Center for Turbulence Research Proceedngs of the Summer Program 00 117 Tools for large-eddy smulaton By Davd A. Caughey AND Grdhar Jothprasad A computer code has been developed for solvng the ncompressble Naver-Stokes equatons for test flows that wll allow the comparson of varous strateges for assessng the accuracy of LES solutons for flows at large Reynolds number, where t s mpractcal to make drect comparsons wth DNS solutons for the same flow. The code ncludes optons for a conventonal Smagornsky subgrd model, as well as hypervscosty dsspatve terms that wll allow a greater separaton of scales for hgh Reynolds number flows. In ths report, the code s valdated for several smple, perodc flows, ncludng the Taylor-Green vortex and decayng sotropc turbulence, and prelmnary results are presented, showng good agreement for (forced, perodc) Kolmogorov flow n the lmt of hgh Reynolds number, on relatvely modest meshes usng the hypervscosty dsspaton. 1. Introducton Large Eddy Smulaton (LES) holds the promse of mproved predcton of turbulent flows at large Reynolds numbers relatve to computatons usng the Reynolds-averaged equatons, wth substantally less computatonal cost than that requred for Drect Numercal Smulaton (DNS) snce only the largest, most energetc, scales need to be resolved. Nevertheless, the computatonal resources requred for LES computatons for all but the smplest flows reman substantal. The requred resources are so great, n fact, that most LES computatons are performed at the lmts of avalable resources, and t usually s not clear to what extent the solutons are resolved. Furthermore, snce the accuracy of the computaton s not assessed n any systematc way, the most frequently used metrc for the performance of LES s comparson wth DNS computatons whch, as a result of the computatonal resources requred for DNS, necessarly lmts these assessments to flows at relatvely low Reynolds numbers. In order for LES to become useful for practcal engneerng problems, tools must be developed that allow one to estmate the accuracy of the LES soluton wthout havng to make a drect comparson wth a DNS soluton for the same flow. The avalablty of such tools would free one from the need to make comparsons only at low Reynolds numbers where DNS solutons are feasble, and permt the evaluaton of LES at the larger Reynolds numbers for whch LES s most attractve, and where the separaton between the energetc and dsspatve scales s large enough that the LES approach has some theoretcal bass. The long-term objectve of the authors work s to develop tools sutable for assessng the accuracy of LES solutons, especally for flows at large Reynolds number. As a frst step n ths process, the authors have developed a computer code for solvng the ncompressble Naver-Stokes equatons, ncludng addtonal hypervscosty dsspaton, for test flows that wll allow the comparson of varous accuracy-assessment strateges. The ncorporaton of hypervscosty dsspaton (based on the bharmonc of the velocty feld) s motvated by the desre to ncrease the separaton of the energetc and dsspatve Cornell Unversty
118 D. A. Caughey & G. Jothprasad scales, so that the hgh Reynolds number lmt of solutons can be studed on meshes of relatvely modest resoluton. In ths report, valdatons of the code are presented for several smple, perodc flows, ncludng the Taylor-Green vortex and decayng, sotropc turbulence, and prelmnary results show good agreement for (forced, perodc) Kolmogorov flow n the lmt of hgh Reynolds number, on relatvely modest meshes usng the hypervscosty dsspaton.. Algorthm In ths secton we descrbe a varant of the fractonal-step method (Chorn (1968), Temam (1969), Km & Mon (1985)) used for the tme-advancement of the ncompressble Naver-Stokes equatons wth a Smagornsky model and added hypervscosty dsspaton. The constant-densty Naver-Stokes equatons, wth an added hypervscosty, and the contnuty equaton u t + (u u j ) = p u 4 u + (ν + ν r ) + ν 4 x j x x j x j x j x j x l x l (.1) u = 0 x (.) are dscretzed on a staggered grd. Veloctes u are defned at the centers of cell faces havng normals n the drecton x, and the pressure s defned at the cell center. The equatons are marched n tme usng an explct approxmaton for the convectve terms and an teratve Alternatng Drecton Implct (ADI) scheme for the vscous (and hypervscous) terms. Such a fractonal-step tme-advancement scheme for (.1) and (.) can be wrtten as û u n [ 3 + t wth where H {u n } 1 H { u n 1 }] + G {φ n } = ν + ν r [D {û } + D {u n }] + ν 4 [D 4 {û } + D 4 {u n }] (.3) u n+1 û { = G δφ n+1 }, t (.4) C { u n+1} = 0 (.5) φ n+1 = φ n + δφ n+1 (.6) H {u} Spatal dscretzaton of (u u j ) x j G {φ} Spatal dscretzaton of φ x D {u } Spatal dscretzaton of u x j x j D 4 {u } Spatal dscretzaton of 4 u x j x j x l x l C {u} Spatal dscretzaton of u x.
Tools for LES 119 The convectve terms H {} are dscretzed to fourth-order spatal accuracy usng central dfferences, and the tme advancement s carred out usng a second-order Adams- Bashforth approxmaton for the convectve terms. The spatal dscretzaton of the convectve terms s desgned to be energy conservng (Mornsh & Mon (1998)) n order to acheve explct control over the dsspaton ntroduced n the numercal scheme. The vscous dffuson terms D {} are dscretzed to fourth order spatal accuracy, whle the hypervscous terms D 4 {} are dscretzed to second order spatal accuracy, both usng central dfferences. These two terms are advanced n tme usng an teratve ADI scheme. It should be noted that φ dffers from the pressure by an O( t) term (Km & Mon (1985)). The scheme has been mplemented for perodc boundary condtons on a unform grd. Equaton (.5) can be used to elmnate u n+1 from the dvergence of (.4), gvng a Posson equaton for δφ n+1 C { G { δφ n+1}} = C {û} t (.7) Further detals of the spatal dscretzaton can be found n Mornsh & Mon (1998). The mplct equatons arsng at each tme step are solved usng an teratve ADI scheme. We frst defne a splttng of the operators D {} and D 4 {} nto dfferences n the x 1, x and x 3 drectons and cross dervatves as follows, D {u } = D x1 {u } + D x {u } + D x3 {u } D xj {u } Spatal dscretzaton of u x j D 4 {u } = D 4x1 {u } + D 4x {u } + D 4x3 {u } + D 4cross {u } D 4xj {u } Spatal dscretzaton of 4 u [ 4 ] u D 4cross {u } Spatal dscretzaton of x + 4 u 1 x x + 4 u x 3 x. 3 x 1 x 4 j Let û [m] denote the approxmaton to û at the m th teraton. Equaton (.3) can be rewrtten as û [m+1] t u n + 3 = t H {u n } 1 H { u n 1 } G {φ n } + ν + ν { } r D û [m+1] ν + νr + D {u n } { } { [D 4x1 + D 4x ν 4 ν { 4 D 4 cross û [m+1] û [m] û [m+1] } ν4 + D 4 {u n } + } { + D 4x3 û [m+1] }] + (.8) The terms wthn angle braces do not change wth teraton m, and can be grouped
10 D. A. Caughey & G. Jothprasad together to form the source term { S u n,u n 1} un t + 3 H {u n } 1 H { u n 1 } G {φ n } + ν + ν r D {u n } + ν 4 D 4 {u n } (.9) Equaton.8 then reduces to ( 1 t ν + ν r [ Dx1 + D + D ] ν 4 [ x x3 D4x1 + D 4 + D ] ) } x 4 x3 {û [m+1] û [m] (.10) { = S u n,u n 1} + û[m] { t R û [m],u n,u n 1} + ν + ν { r D û [m] } + ν { 4 D 4 û [m] } It s easly seen that the rght-hand sde of (.10) s smply the resdual evaluated at û [m]. Multplyng (.10) by t and approxmatng the left-hand sde usng ADI gves ( 1 t [ (ν + νr ) D + ν ] ) ( x1 4D 4x1 1 t [ (ν + νr ) D + ν ] ) x 4D 4x ( 1 t [ (ν + νr ) D x3 + ν 4D 4x3 ] ) {û [m+1] } { (.11) û [m] = tr û [m],u n,u n 1} Equaton (.11) requres nversons only of trdagonal matrces, as opposed to the large sparse matrces requred n (.10). When the coeffcents ν r and ν 4 vary spatally, the left-hand sde of Eq..11 s further approxmated by replacng the coeffcents wth ther correspondng spatal averages, whle the rght-hand sde s computed exactly. It has been found by numercal expermentaton that three teratons are usually suffcent to reduce the resdual by at least four orders of magntude. 3. Results Here we present results of several flows to valdate the computer code descrbed n the precedng secton, and prelmnary results to show ts promse for computng flows at hgh Reynolds number. All flows consdered here are perodc and are solved n a three-dmensonal perodc box of edge sze π. 3.1. Code valdaton The perodc two-dmensonal vortex problem was used extensvely to valdate the spatal and temporal accuracy of the code. The ntal condtons for a perodc vortcal flow n the x 1 -x plane are gven by u 1 = cos x 1 sn x, u = cos x sn x 1, u 3 = 0. (3.1) The numercal solutons obtaned are compared wth the known analytcal solutons. In order to completely valdate the dfference codng for all 3 drectons, solutons for a
Tools for LES 11 perodc vortex n the x x 3 plane also were computed. Although the test problem has some symmetres, any serous error n the code could have been detected usng ths test case. Mesh refnement studes were used to verfy the spatal order of accuracy, whch was confrmed to be fourth-order for the Naver-Stokes equatons (.e., wthout the hypervscosty dsspaton) and second order when the hypervscosty terms are ncluded. Smlarly, the temporal order of accuracy was verfed to be second order. 3.. Taylor-Green Vortex Next, the code was used to carry out a DNS of the Taylor-Green vortex flow (Brachet et al. (1983)). Ths flow s one of the smplest systems n whch one can study the generaton of small scales and the turbulence resultng from three-dmensonal vortex stretchng. The ntal condtons for the Taylor-Green vortex flow are u 1 = 3 sn u = 3 sn u 3 = 0. ( ) π sn x 1 cos x cos x 3, 3 ( ) π cos x 1 sn x cos x 3, 3 (3.) The Reynolds number for ths flow s defned as Re 1/ν. The DNS of the Taylor-Green flow was carred out on a 64 3 mesh for Re = 100, 00, 300 and 400. Snce the smallest scales generated n the Re = 400 smulaton were not resolved on ths mesh, the DNS was repeated on a 18 3 mesh. All computatons were performed at a constant Courant number C = 0.5. The tme hstores of the dsspaton rate are compared wth results of the fully-resolved spectral computatons of Jeanmart (00) n fgure 1. The fgure shows that the tme hstory of the dsspaton rate s well predcted n our 64 3 smulatons for the lower Reynolds numbers, but that there are dscrepances between the present results on the 64 3 mesh and the reference computatons n the vcnty of the maxmum dsspaton for the computaton at Re = 400; however, the present results agree well wth the reference computaton when repeated on the 18 3 mesh. 3.3. Decayng, sotropc turbulence The case of decayng, sotropc turbulence allows comparson of results computed wth varous subgrd-scale models to those of fully-resolved DNS computatons for modest Reynolds numbers. Two dfferent subgrd-scale models were compared: (a) Smagornsky model: An eddy-vscosty model of the form, ν r = (C r ) (S j S j ) 1 (3.3) where S j s the local rate-of-stran tensor, s the mesh spacng, and the Smagornsky coeffcent s taken to be C r = 0.17, followng Llly s analyss (Llly (1967)). (b) Smagornsky-type hypervscosty model: A hypervscosty model of the form, ν 4 = (C 4 ) 4 (S j S j ) 1 (3.4) where S j s the agan the local rate-of-stran tensor, s the mesh spacng, and the (dmensonless) constant C 4 0.78 s chosen such that the flow s fully resolved on the gven grd. (Note: t s clear from Eq. (.1) that the dmensons of ν 4 are length 4 /tme.)
1 D. A. Caughey & G. Jothprasad 0.0 0.015 Dsspaton Rate 0.01 0.005 0 0 4 6 8 10 Tme Fgure 1. Hstory of dsspaton rate for Taylor-Green flow at varous Reynolds numbers. Re = 100: sold lne; Re = 00: broken lne; Re = 400: 64 3 grd s dashed lne, 18 3 grd s dash-dotted lne. Comparson wth spectral computatons of Jeanmart, shown as dotted lnes; dotted lnes are nvsble when overplotted wth current results. Large-eddy smulatons (LES) of decayng, sotropc turbulence were performed on a 64 3 grd usng both these models. An under-resolved DNS was also carred out on the 64 3 mesh. The ntal condtons were provded by Alan Wray from a 56 3 spectral smulaton of forced, sotropc turbulence (Wray 00). The ntal spectral feld was frst truncated to 64 3 spectral modes and then approprately transformed to physcal space to be used as ntal condtons on the 64 3 staggered mesh. The results of the LES were compared wth a drect spectral smulaton for decayng turbulence carred out by Wray wth the full 56 3 spectral modes, usng the same ntal condton. Fgure compares the tme hstory of the turbulence knetc energy for the varous computatons. For the purposes of ths comparson, only the knetc energy n the frst 64 3 modes of Wray s full DNS are plotted. The fgure shows that the results obtaned usng the Smagornsky-type hypervscosty model agree best wth results of the full 56 3 smulaton. Fgure 3 compares the energy spectra from the three dfferent computatons at a relatvely early tme n the computatons. It s seen that n the case where there was no subgrd-scale model there s an accumulaton of energy at the hgher wavenumbers, snce the physcal vscosty s unable to remove energy from the small scales at a suffcently fast rate. It s also seen that the two computatons usng subgrd-scale models were well resolved on the 64 3 mesh. 3.4. Kolmogorov flow Kolmogorov flow s an open flow n a perodc box, drven by a large-scale steady forcng n the x 1 -drecton gven by, f 1 = F sn (κ f x ) ; κ f = 1; F = 0.16 (3.5)
Tools for LES 13 0.4 0.35 0.3 Knetc Energy 0.5 0. 0.15 0.1 0.05 0 4 6 8 10 1 14 16 18 Tme Fgure. Hstory of the knetc energy decay for sotropc turbulence. LES computaton usng: no subgrd model (dotted lne); Smagornsky model (dashed lne); and Smagornsky type hypervscosty model (sold lne) carred out on a 64 3 grd. Reference DNS computaton by Wray usng 56 3 spectral modes ( ). The hgh-reynolds-number lmt of ths flow has been studed by Borue & Orszag (1996) and Shebaln & Woodruff (1997). Breakng of the symmetres mposed by the forcng has been studed by Jeanmart, Carat & Wncklemans (00) (00) n computatons usng box szes larger than the perod of the forcng. In order effectvely to ncrease the extent of the nertal range, we use only hypervscous dsspaton. Snce the Kolmogorov flow s characterzed by the ampltude of the force F, the spatal frequency of the forcng functon κ f, and the hypervscosty ν 4, we defne the followng velocty and tme scales and a Reynolds number based on the hypervscosty (Borue & Orszag (1996)). ( ) 0.5 F U 0 =.5 (3.6) t 0 = Re = U 0 κ 3 f ν 4 κ f 1 (F κ f ) 0.5 (3.7) (3.8) Ths flow s a good test case for the followng reasons: (a) The flow has a statstcally statonary state. (b) The flow s beleved to have a well-defned hgh Reynolds number lmt (Borue & Orszag (1996)), for whch the turbulent ntenstes, energy dsspaton rates, and varous terms n the energy balance equatons have a smple coordnate dependence, a+b cos x. Ths makes Kolmogorov flow a good model to explore the applcablty of turbulent transport approxmatons n open flows. (c) Snce the snusodal forcng has only one spatal frequency (κ f = 1) and a constant
14 D. A. Caughey & G. Jothprasad 1 0.01 Energy Spectrum functon 0.0001 1e-06 1e-08 Energy Spectrum Functon 1e-10 1 10 Wave number, κ Fgure 3. Comparson of the nstantaneous energy spectra from the computatons usng: only molecular vscosty (+), molecular vscosty plus the Smagornsky model ( ), and molecular vscosty plus the Smagornsky-lke hypervscosty model ( ), at very early tmes. ampltude F, the exact amount of energy nput nto the large scale can be calculated. At a suffcently hgh Reynolds number, when a statstcally steady state s reached, the amount of energy nput nto the large scales s equal to the amount of energy dsspated ɛ. Ths s one of the flows n whch ɛ can be estmated exactly and hence enables us to estmate other length and velocty scales more accurately. (d) The boundary condtons are perodc n all three drectons, whch means that the exstng DNS code could be easly modfed to nclude the snusodal forcng. Whle ths flow has been studed extensvely usng DNS, generatng such a flow n the laboratory would be dffcult. However, the flow n the vcnty of the nflecton ponts of the mean profle should be smlar to that for homogeneous shear flow. In the present LES computatons, the flow was ntalzed to the soluton of an analogous lamnar flow problem plus an added sotropc velocty fluctuaton. The fluctuatons were found to elmnate the otherwse long transent tme needed for the turbulent fluctuatons to develop and grow. The flow propertes were averaged over planes contanng the statstcally-homogeneous drectons (x 1 and x 3 ), and over tme usng a varant of movng averages. It was found necessary to average the flow quanttes over a large number of eddy turnover tmes to obtan statstcally relable profles of mean velocty, velocty autocorrelatons, and Reynolds stress. The tme averagng was begun only after the flow had reached a statstcally-statonary state. It has been observed by Borue & Orszag (1996) that the statstcally statonary state has a mean velocty profle that becomes ndependent of the Reynolds number for suffcently large values of Reynolds number. The present smulatons were carred out on both 64 3 and 3 3 grds. Fgure 4 show the tme hstory of turbulence knetc energy and dsspaton rate, whle fgure 5 shows the nstantaneous energy spectra for the flow usng hypervscosty computatons on the 64 3 grd. Fgure 6 compares the mean velocty and velocty correlatons obtaned from the 64 3 hypervscosty computaton wth the analytcal curves found by Borue & Orszag
Tools for LES 15 1 1000 Flow Varable 100 10 1 0.5 0-0.5 Skewness 0.1 0.01-1 0 500 1000 1500 000 Tme Fgure 4. Tme hstory of development of mean quanttes for the Kolmogorov flow hypervscosty computatons on a 64 3 grd. Upper curve s skewness, mddle curve s average energy, and lower curve s dsspaton rate. -1.5 1 Energy Spectrum Functon 0.1 0.01 0.001 0.0001 1e-05 1e-06 1 10 Wave number, κ Fgure 5. Instantaneous mean energy spectra for the Kolmogorov flow hypervscosty computatons on a 64 3 grd. Spectra are plotted at tmes t = 071 (+), 089 ( ), 107 ( ), 14 ( ), and 143 ( ). (1996) to ft the hgh Reynolds number lmt. The agreement s qute good n spte of the relatve coarseness of the grd used here.
16 D. A. Caughey & G. Jothprasad Correlaton -0.4-0. 0 0. 0.4 0.6 0.8 Vertcal Coordnate, (y y0)/(π) 0.4 0. 0-0. U 1 u 1u 1 u u u 3u 3 u 1u -0.4-1.5-1 -0.5 0 0.5 1 1.5 Mean velocty Fgure 6. Mean velocty and velocty correlatons for Kolmogorov flow; symbols are results from hypervscosty computatons on a 64 3 grd wth ν 4 = 0.006, and lnes are hgh-reynolds-number curve fts by Borue & Orszag (1996). Varables plotted are: u 1, + and sold lne; u 1u, and short dashed lne; u 1u 1, and long dashed lne; u u, and dotted lne; u 3u 3, and dash-dot lne. 4. Conclusons As a frst step n developng tools sutable for assessng the accuracy of LES solutons for flows at large Reynolds number, the authors have developed a computer code for solvng the ncompressble Naver-Stokes equatons for test flows that wll allow the comparson of varous accuracy-assessment strateges. The code ncludes a conventonal Smagornsky sub-grd model, as well as hypervscosty dsspatve terms that wll allow greater separaton of the energy-contanng and dsspatve scales for hgh Reynolds number flows. The code s valdated for several smple, perodc flows, ncludng the Taylor-Green vortex and decayng, sotropc turbulence, and prelmnary results show good agreement wth the hgh Reynolds number lmt for (forced, perodc) Kolmogorov flow on relatvely modest meshes usng the hypervscosty dsspaton. The benefts of usng the Kolmogorov flow for these studes are descrbed, and future results wll concentrate on comparsons of methods for accurately predctng the statstcs of the energetc scales for ths flow. REFERENCES Borue, V. & Orszag, S. A. 1996 Numercal study of three-dmensonal Kolmogorov flow at hgh Reynolds numbers. J. Flud Mech. 306, 93-33. Brachet, M. E., Meron, D. I., Orszag, S. A., Nckel, B. G., Morf, R. H. & Frsch, U. 1983 Small-scale structure of the Taylor-Green vortex. J. Flud Mech. 130, 411-45.
Tools for LES 17 Chorn, A. J. 1968 Numercal soluton of the Naver-Stokes equatons. Math. Comput., 745-76. Jeanmart, H. 00 Prvate communcaton. Centre for Systems Engneerng and Appled Mechancs, Unversté Catholque de Louvan, 1348 Louvan-la-Neuve, Belgum. Note: The (unpublshed) results provded by Jeanmart are equvalent to those plotted n fgure 7 of Brachet et al. (1983). Jeanmart, H., Carat, D. & Wnckelmans, G. S. 00 Non unversalty and symmetry breakng n three-dmensonal turbulent Kolmogorov flow. Submtted to Phys. Fluds. Km, J. & Mon, P. 1985 Applcaton of a fractonal-step method to ncompressble Naver-Stokes equatons. J. Comput. Phys. 59, 308-33. Llly, D. K. 1967 The representaton of small-scale turbulence n numercal smulaton experments. Proc. IBM Scentfc Computng Symposum on Envronmental Scences IBM Form No. 30-1951, 195-10. Mornsh, Y., Lund, T. S., Vaslyev, O. V. & Mon, P. 1998 Fully conservatve hgher order fnte dfference schemes for ncompressble flow. J. Comput. Phys. 143, 90-14. Shebaln, J. V. & Woodruff, S. L. 1997 Kolmogorov flow n three dmensons. Phys. Fluds, 9, 164-170. Temam, R. 1969 Sur l approxmaton de la soluton des equatons de Naver-Stokes par la méthode des pas fractonnares. Arch. Ratonal Mech. Anal. 3, 135-153 and 33, 377-385. Wray, A. 00 Prvate communcaton. NASA Ames Research Center.