the Reynolds-stress transport equations: thissection can be regarded as an addendum to the broader-rangmg specialists' review by W C. Reynolds@', and asan introduction to methods potentially suitable for com-plex flows.2. THE MECHANISM OF TURBULENCE2.1. EddiesThe only short but satisfactory answer to the question"what is turbulence?" is that it is the general solution ofthe Navier Stokes equations. The motion is rotationaland the dependent variables U , W and are functionsof all the independent variables x y, z and r. The non-linearity of the equations causes interaction betweenfluctuations of different wavelengths and directions and, asa result, the wavelengths of the motion usually extendall the way from a maximum set by the width of the flowto a minimum set by viscous dissipation of energy. Fromthe physical viewpoint the mechanism that spreads themotion over a large range of wavelengths is vortex stretch-ing(lO). nergy enters the turbulence if the vortex elementsare mainly orientated in the right sense to be stretched bythe mean velocity gradients. Naturally the part of themotion that can best interact with the mean flow is thatwhose wavelengths are not too small compared to themean-flow width, and this larger-scale motion carries mostof the energy and Reynolds stresses in the turbulence.One of the most striking things about a turbulent shearflow is the way that large bodies of fluid migrate acrossthe flow, carrying smaller-scale disturbances with them.The arrival of these large eddies near the interface betweenthe turbulent region and non-turbulent fluid distoris thisinterface into a highly re-entrant shape (Fig. 3) andappears to control the rate of spread of the turbulence(another way of saying that the large eddies carry most ofthe Reynolds stress). Not only do the large eddies migrateacross the flow, but their lifetime is so long that during itthey may travel downstream for a distance many timesthe width of the flow (a rough calculation suggests 306in the case of a boundary layer of thickness 6 . Thereforethe Reynolds stress at a given position depends signific-antly on upstream history and is not uniquely specified bytbe od mean velocity gradient(@ as in a laminar flow.

This is why we need transport equations to describe turbu-lence. Fig. 3 is a sufficient answer to those who hope for asimple description of this complicated phenomenon.Vortex stretching and large-eddy migration betweenthem dominate the Reynolds-stress transport equations:vortex stretching also dominates equations for turbulentlength scales, since it is the mechanism of energy exchangebetween eddies of different sizes. Unfortunately it is notpossible to write the equations solely in terms of vorticityand an effort is needed to remember the connection between1. The term in a transport equation.2. The process represented (eg. energy dissipation).3. The mechanism (e.g. transfer of energy by vortexstretching to viscosity-dependent eddies).2.2. Behaviour of the Revnolds stressesTo avoid proliferation of iensors we will consider only thetransport equation for the Reynolds shear stress in thex, y) plane, pz.ven if the flow does not strictly obeyPrandtl's boundary-layer approximation, shear-stressgradients are usually more important than normal-stressgradients. In a three-dimensional flow the shear stress hasn additional component in the (y, z ) plane, p w . Forv in steady incompressible flow we have, with the usefulDIDt notation,Transport by mean flow

1 ) Generation by interaction with mean flow

2) Redistribution by pressure fluctuations

(3a) Transport by velocity fluctuations

( 3 b ) and pressure fluctuations

4) Transport and destruction by viscous forces

In any two-dimensional flow all the z-wise gradients oftime-average quantities would be zero and in a two-dimen-sional thin shear layer only the terms marked with adouble underline would remain. These simplifications donot eliminate any group completely since each repre-sents an essential physical process: usually the thin-shear-layer terms are larger than the others even if the shearlayer is not strictly "thin". The transport equation for anyother individual Reynolds stress looks similar to 2.1),with appropriate changes in symbols. "Transport" termse.g. 131 can be distinguished from "source/sink" terms

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Figv:r 3 Illuminated cross-section of smoke-filled boundarylayer. Flow right to left.(e.g il]i because they can be written as spatial gradientsof mean or turbulence quantities ("divergences" in thecontext of control-volume analysis-see Fig. 4). The onlytransport equation whose terms have been measured ex-tensivelv':: is the turbulent energy equation for half thesum of the Reynolds normal stresses, f (?+ -T+iF) . Theredhtribution term analogous to [2] sums to zero in theturbulent energy equation, and the other terms are givenspecial names: the left hand side is called "advection"and the terms analogous to [I]. [3] and [4] are called"producrion". "diffusion" and "dissipation" respectively.

Since the prediction of Reynolds stresses is explicitlyor implicirly the solution of the Reynolds-stress transportequation


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Figure 5. oundary layer nomenclature.

caUy on the shear stress transmitted through the layer tothe surface, 7 . If the velocity scale of the turbulenceis independent of y and the length scale dependent only ony we expect turbulent transport of Reynolds stress withinthe layer to be small, so that the exact Reynolds-stressequations like (2.1) reduce to generation = destruction ,all transport terms being zero. This is called a state oflocal equilibrium(ls).Given that the turbulence is specifiedby a length scale y and a velocity scale J(r,lp)=u,,either substitution of the appropriate dimensional formsinto the Reynolds-stress equations or direct dimensionalanalysis based on the argument that the local equilibriummust also include the mean velocity gradient (which comesto the same thing) leads to

where is an absolute constant found experimentally tobe about 0.41. The point of repeating this familiar analysisis to show how it depends on quite subtle and special argu-ments about the turbulence which are emphatically nottrue in general. The arguments break down very near asolid surface where viscous effects are important and theturbulence scales are no longer simply ur and y. Furtherdimensional analysis leads to the law of the wall

of which (3.1) is a special case :fl-- 1 at u,y l v 30, or y 16 0-01 in a boundary layerwith U,O/v=lO'). This does not mean that the flow inthe viscous sub-layer is in local equilibrium, and in factthere is a significant transport of Reynolds stress towardsthe surface. Integration of (3.1) and use of (3.2) gives thewell-known logarithmic profile for u,y v 30

U 1= U7Ylog Cu7 vwhere C is about 5.

If the total shear stress is a function of y, dimensronalreasoning suggests that (3.1) be replaced by

but measurements show that f, =l is a good approxima-tion. Closures of (2.1) usually reduce to (3.4), with f = .if mean and turbulent transport terms are neglected andthe appropriate inner-layer scales inserted. This is ofcourse a warning that (3.4) is not to be trusted if transportterms are likely to be important, e.g. if dr/dx or dr/l)yare large. Equation (3.4) can be integrated in many casesof interest( ). It is usual to rearrange the integral so thatits right hand side is the same as that of (3.3). The con-stant of integration C, representing the effect of theviscous sub-layer, depends significantly on the shear stressdistribution in the sub-layer: for instance it becomes greaterthan five if (v lp u2)d rl dy is more negative than about-0.003, as it is in pipe flow at bulk Reynolds numbers,based on diameter, of less than about 10000.The chances of understanding the behaviour of theviscous sub-layer well enough to predict C in all casesseem small but, since C is usually a function of one ortwo parameters only, correlations of experimental data arenot too difficult to establish. An entirely adequate reasonfor doing experiments on eddy structure in the sub-layer(12,15,'6,17)s that the sub-layer eddies are likely to havea particularly simple shape, close to the most efficientshape for extracting energy from the mean shear and somaintaining themselves against strong viscous damping.Several experimenters have clearly identified bursts oreruptions , in which organised bodies of slowly-movingfluid move away from the surface in a manner reminiscentof Grant's(lB) mixing jets and Kov as ~n ay 's (~ ~)odel ofthe behaviour of eddies near the outer edge of a boundarylayer. The experiments suggest a strong interaction betweenthe motion at different distances from the surface: anembarrassing difficulty is that flow models based on theexperiments are liable to predict that the interaction con-tinues beyond u,y/v=30, which conflicts with the ideasof local equilibrium and viscosity-independence on which(3.1) and (3.4) depend. At present, direct experimentalchecks of (3.1) have made it one of our firmer foundationsfor a treatment of turbulent wall flows and, though onewould be foolish to regard it as exact, one feels thatmodels for the sub-layer and inner layer should be com-patible to a first approximation with 3.1).3 2 The outer layer Figure 6This is the region of a boundary layer y 0.26 approxi-mately, where the stress-carrying eddies are not stronglyaffected by the boundary conditions at the surface. Thefree-stream edges of other flows are qualitatively like theouter layer. The probability distribution of the interfacebetween turbulent and irrotational (non-turbulent) fluid ina boundary layer is roughly Gaussian with a mean of0-856 and a-standard deviation of 0.156 (so that the inter-face occasionally extends as far in as 0.46 or as far outas 1.36). Most of a boundary layer, then, is stronglyatTected by the surface or the interface: the outer layeris dominated by large eddies which transport low momen-tum and high turbulent energy from the outer part of theinner layer and deposit both near the outer edge of theboundary layer. This behaviour can be deduced from.measurements of turbulent transport of turbulent energy(Fig. 6) but it is shown most spectacularly by still photo-graphs or motion pictures in a smoke-filled flow (Fig. 3).Townsends large eddy equilibrium hypothesis ') focused

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attention on the large eddies; although some of the assump-tions of Townsend's theory have been disproved, the verydisproof has enhanced the importance of the large eddies,which are now seen to carry much of the Reynolds stressin any turbulent shear layer. The behaviour of the largeeddies depends on their interaction with the smaller-scalemotion via the vortex-stretching mechanism, and it seemsprobable that the smaller-scale motion is far from thestate of spatial homogeneity assumed in theories such asthat of Kolmogorov (Ref. 20 is the latest of several experi-mental papers on this subject).Some progress has been made in deducing the typicalshape of the L~geddies frommemutefflents of the coeffi-cient of correlation between velocity fluctuations at twop ~ i n t s ' ~ ~ J ~ ) .recentlytonceived technique which seemsideal for studying the large eddies and their interactionswith the small-scale motion is "conditional ampl ling (*,'^.^').Since this technique is likely to become quite common inbasic research, a brief outline may be helpful to all threesections of the community. The standard form of a hypo-thesis, about turbulence or any other phenomenon, is thatsomething (S, say) depends on something else (S', say).If S and S' are simple, like Reynolds stress and meanvelocity gradient, an experiment to test the hypothesis issimple at least in principle. If the hypothesis is, say, thatthe large eddy structure in the outer layer depends on theaforementioned "bursts" in the viscous sub-layer thendifficulties arise because bursts at a given point occur foronly part of the time. To begin with we must choosemeasurable quantities S and S' to represent large eddystructure and sub-layer bursts respectively. Suppose weake S = v 3 at Cj = 0-8 say; v3 appears in the turbulenttransport term in the turbulent energy equation, which issupposed to depend greatly on the large eddies. We canchoose S to be a short-term average of -uv in the sub-layer. Then, we require the average value of u aty 16=0 8 conditional on the short-term average of ain the sub-layer being greater than, say, twice the conven-tional average 7 : that is, we accumulate contributionsto the average of v3 only during periods of high - uvin the sub-layer. If the average value of v3, in the outerlayer during bursts in the sub-layer is significantly differentfrom the conventional average 7 hen there is a signifi-cant connection between sub-layer bursts and transport ofturbulent energy by the large eddies. Which causes which,or whether a third phenomenon causes both, is another

question The hypothesis mentioned above has been seri-ously suggested-though for reasons mentioned at the endof the last section I do not believe it (the experiment hasnot yet been done). Conditional sampling is not necessarilyrestricted to hypotheses about terms in the exact transportequations, but this seems to be its main relevance to thedevelopment of calculation methods.When making hypotheses about eddy behaviour one isalways tempted to seek pattern and order where none exist.

Mollo-Christensen"' has compared theories of turbulencebased on measurements of gross statistical properties withthe sort of theory about road traffic that might be basedsolely upon the statistic that a given group of cars andmotor cycles has an average of 3 6 wheels per vehicle. Ifeel that a greater error is to classify turbulent eddies intotwo wheel and four wheel types when in reality they arcas complicated as a vehicle which does have 3 6 wheels.It is a great pity that of the three concepts of turbulence:I . all eddies small compared to mean flow dimensions(Boussinesq "eddy viscosity" hypothesis, 1877);2. some eddies large but weak (Townsend's hypothesis(u1.1956 ;3. some eddies large and strong, interacting with smallerand weaker eddies (current view);the most recent is both the most realistic and the mostcomplicated. But what else can one expect of a phenom-enon which is defined as the fluid motion of maximumpossible complexity?3.3. Transition to turbulence and turbulent flow at low

Reynolds n u m hThe problem of tracing the growth of given disturbancesand the eventual breakdown to turbulent flow is, of course.more difficult than the calculation of fully-turbulent flow.but similar methods can be used in principlet and someprogress has already been made (Hall"), Donaldsonm').Unfortunately it is rarely possible to specify the detailsof initial disturbances that exist in a real flow, becausethey arise from free-stream turbulence or structural vibra-tion (note that surface roughness cannot itself producetimedependent disturbances). Only in the case of turbo-machines and _certain other types of internal Bow can wehope to know even the statistical properties of the freestream turbulence. Therefore the detailed prediction oftransition, as opposed to the growth of given disturbances.is often an ill-posed problem. The main need

Figure 6. Energy balance n typical retarded boundary layer.Aeronauticel Journal July 972 Bradshaw

at present is for better empirical correlationsbetween the global parameters of the transi-tion region and the statistical properties of thefree-stream turbulence. Theoretical work mayhelp us to choose relevant statistical properties:certainly the length scale as well as the intensityis relevant since large-scale unsteadiness has adifferent effect from small-scale turbulence.Most of the above remarks also apply to theeffect of free stream turbulence on a turbulentboundary layer("25) in this case, if the tur-bulence intensity is only a few per cent of themean velocity the direct effect is confined tothe outer layer and the law of the wall is un-altered.

Another phenomenon whose direct effectseems to be confined to the outer layer and theinterface is the influence of viscosity on turbu-lent flow at low bulk Reynolds number. Coles(")showed that, on the assumption that the lawof the wall is unaltered (an assumption


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disputed by some) the velocitydefect law alters if theReynolds number based on momentum thickness is lessthan about 5000: in compressible flow the effects appearto be stronger. Once more, we need experiments ratherthan hopeful adjustments of calculation methods. Theultimate in low-Reynolds number effects is reverse transi-tion or "re-laminarisation": several local criteria for rever-sion to laminar flow have been devised but Jones andLaunder 7 have suggested that the inner layer ceases tobecome a local-equilibrium region before reverse transitionoccurs so that the phenomenon must be treated by usinga transport equation for Reynolds stress (whose terms willof course depend on Reynolds number. in a way that mabe sensitive to the details of the flow).In summary, the effect of viscosity on turbulent tiowmakes itself felt in several different plicnomena of engin-eering importance, but we must be prepared to acknow-ledge that turbulence influenced by viscosity presents amore difficult problem than ordinary turbulence.3 4 Outstanding questionsThe questions about turbulence posed by our inspectionof the Reynolds-stress transport equations (Section 2 , andthe discussion in the earlier parts of the present section,are mainly questions about the large eddies and their in-teraction with the smaller eddies and with the irrotational-turbulent interface. Let us list these questions as theyapply to a thin shear layer in two-dimensional flow, re-membering that the same questions also apply to moregeneral flows.I How is the turbulent transport, in the y direction, ofturbulent energy or Reynolds stress (term [3] of 2.1))related to the profile of turbulent energy or Reynoldsstress? So far. calculation methods have used onlysimple gradient-diffusion or bulk-convection hypo-theses. The former is inappropriate to turbulencedominated by large eddies: the latter hypothesis, intro-

duced by Townsend, is more plausible though strictlyapplicable only if the large eddies are weak. Eitherhypothesis is adequate for correlating existing experi-mental data and Lawn") has recently found that hisadmirably accurate data are better fitted by the worsehypothesis, so that the situation is fluid.2. How are the length scales of the turbulence related

to the thickness of the shear layer? Since the shear-stress-producing motion is strongly linked to the largeeddies. whose size is strongly linked to the shearlayer thickness, the length scales should be stronglylinked to the shear layer thickness (though the rela-tion will be different in different flows). If the shearlayer thickness is changing rapidly this argument willbe less reliable and we must use a transport equationfor length scale, which poses new questions.3. \I.'hat determines the difference between the differentcomponents of the Reynolds stress tensor-that is, theanisotropy of the turbulence? We know that the ex-change of intensity between the different componentsis effected by pressure fluctuations, but, as mentionedabove. a plausible measurement of one of these"pressure-strain" terms has only recently been made,and further work is needed to obtain measurementsaccurate enough to be used directly in calculationmethods.4. How long is the "memory" of the turbulence for itsupstream history? If the three preceding questionscould be answered, the answer to the present onewould follow because it is given by the balance ofthe generation, destruction and turbulent transport

terms in the exact Reynolds stress equations. Nevertbe-less it is worth asking the question directly, and tryingto answer it by tracing the downstream developmentof the large eddies. The work of Fav re ) , Kovasz-nay(lg) nd their collaborators springs to mind: modernconditional-sampling techniques can be used to followthe motion of selected parts of the turbulence.What are the effects on the larger eddies of rates ofstrain, additional to the simple shear d U / d y found intwo-dimensional thin shear layers? This is a questionthat embraces all the difficulties of flows other thantwo-dimensional thin shear layers and at present it mustbe answered separately for each flow if a t all. Eventhe case of small extra strain rates poses problems."Complex flows"(5) have not yet received adequateattention theoretically or experimentally: the newgeneration of calculation methods is the first that iseven potentially capable of dealing with flows otherthan thin shear layers. When considering threedimen-sional flows we must distinguish three very differentrate-of-strain fields. The first case is the true three-dimensional boundary layer, in the x plane say, inwhich a U / dy greatly exceeds d U / d x or d U / 82 Since,in any part of the layer, the mean flow directionchanges by only a few degrees over a distance in they direction equal to a typical eddy size, we do notexpect great changes in eddy behaviour. Three-dimen-sional boundary layers can be satisfactorily predictedby logical extensions of turbulence models for two-dimensional flow transport-equation models can re-produce the difference in direction between shear stressand velocity gradient that occurs in most flows of thistype. The second case, typified by the flow in thecorner of a duct or a wing-body junction, satisfieswhat may be called the slender shear layer approxi-mation based on the assumption that d U / d x is muchsmaller than i l U / i l y or d U / d z . Nokmal-stress andshear-stress gradients in both the y and z direction ,are important and, because d U / d z is of the sameorder as i I U / a y we cannot expect simple extensionsof two-dimensional models to work. The third caseof three-dimensional flow, in which dU/dx dU1d .vand d U / i l z are all of the same order, is likely todeter all but the boldest developers of calculationmethods for some time to come.T URBUL E NCE M ODE L S A ND CAL CUL AT ION M E T H ODS

We now turn to the representation in calculation methodsof the eddy processes discussed above. This constitutessomething of a discontinuity because only the most recentmethods deserve the title of "representation of eddy pro-cesses", and 1 shall postpone consideration of these toSection 5. Section 4 describes models that are still usedin engineering practice: some at least may be used fora long time to come because they are simple and giveadequate results in simple cases, but in my opinion noneis even potentially suitable for flows other than thin shearlayers. Most of this section refers to boundary layersbecause most calculation methods have been developedonly for boundary layers, but many of the points made alsoapply to other types of thin shear layer or to complexflows.

The main distinction between turbulence models iswhether turbulence properties are related directly to themean flow or obtained from transport equations. Particu-larly in the case of thin shear layers, one must also dis-tinguish between methods based on relations for the tur-bulence properties at each point in the field and those in

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which the relations are based on integral parameters suchas the velocity profile parameter H. The latter distinc-tion is sometimes difficult to make when reading thedescription of a calculation method, because partial differ-ential equations can be converted into ordinary differentialequations for integral parameters by the generalised Galer-kin method, alias GKD method, alias method of integralrelations alias method of weighted residuals see Ref.30, p. 16): the test is whether the original assumptions,rather than the final equations to be solved, are formulatedat each point in the field. Thus the four classes are Mean-flow / Field, Mean-flow / Integral, Transport-equation /Integral and Transport equationjField methods (Fig. 7).Wall jet outer par1 of or igmalbounaarv l ver

Figure 7 Dis t r ibu t ion o f eddy v iscos i ty and mix ing leng thin a wall t beneath a boundary layer. Qua litativ e trendsfor auantitative values see Ref. 334.1. Mean flow/Field methodsOnly two of the early ideas about the behaviour of Rey-nolds stress are still used in a recognisable form: they arethe eddy-viscosity and mixing-length hypotheses, whichare closely-related hypotheses of a connection between theReynolds stress and the local mean velocity gradient. Bous-sinesq's 1877 paper introducing eddy viscosity was actuallywritten in 1872, so we may celebrate its centenary thisYear. The mixing-length hypothesis is perhaps Prandtl'sbest-known contribution to turbulence studies (Ref. 31,P. 714). Therefore this time and this lecture are appro-priate for a reasoned assessment of these two hypotheses.We can define a quantity p having the dimensions ofviscosity, by an equation like

and a length I by an equation like

Further, we can establish empirical correlations for thebehaviour of p or in a given flow.just as we can do for

pztself. The question is whether working with peor gives more accurate or more reliable answers thanworking with pz irectly.

If p and are regarded as given, equations like (4.1)and (4.2) represent gradient diffusion : (4.1), for instance,states that the rate of transport of U-component momen-tum in the direction (p%) is directly proportional to themean gradient of U-component momentum in the ydirection. Gradient diffusion is an accurate model ofmolecular transport of mass, heat or momentum in a gas,where the mean free path between molecular co1lisior.s issmall compared to the flow dimensions but large com-pared to the molecule diameter. Boussinesq and Prandtlhoped that the transport of momentum by turbulent eddiesmight be similar to that of gas molecules but on a largerspatial scale. However. eddies or vortex motions in turbu-lence are not all small compared to the flow dimensions,and the interaction between different eddies is a continu-ous process and not a series of discrete collisions. There-fore p and I are likely to depend strongly on flowconditions (perhaps even more strongly than p ~tself)instead of approximating to constant properties of the Auidlike molecular viscosity or mean free path. G. I. T a y l ~ r ' ~ ) ,describing his own early reasoning on the subject. gives acharacteristically clear and simple demonstration of theinadequacy of results based on a near-constant diffusivity.Any theory of diffusion which is based on a virtualcoefficient of diffusion must predict a mean shape for asmoke plume that is paraboloidal and it was clear to methat near the emitting source the mean outline of a smokeplume is pointed . In some cases, like the wall jet:*' shownin Fig. 7 the apparent eddy viscosity or mixing lengthdoes some very strange things.

How are we to reconcile this unphysical behaviourwith the success of eddy-viscosity and mixing$ength cor-relations in a large number of practical cases? I think thegeneral view could be put as follows: mixing lengththeory in particular, with its concept of momentum trans-fer by fairly well-organised lumps of fluid, is not too bada first approximation to the behaviour of turbulencedominated by large eddies as shown in Fig. 3, and theresult of using it-should therefore be not too bad . How-ever the main reason for the success of mixing length andeddy viscosity is that most of the basic fluid flows used astest cases for Reynolds stress models are nearly in a stateof self-preservation or of local equilibrium, and in thesecases eddy viscosity and mixing length are simply-behavedpurely for dimensional reasons.

self-preserving flow (sometimes rather misleadinglycalled an equilibrium flow) is one in which profiles ofmean velocity (measured from a suitable origin) and ofother quantities like Reynolds stress are geometricallysimilar at all streamwise positions. Turbulent flows usuallytake up self-preserving forms if the boundary conditionsallow. For instance, the outer layer of a boundary layeron a flat plate at high Reynolds number is very close toself-preservation with

and T/T,ET/~u?= f 916) (4.4)where 6 and u are functions of x. These expressions andthe definitions of eddy viscosity and mixing length give,at once,

p,lpur8= f (~16) (4.5)

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(note that, using (4.3), u,8 is proportional t o U,8*, theusual choice for a mean-flow scaling factor for eddy vis-cosity). As mentioned in Section 3, the inner layer of aboundary layer in any pressure gradient and at anyReynolds number is close to local equilibrium (a specialcase of self-preservation) where the only relevant lengthscale is y In this case

(or p pKury if puTz everywhere)and l=Ky (4.8)(note that these formulae correspond to f Ky 16,with f log y / constant and f constant). Now theonly physical input to this analysis is the statement thatthe flow is self-preserving, and there is nothing in whatwe know of these flows to suggest that the turbulencearranges itself so as to approximate to the Prandtl orBoussinesq models: these simple formulae for pe andare the results solely of dimensional analysis. Any dimen-sionally-correct theory with a disposable constant wouldgive the same results irrespective of its physical correct-ness.Closures of the exact Reynolds-stress transport equa-tions reduce to formulae of eddy viscosity type if the trans-port terms are negligible (as in local-equilibrium flow: seeSection 3.1) or if their ratio to the generation or destruc-tion terms is a function of y16 only (as in self-preservingflow). This strongly suggests that eddy viscosity formulaecan be no better than first approximations in non-self-preserving flows where the behaviour of the transport termsis more complicated. All the early tests of the mixingLength and eddy viscosity models were made on self-pre-serving flows, for the simple reason that no computerswere available to solve the partial differential equationsthat appear in more general flows. Only in recent yearshave these models been tested in non-self-preserving flowssuch as boundary layers on aerofoils. Even here, pro-viding that the boundary layer or other flow is not too farfrom self-preservation, fairly accurate predictions can bemade by using values of pe or 1 from self-preserving flow.The fact that f,=constant and f,=constant have bothbeen used successfully in the outer part of a boundarylayer and in wakes and jets is simply a demonstration thatour present standards of accuracy are not high: it caneasily be shown that at the free-stream edge of a flowpc+O and I oo.

One of the major obstacles to the use of more refinedprediction methods is the lingering belief that the successof the eddy viscosity and mixing length models in fairlysimple cases is a general a posteriori justification of thephysical concepts used to derive those models, or at leasta licence to use them in more complex flows. However,the mixing-length concept of identifiable lumps of tur-bulence is approached most closely near a free streamedge while the local-equilibrium justification of themixing-length formula (4.8) is valid only in the. innerlayer of a wall flow . . similar arguments to those aboveapply to the use of eddy-viscosity formulae for Reynoldsstresses other than - p Z , and to eddy-conductivity con-cepts in heat transfer.

We now come to a crucial question for the future: grantedthat (for whatever reasons) the eddy-viscosity and mixing-length formulae are useful first approximation in simplecases, can we use them as the basis of a second approxima-tion for use in more complex cases or where higher accuracyis needed? The only such second approximations suggested

to date are closely related to second-order closure asdescribed in Section 1, being transport equations for eddyviscosity rather than Reynolds stress. Prandtl in 1945 (Ref.31, p. 874) suggested a model in which the eddy viscositywas equated to p qTl where q2= ua va w . The modelwas by implication restricted to thin shear layers, I beingrelated to the shear layer width while 2 was obtainedfrom a closure of the turbulent energy equation. Severallater workers (Refs. 34-36 and Mellor and Herring in Ref.30) have used Prandtl's 1945 model: otherda7*' haveformulated transport equations for 1 in addition to onefor 7 o that the model is nominally able to deal withflows other than thin shear layers. Nee and Kov as~nay@~'have devised a wholly-empirical transport equation forp . good deal of effort has been devoted to thesemethods and numerical solutions have been obtained inquite complicated flows. In view of this I am sorry to saythat I do not think eddy-viscosity transport equations areworth pursuing, for the following three reasons.1. The simple behaviour of eddy viscosity and mixinglength in simple thin shear layers is not maintainedin more complicated cases like three-dimensional flow.

multiple shear layers (see Fig. 7) or flows with signifi-cant extra rates of strain. In cases where the rate ofstrain changes rapidly (in the x or y direction) theReynolds stress will respond slowly and not at onceas implied by eddy-viscosity formulae.2. There is no independent exact equation for eddy vis-cosity or mixing length analogous to the transportequations for Reynolds stress: therefore any indepen-dent eddy-viscosity transport equation must be com-pletely empirical.3. Any transport equation for eddy viscosity or mixinglength can be converted into a transport equation forReynolds stress by substituting for the velocity gradi-ents, obtained by differentiating, the time-average

Navier Stokes equations.The eddy-viscosity transport equations suggested todate look rather like the exact Reynolds-stress transportequations however the empirical Reynolds-stress trans-port equations deduced from them by the process justdescribed in (3) contain second derivatives of mean pres-sure and Reynolds stresses, and velocity-gradient factors,which are absent from the exact Reynolds-stress equa-tions. Assuming that the eddy viscosity remains finite.the shear stress -pz ecessarily goes to zero when tbemean rate of shear strain, dU/dy in a thin shear layer,goes to zero: there are corresponding results for the otherReynolds stresses. This characteristic feature of eddy-viscosity models implies that there is no smooth transitionfrom them to explicit Reynolds-stress closure. Formulationof explicit Reynolds-stress closures in terms of eddy vis-cosity, so that laminar-flow numerical procedures can beused, is unsuitable for the more complicated flows becausethe eddy viscosity is in general a fourth-order tensor whichmay (Fig. 7) take infinite values.

Various modifications of eddy-viscosity formulae havebeen suggested to remove their qualitative deficiencies (e.g.the addition of a term in d2U/dy2 o maintain non-zeroshear stress with zero dU/dy, and the use of an aniso-tropy factor to maintain a difference between the direc-tions of shear stress and of velocity gradient in three-dimensional flow). However these modifications seem totake us further away from the exact Reynolds-stress equa-tions, and I am not convinced that there is any gooddynamical reason for formulating assumptions in termsof eddy viscosity rather than Reynolds stress, except inBradshaw Aeronautical Journal July 187;


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the simple cases where first approximations like (4.5) or(4.6) suffice.In this discussion of eddy viscosity and mixing lengthI have had to criticise the first steps taken in our subjectby great men no longer living. We may be sure that thepioneers, were they still alive, would themselves havesuperseded their earlier theories, so I hope my commentswill not be taken amiss.4.2. Mean flowllntegral methodsBecause of the difficulty of solving partial dser en ti alequations without a computer, the methods developed inthe period 1930 to 1960 for aerofoil boundary layer cal-culations were based on the momentum integral equation.Information about the Reynolds-stress was inserted via anempirical skin friction formula and an auxiliary equationfor the rate of change with x of the velocity-profile shapeparameter H ~ 6 * / 8 r its equivalent. The usual form ofthe auxiliary equation is

The exact equation for dH/dx, derivable from the bound-ary-layer form of the momentum equation, contains aweighted integral of the shear stress profile, so we seethat (4.9) implies that the shear stress profile depends onH and U,8/v and possibly on dU,/dx. If an algebraiceddy-viscosity formula is combined with a velocity-profilefamily of the type U/U,=f3(y18, U,Blv, -andfamilies exist which fit experimental profiles quite accur-ately-then .r/ pU m2 =f4 y18, U,8/v, H which is of theform implied by (4.9). This is an example of the useof the generalised Galerkin technique.The older equations of the type (4.9) were not derivedfrom eddy-viscosity assumptions: instead, f, and f wereadjusted by trial and error so as to optimise predictions ofH. As shown by Rottao) and Thompson(~ esults weregenerally poor, especially in pressure distributions differingsignificantly from those used to find f, and f,. One of thebest wholly-empirical versions of (4.9) is that ofHead(so.u)who did not adjust f, and f, directly but insteadassumed that the rate of entrainment of fluid into a bound-ary layer is equal to U times a function of H

This still leads to an equation of the form (4.9) but is asimple and explicit assumption involving one empiricalfunction of one variable. Good results were obtained foraerofoil-type flows not too different from those used todetermine the function f,. More recently Head and PateFU'have improved the entrainment method, in effect by allow-ing f, to depend on ( B / U d dU,ldx as well as H. Thishelps the method to distinguish between flows in whichH is high because of high shear stress (leading to highentrainment) and those in which H is high because ofstrong adverse pressure gradient. A similar method inwhich an allowance for flow history is made by insertingthe pressure gradient in what is nominally a turbulencefunction is that of W a l ~ c ~ ~ )nd his collaborators, whichmakes use of a great deal of carefully-organised empiricalinformation and is probably the ultimate method of itskind. By making this kind of allowance for flow history,wholly-empirical versions of (4.9) can in principle achievebetter accuracy than methods based on algebraic eddy-viscosity formulae. Strictly speaking, however, the historyis specified .by the whole pressure distribution between theAeronautical Journal July 972 Breduhew

start of the boundary layer and the point considered, andthe local mean pressure gradient does not appear in anyexact Reynolds-stress transport equation in incompressibleflow.4.3. Transport equation/lntegrsl methodsA more formal way of representing the effect of flowhistory on the turbulence in a thin shear layer while stilldealing only with ordinary differential equations is toleave the shear-stress integral in the shape-parameterequation in the exact form, so that (4.9) is replaced by

where c, is a dimensionless form of the shear-stress inte-gral, and the only empirical content is that provided by thevelocity-profile family or its equivalent. We then writean empirical ordinary differential equation for c as

Several calculation methods with partly or whollyempirical versions of (4.12) have been produced and somegive very good results.Methods of this sort would result from the applicationof the Galerkin technique to the partial differential equa-tions for shear stress obtained as closures of the exactReynolds-stress transport equations: the latter will bediscussed in Section 5 The disadvantage of the Galerkintechnique-and indeed of integral methods as a whole-isthe computational difficulty of extending the calculationsto more complicated flows even if the turbulence assump-tions are still sound. For instance, quite large changes,particularly in the profile families , may be needed to

incorporate suction, injection or surface roughness. In afield method extra effects can be incorporated withoutchanging the basic calculation: in the example above onewould simply change the surface boundaiy condition,while to extend the method to compressible flow one wouldadd a density calculation to the existing program. Forthis reason I do not feel very enthusiastic about using theGalerkin technique on closures of the Reynolds-stresstransport equations.A general but apparently little-employed technique'@lis to use calculations by a field method, instead of experi-ments, to derive the empirical functions used in an integralmethod. The technique is most simply applied to functions of one variable, such as occur in Head's method, butt may be attractive for more elaborate integral methods,such as those based on (4.12). It has some relation to themore formal Galerkin technique.

1ype O flurbulencemodel

l y pe of

Mean flow

Figure 8 Classification of turbulence models and numericalprocedures.413

equat,on D E s i f m ~ d i D E s Cm~egral l

1ype Onurnerlcal

Galer mt e c hn~ que

procedure Solutmn o f Soluteon olP a r t m l D E s 3rdlnary OE 5


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4.4. The 1968 Stanford meeting 30)This meeting was a competition between all the availablecalculation methods for twodimensional incompressibleboundary layers, based on predictions of all the availabletest cases. Nearly all the methods in current use were pre-sented. There were eight mean-flow integral methods, 10mean-flowlfield methods, three transport equationlintegralmethods and four transport equation/field methods. Inmany cases the authors' presentations did not make theshear-stress assumption clear: W C. Reynolds' introduc-tory paper A morphology of the prediction methodsis helpful although he counts Galerkin methods as Integralmethods.The sample of calculation methods was too small forthe best type of method to stand out clearly. There weregood and bad examples of each type, and good perform-ance was as much a function of the state-of developmentas of the realism of the assumptions. The evaluation com-mittee which assessed the results emphasized that, for thesereasons, its assessments were not durable. The main com-ments made about future developments were that onlyfield methods were likely to be able to deal with complex(three-dimensional) flows and that methods allowing forturbulence history (transport equation methods) were likelyto give better results. when properly developed, thanmethods which did not allow for turbulence history.' It ismainly field methods that have been extended to takeaccount of three-dimensionality, compressibility, heat trans-fer and other real-life effects, and to flows other thanattached boundary layers, although Head's Integral en-trainment method has also been extended to these cases.Also, most of the new calculation methods developed sincethe Stanford meeting have made allowances for turbu-lence history.

The proceedings of the Stanford meeting are still anexcellent user's guide to well-established methods for pre-dicting turbulent flow but a few shortcomings have to beborne in mind. Quite apart from the restriction of themeeting to two-dimensional incompressible isothermalflow, boundary layers are a less severe test of turbulenceassumptions than, say, jets, because most of the work isdone by the law of the wall and the assumptions made tocalculate the small velocity defect in the outer layer arenot critical except for the cases of strong adverse pressuregradient or strong injection. Near separation, wind-tunnelboundary layers tend to converge laterally because ofboundary layer growth on the side walls, and the omissionof any allowance for this makes it difficult to judge thereliability of the Stanford methods near separation. Ibelieve that a boundary layer calculation method shouldbe able to predict surface shear stress (in a known sur-face pressure distribution) up to a point very close indeedto separation, because failure of the boundary layerapproximation in the outer layer does not affect the flownear the surface. The Stanford meeting did not attemptto assess numerical accuracy or computing times: somemethods showed obvious shortcomings in one or otherarea. I made some comments about comparative runningtimes for the different field methods, and for severalquasi-Galerkin versions of my own calculation method,in Ref. 44. It appears that the better numerical proceduresused in field methods all take very roughly the same com-puting time, that the Galerkin versions of field methodscan be up to half an order of magnitude faster and thata simple method like Head's, programmed without especialcare, is roughly one order of magnitude faster than afield method. Only the user can decide whether an in-tegral method or a slower-running but (perhaps) moreaccurate field method is the better choice for a ph c u l a r

case. We may expect field methods to be used in futurefor most cases unless speed is overwhelmingly important.5 TRANSPORT EQUATIONIFIELD METHODSAt the time of the stanford- meeting, calculation methodsusing partial differential (transport) equations for turbu-lence quantities were mostly in an early stage of develop-ment and their performance as a group was not impres-sive. However most of the new calculation methods (asdistinct from extensions of existing methods to more com-plicated cases) that have appeared since the Stanford meet-ing have been of this type: evidently workers have heededthe remarks of the evaluation committee mentioned above.especially those relating to complex flows. For a special-ist review of individual methods up to 1970 see the paperby W C. Re yn~l ds '~ ). ew methods continue to appearbut it is possible to distinguish the general type of methodthat will receive most attention from developers of calcu-lation methods in the next few years and most attentionfrom users of calculation methods in the few years afterthat. Perhaps I am crowing too long before sunrise butI think it is clear that transport equations for eddy vis-cosity, after flourishing just after the Stanford meeting,are being abandoned by some of their former adherentsand that the newest methods are based solely on equationsderived from the Navier Stokes equations exact trans-port equations for the Reynolds stresses, and in generalfor turbulence length scales, are approximated term byterm, in exactly the way suggested by R~ttac'~)n 1951Moreover, people are now giving Dr. Rotta credit for it,which is almost as big a breakthrough. My guesses at therange of validity of the various assumptions used in thisclosure process are given in Fig. 9.The simplest type of transport-equation closure(u) usesone Reynolds-stress transport equation. necessarily forzand an algebraic relation between length scale and shear-layer thickness. Both of these features restqct it to thin orquasi-parallel shear layers. The basic assumption is thatthe shear-stress profile sufficiently defines the turbulencefield. The next step in sophis ti~ation'~)s to use transportequations for all the non-zero Reynolds stresses (whethertheir gradients affect the mean motion or noti still usingan algebraic length scale. Here the whole set of Reynoldsstresses is used to define the turbulence field. This will bean advance on the first type of closure only if .the latteris limited by the assumption rather than the lengthscale assumption and I am not sure myself that this isso (compare rows two and three of Fig. 9). To deal withcomplex flows, in which the shear layer thickness may bechanged rapidly by mean-flow acceleration, a transportequation for length scale is needed in addition to trans-port equations for all the Reynolds stresses whose gradi-ents are significant. Several of the new methods use length-scale equations but I doubt whether any of them can yetbe relied on to reproduce all the curious effects of extrarates of strain which are the essence of complex flows.However, second-order closure has already shown greatpromise and, when based more firmly on measurements,the new methods should produce results of engineeringaccuracy for a much wider range of flows than could betreated by any of the earlier generations of calculationmethods.5.1. Length scale equationsTransport equations for length scale are less w ll under-stood than the Reynolds-stress transport equations dis-cussed in Section 2 To begin with, there is no agreementon what length scale one should choose to represent theenergy-containing eddies. More than one may be needed(46)

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ation, if not the former, to have some appeal. Fortunatelythe workers who used (5.2) with equal to the pressurehave now abandoned this reductio d absurdurn and in-cluded the pressure-transport terms with the triple-velocity-product terms. Hanjalic and Launder ) have consideredthe exact transport equation for the triple products and,using experimental data and heuristic arguments, reducedit to the form 5.2) with C = up . However their reasonsfor neglecting the explicit or implicit appearance of meanvelocity gradients in the exact equations are not com-pletely convincing.Agreement is in sight on another of the controversialsubjects, whether the empirical representation of the pres-sure-strain terms [2] in (2.1) should contain mean velocitygradients or only turbulence terms. Undoubtedly thePoisson equation for p' (another exact equation derivablefrom the Navier Stokes equations( )) contains meanvelocity gradients and Rotta (1951)( ' suggested that thepressure-strain term should do so as well: Rotta later(1%2)( reverted to assuming that only the turbulenceterms were important and suggested that the pressure-strainterm was directly proportional to the anisotropy1

, 2x +3x G s,,?Donald~on(~~'nd Herring and Mell~r(~~)sed this form.Champagne et a/(51) stated that their Reynolds-stressmeasurements in a homogeneous shear flow supported theanisotropy model, but in fact the discrepancies betweentheir measurement. and the anisotropy model werelarger than one could accept in a calculation method.Bradshaw(*) showed that his shear-stress equation, derivedfrom closure of the turbulent energy equation, impliedthe presence of the mean velocity gradient in the pressure-strain term. More recently Hanjalic and La ~n de r' '~)aveclosely followed Rotta (1951): Daly and Har10w ~)haveproduced a form containing mean velocity gradients butsay that there is evidence (unspecified: Champagne?) thatthey are negligible.The length scale equations proposed in the literaturediffer widely even when one has reduced the equations fordissipation. (energy/length) and vorticity to the explicitlength-scale form (5.1)-for instance the equation of Rodiand S~alding '~~)roves to have a =O and a j is small inrefs. 39 and 48. Some of these equations(-) are intendedfor use with eddy-viscosity formulae of the typepe=cp .\I and therefore form part of an eddy-viscositytransport equation, but they could also be used, in prin-ciple, for true Reynolds-stress closure.

As representatives of the postStanford developmentsI will select the methods of Daly and Harlowca),Donald~ono,'~)nd Hanjalic and Launder' ), which areall true Reynolds-stress closures (Table I). My own 1967method(u), using only one Reynolds-stress equation andan algebraic length scale, is in the same spirit as these butrather less ambitious. Other recent work (e.g. that ofRotta(*), Shamroth and Elrod( ), L~ ndg re n' ~~) .nd Mellorand Herringcjo))has not progressed as far. Earlier work bythe groups to which Harlow and Launder belong usededdy-viscosity transport equations, so their abandonmentof eddy-viscosity can be regarded as significant, thoughLaunder (private communication) considers that at presenthis own method is less computationally flexible than thatof Ref. 39 which uses an isotropic eddy viscosity.So far, only Daly and Harlow's method has beenapplied to a case other than a thin shear layer (they giveresults for a comparatively simple case of turbulence dis-tortion). Hanjalic and Launder have treated the flow in

TABLEComparison of essential features of the latest transport-equation methods

AuthorsReynolds-stresstransponequationsLength scale'

Mean strainrate inpressure-strain?Reynoldsnumberdependent?Turbulenttransport?

Donaldsonus va d V

algebraic

maybe

gradientdiffusion

HarlowlU

Transportequation fordissipationrateyes butassumed zero

gradientdiffusion

Launderv u 2

Transportequation fodissipationrate

gradientdiffusiont'In thin shear layer version: ratio of normal stresses to UPassumed constant.tob tai ne d by firm pruning of transport equation for tripleproducts.an asymmetric duct (in which velocity gradient and shearstress change sign at different points) and Donaldson hascalculated an axisymmetric vortex flow: these are quitesevere tests of the basic concepts, and the asymmetricduct would defeat an eddy-viscosity method. Hanjalicand Launder have done most of the analytical work neededto extend their method to general flows, but in its presentform Donaldson's method is restricted to flows where alength scale is imposed by the boundary conditions-ineffect, to thin shear layers.

Apart from the differences listed in Table the threemethods agree on most general points, though there arenumerous details over which they disagree with each other(and with my own views). Hanjalic and Launder's treat-ment of the dissipation equation seems more realistic thanthat of Daly and Harlow (see Ref. 47). All three methodsare still in progress of development (Harlow's in particu-lar has changed greatly since the publication of Ref. 37)and the empirical content of each may be further refined,by physical reasoning or the use of turbulence data. Atpresent most of the numerical input consists of constantseven where the methods would admit functions of dimen-sionless quantities. For the sake of ending this paper witha positive statement after so many negative and caution-ary ones, I would choose Hanjalic and Launder's methodas the optimum model , the most promising of the post-Stanford generation: it seems to be the be$ compromisebetween flexibility (requiring a refined turbulence model)and tractability (requiring a simple model so as not tooutstrip our empirical knowledge of turbulence).6. CONCLUSIONSIn the last decade, the digital computer has enabled thedevelopers of prediction methods to make fuller use ofour experimental understanding of turbulence in devisingrealistic models of Reynolds-stress behaviodS6). In par-ticular, it is now possible to use the exact Reynolds-stresstransport equations as a framework for calculationmethods, just as they have long been used as a frameworkfor experiments. For the first time, significant interactionis occurring between predictor and experimenter and we

Braushaw Aeronautical Journal July 197;


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may hope for rapid progress in future. The experimenterhas the new techniques of conditional sampling, laseranemometry, and measurement of pressure fluctuationswithin the stream. Solutions of the full time-dependentNavier Stokes equations, though too expensive for engin-eering use even with approximations for the small-scalemotion, can be used for numerical experiments onquantities which are difficult to measure directly. Calcu-lation methods and basic experiments are starting to treatcomplex turbulent flows (flows other than simple shearlayers) which should increase the possibility of interactionbetween research workers and engineers. Fig. 10 showsthe ideal state of turbulence studies, incorporating inter-actions between developers of calculation methods, experi-menters and users. 1 hope this lecture has given some ofthe engineers who use calculation methods an introductionto the latest developments, which should soon start toreplace the methods of the Stanford era(=) in cases wherethe highest accuracy or the greatest flexibility are needed.I hope the lecture may also be a useful introduction t o Dr.Rotta's book@:', which I did not see until after my lecturewas written. I have not had space to deal with geophysicalproblems although I acknowledge their importance andalso the significant contribution made t o our knowledgeby meteorologists and oceanographers. Nor have I dealtwith more than a small fraction of th e exciting experimen-tal work on turbulence that is being done in this countryand elsewhere, but I hope I have made clear my belief thatexperiments are the foundation of our understanding andshould be the foundation of our prediction methods.

m o r e m e a s u r n m t s

Figure 10 The ideal state of turbulence studies.

What would our heroes say to all this, Reynolds whonever saw hot-wire measurements of his turbulknt stresses,Prandtl who never saw computer solutions of his turbulencemodels? Would they be amazed at the spectacular pro-gress we have made? Perhaps they would be amused tofind that with all our hot wires and computers we havestill not achieved an engineering understanding of turbu-lence, and that it is still as important and fascinating anddifficult a phenomenon as when the first steps in studyingit were taken by Reynolds and Prandtl.ACKNOWLEDGMENTSI am grateful to Dr. T. Cebeci, Prof. H. Fernholz, Dr. B E.Launder, Mr. W Rodi and Dr. J C. Rotta for their help-ful comments on a first draft of this lecture. I am indebtedto Messrs. 3. P. O'Leary and G. R. A Sandell for Fig. 3

REFERENCES1.

2.3.4.5.6.7.8.9.

10.11.12.13.14.IS.

16.17.

18.19.

20.

21.22.23.14

25.

26.77.

28.

DEARDORFF,. W.A numerical study of three-dimensionalturbulent channel flow at large Reynolds numbers.Journal of Fluid Mechanics. Vol 41 p. 453 1970. (Thisvolume 1s devoted enhrely to papers on turbulence andis a useful guide to current basic research.)DONALDSON,. duP. Calculation of turbulent shear flowsfor atmospheric and vortex motions. A I A A J o u r a l . Vol10 P . 4 1972.EGOST, J . A. Microscale pressure fluctuations measuredwithin the lower atmospheric boundary layer. Ph.D.Thesis, University of British Columbia, 1970.THOMPSON,. G. J. A critical review of existing methodsof calculating the turbulent boundary layer. ARC R andM 3447. 1964.BRADSHAW,Variations on a theme of Prandtl. AGARDConference Proceedings No. 93 1971.MORKOVIN,. V. Effects of compressibility on turbulentflows. The Mechanics of Twbulcnce. Editor: A. Favre.Gordon and Breach, 1964.R ~ A ,. C. Turbulent boundary layers in incompressibleflow. Progress in Aeronautical Sciences. Vol 2 p. 5. 1962.BRADSHAW,. Turbulent boundary layers. The Aero-nautical Journal o f the Royal Aeronautical Society. Vol72 p. 689. 1968.REYNOLDS, . C. Computation of turbulent flows-State-of-the-art, 1970. Thennosciences Division, Stanford Uni-versity, Report MD-27 1970.BIZADSHAW,. An Introduction to Turbulence and itsMemurement. Pergamon, 197 1.TOWNSEND.. A The Structure o f Turbulent Shear Flow.Cambridge, 1956.GUPTA,A K., LAUFER. and KAPLAN, E Spatialstructure In the vlscous sub-layer. Jo~rrnalof Fluid Mech-anics Vol 50 p. 493 1971.TOWNSEND.A A Equilibrium layers and wall turbu-lence. Journal o f Fluid Mechanics Vol 1 1 p. 97 1961.BRADSHAW, Special topics in turbulent flow. LectureSeries 10, von Karrnan Institute, 1969.KIM, H. T., KLINE S. J and REYNOLDS. C. The pro-duction of turbulence near a smooth wall in a turbulentboundary layer. Journal o f Fluid Mechanics. Vol 50 p.173. 1971.. -WALLACE.. M., ECKELMANN,and BRODK R. S. Thewall region in turbulent shear flow. ax ~Xnck ~nstitutfur Stromungsforschung Ber., 1 19 1971.NARAHARIAO.K. NARASIMHA,. and BADRINARAYANAN.M. A. The bursting phenomenon in a turbulent bound-ary layer. Journal o f Fluid Mechanics Vol 48 p. 3391971.GRANT H. L. The large eddies of turbulent motion.Journal of Fluid Mechanics Vol 4 p. 149 1958.KOVASZN~Y,. S. G. IBENS. and BLACKWELDER..F. Large-scale motion in the intermittent region of aboundary layer. Journal o f Fluid Mechanics. Vol 41. p.283 1970.WYNGAARD,. C and PAO Y H Some measurements ofthe fine structure of large Reynolds number turbulence.Proceedings, Symposium on Statistical Models and Turbu-lence, Springer-Verlag. 1971.MOUO-CHRISTENSEN,. Physics of turbulent flow. AIAAJournal Vol 9 p. 1217 1971.HALL . G. A model for the study of boundary-layerinstability. RAE Tech Memo Aero 1 1 18. 1969.DONALDSON,. dup. A computer study of an analyticalmodel of boundary layer transition. AIAA Paper No 68-38 1968.ROBERTSON,. M. and HOLT. C. F. Stream turbulenceeffects on the turbulent boundary layer. Proc ASCE,J Hydraulics Divn, 1972 (to be published).CHARNAY.., COMTE-BELLOT,. and MATHIEU.. De-velopment of a turbulent boundary layer on a flat platein an external turbulent flow. AGARD Conference Pro-ceedings No 93 1971.COLES,D The turbulent boundary layer in a compressiblefluid. Rand Corp Report R-403-PR 1962 and ARC 244781963.JONES,W.. P. and LAUNDER. E. The prediction oflamlnarlzatlon wlth a two-equat~onmodel of turbulence.international Journal o f Hear and Mass T ransfe r. Vol 15D 301 1972.LAWN C. J. and ELLIOI'T, C. J. Fully-developed flowthrough concentric annuli. Central Electricity GeneratingBoard Report RD/B/N1878. 1971.


16/16

FA\.RE,A. Review on space-time correlation in turbulentfluids. Journal o f Applied Mechanics. Vol 32E, p. 241,1965. (Bibliography of Marseille work up to this date.)KLINE.S. J., MORKOVIN, V., SOVRAN and COCKRELL D. J (Editors). Proceedings, com utation of turbu-lent boundary layers-1968 ~ ~ & ~ - 1 F P S t a n f o r dconference, Vol 1.. Thennosciences Division, StanfordCniv.. 1969.RIECELS,F. W. (Editor). Ludwig Prandrl GesammelteA bhandlungen. Springer 1961.TAYLOR, . I. Some early ideas about turbulence. Journalof Flltid Mechanics Vol 41, p. 3, 1970.~ ~ A C K E R . C. and WHITELAW. H. The turbulencecharacteristics of two-dimensional wall-jet and wall-wakeflows. Journal o f Applied Mechanics Vol 38E, p. 239,1971.G r c s ~ a o . . S. Turbulent boundary layer on a flat platein an incompressible fluid. NASA ?T F-10080. (Trans-lation from Izw. Akad. Nauk, USSR, 1 ~ )SPALDING,. B. Heat transfer from separated flow. Journalof Flrtid Mechanics Vol 27, p. 97, 1967.BECKWITH.. E. and BUSHNEU.D. M. Detailed descrip-tion and results of a method for computing mean andfluctuating quantities in turbulent boundary layers. NASATN D-4815, 1968.H~RLOW,. H. and NAKAYAMA,. I. Turbulent transportequations. Physics of Fluids Vol 10, p. 2323, 1 7.RODI.W. and SPALDING, B. A two-parameter model ofturbulence and its application to free jets. Warme irndSrctPithc~rrrogirripVol 3. p. 85. 1970.SG K. H. and SPALDING, . B. Turbulence model forboundary layer near walls. Physics o f Fluids Vol 15, p.20. 1972.~ F F M A N ,. G A model for inhomogeneous turbulentf l ow Proceedings of the Royal Sociery. Series A, Vol 317,D 417. 1970.HEAD M R . Entrainment in the turbulent boundarylayer. .ARC R M 3152, 1958.HEAD.M. R. and PATEL. V. C. Improved entrainmentmethod for calculating turbulent boundary layer develop-ment. ARC R 8: M 3643, 1968.

BRADSHAW,. Use of a refined boundary-layer calculationmethod to calibrate a simple one. Imperial College AeroReport 71-23, 1971.BRADSHAW,. and FFSS. D. H. Applications of a gen-eral method of calculahng turbulent shear layers. Journalof Basic Engineering Vol 94D, 1972. (ASME Paper NO71-WA/FE-8, 1971.).R ~ A ,. C. Stahstlsche theorie nichthomogener turbu-lenz. Zeitschrifr fur Physik Vol 129, p. 547 and Vol 131,p. 51, 1951.DALY,B. J . andWOW. H. Transport equations inturbulence. Physics o f Fluids Vol 13, p. 2634, 1970.ROD],W. On the equatlon governing the rate of turbulentenergy dissipation. Imperial College Mech Engg ReportTM/ TN/A /14 , 1971.HANJALIC,K. and LAUNDER.. E. A Reynolds-stress modelof turbulence and its application to thin shear flows.Journal of Fluid Mechanics Vol52, p. 609, 1972.DONALDSON.. duP. and ROSENBAUM,. Calculation ofturbulent shear flows through closure of the Reynoldsepuatians by invariant modelling. ARAP Inc Report 127.k 8 .MELLOR. . L. and HERRING,H. J A study of turbulentboundary layer model. Part 11, Mean turbulent field clos-ure. Sandia Corp Reoort SC-CR-70-6125B, 1971.CHAMPAGNE,H.. HARRIS,V. G. and CORRSIN, . EXperiments on nearly homogeneous turbulent shear flow.Journal of Fluid Mech anics. Vol 41. p. 81. 1970.BRADSHAW,. Calculation of three-dimensional turbulentboundary layers. Journal o f Fluid Mechan ics Vol 46,p. 417, 1971.ROITA. I C. Recent attempts to develop a generally ap-plicable calculation method for turbulent shear flowlayers. AGARD Conference Proceedings No 93, 1971.SHAMROTH, J and ELROD,H G. The development ofthe Reynolds stress in a turbulent shear flow. Journal ofBasic Engineering Vol 92D, p. 836, 1970.LUNDOREN,. S. Model equation for non-homogeneousturbulence. Physics o f Fluids Vol 12, p. 485, 1969.SMITH,A. M. 0 A decade of boundary layer research.Applied Mechanics Reviews Vol 23, p. 1 1970.ROT~A. C. Turhulortc Striirnr~rlgcn.Teubner. 1972.

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