Mechanical Engineering Department, Stanford University, STANFORD, CA 94305-3030

This project, supported by AFOSR, Army Research Office, NASA and ONR, was administered by the writer with Prof. Brian E. Launder, University of Manchester, England and Prof. John L. Lumley, Cornell University. Statistical data on turbulent flows, from lab. experiments and simulations, were circulated to turbulence modelers all over the world. This is the first large-scale project of its kind to use the results of simulations (numerically-exact solutions of the three-dimensional, time-dependent Navier-Stokes equations) and for this reason alone is a landmark in the testing of turbulence models. The modelers compared their "predictions" with the data and returned the results to Stanford, for distribution to all modelers and to additional participants ("experimenters"); over 100 participants in all. The object was to obtain a consensus on the capabilities of present-day turbulence models, and to identify the types of model which most deserved support for future development. This has not been achieved, mainly because not enough modelers could produce results for enough test cases within the duration of the project (our modest request was, roughly, 25 test cases in two years). However a clear picture of the capabilities of various modeling groups has appeared, and the interaction has also clarified the outlook of the modelers themselves. The results support the proposition that Reynolds-stress transport closures (second-moment closures) are more accurate/adaptable, but no account has been taken of their greater cost per calculation.

Note that in this World Wide Web version the mathematics has been left in TeX style: "^" denotes a superscript, "_" a subscript, and "\" the start of a typesetting command or special symbol.

After consideration of a 1992 conference at Stanford, as a linearly-extrapolated successor to the 1968 and 1980-81 meetings (Refs. 1 and 2), the first formal proposal envisaged a 4-year "mail order" effort, hopefully long enough for significant improvements to be made in the models. This was subsequently cut, at the funding agencies' request, to a nominal 18 months, with the object of finding "where we are at in turbulence modeling" (without allowing time for improvements) and then extended to just over 2 years. Although a great deal of useful information has been obtained, it has become sadly clear that very few turbulence modeling groups are both able and willing to compute test cases, covering a wide range of turbulent flows, within a reasonable period of time. Unsurprisingly, some time was taken in improvement of models and codes, but this was only part of the reason for the delay.

In brief, the quality of the results obtained seems to be much more closely correlated with the competence of the modeler/modeling group, the personnel available to do the actual ruunning of test cases, and the adequacy of the computer program, than with the intrinsic quality of the turbulence model. It is certainly not possible to say that any one class of turbulence model has conclusively proved its superiority over the others, even when cost of computation is ignored. It does, however, appear that Reynolds-stress-transport methods are a distinct improvement over eddy-viscosity methods in complex flows, though both are wounded by the unsatisfactory state of the dissipation-transport equation. The treatment of the viscous wall region, which is not closely linked to the model type, also influences the results.

Perhaps the most telling result was the large range of predictions of flat-plate skin friction -- even for different implementations of a single model (the popular "2-equation"(k, epsilon) model based on partial differential equations for turbulent energy and dissipation rate): indeed this range was as large as the differences between independent models. Even after considerable pressure from the organizers, low-speed flat- plate skin-friction predictions still fill a 7 percent band (ignoring outliers with serious discrepancies): taking the wetted area of a civil transport aircraft as five times the wing area, this corresponds to 35 drag "counts", or 35 passengers in a large aircraft.) Numerical inaccuracy was only partly to blame: a remarkable cause of discrepancy was the disagreement over the supposedly-universal "law of the wall", discussed in Appendix 1.

The prediction of compressible flow was a primary interest of the funding agencies: most flat-plate results closely followed the Van Driest correlation of experimental data for the ratio of compressible to incompressible skin friction, which is still believed to be the most reliable -- it allows for the effects of mean density variations but ignores compressibility effects (density/pressure fluctuations) as such. (Note that presenting results as the ratio of compressible to incompressible skin friction suppresses the scatter in incompressible c_f discussed above.) The only well-documented flow that shows large Mach-number effects is the mixing layer: "predictions" of the decrease in spreading rate with increasing Mach number either used an ad hoc compressibility correction or gave poor results.

The decision to run this project via interaction by mail, rather than as a conference like the 1968 and 1980-81 Stanford meetings, was taken so that participants would have time to consider their early results, compare them with those of others and make minor improvements in their prediction methods. Although this happened, very usefully (in some cases, program bugs of embarrassingly long standing were uncovered) it seems that only a "drop dead" conference deadline can concentrate the minds of the turbulence modeling community enough to produce results on demand. Each section of the community -- universities, government establishments, consulting companies -- has its own difficulties over manpower, facilities and finance.

Detailed technical results will be discussed in the
following sections: administratively, the main conclusion
is that in
spite of the rather large rate at which papers on turbulence
modeling are being published, some with quite detailed
comparisons with experimental data, few groups can rise to
the challenge of producing comparisons with *
independently-chosen * test data within a reasonable
time frame. A
number
of modelers quoted lack of resources or budget constraints
as a reason for lack of response, but we have not heard that
any sponsors have refused explicit requests for
diversion of funds from the turbulence modeling efforts they
support. The organizers have -- entirely unofficially
--
repeatedly pointed out to modelers that the funding
agencies, and not necessarily only those supporting the
present project, will use the outcome as a guide to how much
support to give turbulence modeling in the next few years,
and to which groups that support should be directed.
Quantity and quality of response has by no means been
proportional to the size of the group. U.S. government
laboratories have contributed practically no results for
the final group of test cases, while two one-man
consulting
companies were among the most competent and helpful
collaborators. Undoubtedly a number of modelers have
dropped out simply because they were not able to predict the
test cases to an accuracy which they wished to demonstrate
in
public, but we have no way of distinguishing these
from
modelers who dropped out through lack of facilities or
simply through lack of motivation. Most
collaborators were from the United States: on a percentage
basis, the enthusiasm and competence of response was no
better, and no worse, from the United States than from other
countries. It was one of the overseas modelers who
pointed out that the research climate has become very much
less favorable since the time of the 1980-81 Stanford
meeting, for which a large number of groups produced
results
for a larger number of test cases than those employed in the
present collaborative effort. Although no spectacular
advances have been made in turbulence modeling in the last
decade, it remains a lively subject, with at least 100
high-quality papers being published each year, apparently
as the
results of basic research rather than deadline-driven
development work. It is difficult to see that the modeling
community's poor response can be attributed simply to lack
of funding.

Details of which modelers attempted which test cases are given in Appendix 4.

(The history of the project has been recorded in the five project Newsletters and their attachments, already distributed to the funding agencies. The following is an outline: the sixth Newsletter is an Appendix to this report, and vice versa.)

After discussions at the "Whither Turbulence" meeting at Cornell University in March 1989 (Ref. 3) a proposal for international collaboration on testing of turbulence models was submitted to U.S. Air Force Office of Scientific Research, acting as coordinator for U.S. Army Research Office, NASA and Office of Naval Research. Invitations to participate were sent out in late August 1989, to all originators of turbulence models known to the organizers, to a number of consulting companies and other organizations likely to have well-developed versions of models originated by others, and to all experimenters identified as likely to be able to contribute data or comments.

In order to calibrate both the models and the modelers (specifically, the time of response of the latter), simple "entry test cases" were distributed in February 1990. The requirement was to predict the skin friction in a turbulent boundary layer in zero pressure gradient at a Reynolds number, based on momentum thickness, of 10,000, in as many as possible of the following cases: (i) low-speed flow, (ii) a Mach number of 5 on an adiabatic wall, (iii) low-speed flow with an absolute wall temperature 6 times the free stream temperature, corresponding approximately to the temperature ratio across the M=5 adiabatic boundary layer. Stanton number (heat transfer) predictions were requested for cases 1 and 3. This set of test cases also had the organizational purpose of identifying modelers who could produce results for compressible flow.

The low-speed high-temperature test case was chosen so
that the
relative importance of density changes, and of
compressibility (Mach number) effects as such, in the
various models could be clarified. Many flat-plate skin
friction * formulas * (as distinct from detailed
prediction methods), use explicit Mach number factors and
would not necessarily do well for a low-speed hot wall.
In fact most models performed as well for the low-speed hot
wall as for the M=5 adiabatic wall, indicating that the
models allowed adequately for density changes: indeed, true
compressibility effects are probably small in boundary
layers up to M=5. It was of course very satisfactory that
this test case turned out to be a non-issue.

The "entry" cases proved to be an invaluable calibration. Few modelers managed to keep to the relatively tight deadline imposed for return of results, even for these almost trivial test cases: moreover, as the results began to come in it became obvious that the range of predictions was rather wide. In many cases predictions were, quite simply, outside the possible bounds of experimental error for these simple cases. (In general the organizers have attempted to keep to the error standards appropriate to the aerospace industry: a discrepancy of 0.0001 in skin-friction coefficient -- about 3 percent --is big enough to worry about.) A great deal of time and effort was spent on interactions with individual modelers and requests for further information. Probably, a few modelers had calibrated their methods against pipe or duct flow rather than boundary layers, but the main explanation of the really large errors seems to be that many models intended for complex or compressible flows had simply not been adequately checked in simple low-speed flows. Another cause of error was inconsistency in choice of logarithmic- law constants or their equivalent (see Appendix 1). The final results can be described only as a computational catharsis: many modelers submitted revised results after cleaning up empirics, numerical resolution, and downright programming errors. It should be remarked here that the modelers whose results were questioned by the organizers were uniformly grateful.

We felt it essential to clean up most of the questions over the "entry case" flat-plate computations before proceeding to the next set of test cases, so that it was not until August 1990 that the first real test cases were sent out. They were intended to cover a wide range of flows, keeping as far as possible to thin shear layers and/or simple geometries. We took it for granted that the turbulence models which were most advanced, or most up-to-date, would probably be imbedded in simplified codes, capable of handling only a limited range of geometries -- perhaps to the point of being restricted to thin shear layers. For this reason, we have concentrated throughout the project on test cases which are geometrically simple but physically general.

The "August 1990" data included some
recent experiments and simulations, but also test
cases from the Stanford 1980-81 conference on complex
turbulent flows and from the AGARDograph compilation of
compressible-flow data by Fernholz
and Finley (Ref. 4). In the case of free shear layers
(plane jets,
round jets, and mixing layers) detailed experimental results
were not given, and modelers were simply asked to compare
predicted growth rates with the consensus of
experimental data. The only flows requiring a full
Navier-Stokes program were the backward-facing steps of
Driver and Seegmiller (Ref. 5). The boundary-layer
simulation of Spalart and the duct simulation of Moin, Kim
and Moser were included, and modelers were asked to compare
the * highest-order quantities* they modeled (e.g.
dissipation or triple products).

Because of the ongoing discrepancies in the incompressible-flow results, the compressible flows in the August 1990 package were deliberately restricted to one real test case, a boundary layer in strong adverse pressure gradient, plus a second set of "entry" test cases, namely the prediction of flat-plate skin friction for Mach numbers of 2, 3, 5 and 8 and temperatures down to 0.2 of the adiabatic-wall temperature. The corresponding "data" were simply the predictions of the Van Driest II skin-friction formula, which experts in the field regard as being still an acceptable data correlation.

One group of modelers who were disadvantaged by our concentration on thin shear layer data was those who use (Reynolds-averaged) Navier-Stokes codes, which do not easily accept the boundary-layer simplification of specified velocity at the boundary layer edge. Their polite reproaches were entirely justified: a boundary-layer calculation is a solution to only half the problem.

The speed of response to these test cases was extremely disappointing, with very few results being returned by the specified deadline. A number of modelers stated that they would be able to produce results, although not within the deadline. Since the object of the collaboration was to avoid the "drop dead" deadline of a conference, we, the organizers, decided to wait until a representative body of results had been returned. Obviously this totally disrupted our plans for handling the data, which envisaged an intensive effort beginning at the deadline and accomodating only a few latecomers. In fact, despite promises, only a very few sets of results were returned after Spring 1991. In August 1991 the assembled results were distributed to all the known collaborators, with a request for comments.

To
keep down the amount of material to be
redistributed, we specified that modelers should return
plots only of key quantities (e.g. in the
backward-facing-step flow, simply the surface shear
stress and the maximum
shear stress at each streamwise position), and although
many
of those modelers who did respond did not complete the full
set of "priority" test cases, the stack of
graphs
distributed was about 1.5" thick. Some useful comments
have
been received, but it is clear that not too many of the
experimenters or modelers who had undertaken to join the
project were able to devote serious effort to assessing
other people's
results. Since this was the main reason for calling the
project a * Collaboration *, the poor response by the
experimenters, coming on top of delays in the computations,
was unfortunate.

The results for the thin shear layers were mixed, with no obvious best model. The Wilcox k, omega and multiscale models gave good and closely similar results: the multiscale model has some of the features of an Algebraic Stress Model -- a type which was otherwise used only for a few test cases -- and this suggests that the improvement shown by ASM-type models over a good two-equation eddy-viscosity model in 2-D thin shear layers may not be significant. The spread of results from the different versions of the k, epsilon model was comparable with, but not closely correlated with, the spread for the "entry" cases. The results for free shear layers showed the usual round-jet / plane-jet "anomaly": few models can predict both flows without some form of special correction factor. Launder's group has recently shown (Ref. 6) that Navier-Stokes calculations for jets produce significantly different results from parabolic ("boundary layer") calculations, partly because of the effect of longitudinal stress gradients, but also partly because of the large effect of longitudinal diffusion of dissipation rate: the modeled transport equation for dissipation is so highly empirical that there may be no physical explanation, but the discrepancy provides a further opportunity for confusion in testing turbulence models.

The number of different models was too small to build up a pattern in the comparisons of "highest-order" quantities with the simulation data. Because of the low Reynolds number of the simulations this was mainly a check on the wall-layer treatments, and the "low-Reynolds-number" versions of the stress-transport models produced tolerably close agreement with the simulations. The simulations show higher dissipation rate in the viscous wall region than do the experiments, and the models seem to have been tuned for the latter. Both experiments and simulations can suffer from errors due to inadequate spatial resolution, most severe near the wall, but on balance the simulations are likely to be more accurate.

Predictions of the backward-facing step flow were surprisingly scattered (even discounting the differences among the k, epsilon models), and most models considerably overestimated the maximum negative skin friction in the recirculating flow, whether they used wall functions or low-Reynolds-number treatments. The simplest explanation is excessive diffusion of momentum into the recirculating flow. Unfortunately no stress-transport models were integrated for the full length of the test flow (32 step heights -- expensive in a Navier-Stokes calculation) so that their potential advantage in representing "history" effects has not been demonstrated in this flow.

Predictions of compressible flat plate skin friction up to
M=8 and of heat transfer on cold walls at M=5 were
mixed. Results for the adiabatic cases were good, with a few
exceptions (thought to be models developed for transonic
flow and not previously tested at hypersonic speeds).
As in the case of low-speed flow, the results depended
strongly on the treatment of the wall region: many models
reproduce the mixing-length formula with a constant
turbulent Prandtl number, and this is of course the basis of
the Van Driest transformation. Recent work, independent of
the present project (Ref. 7), has shown that the k,
epsilon model does * not * reproduce the Van Driest
transformation because of the presence of density gradients
in the diffusion terms: however, many users of k, epsilon
models obtained results in fair agreement with the Van
Driest "predictions".

Results for cold walls showed a wide spread. Confusion
occured when several modelers did not realise that Stanton
number has to be based on the adiabatic wall temperature
actually predicted by the model, *not * that given by a
selected value of recovery factor (if the latter is used,
*St* goes to infinity as the predicted adiabatic wall
temperature is approached).

Only a few modelers reported results for the compressible boundary layer in strong pressure gradient near M=3, and these results were generally satisfactory (this particular flow happens to have almost constant skin friction coefficient and therefore looks uneventful, but it is a reasonably severe medium-Mach-number test case). The user of one of the only "integral" methods presented pointed out, in connection with this case, that a simple model which has been carefully calibrated may out-perform more advanced models on its home ground. This may be the First Law of Turbulence Modeling.

A few modelers argued that their only concern was with compressible flows and they therefore did not wish to bother with incompressible test cases. This seems a shortsighted attitude: obviously if a model is found to be inaccurate in incompressible flow it cannot be relied on in compressible flow. (Modelers with codes that do not run exactly at M=0 were encouraged to run at, say, M=0.4 and M=0.3 and extrapolate to M=0: most compressibility effects vary as M^2 at low M so there is no difficulty of principle here.)

A second set of "real life" test cases was sent out at the same time as the results for the August 1990 set. These test cases were chosen to explore various complex-flow effects, such as reverse transition, streamline curvature, 3-dimensionality and unsteadiness. Again, most of the test cases were thin shear layers; the 3-dimensional flows in fact had only two independent variables; and the time-dependent flow was homogeneous in the horizontal plane (and therefore computable by trivial adaptation of a 2-dimensional space-marching program).

Two cases were specifically intended to test treatments of the viscous wall region. The first, a simulation of sink-flow boundary layers, gives a simple performance index: does the model predict reverse transition at the same value of pressure-gradient parameter as the simulation? The second, a sinusoidally-oscillating time-dependent flow, in principle causes grief to a "wall function" which uses the friction velocity (proportional to the square root of surface shear stress).

Unfortunately, no modeler did the parametric check we requested for the sink flow in sufficient detail to bracket the critical pressure-gradient parameter. To our surprise, two modelers successfully predicted the oscillating flow with wall functions: presumably their finite time steps did not land them too close to the phase angle at which the surface shear stress changes sign. Amazingly, only one modeler reported results for the curved boundary layers, although the curved jet flows generated the best response.

Despite our attempts to maintain geometrical simplicity and avoid excluding modelers without curvilinear Navier-Stokes codes, the response to the second set of test cases has been extremely disappointing. Many modelers have stated an inability to devote more effort to the project: undoubtedly some prompt responders have become impatient of the delays caused by the slow responders. A final deadline of 31 May 1992 was imposed, but a few later results were accepted for good reason -- and were still arriving in December 1992! The outcome is that the only complex flow for which a worthwhile number of predictions has been received is the backward-facing step flow, which is the complex flow most likely to be used for testing a model during development and is therefore not an entirely independent test case.

The Collaboration has not clearly revealed a "best model": in particular, stress-transport models have not demonstrated a large superiority over eddy-viscosity-transport ("two-equation") models although other reviews have come to the conclusion that they do perform consistently better. The main reason for this lack of clear conclusions is a lack of response from the modelers, particularly the developers of stress-transport models.

To minimize numerical difficulties and the time taken to prepare input data, and to avoid excluding models which were implemented only in simple codes, the organizers tried to ensure that the test cases had simple geometries; as far as possible, the cases were thin shear layers accessible to parabolic "marching" programs. Nevertheless, the performance of any given prediction method, and the fraction of the test cases for which results were reported, seemed to depend much less on the model than on the state of development of the code and the care devoted to checks of grid independence and other numerical issues.

The Collaboration raised a number of general questions about turbulence modeling as well as queries about numerical or other shortcomings of particular methods. It has resulted in correction of errors in several models as well as acting as a clearing-house for facts and opinions.

On balance, the Collaboration has been a success, although it would have been much more fruitful if more modelers had been able to devote effort to producing results for more of the 25 test cases, and if more of the "experimenters" had made a serious effort to return comments on the results.

The reader may have detected a tone of irritation in this
report. It should therefore be recorded that modelers'
replies to the organizers' stream of requests and reminders
were always courteous: on the one occasion when a tardy
modeler wrote claiming that "the results are in the mail"
-- they * were*! The project has indeed done a lot to
bring the modeling community together, to point out
discrepancies in generally-satisfactory models and, perhaps,
to establish the principle that a basic requirement of a
model of turbulence (or anything else) is good performance
in test cases chosen by someone other than the modeler.

The hierarchy of available material is:--

Draft journal paper

This Final Report

Newsletters (including graphs)

Data Library (disks and documentation for used and unused data).

The 1980-81 Stanford meeting on Computation of Complex Turbulent Flows was hampered by lack of time for consideration of the results presented at the meeting. The present "mail order" effort was hampered by the failure of modelers to keep to the deadlines. On the assumption that there is an ongoing need for public comparisons of turbulence models, a possible compromise for the mid-1990s would be to have two meetings, with an interval of six months or a year: at the first, initial comparisons with test cases would be presented and briefly discussed (as at the 1980-81 meeting); modelers would then reconsider their results and present updated versions for final discussion at the (longer) second meeting. The volume of simulation data is already quite large and an increasing range of complex flows is being covered, so that simulations (direct or large-eddy) would probably provide the largest part of the data sets in a mid-nineties project.

1. S.K. Kline, M.V. Morkovin, G. Sovran and D.G. Cockrell (Eds., vol. 1); D. Coles and E.A. Hirst (Eds., vol 2). Computation of Turbulent Boundary Layers -- 1968 AFOSR-IFP- Stanford Conference. Mech. Engg Dept. Stanford University, 1969.

2. S. J. Kline, B.J. Cantwell and G.M. Lilley (Eds.). 1980- 81 AFOSR-HTTM-Stanford Conference on Complex Turbulent Flows. Mech. Engg Dept., Stanford University, 1981.

3. J.L. Lumley (Ed.). Whither Turbulence? Turbulence at the Crossroads. Springer, 1990.

4. H.H. Fernholz and P.J. Finley. A further compilation of compressible boundary layer data with a survey of turbulence data. AGARD-AG-263, 1981.

5. D.M. Driver and H.L. Seegmiller. Features of a reattaching turbulent shear layer in divergent channel flow. AIAA J. 23, 163, 1985.

6. A. El Baz, T.J. Craft, N.Z. Ince and B.E. Launder. On the adequacy of the thin-shear-flow equations for computing turbulent jets in stagnant surroundings. UMIST, Manchester, TFD/92/1, 1992. `

7. P.G. Huang, P. Bradshaw and T.J. Coakley. Assessment of closure coefficients for compressible flow turbulence models. NASA TM 103882, 1992.

Flat plate boundary layers, that is, those in zero longitudinal pressure gradient, are one of the most basic test cases for turbulent flows (the others being the 2-dimensional duct or "channel", which is an idealization for which reliable experimental data are scarce, and the circular pipe, for which computations are complicated by the axisymmetric geometry). Because the law of the wall extends from the surface to roughly 15% of the boundary layer thickness, where the velocity has typically risen to 70 percent of the free-stream value, model predictions of flat plate boundary layers are dominated by the assumptions made about the law of the wall. Virtually all turbulence models are compatible with law-of-the-wall scaling, and can thus reproduce a logarithmic velocity profile. This applies whether the region between the surface and the start of the logarithmic law at about y^+=30 is predicted by integrating a "low Reynolds number" version of the model down to the wall, or imposed as a "wall function" boundary condition at y^+=30 (say). A data analysis done by Prof. Donald Coles of the California Institute of Technology for the 1968 Stanford meeting (Ref. 1) recommended K=0.41 and C=5.0 in the standard form of the logarithmic law, and we are not aware of any later review which has specifically challenged Coles' conclusions. A rather wide range of values for K and C is quoted in textbooks, but this scatter is attributable mainly to the difficulty of finding, separately, the slope and intercept of a line which is defined over only a relatively short range (of log y^+): most of the published values of K and C lead to very nearly the same value of U^+ at, say, y^+=100, which is somewhere near the middle of the logarithmic range in a typical laboratory boundary layer.

When the scattered predictions of flat plate skin
friction
from the modelers started to come in, we requested them to
supply details of their treatment of the universal law of
the wall, including their predictions (or assumptions) for
U^+
at y^+=100, hereafter referred to as I_{100}. Figure 1
shows the
values of skin friction (modelers' names not identified)
plotted against the reported value of I_{100}. Now
if a
turbulence model as used in the outer part of the boundary
layer is left fixed, but the value of I_{100} is
changed by
changing the wall treatment, then (U_e^+)-(I_{100} will)
remain very
nearly fixed, so that U_e^, equal to the square root of
c_f/2, changes.
The line on
Figure 1 was produced by Rodi and Scheuerer at
our
request, using their standard k, epsilon model with
different values of C. If the
outer-layer
predictions of all the models shown in Figure 1 were
identical, the results would lie along the curved line, or
at least on at line parallel to it. In fact, there seems to
be very little correlation between the values of c_f and
the
values of I_{100}, in spite of the fact that the version
of the
plot shown in Figure 1 is the latest, after considerable
exhortation of the modelers to clean up their assumptions or
predictions of the logarithmic law. Specific requests to
modelers to justify using logarithmic law constants other
than those recommended by Coles have not produced a very
satisfactory response. If Figure 1 is divided into four
regions by the Rodi-Scheuerer line and a vertical line
I_{100}=16.24,
predictions in the first and third quadrants would be made
* worse* by imposing the Coles values for the log-law
constants,
while predictions in the second and fourth quadrants would
be made * better*.

It is, of course, impossible to compare outer-layer models
logically in the face of these differing assumptions/results
for the law of the wall. Many modelers used the popular
k, epsilon
turbulence model. When the collaboration began we were
warned that there would be no point in having a large number
of modelers all using the same model, but in fact
comparisons
of their results have proved instructive, if not heartening.
Several k, epsilon users have modified the
empirical
constants in the model, but those whose results in Figure 1
are marked with a tail used the "standard" coefficients
with
-- obviously -- different wall treatments. The failure of
the
tailed symbols to lie on a line * parallel * to the
curved line
in Figure 1 can only be contributed to numerical error in
the outer layer
(grid dependence or programming mistakes). (Note that grid
resolution near the wall is not an issue here, since the
law-of-the-wall results are being taken for granted.)

Figure 1, therefore, presents a gloomy picture. The common correlation formulas for flat-plate skin friction agree to within about 2 percent at a momentum thickness Reynolds number of 10,000, and it is clear that many models have simply not been optimized for the flat plate boundary layer. Popular values for the law-of-the-wall constants give values of I_{100} that agree to within 3 or 4 percent, at the outside, but many models use or give results well outside this range. Several models with nominally identical assumptions in the outer layer show discrepancies of several percent in skin friction, even in this numerically-simple flow.

The object is always to predict the Reynolds (turbulent)
stresses. Exact partial-differential
"transport" equations for the stresses can be derived from
the Navier-Stokes equations, but their right-hand sides
include unknown higher-order statistical
quantities. The transport equation for a given Reynolds
stress contains a source term which is more-or-less
proportional to the
mean rate of strain in the plane of that stress,
which implies that the ratio of the Reynolds stress to the
mean rate of strain varies * less * than the Reynolds
stress itself and may therefore be easier to correlate
empirically: this ratio is of course the "eddy viscosity",
which may be different in different planes.
The terms in the Reynolds-stress transport equations all
have dimensions [velocity^3/length], and almost all models,
of
whatever order, assume that the turbulence can be described
by one velocity scale and one length scale. This is a
sweeping assumption, even when one is
concerned only with the larger, Reynolds-stress-producing,
eddies.

Except for one "integral" method, all the methods used partial differential equations for the mean velocity. (Integral methods can in principle be derived by applying the Galerkin technique to PDE models: this is not often done in practice, but it shows that integral methods are not a class by themselves and are not restricted to crude turbulence models.)

Naturally, modelers were asked to use the same model for all test cases, and to repeat the entry test cases if they made any changes to their models. Many of the major modelers have done exactly this and we have no reason to suppose that the results have been significantly confused by unreported changes. Several multi-person groups, and even some individual workers, used entirely different models at different times during the collaboration. The test cases got progressively harder so there was a tendency for simple models to be replaced, or for their users to drop out entirely. The results for simple models (algebraic eddy viscosity or mixing length, and the one integral method) reinforce the conclusion of the 1980-81 meeting that such models, when carefully tuned, are useful in a restricted range of flows.

The "zonal modeling" technique, in which coefficients are altered from flow to flow by logic in the computer program, was explicitly permitted, on condition of full disclosure, but seems not to have been used. We did specify that the same coefficients should be used in the compressible mixing layer and the compressible boundary layer, mainly because a correction to the coefficients which reproduces the observed decrease in spreading rate of a mixing layer also tends to produce an undesired reduction of skin friction in a boundary layer. (Compressibility corrections developed rapidly during the course of the Collaboration and we have not tried to reach a consensus: the present situation is that all the corrections include adjustable constants multiplying quantities of order M^2, and it is difficult to judge the plausibility of the physics.)

Models can be divided into those which assume a direct relation between the Reynolds stresses and the mean velocity field ("eddy-viscosity methods") and those which solve explicit equations for the stresses ("stress-equation methods"). There is a deep hierarchy of eddy-viscosity methods, but they share the feature that if the mean velocity gradients change suddenly, so also do the Reynolds stresses. The exact transport equations for the Reynolds stresses show that a sudden change in mean-velocity gradient merely produces a sudden change in the rate of growth of Reynolds stresses. That is, in reality the mean-velocity gradient occurs as a source term in a differential equation for the stress, rather than as a factor in an eddy viscosity formula.

Since an eddy viscosity is always the ratio of a turbulence quantity (Reynolds stress) to a mean-flow quantity (mean rate of strain), it is obviously determined by a combination of mean-flow scales and turbulence scales and will be well-behaved only when the two sets of scales are proportional (a definition of "local equilibrium" flow). It is clear in practice that relating the eddy viscosity to the turbulence scales gives better results when the mean-flow scales are strongly perturbed, by pressure gradients or otherwise.

The advantage of eddy-viscosity models is that they will usually give smoothly-varying predictions of Reynolds stresses -- obviously, like a laminar flow with a smoothly-varying viscosity. Their disadvantage is identical: they will not reproduce the dependence of Reynolds stresses on mean-flow history and the slow response of Reynolds stresses to sudden changes in mean flow.

Eddy-viscosity models are often classified by the number of partial differential equations used to describe the turbulence:--

- "Zero-equation" or "algebraic" models relate the eddy
viscosity to the velocity and
length scales of the mean flow (typically free-stream
velocity
and boundary layer thickness); those used in the present
Collaboration included mixing length, eddy
viscosity, and one integral method. "Zero-equation" methods
can have some success if they
conform to the law of the wall (or the
skin-friction laws derived from it) and the corresponding
density
correlation for compressible flow (amounting to an
assumption of constant total temperature in an
adiabatic-wall boundary layer). They can be expected to
perform well
in boundary layers in mild pressure gradients -- and,
paradoxically, in short regions of very strong pressure
gradient where the skin friction is determined by the
response of the inner layer and the total pressure in the
outer layer changes little (so that prediction of those
changes by the turbulence model is not critical).
The outer-layer model, for example the assumption that
mixing length is proportional to shear layer thickness,
necessarily relies on the shear-layer thickness being
well-defined. Although eddy-viscosity models such as the
Baldwin-Lomax model have been used in quite complicated
flows, no
predictions other than for boundary
layers have been submitted to the present Collaboration.

- "Two-equation" or "eddy-viscosity-transport" methods
relate the eddy viscosity to the
velocity and length scale of the turbulence. Both make the
gross assumption that only

-\overline{u_i u_j}= c_{\mu}k^{1/2}L(\partial U_i/ \partial x_j + \partial U_j /\partial x_i)

which is nominally a * definition * of c_{\mu} as a
dimensionless tensor with indices i and j. In reality
c_{\mu} is assumed to be a scalar and in all the methods
used in the Collaboration it appears to have been taken as a
constant (except for "low Reynolds number" modifications
in the viscous wall region).
The most popular of the
"two-equation" eddy-viscosity models is the k, epsilon
model (m=0, n=1 in the above classification. Other
models in the (m, n) family which have actually been
implemented
are k, kL (m=5/2, n=-1), k, omega (m=-1, n=1)
and k, tau (m=1, n=-1). It is straightforward to
convert from one (m, n) pair to another but the diffusion
term in the first model converts to a diffusion term
* plus a source/sink term * in the second, because the
diffusivity is a function of the dependent variables. Since
the models are usually formulated without a source/sink term
of this sort, the implication is that there are real
differences between the different (m, n) combinations,
with the further implication that there must be a best (and
worst) choice: m and n do not have to be integers, though
the physics may become obscure if they are not. This
point was not addressed during the Collaboration but
deserves future consideration.

- "One-equation" models either use a partial differential
equation for a velocity scale and relate the length scale to
the shear-layer thickness, or use a single PDE for eddy
viscosity. In the latter case the necessary length scale
comes, in effect, from
the ratio of the velocity scale to a typical mean velocity
gradient. One-equation models (with a PDE for turbulent
energy k) were sometimes used in the wall region where a
two-equation model was used in the outer layer. This is
primarily a numerical simplification, but comparison of
results in the Collaboration suggested that performance of
two-equation methods in boundary layers in adverse pressure
gradient can be improved by using the one-equation model as
far out as possible! (The limit is set by the need to match
the models, and corresponds, in principle, to the onset of
significant
transport terms in the length-scale equation.) The recent
one-equation models of Baldwin & Barth and of Spalart &
Allmaras were not represented in the Collaboration.

These are (usually) term-by-term models of the exact Reynolds-stress transport equations. The object is to avoid using an eddy viscosity for the Reynolds stresses, but gradient-diffusion assumptions are commonly used for the turbulent transport ("diffusion") terms. The Reynolds stresses themselves can yield velocity scales -- k, being a scalar, is the natural choice for most purposes -- but a length (or time) scale is also needed: most main-stream models use the dissipation, obtained from essentially the same equation as in the k, epsilon model. The key parts of the model are the dissipation-transport equation, and the pressure-strain "redistribution" terms in the stress-transport equations themselves. Nearly all the current models are recognizable descendants of the Launder-Reece-Rodi model of 1975, in turn a generalization of the Hanjali\'c-Launder thin-shear-layer model of 1972. The main improvements in stress-equation models since the 1980-81 meeting are the enforcement of "realizability" (no physically-impossible negative values, or correlation coefficients outside the range -1 to +1) and of correct behavior in the two-component limit (e.g. at a solid surface, where v goes to zero faster than u and w). Most of the work has gone into improved modeling of the pressure-strain term, with comparatively little attention to the dissipation equation. There is considerable current interest in the transport equation for omega, a.k.a. epsilon / k, as an alternative to the transport equation for epsilon as such, both in two-equation models and in transport- equation models: this is partly a result of the good performance of Wilcox's models in the current collaboration.

"Algebraic stress models" (ASM), of which one example was used for a few test cases, are stress-transport models with severe simplifications of the mean and turbulent transport terms, resulting in an eddy viscosity model with different values of eddy viscosity for the different Reynolds stresses. A specific advantage over standard two-equation models is that the ASM will at least qualitatively predict stress-induced secondary flow in non-circular ducts. Wilcox's multiscale model falls in the transport-equation family but has some features of the Algebraic Stress Model.

Early in the Collaboration, a paper by
J.H. Ferziger was circulated, offering simple
advice for testing numerical resolution (see Ferziger and Peric,
*Int. J. Num. Methods in Fluids* 23, 1263, 1996). Although some
of the collaborators themselves urged us to impose
any explicit numerical checks, we decided against it,
believing that the more
serious errors would be spotted by the collaborators
themselves when comparing their results with those of
others. In particular, we hoped that the flat-plate test
cases would be sufficient to identify serious failures of
grid independence close to a solid surface, probably the
most critical area in most turbulent flows. Apart from this
the only test case which grid independence is likely to have
been a serious issue is the flow over backward-facing step,
where the singularity in geometry at the top face of the
step requires step lengths in the x- and y-directions
which
are considerably smaller than one wall unit. Again, some
modelers submitted revised results after private querying of
their initial computations. The curved jets (test cases 5.4
and 5.5) are also likely to cause difficulty in
numerics, because of the large angle between the velocity
vector and the axis, so that rectangular meshes could lead
to large false diffusion.

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Last updated 18 September 2007