Analysis of Numerical Errors in Large Eddy Simulation.
We consider the question of "numerical errors" in large eddy simulation. It is often claimed that straightforward discretization and solution using centered methods of models for large eddy motion can simulate the motion of turbulent flows with complexity independent of the Reynolds number and dependence only on the resolution ``$\delta$'' of the eddies sought. This report considers this question analytically: Is it possible to prove error estimates for discretizations of actually used large eddy models whose error constants depend only on $\delta$ but not Re? We consider the most common, simplest, and most mathematically tractable model and the most mathematically clear discretization. In two cases, we prove such an error estimate and try to explain why our technique of proof fails in the most general case. Our analysis aims to assume as little time regularity on the true solution as possible.
Introduction
The laminar or turbulent
ow of an incompressible
uid is modeled by solutions (u; p) of the
incompressible Navier-Stokes equations:
in
r
in
in
@
Z
(1)
Here
is a bounded, simply connected domain with polygonal boundary
d is the
uid velocity, p
R is the
uid pressure,
f(x; t) is the (known) body force, u 0 (x) the initial
ow eld and Re the Reynolds number.
Unfortunately, when Re is large the resulting turbulent
ow is typically so complex that, so
called, direct numerical simulation of (u; p) is not practically feasible.
One conjecture of Leray is that \turbulence" in nature is associated with a breakdown
of uniqueness of weak solutions to (1). It is known that, for example, weak solutions to (1)
are unique for
There numerous generalizations of this basic result, [GHR00], [Lad69]. With this in mind,
solutions u to (1) with kruk are frequently described as \laminar". Thus,
the L p {regularity in time which can be reasonably assumed is of critical importance.
There are numerous approaches to the simulation of turbulent
ows in practical settings.
One of the most promising current approaches is large eddy simulation (LES) in which
approximations to local spacial averages of u are calculated. A spacial length scale - is
selected and the velocity scales of O(-) and larger are approximated directly while the eects
of those smaller than O(-) on the O(-) and larger eddies are modeled. In computational
turbulence studies using LES it is often reported that the resulting computational complexity
is independent of the Reynolds number (but dependent on the resolution sought, -). There
has been little or no analytical support for this observation however. The goal of this report
is to begin numerical analysis in support of this claim.
To be more specic, a smooth, nonnegative function g(x) with
Z
R d
is selected and the mollifer g - (x) is dened in the usual way:
One common example is a Gaussian, where the summation
convention is used. The spacial averaging/ltering operation is now dened by convolution
In LES, approximations to (u; p) are sought rather than to (u; p). The usual procedure is to
rst lter the Navier-Stokes equations:
in
r
in
where the \Reynolds' stress tensor" T is
Closure is addressed by a modeling step in which T is written in terms of u. The resulting
(closed) space ltered Navier-Stokes equations are solved numerically. In this procedure,
there are three essential issues:
1. The \modeling error" committed in approximating T.
2. The \numerical error" in solving the resulting system.
3. Correct boundary conditions for the
ow averages.
In this report, we study the numerical error analytically. Since there are many models
in LES (see, e.g., [IL00], [GL00], [HMJ00], [FP99], [BFR80], [Lia99], [Sag98], [Par92] for
examples) and few analytical studies, we take herein the simplest model commonly in use
presented, for example, in Ferziger and Peric [FP99, Section 9.3].
To describe the model, let D (u) be the deformation tensor associated with the indicated
velocity eld by:
The Reynolds stresses are thought of as a turbulent diusion process based upon the Boussinesq
assumption or eddy viscosity hypothesis that \turbulent
uctuations are dissipative in
the mean", [IL00], [Fri95], [MP94], [Par92]. We will accordingly consider a model of the
where  turb
turbulent viscosity or eddy viscosity. This turbulent
viscosity's determination can be very complex, involving even solutions of accompanying
systems of nonlinear, partial dierential equations. In the simplest case, the turbulent
viscosity depends on the mean
ow u through the magnitude of the deformation
of u;  (D (u)), with a functional dependence. Under the Boussinesq assumption,
r  T should act like a physical viscosity. Following the reasoning of Ladyzhenskaya [Lad70],
thermodynamic considerations imply that the Taylor series of  turb (D ) should be dominated
by odd degree terms. The simplest case is of linear dependence upon jD j
turb (jD
where jD j denotes the Frobenius norm of D . For specicity and for accord with the most
commonly used Smagorinsky [Sma63] model, we take a 0 (-)  0 and a 1 Other
scalings are possible, [Lay96], though less tested, as are many other subgridscale models
typically either chosen to be around 0.1 or taken to be a function
extrapolated as in the \dynamic subgridscale model" of Germano,
[GPMC91].
With the model (2) the resulting system of equations for the approximations (w; q) to
in
in
in
Z
Boundary conditions must be supplied for the large eddies. It is physically clear that large
eddies do not adhere to solid walls. (For example, tornadoes and hurricanes move while
touching the earth and lose energy as they move.) Therefore, in [GL00], [Sah00], (see also
[Par92] for the use of similar boundary conditions in a conventional turbulence model), it
was proposed that the large eddies w should satisfy a no-penetration condition and a slip
with friction condition on @
@
where ~ t is the Cauchy stress vector on , for background information see Serrin [Ser59],
is the friction coe-cient (calculated explicitly in [Sah00]), ^ n the outward unit
normal
j an orthonormal system of tangent vector's on . The friction coe-cient
can be calculated once a specic lter is chosen, [Sah00]. It has the property ([Sah00]) that
no slip conditions are recovered as - ! 0:
A Dirichlet boundary condition
ow on 0 , is appropriate if 0 is an in
ow boundary
upon which u can be calculated by extending the known, in
ow velocity eld upstream.
The Cauchy stress vector ~ t includes the action of both the viscous stresses and Reynolds
stresses and is given by:
Standard properties of convolution operators imply that the
ow averages (u; p) are
in space, have bounded kinetic energy
Z
Z
have no solution scales smaller than O(-) and converge to u as On the
other hand, it is not obvious, nor has it been proven yet, that solutions (w; q) to the large
eddy model approximating (u; p) share any of these properties! Nevertheless, the spacial
regularity of solutions (w; q) we shall consider to be a modeling issue (beyond the scope of
this report studying numerical errors in LES). The time regularity of solutions (w; q) is still
an important consideration. For example, we shall show that solutions of this model satisfy
Z Tkrwk 3
uniformly in Re. One goal is to keep the assumed time regularity as close to L 3 (0; T ) as
possible and below L 4 (0; T ). The fundamental error analysis of Heywood and Rannacher
[HR82] for the Navier-Stokes equations is based, in part, on a laminar-type assumption
62 Weakening this to an assumption of the form ru 2 L 3 (0; T; L
3((as we seek to do herein) is nontrivial.
Preliminaries
This section sets the notation used in the report, describes the function spaces employed and
collects several useful inequalities. The notation used is standard for the most part. The
norms, for p 6= 2, are denoted explicitly as kfk L p . Sobolev spaces W k;p
are dened
in the usual way, [Ada75]. The associated norm is denoted k  k k;p . If the domain in question
is
not
(e.g.,
will be explicitly indicated. If norms will be
for the W
norm and k  k k; for the W k;p () norm and k  k and k  k ,
respectively, for the L
and L 2 () norms. Suppose the polygonal boundary =
@
is
composed of faces
The spaces associated with the boundary conditions (4) are:
d
Z
The boundary condition in X is dened to hold in the sense of the trace theorem on each
is the outward unit normal to = @
The L
and L 2 () inner products are
denoted (; ) and (; ) respectively.
denotes the usual deformation tensor, dened in the introduction. The
denotes an orthonormal system of tangent vectors on . Whenever
it will be understood that the term is to be summed over the two tangent vectors if
For example:
The following dual norms are dened in an equivalent but slightly nonstandard way:
kfk  := sup
kfk W 1;q
R t(f; v)dt 0
R tkD (v)k q
Lemma 2.1 [Inf-Sup Condition]. Let ~
Jg: The velocity-pressure spaces ( ~
X;Q) satisfy the inf-sup condition:
sup
kk
C > 0: (5)
Proof: The trace theorem [Gri92] and Korn's inequality together show that (5) is
implied by the usual inf-sup condition
sup
d
kk krvk  C > 0:
Lemma 2.1 implies that the space of weakly divergence free functions
is a well dened, nontrivial, closed subspace of ~
X. The inf{sup condition (5) is used to get
a bound on kqk once a bound on kD (w)k is proved. The bound on kD (w)k follows from
Lemma 3.1 below. An inf{sup condition with ~
replaced by X and kD (w)k by kD (w)k L 3
would also su-ce to get the bound on kqk.
Remark 2.2 Since is not C 1 discontinuities in  j and ^
have forced modications in the
norms to piecewise denition. For example,
but
The conforming nite element method for this problem begins by selecting nite element
spaces X h  X and Q h  Q, where \h" denotes as usual a representative meshwidth for
satisfying the usual approximation theoretic conditions required of nite element
spaces. The condition that X h  X imposes the restriction that v h  ^
. For intricate boundaries, this could possibly be onerous so it is interesting to
consider imposing
with penalty or Lagrange multiplier methods, following, e.g.,
the work in [Lia99]. Nevertheless, there is already considerable computational experience
with imposing this condition in nite element methods, see, e.g., [GS00], [ESG82], so we
shall not focus on the interesting detail of the treatment of corners. Without these additional
regularizations in the numerical method, it is useful in the analysis to assume that (X h
satises the discrete analogue of (5):
sup
kD (v h
where C > 0 is independent of h. The next lemma shows, in essence, that if the computational
mesh follows the boundary and if the velocity space restricted to no-slip boundary
conditions and the pressure space satisfy the usual inf-sup condition, then (6) holds.
Lemma 2.3 [Discrete Inf-Sup Condition]. If (X
sup
d
then (6) holds.
Proof: By trace theorem [Gri92] and the Poincare-Friedrichs inequality, for any  h 6= 0,
kD (v h
Thus, (6) will be assumed throughout this report. Under (6), the space of discretely
divergence free functions
is a nontrivial closed subspace of X h , [GR86], [Gun89].
We shall frequently use Young's inequality in the form:
ab
a
The generalization of Holder's inequality:
Z
r
is also useful. We shall frequently use the Sobolev embedding theorem, often, but not always,
in the form that in 3 dimensions W
The nonlinear form in the subgridscale term: for v; w 2 W
(jD (w)jD (w); D (v))
is of p-Laplacian type (with Thus, it is strongly monotone and locally Lipschitz
continuous in the sense made precise in the following well-known lemma, see, e.g., [Lay96],
[DG91].
Lemma 2.4 [Strong Monotonicity and Local Lipschitz-Continuity]. There are constants C
and C such that for all
6 d and
(jD
(jD
Korn's inequalities relate L p norms of the deformation tensor D (v) to those same norms
of the gradient for 1 < p < 1, see Galdi [GHR00], Gobert [Gob62, Gob71], Temam [Tem83]
or Fichera [Fic72], and fail if
Theorem 2.5 [Korn's Inequalities]. There is a C > 0 such that for 1 < p < 1
for all v 2 (W
1 d .
Further, if
(v) is a semi-norm on L
which is a norm on the constants, then
holds for 1 < p < 1 and for all v 2 (W
2 d .
As a consequence of Korn's inequality it follows that, taking
for all v 2 fv 2 W 1;p
We will often use Poincare's inequality, which holds since
p. 56.,
kvk
krvk; for all v 2 X:
We shall use the Gagliardo-Nirenberg inequality in W 1;p
\ X to reduce the time
regularity required for w. This inequality [Ada75], [Nir59], [GHR00], [DiB93] states that
provided satises a weak regularity condition (holding in particular for polygonal domains)
and
kvk L q  Ckrvk a
d
where,
for
if
s
In particular, note that taking
kvk L
Ckrvk 2=3
The following combination of this and Korn's inequality will be useful in Section 4.
Lemma 2.6 Let
and
Proof: This follows immediately from (7) and Korn's inequality.
3 Finite Element Formulation
This section develops the nite element method for the LES model. The stability of the
model is also studied. In particular, we show w and w h 2 L
1(uniformly in Re. Lastly, the error in an equilibrium projection is considered.
The variational formulation is derived in the usual way by multiplication of (3) by (v;
(X; Q) and applying the divergence theorem. The boundary integral terms require careful
treatment (following, e.g., [Lia99]) on account of the slip with friction condition on . Let
0 be a constant. The formulation which results is to nd w
(w
and w(x; X. For compactness, dene the nonlinear and trilinear form:
(w
It is a simple index calculation to check that for v 2 X; w 2 V (since such functions have
zero normal components on ) (w  rw; Thus, the variational formulation
can be rewritten as: nd (w;
(w
for all (v;
Using Lemma 2.4, it is easy to prove that the LES model (3), (4) satises the analogue
of Leray's inequality for the Navier-Stokes equations.
Lemma 3.1 [Leray's inequality for the LES model]. A solution of
Z t" J
Proof: in (9) and use Lemma 2.4.
Remark 3.2 1. Because of the slip with friction boundary conditions (4), it is important to
choose the formulation of the viscous terms, as in (8), (9), involving the deformation tensor.
2. Leray's inequality immediately implies stability in various norms (which we will de-
velop) and is the key, rst step in proving existence of weak solutions to (3), (4). This last
question is fully investigated (under dierent boundary conditions) in remarkable papers by
Ladyzhenskaya [Lad67], Pares [Par92] and Du and Gunzburger [DG91].
The continuous-in-time nite element method for (3) uses the variational formulation
as follows. First, velocity-pressure nite element spaces X h  X \ W
satisfying (6) are selected. Next, the least squares parameter   0 is selected.
The nite element approximations to (w; q) are maps (w h ; q h
(w h
for all (v h ;  h is an approximation to w(x;
It is straightforward to verify that Leray's inequality holds for w h as well as w.
Lemma 3.3 [Leray's inequality for w h ]. Any solution of
Using various inequalities in the right hand side, stability bounds for w h follow from
Lemma 3.3.
Proposition 3.4 [Stability of w h ]. The solution w h of (10)
J
Proof: Inequality (11) follows by applying Young's inequality to Lemma 3.3. The
bound (12) follows from the denition of the dual norm and ab  a 3 applied
in the same manner.
For in (10), use Lemma 2.4 and apply the Young's
inequality on the right hand side. This gives:
d
dt
Inequality now follows by using an integrating factor.
In the analysis of the error in the approximation of the time dependent problem, it is
useful to have a clear description of the error in the Stokes projection under slip with friction
boundary conditions, [Lia99]. It is also necessary that any dependence on Re; - and  be
made explicit.
Under the discrete inf-sup condition, the Stokes projection
dened as follows. Let (w;
q), where ( ~
(r  (w ~
((w ~
(r  (w ~
This is equivalent to the following formulation provided Given (w; q),
nd ~
(r  (w ~
((w ~
for all v h 2 V h and  h 2 Q h . Under the discrete inf-sup condition, it is well-known that
q) is a quasi-optimal approximation of (w; q). The dependence of the stability and error
constants upon Re and is important to the error analysis. That dependence is
described in the next lemma and proposition.
Lemma 3.5 [Stability of the projection ~
w]. Let w 2 V be given. Then, ~
w satises if  > 0:
kr  ~
in the second formulation of the Stokes projection. This gives
immediately:
kr  ~
(w
from which the rst result follows. If the term (q; r  ~
w) is bounded by noting that
r  ~
trace (D ( ~
w)) so that
(q; r  ~
Proposition 3.6 Suppose the discrete inf-sup condition (6) holds. Then, ( ~
exists uniquely
in
kr  (w ~
k(w ~
C inf
Proof: The proof follows standard arguments, carefully tracking the dependence of the
constants upon Re and .
Note that the use of least squares penalization of the divergence allows an error estimate
for the Stokes projection whose constants are essentially independent of the Reynolds number
in a suitably weighted norm.
4 The Convergence Theorem
Let us rst note that for standard piecewise polynomial nite element spaces it is known
that, e.g., the L 2 -projection of a function in L p ; p  2, is in L p itself and the L 2 -projection
operator is stable in L
w denote an approximation of w in V h \ W
5 for example,
the L 2 -projection under the conditions of [CT87]. We assume that each norm of ~
w can be
bounded by the same norm of w times a constant which is independent of Re and h. This
assumption is proven for many piecewise polynomial nite element spaces in [CT87].
The error is decomposed as
w) (w h ~
w and
An error equation is obtained by subtracting (9) from (10) and using
the fact that w 2 V . This gives, for any v h 2 V h \ W
and  h 2
This is rewritten, adding and subtracting terms and setting v
The monotonicity lemma (Lemma 2.4) implies that
and with r := maxfkD (w)k L 3 ; kD ( ~
Remark 4.1 If ~
w is taken to be the Stokes projection of (w; q) into V h then, e.g., the term
kD ()k" on this last right hand side does not occur.
Inserting these two bounds in (15) and using the Cauchy-Schwarz and Young's inequalities
d
dt
s r 3=2 kD ()k 3=2
Picking collecting terms gives2
d
dt
This is the basic dierential inequality for the error. Three cases will be considered, revolving
around the treatment of the rst term on the RHS of (16).
Remark 4.2 If completely analogous estimate holds with the pressure term modied
to be either nonuniform in Re (e.g., \C(2Re 1 or nonuniform in -.
Consider the convection terms
The terms containing \" shall be bounded rst. Consider Using
the inequalities in Section 2 appropriately gives
1
1h
The term b(; w h ;  h ) is bounded similarly as follows:
1
Korn's inequality and the stability bounds of Section 3, (12) and (13), immediately imply
that D (w h uniformly in Re so that krw h k 2
uniformly in Re. The
imbedding theorem and Korn's inequality also imply kwk 2
uniformly in
Re. Thus, these bounds su-ce for a later application of Gronwall's inequality.
The rst term containing only  h ; b(w;  h ;  h ), is zero due to skew symmetry. Thus, there
only remains the term Estimating the term is the essential, core
di-culty in obtaining an error bound which is uniform in Re. There are only a few natural
ways to bound this using Holder's inequality and the Sobolev embedding theorem. There
are two cases in which the analysis is successful
very regular, rw 2 L
1(There is one important case in which the analysis fails:
(iii) a 0
3(To highlight subsequent analysis and, hopefully, spur further study, we shall rst present the
case (iii) and explain the failure of the analysis.
4.1 The case rw 2 L 3 (0;
8 and a 0
If we assume only that rw 2 L 3 (0; T; L
84 there is no need to add and subtract terms
since a priori bounds on krw h k L 3 (0;T;L 3 ) have been proven which are uniform in Re. Thus,
we can use Holder's inequality to write:
picking s using the embedding
and Poincare's inequality give
Remark 4.3 Using Lemma 2.6 instead of the embedding of W 1;3 ! L 6 changes the critical
exponent on k h k \3=2" to 12=7 in the rst term of (20) but not the nal conclusion.
Combining (18), (19), (20), with  gives an initial bound on the convection term's
dierence
kk 3=2
Inserting (21) into (16), applying Korn's inequality and collecting terms gives2
d
dt
Thus, pick  such that
d
dt
Consider the three terms bracketed on this right hand side. The rst is approximation
theoretic; the second is an L 1 function multiplying k h (t)k 3=2 the third is an L 1 function
multiplying k h (t)k 2 . Let y(t) := k h (t)k 2 . This inequality may then be written as:
d
dt
(nonnegative terms)  C(t)h
where
The nal step would normally be to apply Gronwall's inequality to deduce y(t) =2
to be bounded by its initial values and approximation theoretic terms. Unfor-
tunately, the term y 3=4 is not Lipschitz, so the argument fails at this last step.
Tracing the inequalities backward, the problem term arises from the steps used to bound
to obtain Re independence. The error analysis in the successful cases (i) and
(ii) centers therefore on alternate bounds for this term. We shall rst consider case (i).
Remark 4.4 If the estimate in (20) is improved as noted in Remark 4.3, the term y(t) 3=4
is changed to y(t) 6=7 but the nal conclusion still holds.
4.2 The case rw 2 L 3 (0;
8 and a 0 (-) > 0
Theorem 4.5 Assume  > 0 and a 0 (-) > 0. Let
a
Then, there is a C independent of Re and h, such that
Further, there is a C independent of Re and h, such that
Then, the error w w h satises for T > 0
~
F(w ~
with
F(w ~
k(w ~
k(w ~
+kr  (w ~
wk 3=2
Proof: This analysis follows closely the previous discussion except for the treatment of
the and the nal application of Gronwall's inequality.
Consider therefore Integration by parts and using the fact that  h  ^
on gives:
Using the embedding H 1=2 ,! L 3 in d = 2; 3 and Young's inequality give
1
Consider now the last term on the above right hand side. By Holder's inequality, we obtain
The Sobolev embedding theorem implies that for any s; 1  s < 1 in 2 or 3 dimensions
C(s;
kD (w h )k 2
This implies that for any r 0 > 2
Consider the last term on the above right hand side. The Sobolev embedding theorem also
implies
(The nal result is not improved by applying here instead the Gagliardo-Nirenberg inequal-
ity.) As r inequality. Thus, picking r close enough to 2
implies, using an embedding inequality and Korn's inequality,
for any t > 0. Thus, for these values of r 0 and s
for any t > 0. For conjugate exponents inequality, we then have
Picking these values of r 0 and s:
Using this bound, (22) and (23) gives nally
Remark 4.6 It appears on rst consideration that this last term (r   h ; w h   h ) can be
agreeably bounded more directly and easily by:
This bound, while certainly true, is not su-cient because of the condition that inevitably
arises from using it that w h or w 2 L
1 . The extra work in the bound we use
reduces the time regularity requirements arising from this term to w h 2 L
1;3((which is bounded uniformly in Re by problem data in Section 3).
Substituting this bound for b( in the derivation of the upper estimate (21) for
the dierence of the convection terms gives
3
To proceed further, (24) is inserted in the right hand side of (16). This yields the
dierential inequality2
d
dt
Pick These choices simplify (25) to:2
d
dt
a
Before applying Gronwall's inequality, let us rst verify that it will indeed give us an
error bound that is uniform in the Reynolds number by considering the coe-cients on the
right hand side of (26).
By the stability estimates kwk 2 uniformly in Re. Thus,
kwk 3=2 kk 3=2
Consider the (critical) bracketed coe-cient of the last term on the right hand side. We
must show this coe-cient is in L 1 (0; T ) uniformly in Re. Indeed, by the stability estimates
uniformly in Re. Since T < 1, L 3 (0; T )
thus the rst factor of the last term is in L 1 (0; T ) uniformly in Re. In addition,
note that r  CkD(w)k L 3.
Hiding all constants in generic C's, Gronwall's lemma now implies for almost all t 2 [0; T
Z tkD (w)k 3=2
Z t1
Note that by the Cauchy{Schwarz inequality in L 2 (0; T ) and the stability estimates
Z tkD (w)k 3=2
C(-)kD ()k 3=2
Now, the essential supremum of t 2 [0; T ] is applied on both sides of the inequality. As
the triangle inequality completes the proof of Theorem 4.5.
4.3 The case rw 2 L 2 (0; T ; L
8 and a 0 (-)  0
We now consider the case of smoother w, i.e.,
9 uniformly in Re;
allowing for the case a 0 (-)  0. This case is primarily of interest because many tests involve
\academic"
ow elds given in closed form (as in Section 5). These are typically smooth and
bounded. In this case Theorem 4.7 gives an error estimate with constants independent of Re
(but depending on - and ). It is noteworthy in this estimate that multiplicative constants
depend on - but the rate constant in the (inevitable) exponential term takes the form
with no explicit dependence on -.
Theorem 4.7 Suppose a 0 (-)  0;  > 0 and w 2 L
9 uniformly in Re. Let
then there is a C (w) such that
be such that
Then, the error w w h satises:
~
F(w ~
with
F(w ~
k(w ~
k(w ~
kr  (w ~
kD (w ~
Proof: In this case, the dierence in the nonlinear terms is decomposed a bit dierently
as:
Consider the individual terms on the right hand side of (27):
1
(w h
Combining these three estimates gives
The term kw h k L 6 is bounded using the Gagliardo-Nirenberg inequality
Since kw h k is bounded uniformly in  and h by (12) or (13), it follows
This bound, together with (28) is now inserted in the right hand side of (16) giving2
d
dt
To apply Gronwall's inequality we need4
in other words w 2 L
The term on the right hand side of this inequality
containing r 3=2 is treated as in the proof of Theorem 4.5. In the nal result of Gronwall's
lemma, we must also verify that the resulting terms containing kD (w h )k L 3 are bounded
uniformly in Re. To this end, apply Holder's inequality:
Z TkD (w h )k 4=3
1. From the stability estimates, we clearly must take q such that 4q=3  3.
Accordingly, take 9. This gives
Z TkD (w h )k 4=3
Similarly, for q and q 0 conjugate exponents:
Z TkD (w h )k 2
kD (w h )k 2
The stated error estimate now follows from Gronwall's inequality and the triangle inequality
as in the proof of Theorem 4.5.
5 A Numerical Example
To give a numerical illustration several decisions must be made, mainly to work on an 'aca-
ow problem with a known exact solution or a more realistic
ow problem containing
the accompanying uncertainties. Since our aim is to illustrate a convergence theorem we have
chosen the former. (To assess a model or study the limitations of an algorithm we would
naturally have chosen the latter.) Accordingly, we have selected the vortex decay problem
of Chorin [Cho68], used also by others, e.g., Tafti [Taf96]. The domain
is
choose
For the relaxation time Re this is a solution of the Navier-Stokes equation consisting
of an array of opposite signed vortices which decay as t ! 1. The right hand side f ,
initial condition and non-homogeneous Dirichlet boundary conditions are chosen so that this
is the closed form solution of (3).
Since we are studying convergence as h ! 0 for - xed and Re varying we have accordingly
chosen
Relaxation time
Vortex conguration
Final time
Eddy Scale
It is signicant that
n so that the vortices are larger than O(-) and hence
should be \visible" to the model. Tables 2 to 4 give the error in L
6 and show
uniformity in Re and in L
5 and show weak dependence on Re (since this term
in the error estimate is scaled by Re 1=2 ).
The least squares constant  is chosen to be zero. Time discretization is by the fractional
step -method with the indicated time steps. The tables show a decrease of the error as h
decreases until it reaches the error introduced by the time stepping procedure. The spacial
discretization is done on a uniform square mesh with the Q 2 =P disc
denoted in
element with the associated meshwidths indicated. The viscous term is treated not
as (rw h ; rv h ) but using the deformation tensor formulation, (D (w h ); D (v h )) as analyzed
herein. Both the Smagorinsky subgridscale model and the convection term are treated
implicitly.
Using the above elements and meshes the calculations involved the numbers of degrees
of freedom listed below in Table 1.
mesh width velocity d.o.f. pressure d.o.f. total d.o.f.
1=8 578 192 770

Table

1: Mesh widths and degrees of freedom (d.o.f.) in space.
These are certainly not extremely large numbers of degrees of freedom. However, their
importance is only relative to the Reynolds numbers chosen
the resolution sought Again, LES is focused on situations in which the number
of degrees of freedom is small relative to Re. Thus, the chosen values of h and Re seem
appropriate.
The

Tables

{ 4 present the L norms of the errors. Note that the trends are
exactly as anticipated by the theory; there is none to minimal degradation in the error as
Re increases from 10 2 to 10 5 and the error plateau's as h decreases at a value which seems
to be the induced error in the time stepping procedure.
Re n h 1/8 1/16 1/32 1/64

Table

2: 0:02. Note the uniformity in Re.
Re n h 1/8 1/16 1/32 1/64

Table

3: observe the uniformity in Re.
Re n h 1/8 1/16 1/32 1/64

Table

4: 0:005: Note the uniformity in Re and the reduction of the
minimal error as t decreases.
The

Tables

{ 7 present the errors in L These are not predicted to be uniform
in Re so some error degradation is expected as Re increases. Very mild degradation is indeed
observed. The degradation is mild, possibly because
are predicted to have the same uniform in Re convergence rates. Thus, the theory forecasts
increase as Re increases until it reaches the (slower) uniform convergence
rate predicted by the kD (e)k L 3 (0;T;L 3 ) bound.
Re n h 1/8 1/16 1/32 1/64

Table

5: kD (e)k L 2 (0;T;L 2
Re n h 1/8 1/16 1/32 1/64

Table

Re n h 1/8 1/16 1/32 1/64

Table

7: kD (e)k L 2 (0;T;L 2



--R

spaces.
Improved subgrid models for large eddy simulation.
Numerical solution for the Navier-Stokes equations
The stability in l p and w 1
Analysis of a ladyzhenskaya model for incompressible viscous ow.
Degenerate parabolic equations.
The implementation of normal and/or tangential boundary conditions in
Existence theorems in elasticity.
Computational Methods for Fluid Dynamics.
Turbulence, the Legacy of A.
An Introduction to the Mathematical Theory of the Navier-Stokes Equations
Fundamental Directions in Mathematical Fluid Mechanics.
Approximation of the larger eddies in uid motion ii: A model for space
Une inequation fondametale de la th
Sur une in
A dynamic subgrid-scale eddy viscosity model
Finite Element Methods for Navier-Stokes equa- tions
Singularities in Boundary Value Problems
Incompressible Flow and the Finite Element Method.
Finite Element Methods for Viscous Incompressible Flows.
Large eddy simulation and the variational multiscale method.
Finite element approximation of the nonstationary navier-stokes problem i: Regularity of solutions and second order error estimates for spacial discretization
An assessment of models in large eddy simulation.
"Al.I.Cuza"
New equations for the description of motion of viscous incompressible uids and solvability in the large of boundary value problems for them.
The Mathemetical Theory of Viscous Incompressible Flow.

A nonlinear
Weak imposition of boundary conditions in the Stokes and Navier- Stokes equation
Analysis of the K-Epsilon Turbulence Model
On elliptic partial di


New perspectives on boundary conditions for large eddy simulation.
Mathematical principles of classical uid mechanics.
General circulation experiments with the primitive equations.
Comparison of some upwind-biased high-order formulations with a second-order central-dierence scheme for time integration of the incompressible Navier-Stokes equations
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--TR

--CTR
Volker John, An assessment of two models for the subgrid scale tensor in the rational LES model, Journal of Computational and Applied Mathematics, v.173 n.1, p.57-80, 1 January 2005
Faisal A. Fairag, Analysis and finite element approximation of a Ladyzhenskaya model for viscous flow in streamfunction form, Journal of Computational and Applied Mathematics, v.206 n.1, p.374-391, September, 2007
navier-stokes equations;turbulence;large eddy simulation;finite element methods