ASME ``ME`98'' Meeting
(a.k.a. ASME Winter Annual Meeting)
This is a transcription of slides for Session FE-4A ``Highlights of Turbulence
Modeling''
The latex2html utility has processed the math. into (mainly) .gif
files 
I can NOT supply hard copies of the papers referenced (except for
my own)!
There is NO ASME written paper!
Anaheim CA, Nov. 1998Highlights of Turbulence Modeling
Or, You Don't Get What You Don't Pay For
Peter Bradshaw
Mech. Engg. Dept., Stanford University
Stanford, CA 94305-3030
bradshaw@vk.stanford.edu
Bradshaw, Wilcox (FE-9) and Ferziger (FE-12) - a Package Deal
Turbulence at advanced graduate level - not for turbulence specialists
Most predictions of turbulence are made by non-specialists
i.e. design or development engineers for whom turbulence is only part of
the problem
The wrong prediction method can give very wrong predictions
This lecture is an introduction to the problem
and the possible solutions
Dr. Wilcox will discuss Reynolds-averaged (time-averaged) modeling. No
model can be trusted completely!
Prof. Ferziger will discuss large-eddy simulation - the larger eddies are
calculated exactly and only the smaller ones are modeled.
LES is much more expensive than Reynolds-averaged modeling but may be the
wave of the future
Turbulence is the most complicated kind of fluid motion
but it is also the most common, on all scales from cream in coffee to the
motion of the Galaxy
including most fluid flows found in mechanical engineering.
The first `bullet' implies that turbulence is the general solution of the
Navier-Stokes (NS) equations - so why not just solve them?
Why turbulence is difficult - a little physics
Why engineers need short cuts - a little economics
Short cuts:- large-eddy simulation, Reynolds-averaged modeling
Hierarchy of Reynolds-averaged models:- eddy viscosity with zero, one,
two
PDEs; stress-transport equations
What of the future?
Flow visualization of turbulence (smoke in air, dye in water) shows a wide
range of length scales in all three axis directions.
The largest motions (``eddies'') nearly fill the flow
while the smallest may be too small for the visualization to resolve.
Turbulence is a 3D tangle of elementary vortices which tend to stretch
each other - the ``drunkard's walk''
The elementary vortex lines and sheets get thinner until viscous diffusion
balances stretching on the average - i.e. statistically.
Mean-flow kinetic energy is transferred to turbulence because mean-flow
distortion (e.g. shear) does work against the turbulent stresses 
The N-S equations lead to a conservation equation for turbulent kinetic
energy per unit mass 
Vortex stretching ``cascades'' TKE to the smallest eddies where viscous
stresses dissipate it into thermal internal energy.
The mean flow transports TKE in space - and so does the turbulence itself
(``diffusion'')
We can produce exact definitions of order-of-magnitude scales
The smallest wavelengths are of order 
, where 
is the rate of dissipation of TKE per unit mass. Close to a solid wall 
is about 1.5 ``wall units''
The large eddies determine the dissipation rate
: the smallest eddies just do the dissipating
The largest wavelengths are of the order of the flow width (they determine
it!). 
is a representative scale - about 
in the outer part of a boundary layer.
The friction velocity
and the thickness 
are useful global scales for wall flows
is a pointwise velocity scale of the larger eddies
but 
is more useful velocity scale. It also varies across the flow
in the inner part of a boundary layer
is typically about twice the friction velocity
Many engineers are concerned with heat transfer rather than momentum transfer
but of course the velocity field must be calculated concurrently
We can construct a heat-transfer analogy of any turbulence (momentum-transfer)
model
The simplest is (eddy conductivity) = (eddy viscosity)/ 
, with a formula for turbulent Prandtl number
Most advances in modeling appear first in momentum transfer and are then
adapted to heat transfer
Incompressible models do well in shock-free attached flows (you need a
coupled heat-transfer calculation for the density)
strictly, they do as well as the data
Shock interactions (or even strong pressure gradients) defeat most models
- heat transfer predictions are bad
Turbulence structure of mixing layers changes greatly with Mach number
(not density ratio) and M-dependent model coefficients are needed
Many engineers are concerned with low-Re turbulent flows
which are greatly influenced by the initial transition from laminar to
turbulent flow
This depends on the background disturbance field, which is almost never
completely known in practice
Surprisingly, erratic transition is not too important in engineering
partly because background disturbances are often so large that transition
occurs quickly
Most engineers use empirical correlations
Reynolds-averaged turbulence models cannot be expected to predict instability
to small disturbances
``Bypass'' or ``breakthrough'' transition with large background turbulence
is a more hopeful case
but the structure of a transitional flow is not much like a fully-turbulent
flow
Wilcox is more optimistic than I am - I would stick with empirical correlation
formulas for now
If the viscosity is small (high Reynolds number) the smallest motions are
very small (though still much bigger than the molecular mean free
path, so the NS equations are valid)
Bad news - if a finite-difference solution of the NS equations, in x,
y, z, and t, is to resolve the smallest eddies
grid size must be orders of magnitude less than flow width.
Things get worse as the Reynolds number increases - ratio of flow width
to smallest-eddy scale (
) is proportional to 
Why turbulence is difficult - a little physics
Why engineers need short cuts - a little economics
Short cuts:- large-eddy simulation, Reynolds-averaged modeling
Hierarchy of Reynolds-averaged models:- eddy viscosity with zero, one,
two
PDEs; stress-transport equations
What of the future?
Solution of the NS equations in x, y, z, and t
for the whole range of eddy sizes is called Direct Numerical Simulation
(DNS).
Computing time and cost restrict DNS to low Reynolds numbers (cost 
Re 
approx., where 3 = 3/4 
4)
The affordable Re rises (slowly) as computing gets cheaper
today the limit corresponds to simple flows at small scales (DNS
is starting to replace lab. experiments)
Computing requirements for DNS at the Reynolds numbers of aircraft or ships
are stupefyingly large
In the absence of an improvement in computer power by factors of 
or so, DNS is not an option for such cases
Today DNS is useful in principle for low-Re flows in engineering
but has the same difficulties as approximate methods in predicting transition
Why turbulence is difficult - a little physics
Why engineers need short cuts - a little economics
Short cuts:- large-eddy simulation, Reynolds-averaged modeling
Hierarchy of Reynolds-averaged models:- eddy viscosity with zero, one,
two
PDEs; stress-transport equations
What of the future?
``Modeling'' of turbulence means replacing unknown terms in some exact
equation(s) by empirical formulas (or equations) calibrated with data from
experiments (or DNS)
The object is to reduce the number of unknowns to equal the number of equations
- i.e. ``close'' the system of equations
Completely- (``Reynolds'')-averaged models are often classified by the
number of turbulence equations
Usually we just want global averaged quantities - skin friction, pressure
drop, boundary-layer thickness and heat transfer
Most of the turbulent transfer (mixing) of mass, momentum and heat is carried
out by the larger eddies
the smallest eddies just dissipate TKE into heat
Computing cost can be greatly reduced by calculating large eddies, as in
DNS, but modeling small ones - this is LES.
Models for the small (``sub-grid-scale'') eddies are usually quite simple
- if the small eddies bear little Reynolds stress, all the SGS model has
to do is dissipate energy
Near a solid surface ALL eddies are small, so either:-
skip=0em (i) LES in this region becomes as expensive as DNS, or (ii)
the SGS model must do the whole calculation, or (iii) a ``boundary'' condition
must be used to terminate the LES at some distance from the surface, or
(iv) (current practice) compromise between (i) and (ii) - use a grid which
is somewhat coarser than needed for DNS (but Re-dependent in the
viscous wall region)
and use a moderately sophisticated SGS model, possibly a self- calibrating
``dynamic'' model skip=8pt
Osborne Reynolds decomposed variables in turbulence into a mean
(e.g. a time average) plus a fluctuation, e.g. u+u'
Taking the mean of the NS equations leaves the mean rates of transfer of
momentum by the turbulence as extra unknowns in the ``RANS'' equations.
Move them to the r.h.s. of the equations and call them gradients of the
(Reynolds) stresses
The apparent stress 
acts in the x direction (for u') on a plane normal to the
y direction (for v'), and so on for all 
Current engineering calculation methods are mostly Reynolds-stress models
(not called ``simulations'')
Turbulence quantities like Reynolds stresses obey exact PDE ``transport
equations'' derivable from the NS equations
but with further unknowns that must be modeled
has dimensions [velocity
/time] or [velocity
/length]
so we need a model transport (or other) equation to give a length scale
or time scale. Common variables are
or 
Why turbulence is difficult - a little physics
Why engineers need short cuts - a little economics
Short cuts:- large-eddy simulation, Reynolds-averaged modeling
Hierarchy of Reynolds-averaged models:- eddy viscosity with zero, one,
two
PDEs; stress-transport equations
What of the future?
Many engineering flows are dominated by boundary layers or other ``thin''
shear flows
dominated by the shear stress in the plane of the mean shear
Turbulent stresses are related to the mean-flow history, but large
mean shear usually implies large local shear stress
and quite simple empirical formulas relating turbulent shear stress
to mean shear 
give adequate predictions
Alas the empirical coefficients have to be changed from one flow to another,
by large amounts in complex flows.
``Eddy viscosity'' is defined as the ratio of a Reynolds stress
to the mean rate of strain in the same plane. It is measurable
Simplest example: 
The full eddy-viscosity formula for constant-density flow is
where 
if i = j and zero otherwise
Alas 
varies from place to place and from flow to flow and is different for different
stresses
sometimes it is negative!
Eddy viscosity is useful if it varies more predictably than the
Reynolds stresses
Reynolds stresses are generally large where mean velocity gradients are
large
(e.g. we said 
tends to be large where
is large)
That is, eddy viscosity usually varies less rapidly than the stresses -
which may mean ``more predictably''
Consider a dog on a long leash - it follows roughly the same path as its
master, but can get up to mischief on the way
Eddy viscosity is the ratio of a turbulence quantity (Reynolds stress)
to a mean-flow quantity (mean rate of strain)
Correlating it entirely in terms of mean-flow scales or entirely in terms
of turbulence scales is justifiable only if the two are proportional
a working definition of a ``equilibrium'' flow
Nevertheless, most turbulence models in commercial use are based on an
eddy viscosity, assumed to be a scalar (same for all stresses)
a.k.a. ``Algebraic Eddy Viscosity'' models
Good results are obtained in simple cases by taking 
near a solid surface, so that the logarithmic ``law of the wall'' is reproduced
while in the outer part of simple boundary layers the empirical relation 
(or equivalent) works well even in non-equilibrium cases (e.g. strong pressure
gradient)
In more complex flows, mean-flow length scales like 
may not be definable, the log law may not work
and most general-purpose models go to the other extreme, parameterizing
eddy viscosity with turbulence scales
We can define a coefficient 
by 
where k and
are TKE and dissipation rate (``typical scales'')
Then modeling and solving the transport equations for k and
gives a ``two-equation'' model for
The catch is to determine
! Remember the mongrel dog
Recall that taking
near a solid surface yields the logarithmic law
and 
is as much as 0.7 at the outer limit of the log. law
Forcing an eddy-viscosity-transport model to reproduce
is a powerful constraint on the empirical coefficients, including
and all the better models have this constraint
Some people might call it cheating!
The typical velocity scale 
and length scale
(or time scale 
)
can be used in modeling (recall the definition of
). (These are most popular scales, not the only ones in use)
Example - the
transport equation itself contains a viscous destruction term, with dimensions
of [
/time]. We model this as 
, where 
is a dimensionless empirical coefficient
Example - the equation for 
has a turbulent transport (``diffusion'') term 
and 
is usually modeled by ``gradient transport'' 
where 
FAMILY - I
Several models use a second variable other than
, but all are of the form
(we need
explicitly in the k equation)
If we transform a model
equation to one for
, say, using the model k equation
the
equation is different for different m and n - in the form
of the terms and not merely in the coefficients. There must be an optimum
m and n, in some sense
The exact transport equations for
or
inspire only fear - so the model equations are almost entirely empirical
The most popular ``two-equation'' model - probably not the best - is the
most straightforward, using k and
FAMILY - II
A few models use fractional powers of k as the first variable
- a k equation can be recovered but has extra terms not found in
the exact k equation, and this is unappealing
Choices for second variable include Wilcox's 
(a constant times
) and its inverse, the time scale
.
Information about the effect of a solid wall on eddy length scale propagates
away from the wall
so if turbulent transport of
is to be modeled by gradient diffusion,
should decrease with increasing y - in practice, n should
be positive
(m=-1, n=1) seems near optimum in several respects
The model transport equations for (e.g.) k and
could be combined to give one equation for 
and hence
There are several wholly-empirical transport equations for
The Spalart-Allmaras
model seems competitive with the best two-equation models and is in principle
cheaper to solve
Even two equations is not many to describe turbulence! Durbin's ``v2f''
model uses a further variable, equal to the normal stress 
near a solid surface, in addition to k and
and also an empirical elliptic equation to represent the effect
of pressure fluctuations (which obey a Poisson equation)
Several nonlinear formulas have been suggested, including products of the
strain rate 
and the ``rotation tensor'' 
up to third order
These formulas are subject to invariance constraints and symmetry requirements
but still include many extra adjustable terms
Obviously the eddy viscosity may behave badly if 
and/or 
becomes large, so products become very large - this worries me!
Six empirical coefficients adequately define my clothes (collar, arm, chest,
waist, leg, foot)
but turbulence is more complicated (a warning to those who buy turbulence
models off the peg?)
The extra coefficients in nonlinear eddy-viscosity models may improve agreement
with experiment even if the extra terms do not correspond to specific phenomena
(e.g. streamline curvature)
which worries me even more!
Eddy viscosity is not a reliable enough concept to carry ``bells and whistles''
like this
These yield nonlinear, anisotropic eddy viscosity but are stress-transport
models with simplified transport terms.
The usual simplification is to assume that the mean-transport and turbulent-transport
terms in each stress-transport PDE are proportional to the stress being
transported
it follows that the transport terms need be modeled for only one stress
equation - always the TKE equation
and the others reduce to (simultaneous) algebraic equations for the stresses
A big publicity point for ASM is the ability to predict stress-induced
secondary flows in non-circular ducts, at least qualitatively
Unfortunately, the exact stress-transport equations show that the assumption
that the mean-transport terms are proportional to the stress being transported
is poor in rapidly-changing flows
Here, large mean transport of 
comes from large generation 
so for a 2D shear layer the generation terms proportional to 
are 
which sum to zero by continuity!
The complete turbulent-transport term is not closely proportional to the
transported stress (not its gradient), especially near free-stream
edges where turbulent transport is largest.
The transport equations for the Reynolds stresses (four nonzero in 2D flow,
six in 3D) can be modeled directly
with a transport equation for
or other quantity giving a length scale
In the late 1960s stress-transport (sometimes called ``second moment'')
modeling seemed to be the wave of the future - solve equations directly
for the quantities you want
However, stress-transport models have proved disappointing - they often
do better than isotropic eddy-viscosity models, but do not deliver engineering
accuracy over a wide range of flows
Most models use an
equation, essentially the same as in two-equation models, to provide a
length scale
bad news?
Parneix and Durbin (1996) showed that the 
dissipation equation does ``surprisingly well'' in a backstep flow
Experience with the equation for 
is mixed
It is difficult to defend using different length-scale equations in 2-equation
and stress-transport models
but disbelievers in eddy-viscosity models are entitled to reject their
(almost-entirely-empirical) length-scale equations and start again!
A major problem is modeling the ``pressure-strain'' (redistribution) term 
in each 
equation (zero in the summed k equation)
The pressure-strain term cannot be reliably measured: DNS results are some
help
Most models parameterize it as a function of local quantities, but p'
is an integral of a Poisson PDE over the whole flow (including the ``image''
flow beneath a solid surface)
Durbin's semi-empirical elliptic PDE for the pressure-strain term is based
on the Poisson equation - a qualitative improvement
Stress-transport models supply the Reynolds stresses to the mean-flow transport
equations as new source terms
whereas eddy-viscosity models supply a factor in existing terms involving
the mean rates of strain
Thus the mean-flow and turbulence equations are less closely coupled in
stress-transport models, and therefore tend to be ``stiff''
More steps, more equations, more money
Also, eddy-viscosity codes need major revision to incorporate stress-transport
models (because the stresses are supplied differently), so users of the
former are reluctant to change to the latter
Near a solid surface, flow structure and optimum model coefficients depend
on 
- a dimensionless distance from the surface and a Reynolds number
Influence of solid surface is partly viscous (from u,w=0),
partly inviscid ``blockage'' (from v=0)
Either way, use
or 
to correlate
Durbin's elliptic equation for the pressure-strain term expresses the blockage
effect
and, given u=v=w=0 at y=0, it seems to avoid
the need for ``damping functions'' (so does 
)
AND WALL FUNCTIONS
``Integration to the wall'' is expensive (small y steps, greater
stiffness) - so as usual we want something cheaper
In 2D simple flows, 
for 
and 
, etc., is constant
Apply these ``law-of-the-wall'' relations as off-the-wall boundary conditions
Catch is that in 3D or non-simple flows these relations can be highly inaccurate
- and the numerics can be a pain
Unfortunately nothing guarantees that a ``low-Re'' model calibrated
to reproduce the law of the wall is accurate where the latter isn't
Why turbulence is difficult - a little physics
Why engineers need short cuts - a little economics
Short cuts:- large-eddy simulation, Reynolds-averaged modeling
Hierarchy of Reynolds-averaged models:- eddy viscosity with zero, one,
two
PDEs; stress-transport equations
What of the future?
No current model can be relied on to produce results of engineering accuracy
over the full range of flows
Usually, the results are more accurate than an expert's guess and are therefore
valuable guides for designers
but computers are nowhere near replacing wind tunnels as promised 25 years
ago
Reynolds averaging is a brutal simplification - s little hope of a model
that gives engineering accuracy for all flows?
Optimizing a model to give good results in the sort of flows that interest
you today is fine - today!
More-elaborate eddy-viscosity relations have even less physical foundation
than linear eddy viscosity
and are surely accidents waiting to happen
Allowances for flow history are more important
and stress-transport models ought to do better than eddy-viscosity
models
It may be worth asking ``Why not?''
LES can already be relied on to predict free turbulent flows to good engineering
accuracy, at any Reynolds number
but the viscous wall region demands either an off-the-wall boundary
condition or a sub-grid-scale model carrying most of the Reynolds
stress, and both involve empirical approximations.
The only alternative is to refine the near-wall grid so that the LES becomes
a DNS - with severe Reynolds-number limitations
This is undoubtedly the ``pacing item'' in LES - and possibly in turbulence
modeling in general
Many people agree that expensive Large-Eddy Simulation will be used only
in difficult parts of the flow
with the ``easy'' parts left to Reynolds-averaged models
Big problem is providing fluctuating inlet data for LES from Reynolds-averaged
statistics alone
perhaps by starting LES upstream of where it is really needed and rescaling
after it has settled down
Most current Reynolds-averaged models are based on PDE ``transport'' equations
for isotropic (scalar) eddy viscosity
but eddy viscosity in real flows is not isotropic and often behaves erratically
Model transport equations for the Reynolds stresses themselves are more
realistic in principle
but sometimes behave erratically when they shouldn't
Wilcox will talk about what works and what doesn't
I have more hope than he has that the reliability of stress-transport models
can be improved
but I fear that Reynolds-averaged models will only improve slowly, eventually
becoming junior partner to LES
Turbulence modeling is important to many mechanical engineers
Current turbulence models are better than nothing
but not as reliable as design tools should be
The user of a turbulence model is more like a test pilot than a sunny Sunday
Cessna flier
Bradshaw's Web page at
http://vonkarman.stanford.edu/tsd/resp_b.html
leads to a bibliography of turbulence and related subjects made up of annual files, totaling about 10000 entries (2MB), each with a one-line abstract, plus the same information sorted into index categories
Files are updated monthly
These are "further reading" references, mainly to review papers. They come from my main bibliography (see above).
..13.0,05.0: nominally elliptic equation for wall effect - still not quite clear that it is d/dy to BL approx.* DURBIN, P.A.* A Reynolds stress model for near-wall turbulence* J. Fluid Mech. 249, 465* 1993. `
..13.0,14.0: review of 2-eq. and stress-transport models -mainly realizability* SHIH, T.-H.* Developments in computational modeling of turbulent flows* NASA CR 198458* 1996. `
..13.0: published version* SO, R.M.C.; LAI, Y.G.; ZHANG, H.S.; HWANG, B.C.* Second-order near-wall turbulence closures - a review* AIAA J. 29, 1819* 1991. `
..13.0: "particle equations" compatible with N-S give Langevin equation - Durbin elliptic relaxation* DREEBEN, T.D.; POPE, S.B.* Probability density function and Reynolds-stress modeling of near-wall turbulent flows* Phys. Fluids 9, 154* 1997. `
..14.0,13.0,05.5: review - mainly free-surface flows* RODI, W.* Impact of Reynolds-average modelling on hydraulics* Osborne Reynolds Centenary Volume, Proc. Roy. Soc. London A451, 141* 1995. `
..14.0,13.0,25.6: useful but somewhat Speziale-oriented* SPEZIALE, C.G.; SO, R.M.C.* Ch. 14. Turbulence modeling and simulation* Handbook of Fluid Dynamics (R.W. Johnson, ed.), CRC Press* 1998. `
..14.0,24.2: viscous modifications of k, omega* WILCOX, D.C.* Simulation of transition with a two-equation turbulence model* AIAA J. 32, 247* 1994. `
..14.0,24.2: Robinson k, zeta model with stability theory for timescale* WARREN, E.W.; HASSAN, H.A.* Alternative to the e^n method for determining onset of transition* AIAA J. 36, 111* 1998. `
..14.0,25.6: extension of 1996 Monterey paper - relaxation model for stress anisotropy as an add-on to k, eps. Still grid-dependent crossover function for RANS to SGS* SPEZIALE, C.G.* Turbulence modeling for time-dependent RANS and VLES - a review* AIAA J. 36, 173 (was AIAA 97-2051)* 1998. `
..14.0: ASM plus dissipation anisotropy equation* JONGEN, T.; MOMPEAN, G.; GATSKI, T.B.* Predicting S-duct flow using a composite algebraic stress model* AIAA J. 36, 327* 1998. `
..14.0: correct asymptotic and near-wall behavior* HWANG, C.B.; LIN, C.A.* Improved low-Reynolds-Number k-eps. model based on direct numerical simulation data* AIAA J. 36, 38* 1998. `
..14.0: extra (dk/dy).(dtau/dy) term in eps. equation, following Ince, improves performance in adverse pressure gradient* GOLDBERG, U.; PEROOMIAN, O.; CHAKRAVARTHY, S.* A wall-distance-free k-eps. model with enhanced near-wall treatment* J. Fluids Engg 120, 457* 1998. `
..14.0: k, zeta where zeta is enstrophy. See Robinson et al. 1995* ROBINSON, D.F.; HASSAN, H.A.* Two-equation turbulence closure model for wall bounded and free shear flows* AIAA J. 36, 109* 1998. `
..14.0: correctly predicts backstep reversed cf, and vortex streets. v^2 is now a free vel. scale* DURBIN, P.A.* Separated flow computations with the k, eps, v^2 model* AIAA J. 33, 659* 1995. `
..14.0: good review including history - lists fundamental weaknesses w.r.t. stress-transport models* APSLEY, D.D.; ET AL.* Non-linear eddy-viscosity modeling of separated flows* J. Hydraulic Res. 35, 723. Was UMIST, Manchester, TFD/97/02* 1997. `
..14.0: iterative solution of an ASM model. See also UMIST TFD 96/05* APSLEY, D.D.; LESCHZINER, M.A.* A new low-Re non-linear two-equation turbulence model for complex flows* Int. J. Heat and Fluid Flow 19, 209 (11th TSF, Grenoble, paper 6-25, 1997)* 1998. `
..14.0: more elaborate eddy-viscosity damping function avoids trouble at reattachment - and review* CHANG, K.C.; HSIEH, W.D.; CHEN, C.S.* A modified low-Reynolds-number turbulence model applicable to recirculating flow in pipe expansion* J. Fluids Engg 117, 417* 1995. `
..18.1,14.0: rho^n - k^m - eps^l. Very slight effect on k, omega (good anyway), but k, eps. etc. improved* CATRIS, S.; AUPOIX, B.* Improved turbulence models for compressible boundary layers* AIAA 98-2696* 1998. `
..24.2: ERCOFTAC special interest group* SAVILL, A.M.* Some recent progress in the turbulence modelling of by-pass transition* Near-Wall Turbulent Flows (R.M.C. So, C.G. Speziale, B.E. Launder, Eds.) Elsevier, p. 829* 1993. `
..25.2,59.2: useful review of real life* COSNER, R.R.* CFD process needs for the next decade* Presented at AIAA 13th Comp. Fluid Dyn. Conf. session 31-CFD-11 (no AIAA paper number - "McDonnell Douglas unpublished paper")* 1997. `
..25.6: useful short review - examples mainly from work by author or CTR. Extended version to appear in Prog. Aero. Sci.* PIOMELLI, U.* Large-eddy simulations - present state and future directions* AIAA 98-0534 1998. `
..25.6: nonlinear model capable of simulating backscatter. Also, useful review* KOSOVIC, B.* Subgrid-scale modeling for the large-eddy simulation of high-Reynolds-number boundary layers* J. Fluid Mech. 336, 151* 1997. `
..25.6: still best review as of 1998* FERZIGER, J.H.* Large eddy simulation* Simulation and modeling of turbulent flows (Gatski, T.B., Hussaini, M.Y. and Lumley, J.L. Eds.),Oxford Univ. Press, p. 109* 1996. `
..44.2,24.2,14.0: correlations for transition onset and intermittency, then Menter SST/k, omega - much better than using models through transition* HUANG, P.G.; XIONG, G.* Transition and turbulence modeling of low pressure turbine flows* AIAA-98-0339* 1998. `
..49.0: as good as or better than full RANS codes* JOHNSTON, J.P.* Review - diffuser design and performance analysis by a unified integral method (data bank contribution)* J. Fluids Engg 120, 6* 1998. `
..59.2,13.0,14.0: code-user-oriented review. 83 refs.* HIRSCHEL, E.H.; STOCK, H.W.; COUSTEIX, J.* Invited lecture. Current turbulence modelling in aircraft design* Engg Turbulence Modelling and Expts 2, (W. Rodi and F. Martelli, Eds.) Elsevier, p. 665* 1993. `
..59.2,48.0: 3-element airfoil. Langley meas. and S-A calcs. Eddy viscosity not good for confluence. Good review* YING, S.X.; ET AL.* Investigation of confluent boundary layers in high-lift flows* AIAA 98-2622* 1998. `