Smagorinsky constant in LES modeling of anisotropic MHD turbulence

Transcription

1 Theor. Compu. Fluid Dyn. DOI.7/s z ORIIAL ARTICLE Anaoliy Vorobev Oleg Zianov Smagorinsy consan in LES modeling of anisoropic MHD urbulence Received: 3 May 26 / Acceped: 7 Augus 27 Springer-Verlag 27 Absrac Turbulen flucuaions in magneohydrodynamic (MHD) flows can become srongly anisoropic or even quasi-2d under he acion of an applied magneic field. We invesigae his phenomenon in he case of low magneic Reynolds numbers. I has been found in earlier DS and LES of homogeneous urbulence ha he degree of anisoropy is predominanly deermined by he value of he magneic ineracion parameer and only slighly depends on he Reynolds number, ype of large-scale dynamics, and he lengh scale. Furhermore, i has been demonsraed ha he dynamic Smagorinsy model is capable of self-adjusmen o he effecs of anisoropy. In his paper, we capialize on hese resuls and propose a simple and effecive generalizaion of he radiional non-dynamic Smagorinsy model o he case of anisoropic MHD urbulence. Keywords Magneohydrodynamic Turbulence Large-Eddy Simulaion PACS d ep Inroducion Magneohydrodynamic (MHD) urbulen flows occur in numerous asrophysical, geophysical, and echnological applicaions. In his paper, we consider he case of low magneic Reynolds number R m ul/η, () ypical for he indusrial and laboraory flows of liquid meals, oxide mels, and oher elecrically conducing fluids (see, e.g., [] or [2]). In (), u and L are he ypical velociy and lengh scales, and η = (σ µ ) is he magneic diffusiviy; σ and µ are he elecric conduciviy of he liquid and he magneic permeabiliy of vacuum. Low-R m ineracion beween a saic magneic field and a urbulen flow is an imporan facor of some meallurgical operaions, such as coninuous seel casing or growh of large semiconducor crysals, where magneic fields are inenionally used o non-inrusively suppress he unwaned developmen of he flow. In oher cases, such as he primary aluminum producion in Hall Héroul or iner anode processes, or in he lihium cooling blanes for magneic confinemen fusion, he ineviably presen saic magneic field has an adverse effec on performance, which has o be minimized hrough opimizaion of he process. The resuls of he presen wor, albei rigorously valid only in he case of low R m, can be exended o he siuaions wih moderae R m and high hydrodynamic Reynolds number, mos noably o he earh dynamo problem. The diffusive cu-off scale of he magneic field is sufficienly large in such cases so ha he magneic Communicaed by R.D. Moser A. Vorobev O. Zianov (B) Deparmen of Mechanical Engineering, Universiy of Michigan-Dearborn, Dearborn, MI , USA zianov@umd.umich.edu

2 A. Vorobev, O. Zianov field can be fully resolved in he DS-lie manner. The as of modeling he urbulen velociy flucuaions a smaller scales reduces o he problem addressed in his paper. We will consider incompressible flows and, for simpliciy, assume ha he applied magneic field is uniform and verical B = Be z.inhelow-r m case, he MHD equaions can be significanly simplified by applying he quasi-saic approximaion [2]. The perurbaions of he magneic field induced by fluid moions are small in comparison wih he imposed magneic field and can be negleced. They can also be approximaely assumed o adjus insananeously o he velociy perurbaions. In he resul of hese approximaions, he Lorenz force is expressed as a linear funcional of velociy. The governing equaions can be represened in a closed form as u + (u )u = ρ p + ν u σ B 2 ρ zz u, u =, (2) where is he reciprocal Laplace operaor ha sands for a soluion of he Poisson equaion for he elecric poenial wih proper boundary condiions. The non-dimensional form of (2) conains wo dimensionless parameers, one of which is he Reynolds number Re ul/ν and anoher is he magneic ineracion parameer σ B 2 L/ρu ha esimaes he raio beween he Lorenz and ineria forces. We focus on MHD urbulence far from he walls and consider he implicaions of he special properies resuling from he presence of he Lorenz force erm in (2) for large-eddy simulaions. Such properies have been relaively horoughly sudied in analyical, experimenal, and numerical wors (see, for example, [3 6]). I has been undersood ha in unbounded flows he direc effec of he magneic field is wofold. Firs, here is an addiional urbulence suppression via Joule dissipaion of induced elecric currens. Second, he flow acquires axisymmeric anisoropy wih flow srucures elongaed in he direcion of he magneic field. The naure of he anisoropy becomes ransparen if we use he Fourier ransform o wrie he rae of he Joule dissipaion of a mode û(, ) as µ() = σ B 2 ρ û(, ) 2 cos 2 θ, (3) where θ is he angle beween he wavenumber vecor and he magneic field B. The Joule dissipaion ends o eliminae he velociy gradiens along he magneic field lines, hus leading o elongaion of flow srucures. The flow approaches 2D form wih zero parallel gradiens a, alhough i has been argued in [7] ha he proper erm is quasi-wo-dimensionaliy due o he ineviable ellipic and shear flow insabiliies of he 2D srucures. Furhermore, pure wo-dimensionaliy is impossible in he presence of walls non-parallel o he magneic field (see, e.g., [8]). Only he anisoropy of gradiens is direcly affeced by he magneic field. Anoher ype of anisoropy referring o inequaliy beween he velociy componens (anisoropy of he Reynolds sress ensor) can follow from he acion of he magneic field indirecly, hrough he nonlinear ineracion mechanism. Oher indirec effecs include he suppression of he nonlinear energy ransfer beween he lengh scales and he associaed increase of he inerial range slope of he energy power specrum (approaching 3 inquasi-2dflowsahigh). The focus of he presen paper is on he efficien LES modeling of anisoropic MHD urbulence. The erm efficiency is undersood in a broader sense o include he model s accuracy, breadh of applicabiliy, and low compuaional cos. More precisely, we would lie o develop a model ha could be applied o MHD flows wih Accuracy comparable wih he accuracy achieved for non-magneic flows a similar Re and numerical resoluion. Applicabiliy range <, i.e., for he enire specrum of MHD flows, from nearly isoropic o quasi-2d. Lowes possible compuaional cos, preferably no exceeding he compuaional cos of models commonly used in he non-magneic case. In he nex secion, we discuss he properies of he anisoropic MHD urbulence and review he earlier modeling aemps. The discussion serves as a basis for Sec. 3, where we invesigae he possibiliy of a simple generalizaion of he Smagorinsy closure model o he MHD case by assuming a funcional dependency of he Smagorinsy consan on a single anisoropy parameer. The final remars are given in Sec Modeling of anisoropic MHD urbulence: resuls of earlier sudies A deailed sudy of he anisoropy of homogeneous MHD urbulence a low R m was conduced in [6]. DS and LES compuaions were performed in a wide range of Re and. In order o achieve a saisically seady flow,

3 Smagorinsy consan in LES modeling of anisoropic MHD urbulence he arificial force ˆf() = α()û() was applied o he large scale modes wih. 3.. The coefficiens α were deermined a each ime sep so as o provide he oal wor by forcing equal o a prescribed consan ɛ according o α() = ɛ /( forced û() 2 ),where forced is he number of he forced modes. This led o saisically seady saes, in which he oal (viscous plus magneic) dissipaion rae flucuaed only slighly around ɛ. In order o reveal he possible effec of he large-scale dynamics, wo ypes of forcing were used, one isoropic wih he energy inpu equally divided among he forced modes and anoher purely 2D wih he energy inpu limied o he modes wih B. Below we idenify hese wo ypes as 3D and 2D forcing, respecively. The main resuls of [6] deal wih he ransformaion of urbulen flucuaions a inermediae and small lengh scales. The scale-dependen gradien anisoropy was esimaed using g() 3τ µ() 2 E() = 3 2 z û û 2 û û, (4) where he sums are over all wavenumber vecors in he shell /2 < + /2, and µ(), E() are he power specra of he magneic dissipaion rae and ineic energy. The scaling facor in (4) uses he Joule damping ime τ ρ/σ B 2 and is chosen so ha g() = inanisoropicflowandg() = inapurely2d flow wih zero magneic dissipaion. I has been found ha a all ouside he range of arificial forcing, g() is a remarably robus funcion of he magneic ineracion parameer. The influence of he lengh scale, Re, and he deails of he largescale dynamics (dicaed by he differen ypes of he forcing) is much weaer. Similar, alhough slighly less pronounced behavior was observed for he scale-relaed anisoropy of velociy componens. Anoher resul of [6] imporan for he presen discussion concerns he meaning of he coefficiens ( ui / z) 2 ( + δ i3 ) ij = ( ui ) 2 ( ), i =, 2, 3, j =, 2, () / x j + δij where sands for he volume averaging. Averaging over he enire compuaional domain was applied in [6] following he assumpion of spaial homogeneiy. I should be replaced by some local averaging in he inhomogeneous flows. The coefficiens () are equal o one in an isoropic flow (see [9]) and zero in a purely 2D flow uniform in he direcion of he magneic field, and are usually close o each oher in anisoropic flows so ha any of hem can be used o evaluae he anisoropy. In our sudy, 22 and 32 are calculaed and averaged o a single coefficien = ( )/2. Formally, is an inegral characerisic of he gradien anisoropy. One could hin of i as relaed o he large energy-conaining scales of he flow. In fac, however, i was demonsraed in [6] ha he dominan conribuion ino is from he inermediae lengh scales smaller han hose where he forcing is applied bu larger han he ypical scale of LES cu-off. As a resul, was found o be close o he scale-dependen anisoropy coefficien g() in he range of is scale invariance and insensiive o he deails of large-scale dynamics, Reynolds number, and he use of LES models. can be considered as a reasonably accurae measure of anisoropy in he inermediae scale range. LES modeling of decaying and forced homogeneous MHD urbulence was conduced in [] and [6], respecively. The Smagorinsy eddy viscosiy closure hypohesis was used in boh cases wih he subgridscale sress ensor τ ij expressed hrough he rae-of-srain ensor of filered velociy field S ij as τ ij = 2 2 S S ij, S = ( 2S ij S ij ) /2, (6) where is he filer widh and is he Smagorinsy consan. The conclusion of boh sudies, achieved hrough he a-poseriori comparison wih he resuls of high resoluion DS, was ha he Smagorinsy model is overdissipaive if used in is simple form wih consan. Subsanially beer agreemen wih he DS resuls was achieved when he consan was deermined using he dynamic procedure [,2]. In he procedure, a second es filer wih he widh > is applied o he compued flow field. I is assumed ha he relaion (6) holds for boh he main and he es lengh scales. This leads o he formula for deermining he Smagorinsy consan based on he flow condiions [2] = Mij Lij / Mij Mij, (7) where L ij = ũ i u j ũ i ũ j is he Leonard ensor, and M ij = 2[ 2 S Sij 2 S S ij ].

4 A. Vorobev, O. Zianov The fac ha he simple Smagorinsy model wih consan is inadequae in he case of MHD flows wih high is quie expecable. The values used for (beween. and.324 depending on he ype and widh of he filer and he srengh of mean shear [3]) do no ae ino accoun he flow ransformaion caused by he applied magneic field (anisoropy, suppressed nonlinear energy ransfer, and seepened energy specrum). All hese facors lead o reducion of he subgrid-scale energy dissipaion and, hus, require reducion of. The beer accuracy of he dynamic model is, oo, no enirely surprising. The model has demonsraed is abiliy o remedy he overdissipaion problem in oher siuaions characerized by reduced SS dissipaion, for example in he laminar-urbulen ransiion. The physical and mahemaical reasons for he accuracy are, however, unclear. Albei he formula (7) can be derived from he assumpion of scale similariy, he assumpion iself is invalid in many cases where he dynamic model demonsraed is superioriy (for example in he ransiional flows or flows a moderae Reynolds numbers). In general, he a-poseriori accuracy of he model can no be convincingly relaed o is accurae represenaion of any paricular aspec of he behavior of subgrid-scale flucuaions [4]. An alernaive, clearly no exhausive explanaion was proposed in [], where he dynamic procedure was shown o minimize he dependency of modeled quaniies on he filer widh. For hese reasons, we refrain from aribuing he accuracy of he dynamic model in he MHD case o some physical feaures of he flow (alhough i can be argued ha he scale invariance of he anisoropy is relevan). Insead, he nex secion presens an aemp o capialize on he demonsraed abiliy of he dynamic model o modify in accordance o he flow ransformaion. We ry o develop a simple formula for ha would mimic he dynamic modificaion. I can be argued ha he isoropic eddy viscosiy relaion (6) is iself invalid in he case of a srongly anisoropic MHD flow since, as was demonsraed in [6], he anisoropies of he ensors τ ij and S ij are very differen. We esed several generalizaions of he dynamic Smagorinsy model based on he eddy viscosiy formulas wih one or wo dynamically adjusable anisoropy correcions. one of hem showed discernibly beer agreemen wih he DS resuls han he convenional dynamic model. I seems ha simple reducion of is sufficien o accoun for he effec of he imposed magneic field. 3 Smagorinsy consan as a funcion of anisoropy parameers For our purposes, he discussion of he previous secion can be reduced o he following wo hypoheses, which, alhough no fully proven, received subsanial facual suppor in compuaions [] and [6].. The dynamic adjusmen of is sufficien o accoun for anisoropy and oher aspecs of he flow ransformaion caused by he imposed magneic field. o oher modificaion of he Smagorinsy closure is needed o eep he accuracy of he model a he same level as in he isoropic case. 2. The variaion of caused by he magneic field can be adequaely described by a funcion of a single scalar parameer, such as he anisoropy coefficien () or he magneic ineracion parameer. In his secion we ry o capialize on he hypoheses and obain he relaions = () and = (). One can speculae on which of he wo dependencies, () or (), is preferable. There is no srong preference in he case of homogeneous urbulence, where, as shown in [6] and confirmed by our compuaions discussed below, can approximaely be considered a one-o-one funcion of. In he real flows, where is evaluaed based on he size of he flow domain and some consan characerisic velociy, is effecive value can change in space (for example because of variaion of B) or ime (for example in he case of decaying flow). The relaion = () seems, herefore, preferable as one based on a universal anisoropy characerisic ha can be evaluaed locally, boh in space and ime. Below we presen he resuls of a series of LES compuaions of homogeneous MHD urbulence in a box of dimensions 2π 2π 4π wih periodic boundary condiions. The dynamic Smagorinsy model is used as a subgrid-scale closure. The numerical mehod is pseudo-specral based on he fully de-aliased fas Fourier ransform. The flow is arificially forced a. 3. wih he 3D and 2D ypes of forcing described in he beginning of he previous secion (see [6] for more deails). Each numerical experimen sars wih non-magneic calculaions ha las long enough o produce a fully developed urbulen flow. Then, a =, he magneic field B is applied and ep consan ill he flow ransformaion is complee, afer which he flow saisics are colleced and averaged over several (a leas, 3) eddy urnover imes T = L( )/u( ).The Reynolds number Re and he magneic ineracion parameer are evaluaed in he isoropic flow a = using he inegral lengh scale L and he rms velociy u. The experimens are conduced a and Re = 7, 2,, and 6,. Two numerical resoluions wih and specral

5 Smagorinsy consan in LES modeling of anisoropic MHD urbulence E Fig. Flows wih 3D forcing, Re = 2,, and numerical resoluion Time evoluions of he resolved oal energy a, global anisoropy coefficien b, and volume-averaged Smagorinsy coefficien c are shown for =,,, and - -/3 E ε, µ -2-4 g Fig. 2 Flows wih 3D forcing, Re = 2,, and numerical resoluion Time averaged specra of energy a, viscous (solid line) and magneic (dashed line) dissipaion raes b, and he local anisoropy coefficien (4) are shown for =,,, and funcions are used. The filering widh of he LES closure is defined as he grid sep. Furher deails of he compuaional procedure can be found in [6]. Figure illusraes he ypical emporal evoluion of he flows. The resuls obained wih 3D forcing a Re = 2, and =,,, and wih he numerical resoluion of are shown. Afer he inroducion of he magneic field, he global characerisics such as he oal resolved energy, rae of resolved viscous dissipaion or he anisoropy coefficien evolve rapidly unil hey sabilize a new levels corresponding o he forced anisoropic flows. The Smagorinsy consan evolves in he same way bu, as can be seen from comparison beween he Fig. a c, is evoluion o new levels is slower. Power specra of he flow energy E(), resolved viscous ɛ() and magneic µ() dissipaion raes were obained a he equilibrium sages of each flow. Several flow fields separaed by more han one isoropic eddy urnover ime from each oher were used for he ime-averaged specra, examples of which are presened in Fig. 2. One can see how he increase in he srengh of he magneic field (growh of ) leads o seeper curves E() and µ() and reducion of viscous dissipaion a small scales. The scale-dependen anisoropy coefficien g() demonsraes he behavior observed and analyzed in [6], i.e., scale-independence ouside he forced range. A he sronges magneic field (a = ) he flow becomes srongly anisoropic bu no purely 2D as shown by small bu non-zero values of g() and non-vanishing magneic dissipaion. Figure 3 shows he effec of he magneic field on he resolved viscous and magneic dissipaion raes ɛ and µ and he subgrid-scale dissipaion rae ɛ SS in fully developed flows. Time- and volume-averaged raes obained a differen Re, ypes of forcing, and numerical resoluion are shown. The numerical experimens are saged in such a way ha he oal dissipaion rae is consan, i.e., ɛ + µ + ɛ SS = ɛ =.. One can see ha a relaively large Reynolds numbers used in he simulaions, he resolved scales are responsible for only a small fracion of he oal viscous dissipaion. ɛ is less han.6ɛ a zero magneic field and decreases wih. For he non-magneic runs, he dissipaion is caused predominanly by he Smagorinsy erm. In he presence of he magneic field, µ increases wih and ɛ SS decreases, respecively, alhough i never drops below.ɛ. I is ineresing ha he role played by he SS model grows if he 2D forcing is applied (see Fig. 3c). The reasons for ha are our requiremen of consan oal dissipaion and he srong anisoropy induced by he

6 A. Vorobev, O. Zianov x64, 3D 32 2 x64, 2D 64 2 x28, 3D 64 2 x28, 2D..4 3D..4 ε..2 µ.3.2 2D ε SS.3.2 2D D Fig. 3 Resolved viscous a, magneic b and subgrid-scale c dissipaion raes as funcions of magneic ineracion parameer. Timeand volume-averaged daa obained in fully esablished flows wih 3D and 2D forcing a wo differen numerical resoluions are shown. Righ riangle Re = 7, diamond Re = 2,, lef riangle Re = 6, x64, 3D 32 2 x64, 2D 64 2 x28, 3D 64 2 x28, 2D g(k max ) g( max ) Fig. 4 lobal anisoropy coefficien a and he scale-dependen anisoropy coefficien aen a he scale of filer widh b as funcions of magneic ineracion parameer; c global versus filer-widh coefficiens. oaion is as in Fig. 3 forcing a he large lengh scales. This resuls in decrease of he magneic dissipaion rae a hese scales and so of he oal dissipaion rae µ leading o he growh of ɛ SS. The flow anisoropy was esimaed using he global coefficien and he scale-dependen coefficien g( max ) aen a he lengh scale of he filer widh. Again, ime-averaging over several eddy urnover imes was applied. One can see in Fig. 4a, b ha boh coefficiens decrease rapidly a small and moderae and somewha slower a sronger fields wih > 3. The flow becomes srongly anisoropic bu remains essenially 3D. An imporan feaure of curves in Fig. 4a, b is heir closeness o each oher. This is quie remarable considering he fac ha he curves are obained for subsanially differen Reynolds numbers, forcing mechanisms, and numerical resoluions. Moreover, as can be seen in Fig. 4c, is nearly equal o g( max ) and, hus, o coefficiens g() a any oher ousideherange offorced lengh scales (see Fig. 2c). Ourcalculaions confirm he conclusions obained in [6] on he basis of he DS and LES compuaions a lower Reynolds numbers. The anisoropy properies of he urbulen flucuaions are nearly scale-independen, insensiive o he large scale dynamics and he magniude of he Reynolds number and are deermined by he magneic ineracion parameer or anoher parameer, such as he inegral anisoropy coefficien. The main resuls are presened in Fig., which shows he dependence of he volume- and ime-averaged Smagorinsy consan on he anisoropy characerisics. As in he previous figures, daa obained a differen Reynolds numbers, forcing mechanisms, and numerical resoluions are ploed. The consan decreases wih he srengh of he magneic field as represened by increasing or decreasing anisoropy coefficiens. The resuls suppor he second of our hypoheses. The curves in Fig. are quie close o each oher. The Smagorinsy consan can be considered a funcion of,,org( max ) wih a reasonable degree of accuracy. One can see in Fig. b ha he agreemen is paricularly good for he funcion = (). Requiring ha aains is isoropic value a = and zero in a purely 2D flow a = we can approximae he daa in Fig. b by a simple linear relaion = (8) shown by he bold line. Here is he Smagorinsy consan corresponding o he isoropic non-magneic flow. I is abou. in our calculaions bu can have differen values in oher cases depending on ype of he flow and ype and widh of he filer.

7 Smagorinsy consan in LES modeling of anisoropic MHD urbulence g( max ) Fig. a Volume- and ime-averaged Smagorinsy consan as a funcion of he magneic ineracion parameer a, global anisoropy coefficien b and coefficien of anisoropy a filer widh c. Bold line in b is for he linear relaion (8). oaions are as in Fig E -3 ε µ Fig. 6 Specra of resolved energy a, viscous b, and magneic c dissipaion raes. Flows wih 3D forcing, Re = 2,, = and, and numerical resoluion by funcions are calculaed using he dynamic model (solid lines), Smagorinsy model wih consan (dashed line), and Smagorinsy model wih consan adjused according o (8) (dashed-doed lines) In order o es he formula (8) we conduced simple numerical experimens. Forced flows wih Re = 2, and numerical resoluion were compued using he dynamic Smagorinsy model, simple Smagorinsy model wih consan =, and he modificaion of he Smagorinsy model wih consan adjused a each ime sep according o (8). Each run sared wih he same isoropic iniial velociy field. Calculaions were performed for = and =. Afer compleion of ransiional periods, he saisics were colleced and ime-averaged for he fully esablished anisoropic flows. The resuls are presened in Fig. 6. One can see ha he adjusmen (8) resuls in almos exac reproducion of he specra of energy and dissipaion raes of flows obained wih he dynamic Smagorinsy model. On he conrary, he specra calculaed wih he Smagorinsy model wih consan are noiceably, albei no very srongly, differen, wih ypical over-dissipaive suppression a large. I mus be sressed ha such an excellen agreemen beween he dynamic model and he model wih adjused is observed only for saisically seady periods of he flow evoluion. The siuaion during he ransien periods is quie differen as illusraed by our es simulaions of decaying urbulence. In he simulaions, a fully developed forced non-magneic flow is used as an iniial condiions. A he momen =, he magneic field is applied, and he forcing is disconinued, afer which he flow is allowed o decay freely for several urnover imes simulaneously developing anisoropy. Figure 7 shows he resuls of he dynamic LES of he process for = and =. One can see in Fig. 7a, b ha boh, which represens / deermined according o (8), and he dynamically deermined / decrease wih ime as he flow becomes anisoropic bu a very differen raes. Iniially, he decay rae for he dynamic is almos zero, whereas he anisoropy coefficien drops rapidly. A laer sages, decays faser han bu remains significanly larger han prediced by (8). This is furher illusraed in Fig. 7c. Similar delay in he modificaion of he Smagorinsy consan is demonsraed by he forced flows during he ransiional sages of heir developmen (see Fig. b, c). One can be emped by a simple explanaion ha he generaion of dimensional anisoropy is an essenially linear process governed by he Joule dissipaion (3). Is ypical ime scale is he Joule damping ime τ ρ/σ B 2. On he oher hand, he variaion of he Smagorinsy consan is associaed wih he nonlinear process of esablishing new correlaions beween he urbulen rae of srain and sress ensors occurring a he ime scale of he eddy urnover ime T = L/u. One can expec slower evoluion of a = T/τ >.

8 A. Vorobev, O. Zianov / Fig. 7 Decaying urbulen flows wih Re = 2,, = and calculaed using he dynamic model. a lobal anisoropy coefficien, which also represens / calculaed according o he linear relaion (8), b Smagorinsy consan scaled by he isoropic value, c, versus The explanaion, albei possibly relevan, is clearly an oversimplificaion. In paricular, i does no explain he delay in he developmen of a = (seefig.7). The simulaions of decaying homogeneous flows did no allow us o clearly separae beween he resuls of he dynamic model, model wih consan and model (8). The curves of E(), ɛ(),andµ() were found o be very close o each oher. More definiive esing is required o idenify he effec of he delay in developmen of on he accuracy of he model. This can be done in simulaions of more realisic evolving flows, for example, of a free mixing layer. 4 Concluding remars We performed a series of LES of forced and decaying homogeneous MHD urbulence a low magneic Reynolds number. Analyzing he flow properies in a wide range of he hydrodynamic Reynolds number and magneic ineracion parameer and a differen large-scale dynamics and filer widhs we found a clear confirmaion ha he effec of he dynamic adjusmen of he Smagorinsy consan can be accuraely approximaed by a simple linear funcion of he global coefficien of gradien anisoropy. Apar from purely heoreical ineres, he relaion has clear pracical value. Calculaion of in he dynamic model approximaely doubles he amoun of compuaions in comparison wih he sandard Smagorinsy model. This is undesirable in simulaions of indusrial processes such as, for example, he Czochralsi growh of large crysals or coninuous seel casing, he ass compuaionally challenging even wih he simples urbulence models. Wih quanified dependence of on one could sill use he sandard model bu avoid is over-dissipaive characer by adjusing o he srengh of he flow anisoropy. We found ha he adjusmen formula canno be used in ransien processes because developmen of anisoropy occurs a a faser rae han he modificaion of he Smagorinsy consan. Furhermore, he formula was obained for homogeneous urbulence and may prove inaccurae in he presence of mean shear or roaion. I is possible ha improved correlaions can be developed for such siuaions bu hey are liely o be also more complex and less universal han our simple formula. Their advanage over he dynamic model is, hus, far from being obvious. Acnowledgmens The wor is suppored by he gran DE F2 3 ER4662 from he U.S. Deparmen of Energy. The auhors are hanful o he referee for careful reading of he manuscrip and useful commens. References. Davidson, P.A.: An Inroducion o Magneohydrodynamics. Cambridge Universiy Press, Cambridge (2) 2. Moreau, R.: Magneohydrodynamics. Kluwer, Dordrech (99) 3. Schumann, U.: umerical simulaion of he ransiion from hree- o wo-dimensional urbulence under a uniform magneic field. J. Fluid Mech. 74, 3 8 (976) This may an overesimaion for some, even saisically non-seady flows. The dynamic evaluaion of is no always required a every ime sep. Accurae resuls are nown o be obained by evaluaing once every few (someimes as many as ) seps.

9 Smagorinsy consan in LES modeling of anisoropic MHD urbulence 4. Alemany, A., Moreau, R., Sulem, P.L., Frisch, U.: Influence of an exernal magneic field on homogeneous MHD urbulence. J. Mech. 8, (979). Zianov, O., Thess, A.: Direc numerical simulaion of forced MHD urbulence a low magneic Reynolds number. J. Fluid Mech. 38, (998) 6. Vorobev, A., Zianov, O., Davidson, P.A., Knaepen, B.: Anisoropy of magneohydrodynamic urbulence a low magneic Reynolds number. Phys. Fluids 7, 2 (2) 7. Thess, A., Zianov, O.: On he ransiion from wo-dimensional o hree-dimensional MHD urbulence. In: Proc. of 24 CTR summer program. Sanford Universiy, pp (24) 8. Sommeria, J., Moreau, R.: Why, how, and when MHD urbulence becomes wo-dimensional. J. Fluid Mech. 8, 7 8 (982) 9. Hinze, J.O.: Turbulence. Mcraw-Hill, ew Yor (99). Knaepen, B., Moin, P.: Large-eddy simulaion of conducive flows a low magneic Reynolds number. Phys. Fluids 6, 2 (24). ermano, M., Piomelli, U., Moin, P., Cabo, W.H.: A dynamic subgrid-scale eddy viscosiy model. Phys. Fluids A 3, 76 (99) 2. Lilly, D.K.: A proposed modificaion o he ermano subgridscale closure model. Phys. Fluids A 4, 633 (992) 3. Pope, S.B.: Turbulen Flows. Cambridge Universiy Press, Cambridge (2) 4. Jiménez, J., Moser, R.D.: AIAA Paper (998). Pope, S.B.: Ten quesions concerning he large-eddy simulaion of urbulen flows. ew J. Phys. 6, 3 (24) 6. Zianov, O., Vorobev, A., Thess, A., Davidson, P.A., Knaepen, B.: Anisoropy of MHD urbulence a low magneic Reynolds number. In: Proc. of 24 CTR summer program. Sanford Universiy, pp (24)