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\centerline{{\bf Collaborative Testing of Turbulence
Models}}
\vskip0.15in
\centerline{{\bf Update of AFOSR 90-0154,}}
\centerline{{\bf 1 February 1990 -- 30 April 1992}}
\vskip0.3in
\centerline{{\bf P. Bradshaw, Principal Investigator}}
\vskip0.2in
\centerline{{\bf Mechanical Engineering Department}}
\centerline{{\bf Stanford University Stanford, California
94305-3030}}
\vskip0.5in
{\bf SUMMARY}
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.
{\bf 1. INTRODUCTION}
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
{\sl 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 {\sl
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.
{\bf 2. HISTORY}
(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.
{\bf 2.1 ``Entry'' test cases (flat-plate boundary layers)}
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 {\sl 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.
{\bf 2.2 August 1990 test data (Cases 3.1-3.5, 4.1-4.3)}
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 {\sl 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.
{\bf 2.3 Results for August 1990 test cases}
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 {\sl 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 {\sl 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, {\sl not} that given by a
selected value of recovery factor (if the latter is used,
$St$ goes to $\pm \infty$ 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.)
{\bf 2.4 August 1991 test data (cases 5.1-5.9)}
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 $\sqrt {\mathstrut}
(\tau_w/\rho)$.
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 (undoubtedly using $\sqrt{\mathstrut}
{|\tau_w/\rho|}$): presumably their finite time steps did
not land them too close to the phase angle at which $\tau_w$
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.
{\bf 3. CONCLUSIONS}
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 {\sl 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.
{\bf Future plans}
This Final Report discharges contractual obligations, but
will be considerably augmented later as an unpublished but
publicly-available report: in addition, a journal paper will
be prepared as an ``executive summary'' and an advertisement
for the report. The hierarchy of available material will
then be:--
\hskip2.0truein Journal paper
\hskip2.0truein Final Report (augmented)
\hskip2.0truein Newsletters (including graphs)
\hskip2.0truein Data disks and documentation (used and unused data).
Stanford University will need to make a handling charge for
this material, but it is hoped that one or more of the data
banks now being planned will also archive the disks.
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.
{\bf REFERENCES}
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.
\vfill\eject
APPENDIX 1
THE ``FLAT PLATE'' BOUNDARY LAYER TEST CASES
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 $y \approx 0.15 \delta$, 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 ($y^+ \approx 30$) 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 $U_e^+ \equiv \sqrt{c_f/2}$ 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
{\sl worse} by imposing the Coles values for the log-law
constants,
while predictions in the second and fourth quadrants would
be made {\sl 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 {\sl 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.
\vskip0.2in
APPENDIX 2
THE MODELS
A2.1 The Process of Turbulence Modeling
The object is always to predict the Reynolds (turbulent)
stresses: the Reynolds stress in the $x_i x_j$ plane is $-
\rho \overline{u_i u_j}$, where the overbar denotes an
average (usually a time average). 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 {\sl 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.
A2.2 Review Of The Models
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.
A2.2.1 Eddy-viscosity models
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:--
$\bullet$ ``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.
$\bullet$ ``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 {\sl one} scalar velocity scale and {\sl
one} scalar length scale suffice to fix the eddy viscosity.
Current two-equation'' models all use the turbulent kinetic energy $k$ as one variable. They
can be classified by the variable $k^m \epsilon^n$ used in the second
equation to provide a length scale $k^{3/2} / \epsilon \equiv L$ and thus an eddy viscosity proportional to $k^{1/2}L$. The six Reynolds stresses are then given by
$$-\overline{u_i u_j}= c_{\mu}k^{1/2}L(\pl U_i/ \pl x_j +
\pl U_j /\pl x_i)$$
which is nominally a {\sl 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 {\sl
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.
$\bullet$ ``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.
A2.2.2 Stress-equation models
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 $\pm1$) 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.
A2.3 NUMERICAL ERRORS
Early in the Collaboration, a paper by
J.H. Ferziger was circulated, offering simple
advice for
testing numerical resolution. 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.
\vskip 0.2in
\vfill\eject
\magnification=\magstep1
\normalbaselineskip=21pt
\def\pl{\partial}
\parskip=8pt
\parindent=0em
\vskip2.5in
\centerline{{\bf FINAL REPORT ON AFOSR 90-0154,}}
\vskip0.3in
\centerline{{\bf ``Collaborative Testing of Turbulence
Models''}}
\vskip0.3in
\centerline{{\bf 1 February 1990 -- 30 April 1992}}
\vskip1.0in
\centerline{{\bf P. Bradshaw, Principal Investigator}}
\vskip0.2in
\centerline{{\bf Mechanical Engineering Department}}
\centerline{{\bf Stanford University Stanford, California
94305-3030}}
\vskip0.5in
December 1992\vfill\eject
\centerline{{\bf CONTENTS}}
\vskip0.5in
{\bf 1. INTRODUCTION} \hfill 1
{\bf 2. HISTORY} \hfill 3
{\bf 2.1 ``Entry'' test cases (flat-plate boundary layers)}
{\bf 2.2 August 1990 test data (Cases 3.1-3.5, 4.1-4.3)}
{\bf 2.3 Results for August 1990 test cases}
{\bf 2.4 August 1991 test data (cases 5.1-5.9)}
{\bf 3. CONCLUSIONS} \hfill 8
{\bf REFERENCES} \hfill 9
{\bf APPENDIX 1 THE ``FLAT PLATE'' BOUNDARY LAYER TEST CASES} \hfill 11
{\bf APPENDIX 2 THE MODELS} \hfill 12
A2.1 The Process of Turbulence Modeling
A2.2 Review Of The Models
A2.2.1 Eddy-viscosity models
A2.2.2 Stress-equation models
A2.3 Numerical Errors
{\bf APPENDIX 3 TEST CASE SUMMARY AND TABULATED RESULTS} \hfill 17
{\bf APPENDIX 4 MODELERS' RESPONSES} \hfill 28
PROJECT Newsletter no. 6 follows page 40