Numerical Simulation of the Eddy System on the Basis of the Boltzmann Equation

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1 Numercal Smulato of the Eddy System o the Bass of the Boltzma Equato V.V.Arstov a, O.I.Roveskaya b a Dorodcy Computg Cetre of Russa Academy of Sceces, Moscow, Russa b Uversty of Ude, Ude, Italy Abstract. The am of preset work s to vestgate the feasblty of applyg a ketc approach to the problem of modelg turbulet or ustable flows. We perform three-dmesoal drect umercal smulatos of the Taylor Gree (TG) type codtos ad decay of sotropc turbulece perodc compressble flow. The smulato s based o the drect umercal solvg the Boltzma ketc equato. For the regular tal codto the results show the fragmetato of the large tal eddes ad subsequetly the full dampg eergy of the system. Depedece of the ketc eergy o the wave umber s obtaed by meas of the Fourer expaso of the velocty compoets. The decay expoet of the ketc eergy spectrum for both problems close to the value -5/3. Keywords: the Boltzma equato, the Taylor Gree codto, sotropc turbulece. INTRODUCTION So far turbulece presets dfferet challeges to computatoal methods, especally for smulato of compressble flows at large Mach umbers. To date, most vestgatos of stablty or turbulece the lterature are based o the cotuum Naver-Stokes (NS) equatos [1, 2]. I recet tmes, there has bee creasg terest computg flows wth more fudametal ketc equatos for applcatos volvg turbulece ad stablty whch ca gve aother opportutes comparso wth cotuum oe [3-6]. It was show that for o equlbrum ad ustable processes the utlty of the NS equato may be lmted due to rarefed gas effects or lack of approprate costtutve or state relatos. These effects ca be potetally resolved at the ketc level of flow descrpto. It s expected that the applcato of the Boltzma ketc equato to the compressble turbulet flows wll gve more geeral ad correct results. The problem of producg small eddes from large oes the vscous flow feld, whch s cosdered to be oe of the basc features of turbulet flows [7]. The ketc eergy exchage appears betwee eddy structures alog wth the dsspatg eergy to a heat. The problem was formulated ad examed by Taylor ad Gree for the compressble flow usg NS equato [8]. I the tal feld the seres of vortex structures were gve ad soluto was represeted the form of a power seres tme. Whe the flow has a fte Reyolds umber, the ketc eergy geerated by velocty shear s dsspated by the smallest scales, whch provdes a smple model for the developmet of a turbulet flow ad the cascade of eergy from larger to smaller scales. STATEMENT OF THE PROBLEM AND NUMERICAL METHOD We cosder 3D gas flow a perodc doma usg the drect umercal soluto of the Boltzma equato: 2π b m f f ξ J ( f, f ) ( f f * f f* ) gbdbdε d * ( f ) f N( f, f ) t ξ, (1) x 0 0 where f f ( t, x, ξ ) s the velocty dstrbuto fucto, x = (x, y, z) ad ξ = (ξ x, ξ y, ξ z ), ξ, ξ *, ξ, ξ * are veloctes of par of partcles before ad after collso, J(f, f) s the collso tegral, g ξ* ξ s the relatve velocty, ad b, are mpact parameters, ν(f)f s the tegral of drect collsos, ν(f) s the frequecy of collsos ad N(f, f) s the tegral of verse collsos. I the rest of the paper the o-dmesoal formulato of the problem s used ad 1

2 the scale quattes are the follows: v* RT* v T / 2 (v T s the thermal velocty), the characterstc sze of the flow L, effectve radus σ eff equals to the radus σ of hard-sphere partcles, characterstc desty ad temperature * 3 ad T * respectvely. Thus, o-dmesoal varables are ξ ξ / v, b b / σ, x x / L, f f / ( v ). The tal codtos are gve for the o dmesoal Maxwell dstrbuto fucto: M 3/ * eff * * f ( t, x, ξ) ρ (2π T ) exp c / 2T, c ξ V 0, V 0 ( u0, v0, w0 ), where ρ 0 s ts desty, T 0 s ts temperature, u 0, v 0, w 0 are ts veloctes, R s uversal gas costat ad x = (x, y, z). TG type tal codtos cosst of a frst-degree trgoometrc polyomal all drectos: u 0 ( x ) A cos(α x )s(α y )s(α z ), v 0 ( x ) B s(α x ) cos(α y )s(α z ), w 0 ( x ) C s(α x )s(α y ) cos(α z ), ρ 0( x ) 1 D s(α x)s(α y)s(α z), T 0 ( x ) 1 E cos(α x ) cos(α y ) cos(α z ), (2) where A, B,, E are costats, α = 2π. For a radom tal codto desty ρ 0 = 1, temperature T 0 = 1 ad magtudes of veloctes u 0, v 0, w 0 obey the sotropc eergy spectrum E(k) as [4] u ( x ) a(,, )s 2π( x y z ), a 2 E / k, ( / 0 ) exp k k E C k (3) where k = 2 π s wave vector, k 0 = 10, k = k, C = cost ad s the radom umber [0, 1] for each combato of = ( 1, 2, 3 ), = 0, ±1, ±2,, = 1, 2, 3. For v 0 ad w 0 the expressos are smlar to (3). The boudary codtos are F(x+1) = F(x), for ay F(x) of x. The Kudse K, Mach M ad Reyolds umbers ca be wrtte as K λ / L, λ=0.5 πμ 2 RT / p, M V / a, Re ρ VL / μ, V A B C, (4) where λ s mea free path, calculated usg the expresso for the hard sphere model, s dyamc vscosty, p s pressure, R s uversal gas costat, = 5/3 s specfc heat rato for moatomc gas, a s the speed of the soud. To dscretze the Boltzma equato, we troduce a uform three-dmesoal Cartesa grd { β } wth equdstat odes velocty space ad a three dmesoal grd {x } physcal space. Itroducg grd values, the obtaed set of equatos for f β (x, t) ca be umercally solved usg tme splttg method, whch s obtaed cosderg, a small tme terval t =t +1 -t, the umercal soluto of the trasport step, approxmated by a secod-order accurate explct coservatve scheme based o the fte volume method (TVD approach): f f t / x F F, * 1 * β, β, β, 1/ 2 β, 1/ 2 f f (5) * β, β, ad the space homogeeous collso step, approxmated by a explct-mplct approach [9, 10]: f f t / K 2π ν( f ) f N( f, f ) 1 1 β, β, β, β, β, β,, f f. (6) * 1 β, β, where = (, j, k) ad are dexes physcal ad velocty spaces, respectvely, Fβ, 1/2 s the umercal flux. For collso tegral, wrtte BGK form, the explct-mplct scheme approxmated the collso step trasforms to the mplct scheme sce desty, velocty ad temperature are costat at the uform relaxato stage ( f f ) / t τ(ρ, V, T ) f (ρ, V, T ) f, β, β, M β, where τ s the ter collso tme. The mplct (or explct-mplct) scheme s umercally stable, hece allows us to crease t ad/or evoluto tme, whle troducg the addtoal approxmato error (tme dervatos of collsos tegrals multpled by t/k), thus the accuracy ca be creased by decrease t. Geerally for the mplct scheme teratos for obtag the collso tegrals at the (+1)-th tme level are requred, thus the explct-mplct scheme ca be cosdered as the frst terato of the teratve process. For some stuatos t s suffcet to use oly frst terato, for example, [5] t has bee show that for the developed turbulet flow ( 2

3 the shear layer of the supersoc jet flows) whch s low from the ketc pot of vew the use of frst terato s suffcet. To uderstad the character of mplct scheme the seres of computato have bee carred out usg the mplct scheme for the BGK approxmato of the collso tegral. It was foud that results from BGK equato are close to the Boltzma oes. Ths gves us a drect evdece of the adequacy of our approach. The Goduov method s appled to the lear advecto equato for calculato of the umercal flux Fβ, 1/2. The secod-order accurate scheme (out of extrema ad dscotutes) s obtaed usg the TVD recostructo procedure wth the mmod-type lmter fucto [11]. The tme step s lmted by the Courat codto: Δt = CFL m(δx/ V max + Δy/ V max + Δz/ V max ), where CFL s the Courat umber, V max s a boudary of the velocty space ad Δx, Δy ad Δz are the mesh szes the x, y ad z drectos, respectvely. The quas - Mote-Carlo method, whch Korobov sequeces are used [12], s employed for evaluato of ν(f) ad N(f, f). Geerally, the post-collso veloctes do ot cocde wth grd odes the velocty space. Therefore, to esure the executo of coservatve laws the procedure of redstrbuto eergy betwee earest odes of the velocty grd (for each collso) s used [13]. Ths approach provdes wth mcroscopc (ketc) coservato for each collsos. For explct mplct scheme the procedure of coservatve correcto should be appled [9, 10] after collso step. If g s the soluto of (6), the the coservatve soluto after the relaxato step s defed as: β, f g (1 P ) g (1 a a a a a ) β β β β2 β, β, β, β, 0 1 x 2 y 3 z 4, β α α β β, β β β, ξ ξ ξ ξ, β2 β2 β 2 β2 x y z ξ f ξ g, α= 0,...,4 (7) where Pβ, s the corrected polyomal, β β β ξ x, ξ y, ξ z are velocty compoets the velocty ode β. The polyomal coeffcets a 0,, a 4 are computed from the coservatve laws of mass, mpulse ad eergy (7). The correcto procedure esures the postve value of the dstrbuto fucto after relaxato stage. Therefore, dstrbuto fucto has acceptable accuracy eve f coarse grd s used. Parallelzato physcal space s made to mprove the effcecy of the algorthm. Each processor s assged ts ow set of pots physcal space. The relaxato stage s calculated depedetly o the varous processors. Before the stage of the free-molecular trasport processors exchage data at eghborg pots. The software code was wrtte C++ wth the use of MPI (Message Passg Iterface). The calculatos have bee coducted usg multcores system cossted of 2 processes wth 4 cores each. The ma dcato of the developmet of vortcal cascade s the dstrbuto of ketc eergy E(q) the wave umber space q (q s the wave vector). It s mportat to determe spectral propertes of the flow. To costruct the spectrum of the ketc eergy of the flow the followg procedure s used [14]. A expaso Fourer seres of the compoets of velocty vector u jk, v jk ad w jk, gve o uform Cartesa mesh, s defed as follows: N 1 y 1 x N Nz 1 πl πm π u u cos x cos y cos z, (8) jk lm jk jk jk l0 m0 0 Lx Ly Lz where = 1,, N x, j = 1,, N y ad k = 1,, N z are umbers of the grd, L x, L y ad L z are szes of the computato doma, x jk, y jk ad z jk are coordates physcal space. For v jk ad w jk the expressos are smlar to (8). Each value of the module of the wave vector q ( l / L ) ( m / L ) ( k / L ), x y z qmax ( N x / Lx ) ( N y / Ly ) ( N z / Lz ) correspods to value of ketc eergy E lm ( u lm v lm w lm ) / 2 defg by dfferet combatos of {l, m, }. The terval (0, q max ) s dvded to p umber of the segmets each wth legth Δq = q max /p. Correspodg values of E lm are summarzed each segmet. The costructed ths way fucto s cosdered as a spectral characterstc of the flow feld obtaed by umercal modelg. The spectral fucto well descrbes all scales for p = 36, for larger p the area of moderate ad small scales the oscllatos appear, possbly due to trgoometrc factor. 3D TAYLOR GREEN PROBLEM Here we cosder the 3D case represeted by tal codto (2) ad boudary codtos (4). We vestgate several cases wth the followg tal ampltudes ad dfferet rarefacto: A = 0.5, B = C = -0.2, D = E = 0.01 for K = 0.01, K = ad K = Reyolds umbers vares from 63.8 to respectvely. For all cases the grd the ut cube s , velocty space s lmted by V max = 5 ad the step velocty space s ξ =

4 The tme step s chose such way that the partcle does ot pass the dstace more tha mea free paths ad to avod tme step effect: t = for K = 0.01, t = 10-3 for K = ad t = for K = The CPU tme was about 128 h (for K = 0.01) for fully damp flow utl t = 1. I the Fg. 1 (a, b) the so-surface ad cotours o the plae of desty ad module of vortcty vector ω = v at the several tme momets are show. At the tal tme momet macroparameters demostrates the costat umber of so-surfaceses. t = 0: (a) ρ = (b) = 3.13 t = 0.67: (a) ρ = 1.01 (b) = 1.9 t = 1: (a) ρ = 0.97 (b) = 1.4 FIGURE 1. Iso-surfaces ad cotours o the plae of desty (a) ad vortcty (b) at t = 0, 0.67 ad 1 for 3D TG flow wth A = 0.5, B = C = -0.2, D = E = 0.01 ad K = Wth tme growth small scales appears, evertheless from some tme momet (t > 0.67) so-surfaceses of the flow parameters rema quas costat (Fg. 1). I Fg. 3 a chagg of eergy spectra s show at several tme momets t = 0.25, 0.67 ad 1. The tal ketc eergy mootoous decays wth tme due to the absece of exteral sources ad eergy trasfer from ketc to teral modes due to vscous acto. Nevertheless, at all tme level spectrum demostrates the slope -5/3 for the ertal terval, correspodg the sotropc turbulece power law characterstc [15]. Spectrum of the ketc eergy for smaller Kudse umber s larger. Of course ths case s far from sotropc turbulet state, but demostrates a ma feature of turbulece the forward cascade. ISOTROPIC TURBULENT FIELD 3D decayg turbulece the space perodc cube wth the tal codto gve by a local Maxwella wth a radom phase flow velocty (3) ad a magtude havg a sotropc eergy spectrum wth C = 0.05, ad a uform desty ad temperature are cosdered. Kudse, Mach ad Reyolds umbers are K = 0.001, M = ad Re = 963 respectvely dvso the ut cube s set ad velocty space s restrcted to Vmax = 5, a velocty step sze s Δ = Computatos have bee carred out wth Δt = 10-3, ad 10-4 to assess tme step effect. Comparso of so-surfaces of the desty ad modules of vortcty ad ther cotours o the plae (see Fg. 2) computed wth last two tme steps at varous tme levels shows that the dfferece betwee results s eglgble. Fg. 2 demostrates that the desty dstrbuto chages quckly to a turbulet state from the tal uform state. At a relatve small tme the both fgures of the desty ad vortcty dstrbutos show patters close 4

5 to the turbulet state. Wth tme growth small scales jo to each other ad vortces become fewer. They are larger ad sparser space, ad they udergo less frequet close ecouters. Sce those close ecouters are the occasos whe the vortces chage through deformato ways other tha smple movemet, the overall evolutoary rates for the spectrum shape ad vortex populato become ever slower, eve though the ketc eergy does ot dmsh. A aalyss of spectra, preseted Fg. 3 b, shows there are umber of pots lay o the lae wth clato 5/3, as expected for the ertal terval of 3D sotropc turbulece [15]. After ths terval graph of ketc eergy drops sharply. Wth tme growth the ertal terval of wave umbers creases. t = 0.01 (Δt = ): (a) ρ = (b) = 5.5 t = 0.01 (Δt = 10-4): (a) ρ = (b) = 5.5 t = 0.05 (Δt = ): (a) ρ = (b) = 1.5 t = 0.05 (Δt = 10-4): (a) ρ = (b) = 1.5 t = 0.3 (Δt = ): (a) ρ = (b) 5 = 0.33

6 t = 0.3 (Δt = 10-4 ): (a) ρ = (b) = 0.33 FIGURE 2. Iso-surfaces ad cotours o the plae of desty (a) ad vortcty (b) at t = 0.01, 0.05 ad 0.3. (a) (b) FIGURE 3. Eergy spectra E k (q) (a) for 3D TG flow wth A = 0.5, B = C = -0.2, D = E = 0.01 at t = 0.25, 0.67 ad 1 ad K = 0.01, (b) for decayg sotropc turbulece at t = 0, 0.01, 0.03, 0.05, 0.1, 0.5 ad 1. CONCLUSIONS The drect umercal soluto of the Boltzma equato based o a explct-mplct approach was used to aalyze the log tme evoluto of the vscous compressble weakly rarefed gas. Two kd of the tal codtos (TG ad radom velocty types) were cosdered. The developmet of tal state was demostrated through the ketc eergy dstrbuto. O the ketc level cosdered process of eergy trasfer from large scales to small oes ad sequetal eergy dsspato to heat. Moreover, computato for the space perodc flow started from a radom velocty ad uform desty ad temperature exhbted a flow feld patter close to sotropc turbulece at the tal tme momet. Dscovered a ertal terval wth the expoet -5/3, predcted for 3D sotropc turbulece, whch growths wth tme. REFERENCES 1. D. Drkaks, C. Fureby, F.F. Grste ad D. Yougs, Joural of Turbulece, 8, 1-12, (2007). 2. R. Samtaey, D.I. Pull ad B. Kosovc, Physcs of Fluds, 13, (2001). 3. G.A. Brd, The Itato of Cetrfugal Istabltes a Axally Symmetrc Flow 20th Symposum o Rarefed Gas Dyamcs, edted by C. She, Pekg Uversty Press, Bejg, Cha, 1997, pp A. Sakura, ad F. Takayama, Physcs of Fluds, 15, (2003). 5. V.V. Arstov, Comput. Math. Math. Phys., 44, (2004). 6. O.I. Roveskaya. Comput. Math. ad Math. Phys., 47, (2007). 7. G.K. Batchelor, The Theory of Homogeeous Turbulece, Cambrdge: Cambrdge Uv. Press, G.I. Taylor ad A.E. Gree, Proc. R. Soc. Lodo Ser. A, 158, (1937). 9. V.V. Arstov, Methods of Drect Solvg the Boltzma Equato ad Study of Noequlbrum Flows, NY: Sprger, V.V. Arstov, F.G. Tcheremsse, USSR Comp. Math. Math. Phys., 20, (1980). 11. A.G. Kulkovsk, N.V. Pogorelov, ad A.Yu. Semeov, Mathematcal Aspects of Numercal Soluto of Hyperbolc Systems, Moscow: Fzmatlt, N.M. Korobov, Expoetal Sums ad Ther Applcatos, New York: Sprger-Verlag, F. Tcheremsse, Drect Numercal Soluto Of The Boltzma Equato 24 th Symposum o Rarefed Gas Dyamcs, edted by M. Captell, AIP Coferece Proceedgs 762, Amerca Isttute of Physcs, Melvlle, NY, 2005, pp S.M. Gara, N.V. Zmtreko, M.Ye. Ladoka, I.G. Lebo1, V.V. Nksh, N.G. Procheva, V.B. Rozaov', V.F. Tshk, Mathematcal modelg, 15, 3-11, (2003) 15. A.N. Kolmogorov, J. Flud Mech., 13, 82-85, (1962). 6