Mathematical Foundations of Large Eddy Simulation
William Layton
Department of Mathematics
University of Pittsburgh
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ABSTRACT
This document gives a more detailed survey of our groups ideas and efforts
studying large eddy simulation from a mathematical viewpoint.

1. Introduction.
Is it possible to predict the motion of larger eddies in fluid flow
problems without complete resolution of all persistent eddies? Is it truly
possible to separate modeling errors from numerical errors in flow
simulations ? How is the error in a flow simulation distributed among
eddies of different sizes? Are the details of the smaller eddies needed to
predict the motion of the larger eddies or are some means or moments
sufficient?
The research program begun by our group at Pitt is to begin addressing
these questions mathematically, in other words, to develop the numerical
analysis of large eddy simulation.
For the forseeable future turbulent flow can hardly be simulated
reliably via direct numerical simulation (DNS) because direct numerical
simulation requires resolution of all persistent eddies.  Assuming
Kolmogorov's description of turbulence valid, the smallest length scale is
expected to be
O(Re^(-3/4)).
Thus, O(Re^9/4) grid points are required for
DNS with a correspondingly large computational effort. Conventional
turbulence models, such as the k-epsilon model, also cannot be the answer
since they are based upon loose analogies, guesswork and data-fitting a
host of parameters to a particular flow. Because of the extensive
'calibration' needed to have conventional turbulence models attain some
minimal levels of accuracy, they can never attain the degree of
universality that would accompany accurate description of the physics of
turbulence.
One approach to modeling turbulent fluctuations is the analogy between
the mixing effects of turbulent fluctuations and molecular diffusion put
forward by Boussinesq in 1877. For example, if u^ denotes either a time or
ensemble average and u' the fluctuations about u^ , u = u^ + u' , closure
in all turbulence models is based on this analogy:

div [ u'u' ]^  ~ div{ mu_t ( grad( u^ ) + grad( u^ )tr ) },
where
mu_t = mu_t ( u^ , k , epsilon , ... )
is  the turbulent viscosity
coefficient. Because it is based upon this analogy, the effectiveness of
turbulence modeling depends ultimately upon the calibration, tuning, data
fitting etc. the problem dependent parameters which occur in mu_t.  This
relation has a certain amount of mathematical support (summarized well in
the book of Mohammadi and Pironneau). Thus , it seems likely that most
conventional turbulence models go wrong mainly in how they compute
approximations to the inputs like k (the kinetic energy in the turbulent
fluctuations)  and in how boundary conditions are selected for the
appended partial differential equations.
Large eddy simulation (LES) is often described as being  based on a
direct approximation of the large vortices, or eddies. The effects of the
small eddies upon the large eddies are modeled by a systematic closure
approximation. (We postpone for a bit discussion of, so called,
sub-gridscale models.) While large eddy simulations are typically 3-d and
evolutionary, hence costly, their advantages include:
1. Storage and computational costs are independent of the Reynolds number.
One of the goals of LES is to approximate turbulent flow with
complexity depending only on the resolution sought (the large eddy scale
and not that of the small fluctuations.)

2. Consistency:
closure is based upon approximation rather than
analogy so consistency is automatic.
Our research effort is concerned with two complementary approaches to
large eddy simulation:
(section 2 )
Numerical analysis of continuum level approximations of
large eddy motion ,

( section 3 )
Direct  simulation of large eddy motion.
( section 4 )
Analysis of Scale-Similarity models.
( section 5 )
Reliability estimation via parallel ensemble calculation.
( section 6 )
Better subgrid scale modelling.
The first, numerical analysis of continuum models of large eddy motion, is
similar in spirit to traditional LES in that the Navier Stokes equations
are filtered, closure is addressed, a continuum model is derived and then
solved approximately. We are developing two improvements in this approach
:
(section 2.1) Mathematically sound continuum models of large eddies,

(section 2.2 ) Accurate boundary conditions for LES.
First while the ideas behind the derivation of common models in LES
are well grounded physically, their mathematical execution is ( in our
view) fundamentally flawed. Since these models are approximations , the
filtered true solution of the Navier Stokes equations, inserted into the
model has small residual in some sense. This does not  imply, of course,
that the solution of the model is close to the filtered true solution of
the Navier Stokes equations. To our knowledge, it has never been proven
that solutions of any LES model or turbulence model are close to the true
desired flow averages. Analytical components of the project are attempting
to give a full mathematical description of the modelling error for a new
family of  LES models. In section 2, our approach is presented.
The difference between the true solution of the eddy viscosity model
and the filtered solution of the Navier Stokes equations satisfies an
equation driven by the aforementioned residual. Thus the conclusion holds
only when the eddy viscosity model is stable in the right sense. While
common LES models do not satisfy this important condition, we proposed a
closure approximation in
[GL98] G.P. Galdi and W. Layton, Approximating the larger eddies in fluid
motion II: a model for space filtered flow, to appear in: Math.Methods and
Models in the Appl. Sciences, 1998.
which leads to a new family of LES models which appear to have the
stability property needed to conclude its solutions closely approximate
the large eddies in the flow field. The next step in our research involves
completing the proof that the solutions of the model closely approximates
the true large eddies and then developing the model further and testing it
in computational experiments.
Problems in the boundary conditions for LES arise because there is a
basic difficult in the filtering process near boundaries. Commonly, either
'no-slip' boundary conditions or an analogous wall law are imposed.
However, it is easy to see in the next figure that , while no-penetration
conditions are correct for the large eddies, Dirichlet  boundary
conditions are not.

Fluid Region
| ------->
| ------>
| ----->
| ---->
| --->
| -->
| ->
_______________*___________ tau ->
///////// Solid Wall ////////
Imagine a ball of positive radius centered at the point *.
Average the flow field over that ball. u is extended by zero outside the flow region.
Clearly, u^.n = 0 but u^ . tau is non-zero!
Figure1.1.
w^ . tau = 0 on the boundary is not the correct
condition even for laminar boundary layers!
Researchers have often reported that LES simulations have difficulties
predicting flow fields near boundaries accurately. These large errors near
boundaries are, in our view , introduced in traditional approaches to LES
due to the imposition of incorrect boundary conditions upon the eddy
viscosity models. This fact is increasingly being recognized in LES. For
example, there is an increasing quantity of work studying models where
delta =delta(x), where delta(x)->0 as x-> walls ,
for example,
delta(x) = min{ delta_0 , dist{x,walls} }.
This causes great practical difficulties even in the formal modelling
steps but it at least allows one to apply no-slip boundary conditions on
u^. However, it also implies all boundary layers must be fully resolves-
obviating many of the practical benefits of LES! Furthermore, there are
serious mathematical reasons to keep delta constant-some are detailed in
section 2 below. Still, if there were no other possibility for more
consistent boundary treatment, this program would have to be followed.
Fortunately, there are at least two other approaches which are simpler and
can be persued in parallel to the work on variable delta's. We describe
these other two approaches herein.
The first of these new procedures for correcting the boundary conditions
for constant delta (!)
imposed in any eddy viscosity model was given in [GL98]. This boundary
treatment has fully developed by a Ph.D. student, Mr. Niyazi Sahin. This
work , including laminar layers and turbulent power law layers and log-law
layers is presented in his report:
[S99] N.Sahin, Boundary conditions for large eddy simulation, tech.
report,1999.
Even the simplest of the new eddy viscosity models lead to interesting
new algorithmic challenges. The last aspect of this approach to LES
studied will be algorithm design and validation for eddy viscosity models.
While the proposed research on continuum models of large eddies
is inspired by 28 years of development of large eddy simulation by the
engineering community, the second, direct simulation of large eddy motion,
is (to the best of our knowledge) a completely new approach to large eddy
simulation. This approach is straightforward and mathematically honest:
the Navier Stokes equations are discretize by a "good" method (such as
streamline diffusion methods or stabilized defect correction methods, e.
g., [ELM98]). The approximate solution (uh, ph) is then post-processed:
Approximate NSE: ( u , p ) |----> ( uh , ph ),
Post-process : ( uh , ph ) |--->( g*uh , g*ph ) approximates (u^ , p^ ).
This approach requires no closure approximation and no elaborate boundary
treatment. However, for it to be computationally feasible there are at
least two requirements:
1. The error in the large eddies should be much smaller than the error in
the flow field. (Otherwise it will require the storage and computational
effort of direct numerical simulation.)
Surprisingly, in initial first steps, we have shown, in
[JL98] V John and W Layton, Approximating the larger eddies in fluid
motion I: direct simulation for the Stokes problem, submitted to SIAM JNA,
1998
that this approximation to the large eddies can possess hidden accuracy
without full resolution of the details of the small eddies.
2. A mesh refinement strategy designed to guarantee the accuracy of the
approximation to the large eddies.
In [JL98] the we also begun this task: two mathematically correct a
posteriori error estimators are developed for estimating the error in the
large eddies. With either it is possible to refine to attain a
prespecified guarranteed acuracy in the flow field's large eddies. The
meshes so designed are locally refined yet much closer to uniform meshes
than when an estimated for the energy norm error is used. Further the
estimated errors do satisfy:
Estimator( || uh^ - u^ || ) < < Estimator( || uh - u || ).
These a posteriori error estimators were proven correct for the Stokes
problem in [JL98]. Mathematical structures for their extension to be fully
nonlinear Navier Stokes equations seem to be in place already in the
P.I.'s work. In this proposed research, the estimators will be further
developed for the 3-d evolutionary Navier Stokes equations.
Computational experiments, tests and their benchmarking will be
performed in conjunction with a joint effort between our group at Pitt and
Professor Lutz Tobiska's research group in Magdeburg, Germany. This joint
effort is to develop a 3-d (object oriented) adaptive, parallel C++ code
for the Navier Stokes equations, including modules handling free surfaces,
turbulence and LES . Other portions of the theoretical and computational
research effort will be performed in collaboration with several strong
Ph.D. students in our group.

2. Continuum Models of Large Eddy Motion.
The usual approach to LES is to derive a continuum model for the motion of
large eddies. This introduces "modeling errors". When the continuum model
is discretized and solved, additional "numerical errors" are introduced.
Our research in [GL98] has  focused on first driving continuum models of
large eddies which better capture the evolution of local flow averages.
Second, mathematical support for the continuum model has been started in
our report:

[GIL99] G.P. Galdi, T. Iliescu and W. Layton, report in preparation.
Third, numerical analysis of algorithms for the new eddy viscosity model
is currently under study, as are experiments comparing its predictions to
both conventional models and benchmarks.
First steps in driving improved models for large eddy motion have all
ready been taken in [GL98]. Recall that the most common definition of
large eddies in LES is the local flow average with a scaled Gaussian:
u^(x, t) = g * u(x, t),
where
g(x) = (1/delta^d) g_0(x/delta),
g_0(x): = a Gaussian distribution.
Here  g*u denotes the usual convolution of g with u. Before proceeding ,
it's worthwhile to note the most important properties of g*u = u^:

1. If delta is CONSTANT then differentiation commutes with averaging. For
any derivative d/dx :
g*( du/dx) = d/dx(g*u).
This is an important property for deriving models.
2. u^ is infinitely differentiable, provided delta is CONSTANT.
3. The kinetic energy in u^ is bounded by that of u, if delta is CONSTANT:
KE( u^(t)  ) <= KE( u(t) ) <= C( f, u(0), Re, t).
4. The smallest scale occuring in u^ is O(min{delta(x) : x in the flow region}.
Suppose the fluid motion is decomposed, as usual (1), into means and
fluctuations: u = u^ + u'. Then, the quadratic term in the space filtered
Navier Stokes equations leads to at least three terms requiring closure:
(uu)^ = (u^u^)^ + (u^u') ^ + (u'u^) ^ + (u'u') ^.
In , e.g. ,  common LES models the right hand side of the above is
approximated using Taylor series in the averaging radius delta by terms
involving the flow means only and the following system results for w which
denotes the approximation to u^.
The Most Usual Model:
div w = 0
dw/dt + div( ww ) - grad q - (1/Re) div grad w  - div K = g*f,
where

K_ij = delta^2 w_i,x_m  w_j,x_m ,
_______________________________________________________
(1) It is interesting to note that this decomposition, intensively
exploited by Reynolds, was suggested by Leonardo da Vinci in 1500's in his
descriptions of the eddies behind a blunt body.
________________________________________________________
The true flow average u^ = g * u not only has eddies smaller than O(delta)
suppressed by the averaging process but high frequencies are suppressed as
well: u^ is  infinitely differentiable. The commonly used closure
approximations based upon Taylor expansions actually have the undesirable
effect of actually increasing the high frequencies in the approximation w.
We have studied closure approximation based upon rational approximations
rather than truncated Taylor series. The new aproximation, when developed
into a continuum model, is of the same formal accuracy in the large eddies
as the usual one. Further, it attenuates high frequencies as required .
The simplest case of the new modeling procedure yields the initial,
boundary value problem:
A Regularized Model:
div w = 0,
dw/dt + div( ww ) - grad q - (1/Re) div grad w  - div H = g*f,
- delta^2 div grad( H_ij )   +   H_ij   =   K_ij,
where, as before,
K_ij = delta^2 w_i,x_m  w_j,x_m ,

Mathematical validation of this model has begun in [GIL90]. We are also
exploring :
Models arising from more accurate rational approximations,
Continuum models arising from space time averaging of the Navier Stokes
equations,
Continuum models based upon a 3 field decomposition into space time
averages, space averages and fluctuations.
3. Direct Simulation of Large Eddy Motion.
Picking a length scale delta, the question of LES is simply how to
approximate u^: = g*u with much greater accuracy and much less cost than
approximating u. The approach taken in [JL98] is to take advantage of some
special features of finite element approximations in the filtering
process. First (u h, p h) is calculated by approximate solution of the
unfiltered Navier Stokes equations. Next the approximate large eddy motion
is obtained through post-processing by spacial filtering:
uh^ := g * uh     ,      ph^: = g * ph.
Since not all spatial scales are resolved in the approximations u h to u
we do not expect high accuracy here. In fact, it is expected (and known
rigorously in model problems at least) that errors in this approximation
will be highly oscillatory. The approximations uh to u might also be
oscillatory due to unresolved solution scales. Heuristically, a local
averaging operator might then kill these unwanted oscilations and hence
the leading order error terms as well. Thus, there are reasons to expect
|| uh^-u^ || to be much smaller than || uh-u || . Indeed, in our work it
was proven that with a degree k piecewise polynomial velocity
approximation the error estimate:

(3.1)       || u^-uh^ || < C (h/delta)^(k-1) h || grad ( u - uh ) ||
holds, where h is a representative meshwidth and || * || denotes the L2
norm. As a first step this has been proven for the Stokes problem. The
present effort is to extend the understanding and experience with this
research to the Oseen problem followed by the full, evolutionary Navier
Stokes equations.
The estimate (3.1) shows that accurate resolution of the flow field
is not needed to resolve the field's large eddies. This is because of the
finite element properties of having highly oscillatory leading order error
terms and the error cancellation properties of spatial filtering with a
symmetric filter.
As research moves to more and more difficult problems, the importance
of adaptivity based on good a posteriori error estimators increases. Thus
an important component of correct simulation of large eddy motion is to
design a mesh refinement strategy, with full mathematical support, which
provides guaranteed accuracy in the large eddies. To this end, two a
posteriori error estimators were given in [JL98] for  || uh^-u^ || . These
estimators allow mesh cells to be marked for refinement only when the mean
motion of the local small eddies influences the target accuracy in the
large eddies. The first global estimator is established by duality
techniques. It reflects the extra accuracy of the a priori estimate (3.1).
The second estimator of this report is more computationally intricate. As
compensation, to sided error bounds are proven for the second estimator.
The sum of the local error indicators is a bound for  || u^-uh^ || above
and  the local indicator provides a lower bound to the local error on a
patch of elements.
4. Scale-Similarity Models.
Scale similarity models are based upon a 'scale similarity assumption,
namely that
unresolved eddies interact with the O(delta) eddies in the
same way O(delta) eddies interact with O(2 delta) eddies.
This postulate
was advanced in 1980 by Bardina, Ferziger and Reynolds. Like every
description in turbulent flow, it is quite successful at times and less so
at other times.  (Perhaps because generally only successes are published
there is a dearth of clear descriptions of when each approach fails. This
knowledge is more valuable than successes! ) Another way to think of the
hypothesis is that is is a sort of extrapolation procedure which work
generally if there's a sort of regular pattern in the data over the range
extrapolated. Let T denote the Reynolds stress tensor in the space
filtered flow equations:
T := (g*u) (g*u) - g*(u u) .
The most straightforward scale similarity assumption is that T is modeled
by

T ~  (g*g*u) (g*g*u) - g*( g*u g*u).
The most basic such model is thus (where w denotes the resulting
approximation for g*u and q for g*p ):
div w = 0 ,
dw/dt + div( w w ) + grad q  - (1/Re) div grad w
- div(  g*w g*w - g*( w w ) )  = g*f.
Actually, this system is seldom solved approximately; authors hint at
catastrophic blow up of solutions over long time intervals. (This is
another case where a clear negative result would be very interesting.)
This sort of growth is typically 'cured' by including a Smagorinski type
subgrid scale model in the system. The included p-Laplacian then gives
powerful control over polynomial nonlinearities-but is it physically
correct?
This reasoning and the theoretical studies in [GIL99] point to the
KINETIC ENERGY in the large eddies as the key functional to track.
Mathematically, Young's inequality implies:
Kinetic Energy( g*u(.,t) ) <= Kinetic Energy( u(.,t) ) < infinity,
for any t>0.
The same cannot be said for solutions w(x,t) of large eddy models! It is
our view that a good model for LES must satisfy the following requirement:
"The kinetic energy in the solution of the model of the large eddies
must be provably bounded for all t>0 without the inclusion of ANY subgrid
scale model. It is even better if the model satisfies a kinetic energy
equation similar to the one for he Navier Stokes equations."

This simple test of physical and mathematical reasonableness apparently is
seldom checked in LES.
Using the scale similarity hypothesis is a different way, we have
developed a first scale similarity model satisfying this requirement in:
[L99] W. Layton, Approximating the large eddies in fluid motion IV:
Kinetic energy balance of a new scale similarity model, technical report,
1999.

5. Calculation of ensembles.
One fundamental, persistent and intractable problem in computational
fluid dynamics is to evaluate the reliability of a calculation. A
posteriori error estimators are a very important approach in increasing
reliability. Until [JL98] only energy norm and L2 norm error estimators
have been available. These are of little practical use in turbulent flow
problems. The estimators of [JL98] and under development in this proposal
are an important step towards ensuring reliability of flow calculations.
All error estimators, however, lead to the problem of evaluating the
sensitivity of the continuous problem to data perturbation (i. e., the
constant
||DFh(uh,ph)^(-1)||L(Xh,Xh)
in, for example,
[ELM98] V. Ervin, W Layton and J. Maubach, Adaptive defect correction
methods for high Reynolds number flow problems, to appear in SIAM JNA).
In engineering practice a calculation is often accepted when the same
qualitative flow features appear on two successive meshes. Of course,
"benchmarking" is done on CFD codes; usually this is done, however to
calibrate the current rather than to evaluate it. For example, in the
recent comparative benchmark tests sponsored by the DFG , the initial
results of the codes were all over the place; once a benchmark interval
was published, suddenly each codes results were square in the middle of
the published benchmark interval.
One technique for enhancing reliability, which hasn't yet been fully
exploited in the Mathematical CFD community, is to compute ensembles:
several flow fields are simultaneously computed which correspond to
perturbations of the problem data. The deviations of the required
functionals of the flow field are then examined vis-a-vis the sizes of the
data perturbations.
Our research is examining questions related to enhancing reliability by
ensemble computations. Some aspects include:
1. Compute ensembles with negligible computational effort over a single solution.
2. Use ensemble calculation to provide information on approximate
stability constants.
3. Using ensemble algorithms to simultaneously compute fluid sensitivities
and flow fields.
There has been some effort in numerical analysis on solving linear and
nonlinear algebraic systems with multiple right hand sides. The first
research step has been to adapt this work from generic problems to the
discretized Navier Stokes equations. Next, using ideas that occur already
in condition number estimation in linear systems, we are studying using
the ensemble calculations to estimate the stability constant inside the
adaptive algorithms refinement test and termination test. Including an, on
the fly, estimation of this important constant will be an important step
toward fully reliable turbulent flow simulation.
6. Better Subgrid Scale Modelling.

Recall the Boussinesq model of turbulent fluctuations:
(6.1)     div [ u'u' ]^  ~ div{ mu_t ( grad( u^ ) + grad( u^ )tr ) }.
This model is based on an analogy between the interaction of small eddies
and elastic collisions of billiard ball like molecules. In spite of this
tenuous analogy, mathematical support for (6.1) has been provided for
models like convection-diffusion problems. (This is presented well in the
book of Pironneau and Mohammadi.) If we accept the model (6.1) , and there
seems to be nothing better supported available in turbulence studies, then
clearly the turbulent viscosity must depend only on:
i)the kinetic energy in the small eddies, k',
ii)the length scale of the averaging process, delta.
Of course, finding an approximation for k' in terms of u^ is challenging.
In the report
[IL99] T Iliescu and W Layton, Approximating the larger eddies in fluid
motion III:the Boussinesq model for turbulent fluctuations, to appear ,
1999,
this was done. Using this formula for k' and the Prandtl-Kolmogorov
relation we develop three new subgrid scale models:
mu_t = C delta^2 |g* (div grad u ) |
mu_t = C delta^2 | u - g*u |
mu_t = C delta^2 | div grad u |.
The implementation of the first and last is unclear for C0 finite
elements. We show how this is accomplished using edge jumps and prove that
at its largest, it is bounded by the Smagorinski model. It vanishes for
laminar flows for which the Smagorinski model is overly diffusive. Lastly,
an existence theory is presented for the new model.
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