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Analysis of Discretization Errors in LES

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Analysis of Discretization Errors in LES Center for TurbulenceResearch Annual Research Briefs Analysis of discretization errors in LES By Sandip Ghosal Motivation and ob jectives All numerical simulations of turbulence DNS or LES involve some discretization errors The integrityofsuch simulations therefore dep end on our ability to quantify and control such errors In the classical literature see eg Chu on analysis of errors in partial dierential equations one typically studies simple linear equations such as the wave equation or Laplaces equation The qualitative insight gained from studying such simple situations is then used to design numerical metho ds for more complex problems such as the NavierStokes equations Though suchan approachmay seem reasonable as a rst approximation it should b e recognized that strongly nonlinear problems such as turbulence have a feature that is absent in linear problems This feature is the simultaneous presence of a continuum of space and time scales Thus in an analysis of errors in the one dimensional wave equation one may without loss of generality rescale the equations so that the dep endentvariable is always of order unity This is not p ossible in the turbulence problem since the amplitudes of the Fourier mo des of the velo city eld havea continuous distribution The ob jective of the present researchistoprovide some quantitative measures of numerical errors in such situations Though the fo cus of this work is LES the metho ds intro duced here can b e just as easily applied to DNS Errors due to discretization of the timevariable are neglected for the purp ose of this analysis Accomplishments In this rep ort analytical expressions for the p ower sp ectra of errors due to the spatial discretization of the NavierStokes equations are derived In x an ex pression for the numerical error is presented as the sum of nitedierencing aliasing and mo deling errors that have dierent origins In x expressions for the p ower sp ectra of the rst two kinds of errors are derived as well as the corre sp onding expressions for the subgrid and total nonlinear terms The essential to ol that makes the derivation of such an analytical expression p ossible is the joint normal hyp othesis for turbulentvelo cities The essential technique is identical to that used by Batchelor in his derivation of the pressure sp ectrum of turbulence from the energy sp ectrum Batchelor These results are applied to the LES of turbulence in x to obtain some measure of numerical errors in nite dierence schemes which are increasingly b eing used in turbulence computations on ows with complex b oundaries This rep ort summarizes the essential results the details of the mathematical development will b e presented elsewhere Ghosal henceforth referred to as pap er Present address CNLS MSB LANL Los Alamos NM S Ghosal Calculation of discretization errors Any representation of the true velo city eld in a turbulentow on a nite grid is necessarily approximate One must b e careful to distinguish b etween errors due to the niteness of the representation and the discretization error of a numeri cal scheme In a numerical simulation the velo city eld at any timestep can b e regarded as an elementofavector space with a nite numb er of dimensions N where N is the number of variables retained in the computation This is an ap proximate representation in a subspace of the larger vector space that contains the true solution The b est p ossible approximation to the true solution in the subspace is the pro jection onto the subspace in fact that is the denition of a pro jection op erator see eg Helmb erg The ideal or b est approximation to the NavierStokes op erator in the nite subspace is that op erator that ensures that the numerical solution remains lo cked to the pro jection of the true solution at all times as b oth vectors move around in their resp ectivevector spaces It maybe shown see pap er that this condition is achieved by sp ectral metho ds or prop erly dealiased pseudosp ectral metho ds in the absence of subgrid mo deling errors By discretization error Eofa numerical metho d we mean the deviation of the right hand side evaluated with the metho d from what would have b een obtained if the righthand side of the full NavierStokes equation were pro jected into the computational subspace Thus for a sp ectral metho d used in conjunction with an exact subgrid mo del E In order to evaluate the formal expression for the error E one needs to intro duce a basis The most advantageous choice is the D Fourierbasis since in Fourier space dierentiations reduce to multiplication bywavevectors and numerical dier entiation reduces to multiplication by mo died wavevectors see eg Vichnevetsky Wenow restrict our attention to ows in a p erio dic cubical b ox Further while considering nitedierence metho ds the grid will b e assumed uniform in ev ery direction Let E k denote the comp onents of E in the Fourierbasis with i i to corresp onding to the x y and z directions resp ectively Then E kcanbe i written as FD alias mo del E kE kE kE k i i i i The rst term arises b ecause of the inability of the nitedierencing op erator x to accurately compute the gradient of shortwavelength waves We call this k the nitedierencing error It vanishes for a sp ectral metho d that can dierenti ate waves of all wavelengths exactly The second term arises due to the metho d of computation of the nonlinear term by taking pro ducts in physical space on a dis crete lattice This is called the aliasing error and is well known in the literature on pseudosp ectral metho ds Canuto et al Rogallo The last term is the dierence b etween the true subgrid force and that computed using a subgrid mo del We call this the mo deling error In the following analysis the magnitude of the error E will b e characterized by statistical prop erties such as its p ower sp ectral density Such statistical measures can b e precisely dened only in the limit where the wavevector can assume a con tinuum rather than a discrete set of values In physical space this implies that we Discretization errors in LES are considering the grid size and some characteristic scale of turbulence xed and taking the limit as the size of the b ox L In actual simulations of course the b ox size L is nite However L is taken much larger that or so that smo oth power sp ectra can b e dened and computed statistical quantities are not changed when the b ox size is increased further This ensures that the computed quantities are indistinguishable from the ideal limit L For the purp ose of theoretical analysis it is advantageous to take the limit L rst rather than at the end of the computation Thus in the Fourierbasis the exact solution will b e characterized by a continuum of wavevectors k R and the numerical solution will b e character max max max max max max ized by the subset k B where Bk k k k k k x x y y z z We will assume for simplicity that the grid length is the same in all three di max max max rections so that k k k k Further we will consider m x y z the LES lterwidth and the grid length to b e identical This condition will b e relaxed in x In the limit of innite b ox size the discrete Fourier transform and its inverse take the form a factor of L is absorb ed in the denition of the transform Z X k x exp ik x x dkk exp ik x B x where the summation is over all lattice p oints over the innite cubic lattice of spacing and the integration over wave space ranges over all vectors k BThe following useful identity is readily derived by taking the limit of innite b oxsize X X expiK x K a x a where is the Dirac delta function is the set of wavevectors of the form pk qk rk where p q and r are integers p ositive negative or zero and K m m m is anyvector not necessarily restricted to B This relation is familiar in solid state physics see eg chapter pg of Jones March where the set go es by the name Recipro cal Lattice When the lattice spacing the summation over lattice p oints in b ecomes an integral over space and the usual continuous Fouriertransformisrecovered In this limit the right hand side of relation b e comes simply K and reduces to the familiar expansion of the deltafunction in terms of exp onentials Let us rst consider the eect of pro jecting the exact righthand side of the Navier Stokes equation onto the Fourierbasis with wavevector k The ith comp onent is given by Z Z dk dk k k ku k u k k k u k iP k m n mn i imn B B where P has its usual meaning see eg Lesieur and is the exact imn mn subgrid stress in Fourierspace The Einstein summation convention for tensor S Ghosal indices is implied throughout this rep ort except where otherwise noted If the exact derivative op erator x is replaced by the numerical dierentiation x k k multiplication bywavevectors k are replaced bymultiplication by the corresp onding mo died wavevectors kThus we obtain Z Z h i FD E ki P k P k dk dk k k ku k u k imn imn m n i B B i h M M k k k u k k P k i P kT i imn imn mn mn M where T k is the mo died subgrid mo del obtained by replacing all multipli mn M cation bywavevectors if any in the subgrid mo del kby the corresp onding mn multiplication by mo died wavevectors To obtain the aliasing error we consider the eect of evaluating the nonlinear term in physical space M iP k u xu xT k imn m n mn On using the denition of the discrete Fourier transform wehave X u xu x exp ik x u xu x m n m n x When u x and u x in are expanded in the Fourierbasis weget m n Z Z X dk dk
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