Divergence-free and curl-free wavelets in two dimensions and three dimensions: application to turbulent flows

Transcription

1 Journal of Turbulence Volume 7 No. 3 6 Divergence-free an curl-free wavelets in two imensions an three imensions: application to turbulent flows ERWAN DERIAZ an VALÉRIE PERRIER Laboratoire e Moélisation et Calcul e l IMAG BP Grenoble Ceex 9 France We investigate the use of compactly supporte ivergence-free wavelets for the representation of solutions of the Navier Stokes equations. After reviewing the theoretical construction of ivergencefree wavelet vectors we present in etail the bases an corresponing fast algorithms for two an three-imensional incompressible flows. We also propose a new metho for practically computing the wavelet Helmholtz ecomposition of any (even compressible) flow; this ecomposition which allows the incompressible part of the flow to be separate from its orthogonal complement (the graient component of the flow) is the key point for eveloping ivergence-free wavelet schemes for Navier Stokes equations. Finally numerical tests valiating our approach are presente. Keywors: Divergence-free wavelets; Curl-free wavelets; Turbulent flows. Introuction The preiction of fully evelope turbulent flows represents an extremely challenging fiel of research in scientific computing. The irect numerical simulation (DNS) of turbulence requires the integration in time of the full nonlinear Navier Stokes equations that is the computation of all scales of motion. However at large Reynols numbers turbulent flows generate increasingly small scales; to be realistic the iscretization in space (an also in time) ought to hanle a huge number of egrees of freeom. This is impossible with currently available computers in three imensions. Many attempts have been mae or are uner way to overcome this problem: among these are vortex methos which are able to generate very thin scales or large ey simulations (LES) an subgri-scale techniques that separate the flow into large scales which are explicitly compute from small scales that are parameterize or compute statistically. In this context wavelet bases offer a ifferent approach. They provie an alternative ecomposition allowing the intermittent spatial structure of turbulent flows to be represente with only a few egrees of freeom. This property comes from the goo localization in both physical an frequency omains of the basis functions. The wavelet ecomposition was introuce at the beginning of the 99s for the analysis of turbulent flows [ 3]. Wavelet base methos for the resolution of the two-imensional (D) Navier Stokes equations appeare later [4 9] an very recently for three-imensional (3D) omains [ ]. They have also been use to efine LES-type methos such as the coherent vortex simulation Corresponing author. Erwan.Deriaz@imag.fr Journal of Turbulence ISSN: (online only) c 6 Taylor & Francis DOI:.8/

2 E. Deriaz an V. Perrier (CVS) metho [ 37] an aaptive LES methos [3] or to erive 3D moels of turbulence [4]. Most of these cite works use a Galerkin a Petrov Galerkin or a collocation approach for the vorticity formulation in imension two with perioic bounary conitions [39]. However these approaches are not appropriate for the 3D case with non-perioic bounary conitions. An alternative approach was at the same perio consiere first by Urban an then investigate by several authors. They propose to use the ivergence-free wavelet bases originally esigne by Lemarié-Rieusset [5 38]. Divergence-free wavelet vectors have been implemente an use to analyse D turbulent flows [6 8] as well as to compute the D 3D Stokes solution for the riven cavity problem [9 ]. Since ivergence-free wavelets are constructe from stanar compactly supporte biorthogonal wavelet bases they can incorporate bounary conitions [ ]. The great interest of ivergence-free wavelets is that they provie bases suitable for representing the incompressible Navier Stokes solution in two an three imensions. Our present objective is thus to investigate their practical feasibility an amenability. In orer to eliminate the pressure we project the equations on to the space of ivergence-free vectors. This (orthogonal) projection is the well-known Leray projector an can be compute explicitly in Fourier space for perioic bounary conitions. Unfortunately as alreay note by Urban [] the Leray operator cannot be represente simply in terms of ivergence-free wavelets since they form biorthogonal bases (an not orthogonal). The goal of the present paper is to investigate the use of ivergence-free wavelets for the simulation of turbulent flows. First in section we review the basic ingreients of the theory of compactly supporte ivergence-free wavelet vectors evelope by Lemarié-Rieusset [5]. In section 3 we present in etail the bases that we implement in two an three imensions: isotropic bases as presente in the previous work [5 6 9] but also anisotropic bases which are easier to implement. We shall see that the choice of the complement wavelet basis is not unique an this choice inuces the values of ivergence-free coefficients for compressible flows. We iscuss the algorithmic implementation of ivergence-free wavelet coefficients in two an three imensions leaing to fast algorithms (in O(N) operations where N is the number of gri points). Section 4 is evote to the Helmholtz ecomposition of compressible fiels in a wavelet formulation; the metho that we present uses the biorthogonal projectors both on ivergence-free an on curl-free wavelets. Our metho is an iterative proceure an we shall experimentally prove that it converges. Section 5 aresses the main ingreients of a Galerkin metho for the Navier Stokes equations base on ivergence-free wavelets. Finally the last section presents numerical tests that valiate our approach: nonlinear compressions of D 3D incompressible turbulent flows an the wavelet Helmholtz ecomposition of several examples such as the computation of the ivergence-free part an the pressure arising from the nonlinear term of the Navier Stokes equations.. Theory of ivergence-free wavelet bases In this section we review briefly the construction of ivergence-free wavelets. Compactly supporte ivergence-free vector wavelets were originally esigne by Lemarié-Rieusset in the context of biorthogonal multiresolution analyses (MRAs). For the efinition an properties of biorthogonal MRAs an associate wavelets we refer the reaer to appenix A an to the textbooks in [3 6]. For the theory of ivergence-free wavelets see [5 ] an the book by Urban [8]. We illustrate the construction with the explicit example of splines of egree an.

3 Wavelets in two an three imensions 3. Theoretical grouns for the ivergence-free wavelet vectors Let us introuce H iv (R n ) ={f [L (R n )] n ; iv f L (R n ) iv f = } the space of ivergence-free vector functions in R n. The ivergence-free wavelets in [L (R n )] n efine by Lemarié-Rieusset [5] provie the Riesz bases of H iv (R n ). Their construction is base on the existence of two ifferent oneimensional MRAs of L (R) relate by ifferentiation an integration. Let (V j ) j Z be a one-imensional (D) MRA with a ifferentiable scaling function φ (meaning that V = span{φ (x k) k Z}) an a wavelet ψ ; one can buil a secon MRA (V j ) j Z with a scaling function φ (V = span{φ (x k) k Z}) an a wavelet ψ satisfying φ (x) = φ (x) φ (x ) ψ (x) = 4 ψ (x). () Example: spline scaling functions an wavelets of egree an : Biorthogonal splines provie wavelet bases which are regular compactly supporte an easy to implement. The scaling functions of the associate MRA are stanar B-spline bases an the wavelets are constructe easily by linear combinations of translate B splines. An example of an MRA satisfying equation () is given by splines of egree (Vj MRA spaces) an splines of egree (Vj MRA spaces). In both cases we raw the scaling functions φ an φ an their associate wavelets ψ an ψ with shortest support (figure ). The isotropic ivergence-free wavelets in R n are then obtaine by suitable combinations of tensor proucts of the functions φ ψ an φ ψ fulfilling equation (). Following [5] there exist (n )( n ) vector functions ivi ε H iv (R n )(ε n of carinality n i I ε of carinality n ) compactly supporte such that every vector function u H iv (R n ) can be uniquely expane: u = j Z ε n i I ε k Z n ε ivi jk ε ivi jk with ivi ε nj jk (x) = ivi ε ( j x k). The generating wavelets ivi ε take the general following form in Rn. Let ε n ={ }n \ {(... )} be given. We have to fix an integer i {.. n} such that ε i =. Then for every Figure. Scaling functions an associate even an o wavelets with shortest support for splines of egree (left) an egree (right).

4 4 E. Deriaz an V. Perrier ( [ ] ) i {...n}\{i } the vector function ivi ε ε = ivi [ ε ivi ] l = l l=n ifl/ {i i } γ ε ((i)) if l = i η... η n if l = i can be written () where γ ((i)) ε = θ ε... θ εi θ εi θ εi+... θ εn an η j = θ ε j j i i η i = 4 ψ with the notation η i = x (θ ε i ) { φr if ε j = θ rε j = ψ r if ε j = r being equal to or. Example: The D ivergence-free vector scaling function takes the form φ (x )φ (x ) iv (x x ) = φ (x )φ (x ) = φ (x )[φ (x ) φ (x )] [φ (x ) φ (x )] φ (x ) an the corresponing isotropic vector wavelets are given by the system () iv (x 4 x ) = ψ (x )[φ (x ) φ (x )] ψ (x )φ (x ) () iv (x x ) = () iv (x x ) = φ (x )ψ (x ) 4 [φ (x ) φ (x )]ψ (x ) ψ (x )ψ (x ) ψ (x )ψ (x ). We isplay in figure the three generating vector wavelets in the case of spline generators of egree an of figure. These wavelets have alreay been stuie by several workers for the analysis of D turbulent flows [6 7] an also to solve the Stokes problem in two an three imensions [9 ]. From now on we shall focus on the D an the 3D case an we shall present the associate fast algorithms. We shall then point out that the expansion of compressible flows following the ivergence-free wavelet bases is not uniquely given. Moreover we shall introuce D an 3D anisotropic ivergence-free wavelets that are easier to implement.

5 Wavelets in two an three imensions 5 Figure. Example of isotropic two-imensional generating ivergence-free spline wavelets () iv (left) () iv (centre) an () iv (right). 3. Isotropic an anisotropic ivergence-free wavelets: practical implementation Isotropic D 3D ivergence-free wavelet transforms have alreay been implemente by Urban [8 ] from ivergence-free scaling coefficients. Since the computation of ivergence-free scaling coefficients requires the solution of a linear system we shall propose a ifferent metho. We present in etail our D 3D ivergence-free wavelet ecomposition algorithm; it is base on the construction of a non-unique complement vector space. We also introuce anisotropic ivergence-free wavelet bases an their corresponing ecomposition algorithms which iffer from previous stuies. In the following we suppose that we are given two D MRAs (V j ) an (V j ) an φ ψ an φ ψ their associate (D) scaling functions an wavelets satisfying conition (). 3. Isotropic ivergence-free wavelet transforms 3.. The two-imensional case. The starting point of the construction is a D MRA of [L (R )] : ( V j Vj ) ( V j Vj ) whose D vector scaling functions an are given by φ (x )φ (x ) (x x ) = (x x ) = φ (x )φ (x ). In the isotropic case the six canonical generating D vector wavelets ε i of this MRA are () (x x ) = () (x x ) = () (x x ) = ψ (x )φ (x ) φ (x )ψ (x ) ψ (x )ψ (x ) () (x x ) = () (x x ) = () (x x ) = ψ (x )φ (x ) φ (x )ψ (x ) ψ (x )ψ (x ). (3)

6 6 E. Deriaz an V. Perrier Following wavelet theory the family { ε i jk (x x ) = j i ε ( j x k j )} x k with j Z k = (k k ) Z ε {( ) ( ) ( )} i = forms a basis of [L (R )]. Then a velocity fiel u = (u u )in[l (R )] has the following wavelet ecomposition: ( u = () jk () jk + () jk () jk + () jk () jk j Z k Z + () jk () jk + () jk () jk + () jk () jk). (4) Note that the first line of the ecomposition represents the wavelet ecomposition in the MRA (Vj Vj )ofthe first component u whereas the secon line concerns the wavelet ecomposition of u in the MRA (Vj Vj ). Isotropic generating ivergence-free wavelets are then constructe by linear combination of the i ε. More precisely for each ε {( ) ( ) ( )} the ivergence-free wavelet ε iv is given uniquely (an follows the general form ()) whereas one has to buil a complement function n ε such that span { ε ε } = span { ε iv } span { ε n }. The choice of the functions n ε is not unique. Moreover they cannot be constructe such that the space span{ n ε jk (x) = j n ε( j x k) εj k} is orthogonal to H iv (R ). We propose a choice for n ε escribe in appenix B for which the computation of ivergence-free wavelet coefficients iv ε jk is reuce to a very simple linear combination of the stanar wavelet coefficients i ε jk. Now the expansion (4) of the vector function u can be rewritten u = ( () iv jk () iv jk + () iv jk () iv jk + ) () iv jk () iv jk j Z + j Z k Z k Z k Z ( () n jk () n jk + () n jk () n jk + () n jk () n jk) (5) where the new coefficients are irectly expresse from the original coefficiants by equation (B) in appenix B. Appenix C summarizes the algorithm in pseuocoe. Note that the first line of the above ecomposition represents a ivergence-free part of u whereas the secon line is a complement vector function not orthogonal to the first. Remark: iv( n () ) iv( n () ) an iv( n () ) are generating functions of the scalar space V V. Moreover we have iv u = [ () n jk iv( () ) n jk + () n jk iv( () ) n jk + () n jk iv( () )] n jk j Z Then the incompressibility conition iv u = isequivalent to n ε jk = for all j kε. For incompressible flows since the biorthogonal projectors onto the spaces (Vj Vj ) (Vj Vj ) commute with partial erivatives [5] the ivergence-free wavelet coefficients iv ε jk are uniquely etermine by the formula ( iv) inequation (B) appenix B. Difficulties arise when we want to compute the ivergence-free part of a compressible flow. Because of the non-orthogonality between the ivergence-free basis ( iv ε jk ) an its complement ( n ε jk ) the values of the ivergence-free wavelet coefficients epen on the choice of

7 Wavelets in two an three imensions 7 the complement basis. We aress this problem in section 4 in orer to provie a wavelet Helmholtz ecomposition of any flow. 3.. The three-imensional case. The construction an fast algorithms corresponing to 3D ivergence-free wavelet bases are obtaine in a similar fashion as for the D case except that one has to start with the following vector MRA of [L (R 3 )] 3 : ( V j Vj Vj ) ( V j Vj Vj ) ( V j Vj Vj ). From the canonical generating 3D vector wavelets { i ε i = 3 ε} one constructs 4 generating ivergence-free wavelets an seven complement functions n ε. Appenix B inicates their exact forms (which for symmetry reasons iffer from the general form ()). As for the D case the computation of ivergence-free wavelet coefficients of any 3D vector fiel is given by a short linear combination of stanar biorthogonal wavelet coefficients arising from fast wavelet transforms. 3. Anisotropic ivergence-free wavelet transforms In this section we construct anisotropic wavelets that are ivergence free. Since we start from D wavelets ψ an ψ verifying ψ = 4ψ weerive easily ivergence-free wavelet bases by tensor proucts of D wavelets. We etail in the following the construction of such bases in the D an 3D cases. 3.. The anisotropic two-imensional case. Unlike the isotropic case anisotropic ivergence-free wavelets are generate from a single vector function an iv (x x ) = ψ (x )ψ (x ) ψ (x )ψ (x ) by anisotropic ilations an translations. The D anisotropic ivergence-free wavelets are given by an iv jk (x x ) = j ψ ( j x k )ψ ( j x k ) j ψ ( j x k )ψ ( j x k ) where j = ( j j ) Z is the scale parameter an k = (k k ) Z is the position parameter. When the inices k an j vary in Z the family { iv an jk } forms a basis of H iv(r ). We introuce as complement functions: an n jk (x x ) = j ψ ( j x k )ψ ( j x k ) j ψ ( j x k )ψ ( j x k ) since they verify n an jk is orthogonal to an iv jk ( j k being fixe). The anisotropic ivergence-free wavelet transform of a given vector function u works similarly to the isotropic transform. Starting from anisotropic wavelet ecomposition of u in

8 8 E. Deriaz an V. Perrier the MRA (Vj Vj ) (V j Vj ) (see appenix A) u = ( an jk an jk + an jk an jk) j Z where k Z an jk (x ψ ( j x k )ψ ( j x k ) x ) = an jk (x x ) = ψ ( j x k )ψ ( j x k ) for j k Z are the anisotropic canonical wavelets. Note that for more simplicity the ilate functions are not normalize in the L norm. u can be expane on to the new basis u = ( an iv jk an iv jk + an n jk an n jk) (6) j Z k Z with the corresponing coefficients an iv jk = j j + j an jk j j + j an jk an n jk = j j + j an jk + j j + j an jk (7) 3.. The anisotropic three-imensional case. In the same way the anisotropic 3D ivergence-free wavelets take the form iv an jk (x x x 3 ) = j 3 ψ ( j x k )ψ ( j x k )ψ ( j 3 x 3 k 3 ) j ψ ( j x k )ψ ( j x k )ψ ( j 3 x 3 k 3 ) j 3 ψ ( j x k )ψ ( j x k )ψ ( j 3 x 3 k 3 ) iv an jk (x x x 3 ) = j ψ ( j x k )ψ ( j x k )ψ ( j 3 x 3 k 3 ) j ψ ( j x k )ψ ( j x k )ψ ( j 3 x 3 k 3 ) iv3 an jk (x x x 3 ) = j ψ ( j x k )ψ ( j x k )ψ ( j 3 x 3 k 3 ) with j = ( j j j 3 ) k = (k k k 3 ) Z 3. Unlike the D case we have to choose two functions from the three above to generate the ivergence-free basis. As complement basis we introuce a function that is most orthogonal to the previous functions: j ψ ( j x k )ψ ( j x k )ψ ( j 3 x 3 k 3 ) an n jk (x x x 3 ) = j ψ ( j x k )ψ ( j x k )ψ ( j 3 x 3 k 3 ). j 3 ψ ( j x k )ψ ( j x k )ψ ( j 3 x 3 k 3 )

9 Wavelets in two an three imensions 9 The operations to compute ivergence-free coefficients an complement coefficients are similar to the D case. 4. An iterative algorithm to compute the wavelet Helmholtz ecomposition 4. Principle of the Helmholtz ecomposition The Helmholtz ecomposition [7 8] consists in splitting a vector function u [L (R n )] n into its ivergence-free component u iv an a graient vector. More precisely there exist a potential function p an a stream function ψ such that u = u iv + p an u iv = curl ψ. (8) Moreover the functions curl ψ an p are orthogonal in [L (R n )] n. The stream function ψ an the potential function p are unique up to an aitive constant. In R the stream function is a scalar-value function whereas in R 3 it is a 3D vector function. This ecomposition may be viewe as the following orthogonal space splitting: [L (R n )] n = H iv (R n ) H curl (R n ) where H iv (R n )isthe space of ivergence-free vector functions an H curl (R n ) ={v (L (R n )) n ; curl v [L (R n )] n curl v = } is the space of curl-free vector functions (if n = wehave toreplace curl v (L (R n )) n by curl v L (R )inthe efinition). For the whole space R n the proofs of the above ecompositions can be erive easily by means of the Fourier transform. In more general omains we refer the reaer to [7 8]. Note that one can also prove that H iv (R n )isthe space of curl functions whereas H curl (R n )isthe space of graient functions. The objective now is to generate a wavelet Helmholtz ecomposition. Since in the previous sections we have constructe wavelet bases of H iv (R n ) we have to work analogously to carry out wavelet bases of H curl (R n ). 4. Construction of a graient wavelet basis A efinition of wavelet bases for the space H curl (R n )(n = 3) has alreay been provie by Urban [] in the isotropic case. We shall focus here on the construction of anisotropic curl-free vector wavelets in the D case (it is similar in the n-imensional case). This construction is very similar to the ivergence-free wavelet construction espite some crucial ifferences. The starting point here is to search wavelets in the MRA (Vj Vj ) (Vj Vj ) instea of (V j Vj ) (V j Vj ) where the D spaces V an V are relate by ifferentiation an integration (section.). Since H curl (R )isthe space of graient functions in L (R ) we construct graient wavelets by taking the graient of a D wavelet basis of the MRA (Vj Vj ). If we neglect the L normalization the anisotropic graient wavelets are efine by curl an jk (x x ) = 4 [ψ ( j x k )ψ ( j j ψ ( j x k )ψ ( j x k ) x k )] = j ψ ( j x k )ψ ( j x k ). Thus when j = ( j j ) k = (k k )vary in Z the family { an curl jk } forms a wavelet basis of H curl (R ).

10 E. Deriaz an V. Perrier The ecomposition algorithm on curl-free wavelets { curl an jk } works similarly to that on anisotropic ivergence-free wavelets. Starting from the anisotropic wavelet ecomposition of avector function v in the MRA (Vj Vj ) (V j Vj ) v = ( an jk an# jk + an jk an# jk) j Z k Z where the canonical anisotropic vector wavelets are an# jk (x x ) = ψ ( j x k )ψ ( j x k ) an# jk (x x ) = ψ ( j x k )ψ ( j x k ). By applying the change in basis { an# we obtain jk an# jk v = j Z k Z { an curl jk = j an# jk + j an# jk N an jk = j an# jk j an# jk ( an curl jk an curl jk + an N jk an N jk) (9) where the curl-free wavelet coefficients are obtaine from the stanar coefficients by curl an jk = j an j + j jk + j an j + j jk () an have associate complement coefficients an N jk = j j + j an jk j j + j an jk. () 4.3 Implementation of the Helmholtz ecomposition in the wavelet context From now on our objective is to compute the wavelet ecomposition of a given vector function v; this means that we want to fin a ivergence-free component v iv an an orthogonal curl-free component v curl such that where v = v iv + v curl v iv = jk iv jk iv jk v curl = jk curl jk curl jk are the wavelet expansions on to ivergence-free an curl-free wavelet bases constructe previously (sections 3.. an 4.). We shall focus here on D anisotropic wavelet bases (an we shall omit the superscript an in the notation of the basis functions). To provie such ecomposition we have to overcome two problems. First the ivergence-free wavelets an curl-free wavelets form biorthogonal bases in their respective spaces an as alreay notice by Urban [] they o not give rise in a simple way to the orthogonal projections v iv an v curl of v. Asasolution we propose to construct in wavelet spaces two sequences (v p iv ) an (v p curl ) that converge to v iv an v curl.

11 Wavelets in two an three imensions The secon ifficulty is that ivergence-free wavelets are in spaces of the form (VJ V J ) (VJ V J ) whereas curl-free wavelets are in (Ṽ J Ṽ J ) (Ṽ J Ṽ J ) where V V an Ṽ Ṽ are couples of spaces relate by ifferentiation an integration. These spaces are ifferent an in orer to construct our approximations (v p iv ) an (v p curl ) we have to efine a precise interpolation proceure between them. In particular the spaces Ṽ Ṽ can be suitably chosen from V V Iterative construction of the ivergence-free an curl-free parts of a flow. Let v = (v v )beavector function an suppose that v is perioic in both irections an known on J J gri points that are not necessarily the same for v an v.inthe following we enote by I J v an approximation of v in the space (VJ V J ) (V J V J ) given by some interpolation process an by I # J v an approximation of v in the space (Ṽ J Ṽ J ) (Ṽ J Ṽ J ) also given by some interpolation process. We now efine the sequences v p iv (V J V J ) (V J V J ) satisfying iv v p iv = an v p curl (Ṽ J Ṽ J ) (Ṽ J Ṽ J ) satisfying curl v p curl = as follows. We begin with v = I J v an we compute v iv the ivergence-free wavelet ecomposition of v an its complement v n byformula from equations (6) an (7): I J v = v iv + v n = iv jk iv jk + n jk n jk. jk jk Then we compute the ifference v v iv at collocation points. Seconly we consier I # J (v v iv ) an we apply the curl-free wavelet ecomposition (9) () leaing to a curl-free part an its complement: I # J (v v iv ) = v curl + v N = curl jk curl jk + N jk N jk. jk jk Finally we efine v pointwise by v = v v iv v curl. At step pbyknowing v p at gri points we are able to construct a ivergence-free part v p iv of I J v p from equation (6) an v p curl the curl-free component of I# J (v p v p iv ) from equation (9) (v p v p iv being compute at gri points). The next term of the sequence is again efine pointwise: We iterate this process until v P l <ɛ an we obtain v ɛ P P v p iv + p= = jk ( P v p+ = v p v p iv v p curl. () v p curl p= p iv jk p= ) iv jk + ( P jk p= p curl jk ) curl jk where the right-han sie is an approximation of v which interpolates the ata up to an error ɛ (ɛ being given). Ieally the iteration converges as inicate on figure 3. However we are not able to prove the convergence of the sequence (v p ). We shall emonstrate it experimentally in section 6.4 on arbitrary fiels. Nevertheless we outline some remarks. The convergence rate epens on the choice of complement functions n jk N jk. The smaller the L scalar proucts iv jk n j k an curl jk N j k the faster the sequences converge.

12 E. Deriaz an V. Perrier Figure 3. Iealistic schematization of the convergence process of the algorithm with H N = span{ N jk } an H n = span{ n jk }. Ieally we woul like the convergence rate to be inepenent of the interpolating operators I J an I # J.Wepropose below a choice for these operators base on spline quasi-interpolation which is satisfactory at relatively slow convergence rate Helmholtz-aapte spline interpolation. In this section we shall etail our choice of operators I J an I # J inthe context of the spline spaces of egree (V j ) an egree (V j ) that we introuce earlier. Let us suppose that the components v an v of a velocity fiel v are known at knot points J (k + k ) an J (k k + ) respectively for k k = J. This choice of gri is inuce by the symmetry centres of scaling functions φ of V an φ of V (figure 4). Figure 4. The two scaling functions of V an V an their symmetry centres.

13 Wavelets in two an three imensions 3 For J given I J is chosen as a quasi-interpolation operator (similarly to section 6..) in the spline space (V J V J ) (V J V J ): I J v = k c k Jk + k c k Jk where an are the vector scaling functions introuce in section 3... The secon operator I # J provies a quasi-interpolation of vector functions on to a new spline space (ṼJ Ṽ J ) (Ṽ J Ṽ J ). Uner interpolation consierations we efine Ṽ = { v ; v(x /) V } = span{φ (x / k) ; k Z} Ṽ = { v ; v(x /) V } = span{φ (x / k) ; k Z} Hence we can write I # J v = k c # k Jk + k c # k Jk where Jk an Jk are the D vector scaling functions of the MRA (Ṽ J Ṽ J ) (Ṽ J Ṽ J ). 5. A ivergence-free wavelet metho for the Navier Stokes equations We present in this section the basics of a ivergence-free wavelet numerical metho for the resolution of the incompressible Navier Stokes equations written in velocity pressure formulation (without forcing term) as { u + (u )u + p ν u = t (3).u = with perioic or Dirichlet bounary conitions in a square (or cubic) omain. Our objective is to erive a Galerkin metho base on finite-imensional spaces of ivergence-free wavelets. Galerkin (or Petrov Galerkin) methos are variational methos for DNS of turbulence. In the context of wavelet Galerkin or Petrov Galerkin methos (incluing collocation methos) in [4 7] numerical methos were propose for the resolution of the D Navier Stokes equations in vorticity stream function formulation with perioic bounary conitions. However these methos cannot exten in a simple way to the 3D case nor to Dirichlet bounary conitions. In the (u p) formulation with classical iscretizations like spectral methos (in the non-perioic case) [9] finite-element methos [7] or (nonivergence-free) wavelets [9 ] one has to aapt the iscretization bases for velocity an pressure in orer to satisfy some inf sup conition (also calle 9 conition) or one has to introuce some stabilization term to avoi spurious moes in the computation of the pressure. Then system (3) leas to a sale-point problem. Depening on the chosen formulation (Galerkin collocation etc.) this problem is usually solve by the Uzawa algorithm or by a splitting metho. In any case numerical ifficulties arise in the computation of the pressure which requires solution of the ill-conitione linear system of Schur complement or a Poisson equation. When using ivergence-free bases this ifficulty is totally avoie; inee the pressure isappears by projecting the first of equations of (3) on to the ivergence-free vector functions: u t + P[(u )u] ν u = (4)

14 4 E. Deriaz an V. Perrier where P enotes the Leray projector. The solution u then has the form u(x t) = u iv jk (t) iv jk (x). jk After projecting equation (4) onto a finite imensional wavelet subspace span{ iv( j j )k ; j j < J} Equation (4) is simply reuce to a system of orinary ifferential equations which can be solve by a classical finite-ifference or Runge Kutta scheme. The main ifficulty in this approach is the computation of P[(u )u]. However the Helmholtz ecomposition of the nonlinear term yiels (u. )u = P[(u. )u] p where p is the flow pressure. The wavelet Helmoltz ecomposition presente in section 4 allows us to write (u. )u = P[(u. )u] + [(u. )u] curl = iv jk (t) iv jk (x) + curl jk (t) curl jk (x). jk jk Then we get the ivergence-free wavelet ecomposition of P[(u. )u] = jk iv jk(t) iv jk (x). The secon term gives the pressure from the curl-free coefficients of p as follows. 5. Computation of the pressure Remember that curl-free wavelets are constructe by an curl jk (x) = 4 [ψ ( j x k )ψ ( j x k )] j = ( j j ) k = (k k ). From the equalities p = [(u. )u] curl = jk curl jk (t) curl jk (x) = jk curl jk (t) 4 [ψ ( j x k )ψ ( j x k )] one irectly obtains by integration 4p = jk curl jk (t) ψ ( j x k )ψ ( j x k ). Thus the computation of the pressure is no more than a stanar anisotropic wavelet reconstruction in VJ V J from the curl-free coefficients of the nonlinear term obtaine through the wavelet Helmoltz ecomposition.

15 Wavelets in two an three imensions 5 6. Numerical experiments In this section we present the numerical results of the application of the ivergence-free wavelet ecomposition. We begin with the analysis of perioic numerical incompressible velocity fiels in two an three imensions generate by pseuospectral coes. First we have to take care of the initial interpolation of such fiels in orer not to violate the incompressibility conition satisfie in Fourier space. Then after the vizualization of the ivergence-free wavelet coefficients we shall stuy the compression factor obtaine through the wavelet ecomposition. Finally we investigate an numerically emonstrate the convergence of the algorithm presente in section 4.3 which provies the wavelet Helmholtz ecomposition of any flow. In orer to valiate the approach that we propose in section 5 we compute in wavelet space the ivergence-free component of a nonlinear term of the Navier-Stokes equations an we extract the associate pressure. In all the experiments we use ivergence-free wavelets constructe with splines of egree an egree. 6. Approximation of the velocity in spline spaces Usually the velocity fiels are given by gri point values. The first step of the wavelet ecomposition consists in interpolating these velocity ata on a suitable B-spline space. The problem is that this approximation may not conserve the ivergence-free conition a conition that was satisfie in Fourier space when velocity ata come from a spectral coe. The spline approximation of ata obtaine through spectral methos introuces a slight error for the ivergence-free conition. This ifference may be not negligible. For the turbulent fiels that we stuie (D an 3D) the error is about % of the L norm of the velocity. Thus we propose two ways to overcome this problem. The first way is to interpolate the velocity in the Fourier omain an to compute exactly its biorthogonal projection on wavelet spaces. The secon way is to interpolate on the ivergence-free B-spline spaces with the wavelet Helmholtz ecomposition etaile in section By using the iscrete Fourier transform. Since they are highly accurate spectral methos are often consiere as a reference technique for simulating incompressible turbulent flows. For perioic bounary conitions on the cube [ ] the iscrete Fourier transform (DFT) is use to ecompose the velocity u. If û k means the (vector) iscrete Fourier coefficients of u on a N regular gri given by û k = ( ) n e iπk n/n N N n {...N } u the velocity expansion in the Fourier exponential basis is u(x) = û k e iπk x. (5) k {...N } In this context the ivergence-free conition iv u = is k û k = k {...N }. (6) Assume now that the velocity fiel u to be analyse (assume to be -perioic in both irections) comes from a spectral metho an satisfies the incompressibility conition in the Fourier omain (6). To compute its ecomposition in a ivergence-free wavelet basis of R wehave first to approximate u = (u u )inthe suitable space (VJ V J ) (V J

16 6 E. Deriaz an V. Perrier V J ) introuce in section 3.. where J correspons to N = J. Then we search for an approximate function u J = (u J u J ) such that u J = u J = J J n = n = J J n = n = c Jn n φ Jn φ Jn c Jn n φ Jn φ Jn. For the choice of functions φ an φ verifying equation () the incompressibility conition iv u J = takes the iscrete form on the coefficients c i Jn n : c Jn n c Jn +n + c Jn n c Jn n + = (n n ). (7) To conserve the incompressibility conition satisfie by u asolution consists in consiering u J as the biorthogonal projection on to the space (VJ V J ) (V J V J ) since we know that this projector commutes with partial erivatives [5]. This is equivalent to consiering that c Jn n = u φ Jn φ Jn c Jn n = u φ Jn φ Jn. Replacing u by its Fourier expansion (5) it follows that c Jn n = eiπk x φ R Jn (x ) φjn (x )x x k {...N } û k = J k {...N } û k ˆφ ( J πk ) ˆφ ( J πk )e iπk n/ J where ˆφ ˆφ enote the (continuous) Fourier transforms of the ual scaling functions φ φ. Finally we obtain an explicit form for the DFT of the coefficients c Jn n (an in the same way for c Jn n ): DFT ( c ) Jn k = û k J ˆφ ( J (πk )) ˆφ ( J (πk )) DFT ( c ) Jn k = û k J ˆφ ( J (πk )) ˆφ ( J (πk )). (8) This means that the DFT of coefficients c i Jn n is given by the DFT of u multiplie by tabulate values on [ π]ofthe Fourier transform of the uals ˆφ ˆφ.Inpractice we o not know the explicit forms of these functions except by the infinite prouct: ( ) sin(ξ/) ˆφ (ξ) = ˆφ (ξ) j> [ cos(ξ j )] = j>[ cos(ξ j )] ξ/ ( e ˆφ (ξ) = ˆφ iξ ) ( ) (ξ) sin(ξ/) = e iξ/ ˆφ iξ (ξ) ξ/ Nevertheless the infinite prouct converges rapily which allows us to obtain point values of ˆφ an ˆφ with sufficient accuracy. In three imensions one procees similarly by consiering the biorthogonal projection of a3dvector fiel u on to the space (V J V J V J ) (V J V J V J ) (V J V J V J ).

17 Wavelets in two an three imensions By quasi-interpolation. The spline quasi-interpolation is a goo compromise when we have to eal simultaneously with spline approximations of even an o egrees. In this context the orer of approximation is n + by using B splines of egree n [3]. An avantage of the proceure is that it may be applie for any bounary conitions. Let b be a B-spline scaling function (b = φ or φ ). Given the sampling f (k/n)(n = J ) we want to compute scaling coefficients c k ofaspline function f N that will nearly interpolate the values f (k/n): f N (x) = k Z c k b(nx k). (9) f N is an interpolating function if c k b(l k) = f k Z ( ) l N l Z. For example if we consier b = φ (spline of egree ) the previous conition implies that ( ) l f N = ( ) l N (c l + c l ) = f l Z. N In orer to avoi the inversion of a linear system the quasi-interpolation introuces instea of c l c l = 5 [ ( ) ( )] l l + f + f [ ( ) ( )] l l + f + f l Z. 8 N N 8 N N By replacing c l by c l in equation (9) we obtain the following error at each gri point: / ( c ( ) ) l l + c l f = [ ( ) l f N 6 N ( ) ( ) ( ) ( )] l l l + l f 6 f + 4 f f N N N N = ( ) 48N f (4) (θ) + O 4 N 6 [ l with θ N l + ] N Therefore the pointwise error of quasi-interpolation is of orer 4 for a sufficiently regular function. 6. Analysis of two-imensional incompressible fiels We focus in this section on the analysis of D ecaying turbulent flows. The first numerical experiment that we present analyses the merging of two same-sign vortices. It concerns free ecaying turbulence (no forcing term). The experiment was originally esigne by Schneier et al. [7] an has often been use to test new numerical methos [4 9]. The experiment of [4] was reprouce here by using a pseuospectral metho solving the Navier Stokes equations in a velocity pressure formulation. The initial state is isplaye in figure 5 left. In a perioic box three vortices with a Gaussian vorticity profile are present; two are positive with the same intensity an one is negative with half the intensity of the others. The negative vortex is here to force the merging of the two

18 8 E. Deriaz an V. Perrier Figure 5. Vorticity fiels at times t = t = t = an t = 4 an corresponing ivergence-free wavelet coefficients of the velocity. positive vortices. The time step was δt = an the viscosity ν = 5 5. The solution is compute on a 5 5 gri. The vorticity fiels at times t = t = t = an t = 4 are isplaye in figure 5. The secon row of figure 5 isplays the absolute values of the isotropic ivergence-free wavelet coefficients of the velocity fiel at corresponing times renormalize by j at scale inex j. Asone can see ivergence-free wavelet coefficients concentrate on strong energy zones which correspon to a region of strong variations in the velocity that is aroun or in between vortices or along vorticity filaments. The secon experiment eals with a ecaying D turbulent fiel obtaine with an initial state of the ranom phase spectrum. This vorticity fiel was compute with a spectral coe at a resolution 4 4 (see [3] for more etails). As notice in [3] there is a Newtonian viscosity such that the Reynols number is This fiel has been kinly provie to us by Lapeyre [3] an was publishe in [3]. After 4 turnover timescales of the preominant eies for a timescale base upon the total enstrophy of the flow the vorticity fiel exhibits the emergence of coherent structures together with strong filamentation of the flow fiel outsie the vortices (figure 6 left). We show in figure 7 the isotropic an anisotropic ivergence-free wavelet coefficients (in the L inf -norm) issue from the ecomposition of the velocity fiel isplaye in figure 6 (right). We must emphasize that before computing the ivergence-free wavelet coefficients we first ha to compute the velocity fiel from the vorticity fiel (this is one in the Fourier omain). As expecte the wavelet coefficients give insight into the energy istribution over the scales of the flow. As one can see in figure 7 (left) the energy on the smallest scale (or highest wave numbers) is localize along the strong eformation lines an fits the filamentation between vortices or with strong changes in vortices. The top right square corresponing to vertical isotropic wavelets ( () iv jk )exhibits vertical structures whereas the bottom left square corresponing to horizontal wavelets ( () iv jk )exhibits horizontal eformation lines.

19 Wavelets in two an three imensions 9 Figure 6. Vorticity fiel for a 4 4 simulation of ecaying turbulence (left) an the corresponing velocity fiel (right). Now we investigate the compression properties of the ivergence-free wavelet analysis: as preicte by the nonlinear approximation theory [6]; the compression ratio in the energy norm is governe by the unerlying regularity of the velocity fiel in some Besov space. Let u be an incompressible fiel its ivergence-free wavelet expansion can be written u = u + j k Z ( () iv jk () iv jk + () iv jk () iv jk + () iv jk () iv jk). The nonlinear approximation of u relies on computing the best N-term wavelet approximation by reorering the wavelet coefficients ε iv j k > ε iv j k > > ε N iv j N k N > Figure 7. Isotropic (left) an anisotropic (right) ivergence-free wavelet coefficients of the velocity fiel of figure 6 (right).

20 E. Deriaz an V. Perrier an introucing Then we have if the quantity u q Bq sq belongs to the Besov space B sq N (u) = u + u N (u) L N i= ε i iv j i k i ε i iv j i k i. () ( ) s/n < C u sq N B () q = ε jk ε iv jk q is finite with /q = / + s/n (this means that u q ). As state in [6] the evaluate regularity s cannot be larger than the orer of polynomial reprouction in scaling spaces plus one (which equals the number of zero moments of the ual wavelet). In our experiment the ual spline wavelets ψ an ψ (see appenix A) have respectively two an three zero moments which allows us to evaluate regularities only smaller than two. Figure 8 shows the nonlinear compression of ivergence-free wavelets provie on the 4 turbulent fiel. The curve represents the L error u N (u) L versus N in a log log plot. The convergence rate measure on the curve is s (= p with p the slope of the curve).35 which shows that the velocity flow belongs to the corresponing Besov space B sq q with q =.85. When looking at the compression curve on figure 8 we observe three ifferent zones. First large-scale wavelets capture large-scale structures of the flows. Consequently the compression progresses slowly an irregularly. Then we observe a linear slope that represents the nonlinear structure of the turbulent flows. In this region we are able to evaluate the regularity of the fiel. The last region correspons to an abrupt ecrease because the ata are iscrete. One can also remark that in figure 8 only.% of the coefficients recover about 99% of the L -norm. The same experiment was carrie out on the three interacting vortices but is not reporte here since the slope of the curves saturates at s = because of the small number of vanishing moments of the wavelets we use (equal to in our experiment); this means that these fiels are more regular an suggests using wavelets with more vanishing moments to optimize the compression factor. Figure 8. L error provie by the nonlinear N best terms of the wavelet approximation (); the log log plot shows the L error () versus N for a D turbulent flow.

21 Wavelets in two an three imensions Figure 9. [33]. Isosurface.45 of vorticity magnitue after five large-ey turnovers provie by a spectral metho 6.3 Analysis of a three-imensional incompressible fiel In this part we consier a 3D perioic fiel generate from DNS (spectral metho) of freely ecaying isotropic turbulence kinly provie to us by G.-H. Cottet an B. Michaux. The experiment has been etaile in [33] an was a Gaussian initial conition for the velocity an 8 3 collocation points. Figure 9 isplays the vorticity isosurfaces corresponing to about 4% of the maximum vorticity at five turnover times (time being expresse in terms of largeey turnover time units L /u where L is the initial integral scale an u = /3u() ). The Reynol number base on the Taylor microscale is initially 98 an ecreases to 6 at t = 8 (see [33] for more etails). The isotropic ivergence-free wavelet ecomposition of the corresponing velocity fiel is compute an isplaye in figures an. As explaine in section 3.. the isotropic 3D ivergence-free wavelet ecomposition provies 4 generating wavelets ε iv jk ε iv jk. Figure. Isosurface. ofivergence-free wavelet coefficients associate with iv ε jk (left) an to ε iv jk (right) in absolute value.

22 E. Deriaz an V. Perrier Figure. Isosurface.6 of ivergence-free wavelet coefficients associate with () iv jk (right) in absolute value. (left) an to () iv jk Figure left shows the corresponing renormalize coefficients j iv jk whereas figure right shows j iv jk upto j = 6. The smallest scale ( j = 7) wavelet coefficients are isplaye in figure for two ifferent generating wavelets; we choose () iv which correspons to horizontal structures an () iv which exhibits vertical structures. Figure isplays the nonlinear compression error: unlike the D case owing to the low resolution (8 3 instea of 4 )ofthe fiel it is ifficult to etect a linear part in the curve. Nethertheless in an intermeiate region the curve nearly presents a slope of s (= 3 p) Wavelet Helmholtz ecomposition of velocity fiels 6.4. Convergence of the metho. The wavelet Helmholtz ecomposition presente in section 4. is teste in orer to emonstrate numerically that our algorithm converges. In this part we apply the metho of section 4.3 to ranom D vector fiels v (efine as mathematical functions whithout physical meaning) an we stuy the behaviour of the resiual v P l (see section 4.3). These vector fiels are constructe with a uniform ranom law at each collocation Figure. L error provie by the nonlinear N best terms of the wavelet approximation (); the log-log plot shows the L error () versus N for a 3D turbulent flow.

23 Wavelets in two an three imensions 3 Figure 3. Convergence curves of the iterative wavelet Helmholtz algorithm. point. Figure 3 isplays the l norm of the resiual v P l interms of the number of iterations p infour ifferent situations (ifferent sizes for v; ifferent zero moments for the wavelets). We make the following conclusions. (i) For all functions that we have teste the metho converges an the curves show that except at the very beginning the convergence is exponential. (ii) The slope of the curve harly epens on the number of gri points (it is steeper with fewer gri points). (iii) The convergence rate increases with increasing number of vanishing moments of the ual wavelets. Now if we want to estimate the numerical cost of the wavelet Leray projection (ivergencefree part) of a given (compressible) vector fiel we have to compare it with the cost of the Leray projection in the Fourier omain namely O(N log N) if N is the number of gri points. Since the price to pay for a fast wavelet transform is O(N) operations an since in numerical experiments the wavelet Helmholtz algorithm converges in about iterations to reach an error of 6 the whole cost is O(N) which is asymptotically better. In the future we shall investigate the influence of the wavelet bases an of the interpolation projectors on the convergence rate Wavelet Helmholtz ecomposition of the nonlinear term of the Navier Stokes equations. Since our main objective for further research is to use ivergence-free wavelets to solve numerically the Navier Stokes equations (see section 5) we have to fin the wavelet Helmholtz ecomposition of the nonlinear term (u )u. Toillustrate the feasibility of our approach we consier the D turbulent fiel isplaye in figure 6 an we compute the ivergence-free an curl-free wavelet components of its associate nonlinear term (u )u using the wavelet Helmholtz ecomposition; figure 4 shows the anisotropic wavelet coefficients (L normalize) of the ivergence-free part (left) an of the curl-free part (right) arising from this ecomposition. The wavelet coefficients are isplaye as inicate in figure A of appenix A. One can see the appearance of small-scale wavelet coefficients (at the bottom

24 4 E. Deriaz an V. Perrier Figure 4. Anisotropic wavelet coefficients corresponing to the wavelet Helmholtz ecomposition of (u )u: ivergence-free coefficients (left) an curl-free coefficients (right). right of each square of wavelet ecomposition) in the ecomposition of the ivergence-free part by comparison with the ecomposition in figure 7; it is obvious that the nonlinear term contributes to the creation of small scales. From the ivergence-free wavelet coefficients in figure 4 we reconstruct the ivergencefree part of (u )u an isplay in figure 5 (left) the vorticity fiel associate with this Figure 5. Vorticity (left) an ifferential pressure (right) erive from the wavelet Helmholtz ecomposition of the nonlinear term (u )u with u isplaye in figure 6.

25 Wavelets in two an three imensions 5 Figure 6. Relative error between the ivergence-free part of (u )u (obtaine through Fourier transform) an the part provie by the wavelet Helmholtz ecomposition. velocity. This visualization confirms the creation of small-scale structures in the nonlinear term. Figure 5 (right) isplays the reconstruction of the pressure from curl-free wavelet coefficients as explaine in section 5. As expecte low pressures correspon to coherent vortices. To control our results we have compare the pressure obtaine by the wavelet Helmoltz ecomposition to that compute in the Fourier omain an we have foun a relative error of.5 4 in the L norm an error which probably arose from the interpolation proceure. On the other han the ifference between the Leray projection (in Fourier space) an the wavelet projection on to the ivergence-free space represents % of the L norm. Figure 6 isplays the localization of this error. As one can see the error is localize aroun strong graients of the fiel which are zones where the Fourier interpolation an the spline interpolation (preliminary step before projecting in both cases) o not give the same result. 7. Conclusion an perspectives We have presente in etail the construction of D an 3D ivergence-free wavelet bases an a practical way to compute their associate coefficients. We have introuce anisotropic ivergence-free an curl-free wavelet bases which are easier to hanle. We have shown that these bases make possible an iterative algorithm to compute the wavelet Helmholtz ecomposition of any flow. Thus numerical tests prove the feasibility of ivergence-free wavelets for simulating turbulent flows in two imensions an three imensions. A ivergence-free wavelet-base solver for D Navier-Stokes equations in (u p) formulation is uner way an will be reporte in a forthcoming paper.

26 6 E. Deriaz an V. Perrier An important issue that must be aresse is the flexibility of the metho; although all numerical tests have been presente in the perioic case the metho extens reaily to nonperioic problems in square or cubic omains by using wavelets incorporating homogeneous bounary conitions such as those in [34 35]. Inee the construction of ivergence-free wavelets with non-perioic bounary conitions has alreay been carrie on by Urban []. Another point is that since we consier the (u p) formulation for the Navier Stokes equation an since we are able to compute the Leray projector in the wavelet omain the metho extens easily to the 3D case. Finally we conjecture that this metho shoul be competitive with finite-element methos an spectral methos in the non-perioic case since in this case classical methos involve a sale-point problem an since wavelet methos take avantage of the compression properties of the wavelet bases both for functions an for operators (in the perioic case the comparison woul be unfair since the spectral methos are clearly optimal). Appenix A: Multiresolution analysis A. Introuction MRAs are approximation spaces allowing the construction of wavelet bases an were introuce by Mallat [5]. We recall here some efinitions. Definition: A multiresolution analysis (MRA) of L (R)isasequence of close subspaces (V j ) j Z verifying the following. (i) j V j V j+ j Z V j ={} j Z V j is ense in L (R). (ii) (Dilation invariance) f V j f (.) V j+. (iii) (Shift invariance) There exists a function φ V such that V = span {φ( k); k Z}. φ is calle a scaling function of the MRA. j enotes the level of refinement. By virtue of equation (ii) one has V j = span {φ jk (x) = j/ φ( j x k); k Z}. Wavelets appear as bases of complementary spaces W j such that V j+ = V j W j where the sum is irect but not necessarily orthogonal. In this context (calle the biorthogonal case introuce in [36]) the choice of spaces W j is not unique. In each space W j one can construct a function ψ calle a wavelet such that W j = span{ψ jk = j/ ψ( j k); k Z}. Then the wavelet space ecomposition of L (R)iswritten L (R) = V + j= W j an any function f L (R) has the following wavelet expansion: f (x) = c k φ(x k) + jk ψ jk (x). k Z j> k Z (A) A. Biorthogonal basis As (φψ)isfixe we can associate a unique ual pair (φ ψ ) such that the following biorthogonality (in space L ) relations are fulfille: k Z an j > φ φ k =δ k φ ψ jk = ψ ψ jk =δ jδ k ψ φ k =