On the Large Eddy Simulation of the Taylor-Greenvortex

Luigi C. Berselli

Abstract. In this paper we analyze the role of the Taylor-Green vortex in the study of

some Large Eddy Simulation models. More precisely, we consider such a particular flow

in connection with the Gradient model and the Rational model. In particular, we study:

1) the mechanism of small scale generation; 2) the symmetries of the flow and their role

in the Spectral Galerkin approximation; 3) the connection between pressure fluctuations

and quadratic mean velocity.

Mathematics Subject Classification (2000) 76F65, 76D05, 65M70

Keywords. Large Eddy Simulation, Taylor-Green vortex, spectral methods.

Dedicated to Hugo Beirao da Veiga in occasion of his 60 th birthday

1 Introduction

In this paper we consider a couple of Large Eddy Simulation (LES) models, derivedfrom approximation by wave number asymptotics.

In a turbulent flow the velocity field u = (u, v, w) is governed by the well-knownNavier-Stokes equations (NSE) for viscous and incompressible fluids

∂tu +∇p +∇ · (u⊗ u)− 1Re

∆u = f,

∇ · u = 0,

u(x, 0) = u0(x),

(NSE)

for x ∈ Ω ⊂ R3 and t ∈ [0, T ].The Direct Numerical Simulation (DNS) is unsuited for practical application

since Kolmogorov’s 1941 theory of homogeneous, isotropic turbulence predicts thatsmall scales exist down to O(Re−3/4), where Re > 0 is the Reynolds number.Thus, in order to capture them on a mesh, we need a mesh-size h ≈ Re3/4, andconsequently (in 3D) N = Re9/4 mesh points. This is unacceptable since, forinstance, for airplanes Re ≈ 2 · 107, requiring N ≈ 2 · 1016 mesh points for a DNS,while a challenging geophysical flow with Re ≈ 1020 requires N ≈ 1045 mesh points.

To overcome this limitation many LES models have been proposed. The mainidea underlying LES is that the velocity u may be decomposed in a smooth part (the

2 L.C. Berselli

mean flow -denoted by u- that is the only part that we really need to compute) andin a highly oscillating part (the turbulent fluctuations, denoted by u′). Note thatthis way to describe fluid motion appeared for the first time in the Codice Atlanticoof Leonardo da Vinci (XV Ith Century, about 300 years before the derivation ofthe NSE).

LES models are systems of partial differential equations modeling the equationssatisfied by the smooth part u of the velocity. Generally u is defined through aconvolution u(x, t) = gδ(x)∗u(x, t), where gδ(x) is a Gaussian kernel (and differentmodels may be derived by using different kernels or also different expansions of thekernel):

gδ(x) =(

6π

)3/2 1δ3

e−6|x|2

δ2 .

The positive number δ represents the width of the smallest scale resolved by theequations. In practice the LES model describe the behavior of eddies bigger thanδ, while the additional terms model the effect of the unresolved scales.

Each LES model requires some modeling. This modeling, necessary from themathematical point of view to derive equations involving only the variables (u, p)is also necessary, in principle, to describe/model the effect of the unresolved scaleson the resolved ones.

1.1 A couple of LES models

Since in the derivation of LES models it is necessary to perform some modeling,one possible approach that is to pass to the Fourier space, and to make formaldevelopments (in order of δ) of gδ(k), the Fourier Transform of the filter gδ(x). Inthis way it is possible to derive the partial differential equations satisfied by

U = (U1, U2, U3) = u + terms that are O(δ4),

see Aldama [1] for further details. Among these models we will consider the Gra-dient model (derived by Leonard [23] and analyzed for instance in Clark et al. [16]and Coletti [14]):

∂tU +∇Q +∇ · (U ⊗ U)− 1Re

∆U+∇ ·[

δ2

12∇U∇UT

]= f,

∇ · U = 0,

U(x, 0) = U0(x).

(GRAD)

The additional (with respect to the NSE) term on the left hand-side, coming froma Taylor series expansion of gδ(k) up to terms of the order of δ4, is defined by[

∇U∇UT]αβ

:=∂Uα

∂x

∂Uβ

∂x+

∂Uα

∂y

∂Uβ

∂y+

∂Uα

∂z

∂Uβ

∂z, α, β = 1, . . . , 3. (1.1)

On the LES of the Taylor-Green vortex 3

This term approximates the subgrid-scale stress tensor and even if the above modelis very common, it suffers of problems of stability in the numerical implementa-tion. Several way to improve its performances are known (for instance adding aSmagorinsky/van Driest damping; by using the dynamic method of Germano, oralso the selective model introduced by Cottet et al.), for a review see [29, 9].

As a possible improvement of the Gradient model, Galdi and Layton [17] in-troduced the Rational LES model (RLES) in which the gaussian kernel is approx-imated through the “subdiagonal Pade” approximation of the exponential. TheRational model reads

∂tU +∇Q +∇ · (U ⊗ U)− 1Re

∆U

+∇ ·(

I− δ2

24∆)−1 [

δ2

12∇U∇UT

]= f,

∇ · U = 0,

U(x, 0) = U0(x).

(RLES)

The derivation of this model seems closer to the physics of the problem and we willfocus our attention on the (RLES) model, since it turns out to be very robust andstable and it allows a reasonable backscatter of energy, too. This model has beenstudied also from the numerical point of view and the results are very promising.See the comparison with other models in Iliescu et al. [19] and the error analysis andthe numerical results collected in John [20]. Further discussion on LES models withtheir physical meaning and surveys on several methods can be found in Sagaut [29]and also in the forthcoming [9].

1.2 Spectral approximation

For the above LES models there are some rigorous analytical results, that we willrecall in Section 2. The mathematical theory is known essentially in the spaceperiodic setting and in the sequel all the functions should be considered as 2π-periodic and defined on the cube ]0, 2π[3. For periodic functions it is not veryexpensive to calculate the inverse of(

I− δ2

24∆)

.

Remark. For the RLES model, in the case of the Dirichlet boundary conditionson U, it is also necessary to assign suitable boundary conditions to the aboveoperator. Generally

(I− δ2

24∆)

is equipped with vanishing Neumann data. On theother hand the problem of boundary conditions for LES models is completely open

4 L.C. Berselli

and we focus just on the periodic setting, that decouples the boundary effects fromthe flow.

The present paper mainly involves the study of a particular class solutions, theso-called “Taylor-Green (TG) solutions.” These solutions are a common bench-mark for numerical codes and to perform significant test on behavior of the ve-locity/pressure properties. They are also used in the numerical detection of thepossible singularities in incompressible flows, since

the fluid dynamic interest in the analysis of the TG-vortex is that itis a very convenient flow on which debug and test sophisticated three-dimensional codes (Orszag 1971).

The particular form of the RLES model makes very appealing the analysis withspectral methods. By defining the Fourier expansion u(k, t) of the field u as

u(k, t) =∫

]0,2π[3u(x, t)e−ik·x dx, k ∈ Z3,

where i denotes the imaginary unit, we can write the equation for the wavenumbercomponents U(k, t) = (U1, U2, U3) of the RLES model as follows:

d

dtUα(k, t)− 1

Re|k|2Uα(k, t) = Pαγ

∑k′+k′′=k,

Uβ(k′, t) k′′

β Uγ(k′′, t)

+δ2

12Pαγ

∑k′+k′′=k

kβ

1 + δ2

24 |k|2k′

l Uβ(k′, t) k′′

l Uγ(k′′, t),

kαUα(k, t) = 0,

(1.2)

where we used the Einstein convention of summation over repeated indices, while

Pαγ =i

8π2

(δαγ −

kαkγ

|k|2

)is the projection operator on divergence-less functions. In this case it that can bewritten explicitly since we are working in the periodic setting. Furthermore, sincethe velocity u is real, then U is conjugate symmetric:

U∗(k, t) = U(−k, t).

1.2.1 Galerkin procedure

The pure Fourier Galerkin method is obtained by considering the equation (1.2)for the wave number U(k, t) and truncating it with the finite cutoff |kα| < K,

On the LES of the Taylor-Green vortex 5

where K ∈ N is a given number. In this way the PDE becomes a system ofordinary differential equations for the unknown Uα

K ∈ R2K−1, that is obtained bydisregarding the frequencies bigger than the cutoff K. The equation for the Fouriercoefficients are:

d

dtUα

K(k, t)− 1Re|k|2Uα

K(k, t) = Pαγ

∑k′+k′′=k,

|k′|, |k′′|<K

UβK(k′, t) k

′′

β UγK(k′′, t)

+δ2

12Pαγ

∑k′+k′′=k,

‖k′‖,‖k′′‖<K

kβ

1 + δ2

24 |k|2k′

l UβK(k′, t) k

′′

l UγK(k′′, t),

(1.3)

Spectral method have some advantages over finite difference/element method.Among the others, they may have bigger accuracy properties and the particularform of the basis functions it allows for full utilization of the symmetries of theflow to be simulated. The most expensive part in a spectral code is the evaluationof the right hand side. In particular, the direct evaluation of a convolution sum ofthe type

w(k) =∑

k′+k′′=k

u(k′)v(k′′) (1.4)

is in practice unsuitable for real life simulations. Due to its non-locality, a con-volution sum as in (1.4) makes very difficult to use efficiently multi-processormachines and parallel algorithms, too. In fact, for each value of k there areN(k) =

∏3i=1(2K − 1 − |ki|) values of k′ that contribute to the sum on the right

hand side. A direct evaluation method of (1.4) would require approximatively∑|k|<K N(k) ' 3

4 (2K − 1)2. In the case of the NSE this is the computational costfor each scalar component so the final computational requires approximatively(

34

)3

(2K − 1)6

arithmetic operation at each evaluation of the right hand side. Note that, in the caseof the RLES model, the second term on the right-hand side of (1.2) is very similarto the one deriving from the Fourier transform of the convective term ∇ · (u⊗ u).These two terms are similar in the sense that essentially they requires the sameamount of expensive (convolution) operations, see also the analysis in Section 4.2.Note also that the equations satisfied by the Fourier components of the Gradientmodel are very similar to (1.2), since it is enough to remove the ratio with 1 + δ2

24 |k|2

and the number of operations essentially does not change.

6 L.C. Berselli

Plan of the paper. In this paper we will develop the classical approach of Taylorand Green to make a starting point in the theoretical/computational validation ofsome LES models. In Section 2 we review several analytical results. In Section 3we qualitatively compare the generation of small scales for the NSE and some LESmodels. In Section 4 we consider series expansion of the Taylor-Green vortex,together with the analysis of some symmetries, that make the spectral approxima-tion competitive. Finally, in Section 5 we compare the quadratic mean value ofthe pressure (and of its gradient) to the quadratic mean velocity, to detect possiblediscrepancies of the approximate models with respect to the full NSE.

The scope of this paper is to furnish the basis for a cheap numerical implemen-tation of the RLES model after having shown that the TG-vortex is a good (forseveral reasons) benchmark problem. The next, absolutely necessary, step is anadequate numerical study, with a state-of-the-art LES simulation of Taylor-Greensolutions. This will be the object of a future paper [7].

2 A brief review of some results

In this section we recall the basic analytical results that ensure the local (in time)well posedness of the two LES models. We also show some classes of exact solutionsand we introduce the Taylor-Green vortex and its role in incompressible fluids.

2.1 Analytical results

Regarding the (GRAD) model the local existence, uniqueness, and stability of weaksolutions1 have been proved by Coletti [14].

Theorem 2.1 If U0 ∈ W 1,30 , with ∇ · U0 = 0, and if f, ∂tf ∈ L2(0, T ;L2), then

there exists a unique weak solution U ∈ L∞(0, T ;L2)∩L3(0, T ;W 1,30 ) to the gradient

model (GRAD), provided an additional (Smagorinsky-like) dissipative term

−Cs∇ · (|∇U |∇U), with Cs ≥δ2

6,

is added on the left-hand side. Furthermore, if U ′ and U ′′ are solutions correspond-ing to the same external force f, then the following stability estimate holds:

∃ c > 0 : ‖U ′ − U ′′‖L∞(0,T ;L2) ≤ ‖U′

0 − U′′

0‖L2 exp(c‖∇U ′‖2L2(0,T ;L3)

)1The symbol Lp denotes the usual Lebesgue space, while W 1,3

0 denotes the customary Sobolevspace of functions vanishing on the boundary and belonging to L3, together with their first orderdistributional derivatives.

On the LES of the Taylor-Green vortex 7

For the (RLES) model analytical results have been proved in the space periodiccase, i.e., the solution can be written as

u =∑

k∈Z3

ckeik · x, with ck = c−k, c0 = 0, and k · ck = 0. (2.1)

We consider the space Hsper, s ∈ R, of functions satisfying (2.1), for which the

functional ‖u‖2Hsper

=∑

k∈Z3 |k|2s|ck|2 is finite. In this setting it is not necessary toadd extra-dissipative terms and the main results of existence of smooth solutionshave been proved in [6] and improved in [3, 8].

Theorem 2.2 Let be given U0 ∈ H1per, with ∇ · U0 = 0, and f ∈ L2(0, T ;L2

per).Then, there exists a strictly positive T ∗ = T ∗(‖∇U0‖L2 , Re, f) such that thereexists a unique strong solution U to (RLES) in [0, T ∗) satisfying

U ∈ L∞(0, T ∗;H1per) ∩ L2(0, T ∗;H2

per).

Furthermore, a) if ‖U0‖H1 is small enough, then T ∗ = ∞; b) if the external forcef is smooth, then U ∈ C∞(]0, T ∗[×]0, 2π[3). Note that the positive number T ∗ isindependent of δ

For the Rational model in [8] we also proved the “consistency”, i.e., the convergenceto the solutions of the NSE, as the radius of the gaussian kernel vanishes.

Theorem 2.3 Let U be a strong solution (we do not write it explicitly, but itdepends on δ) to (RLES), while let u be a strong solution to the NSE, in a commontime interval [0, T ]. Let us suppose that both the initial data are smooth (say U(x, 0)and u(x, 0) belong also to H2

per) and that

∃ c1 > 0 : ‖U(x, 0)− u(x, 0)‖L2 ≤ c1δ2.

Then we have, for some c2 > 0,

supt∈[0,T ]

‖U(x, t)− u(x, t)‖L2 ≤ c2 δ2.

2.2 Some exact solutions

In view of comparing the computational results we investigated into analyticalsolutions that can be used to benchmark the results of numerical simulations.In [3] we discovered that in two dimensions the well-known “Poiseuille flow” andthe “2D-Taylor Solutions” are not only exact solutions to the NSE, but they alsosolve both (GRAD) and (RLES). This turns out by an explicit calculation ofthe additional term (1.1). We also investigated into the more interesting case ofthree-dimensional flows. It is possible to construct explicit solutions to the NSE

8 L.C. Berselli

by following the generalization of the Taylor solution given by Ross Ethier andSteinman [28]: produce a vector-valued generalization Ψ of the stream function,i.e., a vector field Ψ = (Ψyz,Ψzx,Ψxy) such that:

∆Ψ(x, y, z) = λ2Ψ(x, y, z).

Then the field V (x, y, z)T (t) defined by;

u(x, y, z, t) := V (x, y, z)T (t) = ∇×Ψ(x, y, z) eλ2t, (2.2)

where λ is a complex constant, results an exact solution to the NSE. This solutioncan be expressed as follows

Ψyz = f(y)g(z)h(x), Ψzx = f(z)g(x)h(y), Ψxy = f(x)g(y)h(z),

withf ′′ = l2f, g′′ = m2g, h′′ = n2h,

and l2 + m2 + n2 = λ2. The flow obtained is a particular case of the generalizedBeltrami [4] flows. These are flows such that∇×

(V×(∇×V )

)= 0 and consequently

the term V ×(∇×V ) can be expressed as a gradient and absorbed into the pressureterm. Note that the 2D Taylor solution is recovered by setting Ψyz = Ψzx = 0 andh = 1.

We consider a particular form of (2.2), namely

u = [sin(z) + cos(y)] e−t/Re

v = [sin(x) + cos(z)] e−t/Re

w = [sin(y) + cos(x)] e−t/Re

p = −[cos(x) sin(y) + sin(x) cos(z) + sin(z) cos(y)] e−2t/Re.

(2.3)

The flow (2.3) is the viscous counterpart of the classical ABC/Arnold-Beltrami-Childress flow, that has been studied by Arnold [2] and Childress [13] in connectionwith problems of stability and breakdown of smooth solutions for the Euler equa-tions.

In solution (2.3) “current lines” coincide with “trajectories,” while the modulusof velocity is not constant in time. The qualitative behavior is similar to that ofTaylor solutions. The main difference is that in the Taylor solution, the vorticity isperpendicular to the velocity, while in Trkal solution the vorticity is aligned withthe velocity field. Straightforward calculations show that (2.3) is an exact solutionof the Navier-Stokes equations (NSE) and that it belongs to the class of Trkal flows,i.e., flows such that ∇× u is aligned with u.

On the LES of the Taylor-Green vortex 9

Explicit calculations for the “turbulent part” of the stress tensor show that if uis defined by (2.3), then

div(

I − δ2

24∆)−1

δ2

12(∇u∇uT

)≡

000

. (2.4)

This holds since if we set x1 = x, x2 = y, x3 = z, then

(∇u∇uT )α,β = Fαβ(xγ) e−2t/Re, for α, β, γ = 1, 2, 3, with γ 6= α, γ 6= β

where Fαβ(xγ) are functions depending only on xγ .The latter identity implies that div(∇u∇uT ) = 0. In the case of the RLES

model, the fact that the solution of the problem with the inverse of I − δ2∆/24is to be found in the space of periodic functions easily implies (2.4). This provesthat (2.3) is also an explicit solution to both the 3D (GRAD) an the (RLES) model.

From the numerical point of view in [3] we detected numerical stability of the 2DTaylor solution for the RLES model. On the other hand the instability of Gradientmodel (without additional damping) destroys very soon the 2D Taylor flow and thevorticity is spread out all over the computational domain. Furthermore, over longertimes, solutions to (GRAD) produce also anomalous increase and concentration ofthe vorticity field, if a Smagorinsky dissipative term is not added.

2.3 The Taylor-Green vortex

The solution (2.3) is of interest since it shows that some LES models have anadditional term with a structure that allows the persistence of wide classes of exactsolutions. Furthermore, starting from this common basis of explicit solutions, it ispossible to perform an unitary treatment of the NSE and of some classes of LESmodels.

Nevertheless, the solution (2.3) is trivial from the point of view of benchmarkingnumerical results or to study the dynamical behavior of turbulent flows, since -essentially- (2.3) involves only one mode. In this respect we quote G.I. Taylor(1937):

“The largest class of three-dimensional motions which have been solvedis the irrotational motions of a non-viscous fluid. Here again there areno vortex lines, so that no motion of this type are significant in con-nection with turbulence. Another class of motions which can be treatedby existing methods is deviations from states of rest of steady motion.Such motion are only significant in discussing the first beginning of tur-bulence arising in a steady flow”.

To perform physically meaningful calculations we will follow the classical approachof Green and Taylor [18] and we will consider the flow that develops from a very

10 L.C. Berselli

simple initial datum:u(x, y, z, 0) = A cos(ax) sin(by) sin(cz),

v(x, y, z, 0) = B sin(ax) cos(by) sin(cz),

w(x, y, z, 0) = C sin(ax) sin(by) cos(cz),

(2.5)

where to satisfy the divergence-free constraint it is necessary to require that aA +b B + cC = 0. In the next section we analyze how this initial datum, with only onefrequency (in each space direction), may generate a very complex flow. This kindof analysis have been introduced in [18] with the idea that:

“It appears that nothing but a complete solution of the equations of mo-tion in some special case will suffice to illustrate the process of grindingdown of large eddies into smaller ones...”

We also note that the initial condition (2.5), or the similar one (the “one-mode”that can be obtained by a shift of the origin of z by π/2 and by setting a = b = 1and B = −A = −1)

u(x, y, z, 0) = cos(x) sin(y) cos(z),

v(x, y, z, 0) = − sin(x) cos(y) cos(z),

w(x, y, z, 0) = 0,

(2.6)

has been used by several authors to study the possible generation of singularitiesin incompressible viscous and ideal fluids, see for instance the review in Majda andBertozzi [24] §5.

Note that the initial datum (2.6) is of interest since it also allows the mecha-nism of vortex stretching. At t = 0 the vortex-lines are bended (see Figure 1) andthey naturally tend to generate intricate geometries. This behavior is shown inBrachet et al. [10, 11], where pseudo-spectral methods have been used to numer-ically predict the formation of singularities. Later Brachet [12] studied the sameproblem with “state of the art” supercomputers and monitored the accumulation ofvorticity as in the Beale-Kato-Majda criterion: if T∗ < ∞ is the maximal intervalof existence of smooth solutions for the Euler equations, then

lim supT→T∗

∫ T

0

supx|∇ × u(x, s)| ds = +∞.

These new experiments showed a bounded accumulation of vorticity and no evi-dence for the breakdown conjecture. In any case this may happen even if the flowis very complicated and complex small-scale-structures develop. The reader may

On the LES of the Taylor-Green vortex 11

-2

0

2

-4

-2

0

-1

0

1

-2

0

2

-4

-2

0

Figure 1: Some vortex-lines at t = 0.

also consider the simulation performed by Kerr [21] with highly anisotropic layerof intense vorticity peaking: the numerical method shows a factor 20 amplificationof the initial vorticity and the potential singularity is conjectured by extrapolatingthe physical structure and the trends of solutions. In this case the geometry of thesolution, together with the small scale structure analysis, seems one of the mostimportant points in the modern study of turbulent flows.

3 On the generation of small scales

The generation of small scales is a classical phenomenon in non-linear problems. Apreliminary analysis of this effect is the one proposed for the NSE in [18] by meansof a Taylor series development with respect to the time variable. We will followthe same guidelines to study the role of big eddies in LES models. For the reader’sconvenience we sketch out the result obtained by Green and Taylor.

Notation. In the sequel we will use the following symbol∑σ

. It means that in the

summation we must add to the written term also those coming out from the twocyclic permutations

1)

a → b b → c c → a

A → B B → C C → A

x → y y → z z → x

and 2)

a → c b → a c → b

A → C B → A C → B

x → z y → x z → y.

This compact notation allows to write short formulas for the terms we are interestedin.

12 L.C. Berselli

3.1 First order approximation

Throughout this section we suppose, for simplicity, that the external force f van-ishes identically. We make this assumption since the presence of a smooth externalforce do not influence significantly the results and we are mainly interested in thefree evolution of a fluid, to focus on the nonlinear behavior dictated by the equa-tions.

The derivation of the Taylor-series-expansion with respect to time is deducedby using the following strategy: given the (NSE) if the initial value p(x, y, z, 0) ofthe pressure is known, it is possible to calculate ∂tu(x, y, z, 0) and then to calculateu with a first order approximation in time:

u(x, y, z, t) ' u0(x, y, z) + t ∂tu(x, y, z, 0).

The natural way to calculate the value of the pressure is to take the divergence of(NSE) to obtain

−∆p =(

∂u

∂x

)2

+(

∂v

∂y

)2

+(

∂w

∂z

)2

+ 2(

∂v

∂x

∂u

∂y+

∂w

∂y

∂v

∂z+

∂u

∂z

∂v

∂x

).

Then, by using (2.5) to substitute (u0, v0, w0), one easily gets

−∆p(x, y, z, 0) =12

∑σ

A2a2 [cos(2by) cos(2cz)− cos(2ax)] .

In the periodic setting the inversion of the Laplace operator gives

p(x, y, z, 0) =18

∑σ

A2

[(a2

b2 + c2

)cos(2by) cos(2cz)− cos(2ax)

]. (3.1)

Substituting into (NSE) the value of ∇p(x, y, z, 0) we have the initial value for ∂tuand Taylor and Green obtained the following formula (for the first component ofut):

ut(x, y, z, 0) =− θ1

ReA cos(ax) sin(by) sin(cz) +

A3

asin(2ax) cos(2by)

− A2

asin(2ax) cos(2cz),

(3.2)

where θ = a2+b2+c2, while A1, A2, and A3 are defined through cyclic permutationsfrom

A1 :=b

4

[A2b

a2

b2 + c2+ ABa

]. (3.3)

On the LES of the Taylor-Green vortex 13

The first order approximation is then the following

u(x, y, z, t) =A(1− θ1

Ret) cos(ax) sin(by) sin(cz) +

A3 t

asin(2ax) cos(2by)

− A2 t

asin(2ax) cos(2cz),

with corresponding values for v and w, obtained again by cyclic permutations.The most interesting result for our intent is that a very simple initial datum

will generate smaller and smaller scales in a very precise manner. This is one ofthe main features present in non linear phenomena and in particular in fluid flows.In particular, the high-order analysis presented in [18] shows how the effect is notsimply a period-doubling (cited in Landau and Lifshitz [22] as the source of chaoticbehavior, even if this hypothesis has been reconsidered) but also a specific way inwhich higher order terms are organized, see Section 4 for further details in thisdirection.

3.1.1 Some remarks on the mean value of the vorticity

Together with this expansion analysis, it is interesting also to capture part of thebehavior of the solution by considering the role of “mean quantities”. Given avector field V we denote by V its mean value

V :=1

8π3

∫]0,2π[3

V (x, t) dx.

G.I. Taylor was interested in the behavior of various mean quantities, as velocity,vorticity, and pressure. We can observe that this concept is very similar to that ofLES, since a mean quantity may be be a first, rough description of the macroscopicbehavior of a fluid (it is very similar to the action of a Box Filter). Recall also thatReynolds [27] proposed a decomposition of the velocity in smooth part/turbulentfluctuations, by mean of a time averaging, while the idea of LES is a space average,with a function that acts as a low-pass filter. In Section 5.1.2 we will analyze thebehavior of mean quantities, in connection with the pressure fluctuations.

In [18] it is also studied the evolution of the vorticity ω = ∇ × u = (ξ, η, ζ).In particular, in the periodic setting it holds the following relation between themean-square vorticity ω 2(t) and the rate of kinetic energy decay ε(t) :

ε(t) = − d

dt

12

u 2 =1

Reω 2(t).

Taylor an Green found also the following expansion for the mean quadratic vortic-ity:

ω 2(t) =34

[1− 6t

Re+(

548

+18

Re2

)t2 −

(53

+36

Re2

)t3

Re+ . . .

]. (3.4)

14 L.C. Berselli

It is clear that finite order truncation cannot remain valid for “big t”, but exami-nation of (3.4) suggests that, as Re → +∞ the time series diverges, if t ≥ 3.

Later, Goldstein performed a perturbation analysis and developed the flow inpowers of the Reynolds number, to obtain the following expansion for the samequantity:

ω 2(t) =34

[e−6t/Re − Re2

384(e−6t/Re − 20e−12t/Re + 35e−14t/Re − 16e−16t/Re + . . .

]Again the series seems to diverges for t/Re ≥ 1 and Re ≥ 20, see Orszag [26].

3.2 Small scale generation for the Rational LES model

In this section we follow the same calculations of Taylor and Green, to show thatthe (RLES) model replicates the generation of small scales, and in particular byshowing that the first-order approximation of the velocity involves O(δ2)−termsthat are, qualitatively, the same appearing in the NSE.

The technique is the same: calculate the initial value of the pressure by takingthe divergence of (RLES)

−∆Q = −∇ ·

∇ · (U ⊗ U) +∇ ·

(I− δ2

24∆)−1 [

δ2

12∇U∇UT

]

and calculate the initial value of the pressure Q ' p. After several simplification offifty-four additional terms (additional with respect to expression (3.1) derived forthe NSE) we easily get

Q(x, y, z, 0) =18

∑σ

A2a2

[cos(2cz) cos(2by)

c2 + b2− cos(2ax)

a2

]+

+∑

σ

δ2A2(a2 − b2 − c2)16(6 + a2δ2)

cos(2ax)−

−∑

σ

δ2c2C2(a2 + b2 − c2)16(a2 + b2)(6 + a2δ2 + b2δ2)

cos(2ax) cos(2by).

With the above formula and after several simplifications coming from the constraintaA + bB + cC = 0 we get the following initial value for the time derivative (of the

On the LES of the Taylor-Green vortex 15

first component of U):

∂tU1(x, 0) =− θ

1Re

A cos(ax) sin(by) sin(cz) +(

A3

a+ B3 δ2

)sin(2ax) cos(2by)−

−(

A2

a−B2δ

2

)sin(2ax) cos(2cz),

where

B3 :=b[−(a2 A B

)+ A b2 B + a b

(A2 −B2

)] (a2 + b2 − c2

)8 (a2 + b2) (6 + a2 δ2 + b2 δ2)

B2 =c[−(a2 A C

)+ A c2 C + a c

(A2 − C2

))](a2 − b2 + c2

)8 (a2 + c2) (6 + a2 δ2 + c2 δ2)

Note that B2 is obtained by B3 with the transpositionb → c, B → C

.

In the above formula the terms Ai are the same appearing in (3.3). The other com-ponents ∂tU

2(x, 0), ∂tU3(x, 0) may be obtained -as usual- by cyclic permutations

of the various indices.Then, we have shown that:

1. The (RLES) models introduce the same new doubling frequencies as the NSE.The behavior is very similar and the growth of new frequencies is replicated.

2. The additional terms are of the order of δ2. In practical calculations δ shouldbe of the same order of a) the mesh-size h, in finite elements or finite dif-ferences methods; b) 1/K where K is the cutoff frequency in the spectralapproximation. The additional terms are then reasonably small perturba-tions of those appearing in the full set of NSE.

3.2.1 The isotropic case

To better compare the results for the NSE with those regarding the RLES modelwe can consider the very special case of initial data (2.6), i.e., we set:

a = b = c, B = −A, and C = 0.

With this choice of the initial datum we get for the NSE the following first orderapproximation

u(x, y, z, t) ' A(1− 3a2 1Re

t) cos(ax) sin(ay) sin(az)− A2a t

8sin(2ax) cos(2az),

16 L.C. Berselli

while for the RLES model we get the following expression for the approximatevelocity

U1(x, y, z, t) 'A(1− 3a2 1Re

t) cos(ax) sin(ay) sin(az)

− A2a t

8

(1− aδ2

2(6 + 2a2δ2)

)sin(2ax) cos(2az),

and it is now very clear what is the effect of the additional term, together with itsasymptotic behavior in δ.

3.2.2 Small scales for the Gradient model

We only write the expression for the gradient model since its derivation is essentiallythe same:

Q(x, y, z, 0) =18

∑σ

A2a2[cos(2cz) cos(2by)

c2 + b2− cos(2ax)

a2]+

+∑

σ

A2δ2(a2 − b2 − c2

)96

cos(2ax)−

−∑

σ

δ2c2C2(a2 + b2 − c2

)96 (a2 + b2)

b cos(2ax) cos(2by).

∂tU1(x, 0) =− θ

1Re

A cos(ax) sin(by) sin(cz) +(

A3

a+ B3 δ2

)sin(2ax) cos(2by)

−(

A2

a+ B2 δ2

)sin(2ax) cos(2cz),

where

B3 :=b[−(a2 A B

)+ A b2 B + a b

(A2 −B2

)] (a2 + b2 − c2

)4 (a2 + b2)

,

while B2 is obtained by B3 with the transposition b → c and B → C; the othercomponents are again obtained through cyclic permutations of the indices.

On the LES of the Taylor-Green vortex 17

From this point of view we can see that also the (GRAD) model replicates thesmall scale generation of the full system (NSE). It will be a further analysis of thepressure properties that will reveal other properties and differences in these twomethods.

4 High order approximations and numerical meth-ods

The main point is that in the expansion (3.2) -or the analogous for LES models-all the terms are of the same type as those in the original velocity (we follow thenotation of Green and Taylor [18]) namely:

cossin

lax

cossin

mby

cossin

ncz, (4.1)

where l,m, and n are integers. The same process may be applied to find u to asecond order approximation involving the same terms, and so on. In this way thesolution will consist entirely of terms of the type of (4.1), multiplied by a powerseries, in the time variable t. In fact, if a solution of the type

uvw

=

∑k∈N

kAcsclmn tk

∑k∈N

kBcsclmn tk

∑k∈N

kCcsclmn tk

cossin

lax

cossin

mby

cossin

ncz

is assumed, the procedure applied in the previous paragraph may be used to calcu-late the coefficients kAcsc

lmn, kBcsclmn, and kCcsc

lmn, where the upper suffices are used toshow which of the two alternatives sine or cosine occur in the Fourier development.

4.1 Symmetries of the flow

Instead of going into these calculations (in the literature you can find the calcula-tion, for the NSE, up to very high orders), we prefer to analyze another discovery,made by Orszag [26], that is very important for the numerical approximation byusing spectral methods. The main observation is that the flow developing from theinitial condition (2.6) has several symmetries, namely:

u(x) = −u(−x) (4.2)

18 L.C. Berselli

u(x1, x2, x3) = u(x1, x2,−x3) (4.3a)v(x1, x2, x3) = v(x1, x2,−x3) (4.3b)w(x1, x2, x3) = −w(x1, x2,−x3) (4.3c)

v(x, y, z) = u(y,−x, z) (4.4a)w(x, y, z) = w(y,−x, z) (4.4b)∫

]0,2π[3u(x) e ik·x dx = 0, if ki are of mixed parity (4.5)

u(x, y, z) = u(x + π, π − y, z + π) (4.6a)v(x, y, z) = −v(x + π, π − y, z + π) (4.6b)w(x, y, z) = w(x + π, π − y, z + π) (4.6c)

Each one of these properties may be used to reduce the computational cost of anumerical method and the memory storage amount. In particular, properties (4.2)-(4.3)-(4.6) speed up the calculation about a factor 2; property (4.4) gives a factor3/2, while (4.5) gives a factor 5. The theoretical increase of efficiency is then 48.We do not show the full details of the implementation, since the reader can findthem in [25], Section 5. In any case in Section 4.2 we outline the main ideas.

4.1.1 Series expansion and LES models

From the above symmetries (4.2)-(4.6) it is possible to derive several symmetriesalso for the Fourier Components. In fact, they imply, respectively

u(k) = −u(−k) (4.7)

u(k1, k2, k3) = u(k1, k2,−k3) (4.8a)v(k1, k2, k3) = v(k1, k2,−k3) (4.8b)w(k1, k2, k3) = −w(k1, k2,−k3) (4.8c)

u(k) = 0 unless k1 ≡ k2 ≡ k3 (mod 2) (4.9)

u(k1, k2, k3) = u(k1,−k2, k3) if ki ≡ 0 (mod 2) (4.10a)v(k1, k2, k3) = −v(k1,−k2, k3) if ki ≡ 0 (mod 2) (4.10b)w(k1, k2, k3) = w(k1,−k2, k3) if ki ≡ 0 (mod 2) (4.10c)u(k1, k2, k3) = −u(k1,−k2, k3) if ki ≡ 1 (mod 2) (4.10d)v(k1, k2, k3) = v(k1,−k2, k3) if ki ≡ 1 (mod 2) (4.10e)w(k1, k2, k3) = −w(k1,−k2, k3) if ki ≡ 1 (mod 2) (4.10f)

On the LES of the Taylor-Green vortex 19

The symmetries give also the following simplified expression for a series develop-ment of the Taylor-Green vortex: u

vw

=∞∑

m=0

∞∑n=0

∞∑p=0

a(1)mnp cos[(2m + 1)x] sin[(2n + 1)y] cos[(2p + 1)z]

a(2)mnp sin[(2m + 1)x] cos[(2n + 1)y] cos[(2p + 1)z]

a(3)mnp sin[(2m + 1)x] sin[(2n + 1)y] sin[(2p + 1)z]

+

+∞∑

m=0

∞∑n=0

∞∑p=0

b(1)mnp sin[(2m)x] cos[(2n)y] cos[(2p)z]

b(2)mnp cos[(2m)x] sin[(2n)y] cos[(2p)z]

b(3)mnp cos[(2m)x] cos[(2n)y] sin[(2p)z]

.

The important fact, that makes the Taylor-Green vortex an efficient benchmarkfor the (GRAD) and (RLES) models, is that the smooth solutions of these modelsshare the same symmetries! This can be checked by direct computation and, forbrevity, we justify the validity of (4.9), leaving the others to the reader.

We consider the evolution of a wavenumber U(k, t), for a fixed k ∈ Z3. Theevolution is driven by the system (1.2) (we consider it, but the same holds alsofor the Galerkin approximation (1.3) and this is crucial for the implementation ofthe symmetries). In particular, we can construct the unique smooth solution, thatexists for some time interval [0, T ∗), by a limit of an explicit finite difference (intime) “Euler scheme”. Let us suppose, for instance, that k1 is even, while k2 isodd. We note that U(k, 0) ≡ 0, due to the particular form of the initial datum. Wewill understand that the second term on the left-hand-side of (1.2) does not givea contribution, since it involves U(k, t) and does not contain any other k 6= k. Apossible contribution may come from the convolution terms on the right-hand-side.Since they come from convolution, they are sum of terms

F (k, δ) U(k′, t)U(k

′′, t), with k = k

′+ k

′′.

The four possible choices of(

k′

k′′

)=(

k′1 k′2 ∗

k′′1 k′′2 ∗

)are then(

e e ∗e o ∗

) (e o ∗e e ∗

) (o e ∗o o ∗

) (o o ∗o e ∗

)where e and o stay for even and odd, respectively, while ∗ denotes any possiblechoice of the last component. This shows that, for any possible choice of k3 thevector k must be the sum of terms in which at least one is of “mixed parity”. Notethat this argument does not distinguish between the first and the second term onthe right-hand-side of (1.2) and this is the reason why the properties for the NSEare transported to the (RLES) model. Then, we divide the interval [0, T ∗) withthe points tn = nh, being h = T ∗/N. We can write, for n = 0, . . . , N − 1

U(k, tn+1) ' U(k, tn) + h

[|k|2

ReUα(k, tn) + r.h.s. of (1.2) evaluated at t = tn

]

20 L.C. Berselli

and, by recurrence all the terms on the right hand side of the above expressionvanish identically. This holds because each term of the convolution is the productof expressions in which at least one is vanishing, being of mixed parity. Passing tothe limit with respect to N, and using uniqueness of the solution, we obtain thedesired result.

4.2 Discrete Fourier Transform

In this section we briefly recall a well known method to calculate in a fast mannera convolution sum as in (1.4) and how it should be modified in order to perform asimulation of the TG vortex by means of a DNS or a LES. We shall use the DiscreteFourier Transform (DFT) of a set of N complex numbers

Zj =∑

|k|≤1/2N

z(k) e ikxj with j = 0, 1, . . . , N − 1, for the grid points xj =2πj

N,

and |k| < K means −1/(2N) ≤ k < 1/(2N). The simplest way to calculate theconvolution sum in (1.4) is to introduce the physical-space transform Uj , Vj , of thespectral quantities u(k), v(k), by means of the DFT with N = 2K (set u(−K) =v(−K) = 0 to make the expression meaningful). Then, for j = 0, . . . , 2K − 1 wedefine Wj := Uj Vj . It is easy now to compute w(k), the inverse DFT of Wj

w(k) =∑

|k′|≤K, |k′′|≤K

u(k′)v(k′′)1N

N−1∑j=0

ei(k′+k′′−K)xj

obtaining

w(k) = w(k) + w(k + 2K) + w(k − 2K), for |k| ≤ K.

In the last equality two of the three terms on the right hand side are due toaliasing and at most one can be non zero. The standard technique to “de-alias”and to compute w(k) exactly is to append zeroes to u(k) and v(k). By extendingu(k) and v(k) to zero for K ≤ |k| ≤ 2K one can compute U j and V j , (they areDFT with with N = 4K points) and then, computing the inverse 4K−points DFTof W j := U jV j , it follows that

w(k) = w(k) + w(k + 4K) + w(k − 4K).

Consequently, w(k) = w(k) for |k| ≤ 2K. This algorithm requires three 4K−pointsDFT transforms and 4K local multiplications.

The disadvantage of using the standard de-aliasing is that the usual transformmust be performed on 4K points rather than 2K. Another standard tool is tocompute this 4K-point transform as “two 2K-points DFT transforms, by usingthe Cooley-Tukey [15] factorization. In the case of the Taylor-Green vortex it isuseful to define special grids that can improve the performances of the method, bytaking into account the symmetries (4.7)-(4.10).

On the LES of the Taylor-Green vortex 21

4.2.1 Approximation of the Taylor-Green vortex

The facts we recall in this section are well-known, see Section 3 in [25]. It is inter-esting to note that the same properties can be transferred to numerical simulationof the LES models we consider. To this end we define the set of points

xj−1/4 =2π(j − 1/4)

2Kxj+1/4 =

2π(j + 1/4)2K

and the transformed fields(Uj

Vj

)=

∑|K|≤K

(uj

vj

)eikxj−1/4

(Uj

Vj

)=

∑|K|≤K

(uj

vj

)eikxj+1/4

By defining Wj := Uj Vj and Wj := Uj Vj and by taking the 2K−points DFTtransforms it follows that

w(k) = e−iπk/(4K)[w(k)− iw(k + 2K) + iw(k − 2K)]

w(k) = e iπk/(4K)[w(k) + iw(k + 2K)− iw(k − 2K)(4.11)

Hence it is possible to compute w(k) exactly by means of four 2K-points-DFT,two inverse DFT and six 2K multiplies. This Fast Fourier Transform tool al-lows a speedup of a factor K/[2 log2(2K)]. Furthermore, since the vector fields areconjugate-symmetric, i.e., they satisfy

u∗(k) = u(−k)

then Uj , Vj , Uj , Vj are all real and it is possible to save half-memory storage and tospeed up the algorithm of a factor 2. The total computation requires 3K log2(2K)complex operations in contrast to the 3K2 required for the direct evaluations.

The latter properties are common to each couple of grids with 2K-points. Toarrive to the increase of a factor 48 claimed before, it is necessary to use the specialgrids xj±1/4 that allow an implementation of the symmetries. Details can befound in [25], Section 5. Just to give the feeling of this result suppose that u and vare symmetric with respect to the origin, (as the components of the velocity field).Then, condition (4.7), together with the conjugate-symmetry, implies u, v havevanishing real part. Therefore w(k) is real for all k. Then (4.11)1 gives

w(k) = Real part of[e iπk/(4K)w(k)

]so it is not necessary to calculate w(k), and the number of computations is halved.

22 L.C. Berselli

5 On the role of the pressure

In this section we perform another check on a simple flow and we follow very closelythe approach in Taylor [32]:

That variations in pressure must accompany turbulent flow is a dynam-ical necessity. . . . . . it seems worth while therefore to discuss, in somevelocity field which can be defined mathematically, the relationship be-

tween the mean variation of pressure√

p2 and the mean values of thecomponents of the velocity.

An appropriate class of fields of flow for the NSE is (2.5), that we widely used asan initial datum in the previous sections. The discussion between the similarity ofGradient and Rational models and the NSE makes plausible to use the same flowin computations involving our models. The fact that we have common periodicsolutions (similar, but simpler than the flow in (2.5)) and the same mechanism ofsmall-scale generation give the necessary justification to our computations. Againthe results we obtain must be considered as a first, very preliminary step to under-stand if a model predicts, or not, reasonable results.

5.1 Pressure mean values

In this section we evaluate the mean quadratic value of the pressure and of themean velocity to deduce properties satisfied by the flow obtained by the (RLES)and (GRAD) model.

5.1.1 Pressure 2D

We start with the two dimensional case that is easier to tract. In this case thesignificant expression is

K :=

√p2

0.5(u2 + v2),

where · denotes the space averaging. Taylor conjectured that The quantity Kshould have a value near to the unity, resembling the Bernoulli theorem for idealflows. By using the same procedure applied in the study of generation of smallscales, we calculate the pressure for the system

u = A cos(ax) sin(by) v = B sin(ax) cos(by)

to get the following expression for the ratio in the case of the (RLES) model:

Kδ =

√(36 A4 B4+12 a2 A6 B2 δ2+a4 A8 δ4+B8 (6+a2 δ2)2) (a2 A2 δ2+B2 (12+a2 δ2))2

B4 (6+a2 δ2)2 (6 B2+a2 A2 δ2)2

√2 (A2 + B2)

. (5.1)

On the LES of the Taylor-Green vortex 23

In the case δ = 0 we recover the expression

K0 =√

2√

A4 + B4

A2 + B2(5.2)

found in [32] for the NSE. The value of the expression in (5.2) varies from 1 to√

2.Note that also in the case of the RLES model the value of the expression in (5.1)equals 1, when B = −A.

If we set z = b/a we can better study the behavior of Kδ, deriving the followingexpression

Kδ(z) =

√(36 z4+12 z2 δ2+δ4+z8 (6+δ2)2) (δ2+z2 (12+δ2))2

z4 (6+δ2)2 (6 z2+δ2)2

√2 (1 + z2)

The limit as z → +∞ is

Kδ(∞) =

√(12 + δ2)2

6√

2

and since the function Kδ(z) is monotonic increasing, then we have 1 ≤ Kδ(z) <Kδ(∞) showing a reasonable consistency with the result known for the NSE. In thecase of the Gradient model the results are slightly different, but we will analyzethem directly in the 3D case.

5.1.2 Pressure 3D isotropic

In the case of a three dimensional motion, the value of

K :=

√p2

0.5(u2 + v2 + w2)

should depend on the ratios a : b : c and A : B : C but, it appears that K mustin all cases lie between certain definite limits. Taylor conjectured that these limitsare the same as in the the 2D case and he focused on the simpler case of a = b = c :

“This is of interest because free turbulence tends to become isotropicand the case is that for which the dimension of eddies are the same inthree-perpendicular directions, so that it resembles isotropic turbulencemore closely than any other motion included in the formula (2.5).”

24 L.C. Berselli

In the case a = 1, b = 1, c = 1, B = zA, and C = −(1 + z)A we obtain thefollowing results

for (RLES) : Kδ =3√

32

√(4+δ2)2 (108+68 δ2+11 δ4)

(3+δ2)2 (6+δ2)2

8

for (GRAD) : Kδ =√

432 + 56 δ2 + 3 δ4

8√

6.

Note that the result is independent of z. In the case of the NSE the result is similarand the value obtained by Taylor is K0 = 1.06 . . . In this case the differences

0 0.2 0.4 0.6 0.8 11.06

1.07

1.08

1.09

1.1

1.11

1.12

1.13

RLES

Gradient

Figure 2: Ratio between quadratic mean pressure and quadratic mean velocity

between (GRAD) and (RLES) are very small and, especially for 0 ≤ δ ≤ 1, thevalues of Kδ are very close each other. In particular, for significant values of δ, i.e.,0 ≤ δ ≤ 0.1 the difference is smaller than 10−6.

In the case of the RLES, if we choose a different value of a the value of Kδ(z)changes, but what is important is that it is always between

32√

2' 1.06 . . . and

3√

332

8' 1.52 . . .

Gradient modelIn the case of the (GRAD) model the value of Kδ is not independent of a but ismonotonic increasing in a and it grows indefinitely with a, not remaining boundedfor any value of a 6= 0. The explicit expression is

Kδ =√

432 + 56 a2 δ2 + 3 a4 δ4

8√

6

On the LES of the Taylor-Green vortex 25

This is another interesting point to keep into considerations to analyze the effec-tiveness of this method. In particular the behavior of the pressure may reveal thewell known instabilities of this model.

Remark. The time series expansion did not reveal any substantial differencebetween these two models. We conjecture that the design of advanced adap-tive/selective LES methods may be performed also by taking into account of thepossibly singular behavior of the pressure, as a measure of the turbulent state ofthe flow. This may open new modeling directions incorporating also the theoreticalresults (recently obtained in [5] for the NSE) that link pressure and regularity ofweak solutions.

5.2 Fluctuations in pressure gradient

Many experiments were performed, since the beginning of the 20th−century, bystudying the diffusion and turbulent phenomena produced in an air-stream. Thesimplest way to produce the turbulent behavior with a definite scale is to forcethe stream to pass through an honeycomb or regular screen. These experimentscan be used to define a length which is related to certain measurable properties ofthe flow and that turns out to be a definite fraction of the mesh-length, M , of theturbulence producing screen.

The Lagrangian conception leads to a length, which is the analogous to theMischungsweg of Prandtl. Experiments show that this length is 0.1M. Completesets of measurements (see [30, 31]) allow also to define a length λη which may beregarded as a measure of the ‘smallest size of eddy’ in the Lagrangian system; thisis connected with the average change in pressure by(

∂p

∂y

)2

=√

2(v2)2

λ2η

.

that becomes

λη = 2u2 + v2 + w2(

∂p∂x

)2

+(

∂p∂y

)2

+(

∂p∂z

)2

(5.3)

for isotropic turbulence.On the other hand the Eulerian theory gives a length that is 0.2M and the

Eulerian conception contains an implicit definition of λ, the ‘average size of thesmallest eddies’ which are responsible for the dissipation of energy by viscosity. Forisotropic turbulence we have

λ =√

5

√u2 + v2 + w2√ξ2 + η2 + ζ2

, (5.4)

26 L.C. Berselli

where (ξ, η, ζ) = ∇× (u, v, w) is the vorticity field.

5.2.1 Pressure gradient and the RLES model

Again, we apply the same procedure of the previous section to the flow (2.5) and,after integration of 2736 terms, we get an expression for the value of λη with 615terms. To obtain something significant, we consider simpler case in which a = b = c,so that the motion is divided into cubical partitions. In the isotropic case we arriveto the following result:

λ2

λ2η

=15(4 + a2 δ2

)2 (180 + 108 a2 δ2 + 17 a4 δ4)

64 (3 + a2 δ2)2 (6 + a2 δ2)2(5.5)

Clearly the limit δ → 0 is consistent with the value

λ

λη= 1, 44 . . .

calculated by Taylor in [32] for the NSE.In the case of the RLES model (namely in (5.5), δ 6= 0) we do not have an

expression independent of a, but it is worth noting that the lower and upper boundsare independent of δ,

infa

λ

λη= 1, 44 . . . sup

a

λ

λη=√

2558

= 1.996 . . .

So the ratio between λ and λη given by (5.4) and (5.3), is consistent with the

20 40 60 80 100

1.2

1.4

1.6

1.8

2

2.2

2.4

Figure 3: Values of λ/λη, for δ = 0.1 and the RLES model

measured value λ/λη = 2. In any case note also that λ ≥ λη.

On the LES of the Taylor-Green vortex 27

5.2.2 Pressure gradients and the GRAD model

In the case of the Gradient model, this behavior is violated, in fact in the isotropiccase the value of λ/λη is

λ

λη=

5(720 a2 + 72 a4 δ2 + 5 a6 δ4

)1728 a2

The limiting value δ = 0 is consistent with NSE but, for each δ > 0, it follows

lima→∞

λ

λη

∣∣∣∣δ 6=0 fixed

= +∞

In this way it is easily seen that the model may control the frequencies of the orderof 1/δ2 and not the bigger ones. This shows again the necessity for the (GRAD)model of a dissipative subgrid-scale term to stabilize the method.

5.2.3 Calculation of B

Another parameter introduced in [32] is B defined as follows

9B2 =

(∂p∂x

)2

+(

∂p∂y

)2

+(

∂p∂z

)2

u2(

∂u∂x

)2+ u2

(∂v∂y

)2

+ v2(

∂w∂z

)2 .

Experimentally it is deduced that B = 0.94. In the case of the (RLES) model wefind

BRLES =

√(4+a2 δ2)2 (180+108 a2 δ2+17 a4 δ4)

(3+a2 δ2)2 (6+a2 δ2)2

4√

2,

withlim

a→∞BRLES = 0.728869,

and this can be compared also with the theoretical value 0.53 calculated for theNSE in [32]. For the Gradient model we found

BGRAD =

√360 + 36 a2 δ2 + 5 a4 δ4

2

36

and we have again divergence for any fixed δ 6= 0 as a →∞. The result of the lastsection shows that the possibly singular behavior of the pressure gradient may beof interest, too. Note that ∇p enters in the equations with the same role of thevelocity, since dimensionally speaking it has the dimensions of velocity.

28 L.C. Berselli

Conclusions

The calculations we reported are not conclusive at all, but they shed now lightinto some points that need a further analysis, after precise and delicate numericalsimulations. In particular, the effect of the pressure may have a big role in theselection of the LES model to be used. Recall also that, even if the pressuredisappears in the weak formulation of the NSE, in any DNS the stability of thepressure itself is crucial for good performances of numerical methods. It is clearthat the analysis performed on only one flow may be inconclusive. On the otherhand, since the early seventies, the studies on the Taylor-Green vortex suggestthat it may be a good playground to perform preliminary tests. Even if it the TGvortex is merely a particular flow, it is significant since it allows to guess a moregeneral behavior of turbulent incompressible flows. The analogies stressed in thepaper show that the same justification and the validation of this analysis can betransferred to the Rational model, too.

AcknowledgmentsI would like to thank C.R. Grisanti and M. Gubinelli for their help in performingthe symbolic calculations needed in the paper.

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Luigi C. BerselliDipartimento di Matematica Applicata “U. Dini”V. Bonanno 25/b I-56125 Pisa, ITALY.berselli@dma.unipi.it