Lattice Boltzmann models for rarefied flows

Transcription

1 Lattice Boltzmann models for rarefied flows Robert Blaga, Sergiu Busuioc, Victor E. Ambruș West University of Timișoara Collisionless Boltzmann (Vlasov) Equation and Modelling of Self-Gravitating Systems and Plasmas Marseille, October 30 November 3, 2017

2 Section 1 Introduction R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 2 / 44

3 Mesoscale approach Macroscopic: n, u, T (Navier-Stokes-Fourier) Mesoscopic: f (x, p, t) (Boltzmann) Microscopic: (xi, pi ) (MD) R. Blaga, S. Busuioc, V.E.Ambrus, (WUT) LB models for rarefied flows 3 / 44

4 Rarefied flows: Knudsen number (Kn) Kn = λ/r p (λ mean free path; r p characteristic channel length). Hydordynamic regime (NSF): Kn 0. Ballistic regime (Vlasov): Kn. R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 4 / 44

5 Lattice Boltzmann (LB): main ingredients 1 Discretisation of the momentum space (Gauss quadratures); 2 Polynomial representation of f (eq) in the BGK collision term; 3 Replacement of p f using a suitable expression; 4 Numerical method for time evolution and spatial advection (RK-3 + WENO-5); 5 Boundary conditions. The LB method ensures the exact recovery of the conservation eqs. for n, ρu and E (for thermal models). R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 5 / 44

6 Section 2 Planar shocks R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 6 / 44

7 Non-relativistic case: equation Let us consider the Sod shock tube problem: f(z, t = 0) = { f (eq) L, z < 0 (n L, P L, u L ) = (1, 1, 0) f (eq) R, z > 0 (n R, P R, u R ) = (0.125, 0.1, 0). The Boltzmann equation reduces to: t f + p z m zf = 1 (eq) (f f M B τ ), f (eq) M B = n (2πmT ) 3/2 exp [ ] (p mu)2, 2mT where the BGK relaxation time approximation was used for J[f]. 1 The p x and p y degrees of freedom can be integrated: ( g h) = dp x dp y f ( ) ( ) 1 g p 2 x +, t p2 y h + p ( ) z g m z h = 1 τ ( ) g g (eq) h h (eq), where h (eq) = 2mT g (eq) and g (eq) = [ n exp (p ] z mu z ) 2. 2πmT 2mT 1 P. L. Bhatnagar, E. P. Gross, M. Krook, Phys. Rev. 94, (1954). R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 7 / 44

8 Non-relativistic case: Discretisation p z are discretised using the Gauss-Hermite quadrature method: 2,3 dp z g P s (p z ) = Q k=1 g k P s (p z,k ), g k = w k ω(p z ) g(p k), where H Q (p z,k ) = 0, w k = Q!/[H Q+1 (p z,k )] 2 and ω(p z ) = e p2 z /2 / 2π. g (eq) k is truncated at order N < Q w.r.t. the Hermite polynomials: g (eq) k = w k N H l (p k ) l=0 l/2 s=0 1 2 s s!(l 2s)! (mt 1)s (mu) l 2s, h (eq) k = 2mT g (eq) k. The space is discretised according to z i = Z (i 0.5) (1 i Z). The following boundary conditions are imposed: 4 g 2,k = g 1,k = g 0,k =g (eq) k,l, g Z+1,k = g Z+2,k = g Z+3,k =g (eq) k,r, and similarly for h i,k. 2 X. W. Shan, X. F. Yuan, and H. D. Chen, J. Fluid. Mech. 550, 413 (2006). 3 V. E. Ambruș, V. Sofonea, J. Comput. Phys. 316 (2016) Y. Gan, A. Xu, G. Zhang, Y. Li, Phys. Rev. E 83 (2011) R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 8 / 44

9 Non-relativistic: n n z τ=10-5 τ=5x10-4 τ=5x10-3 τ=0.05 τ= Analytic Solution in the collisionless regime: 5 n(z, t) = n ( ) L z m 2 erfc + n ( R t 2T L 2 erfc z ) m. t 2T R 5 Z. Guo, R. Wang, K. Xu, Phys. Rev. E 91 (2015) R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 9 / 44

10 Non-relativistic: u u τ= τ=5x τ=5x10-3 τ= τ= Analytic z Solution in the collisionless regime: ρu = n L mtl 2π exp ( mz2 2t 2 T L ) n R mtr 2π exp ( mz2 2t 2 T R ). 5 Z. Guo, R. Wang, K. Xu, Phys. Rev. E 91 (2015) R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 10 / 44

11 Non-relativistic: P P z τ=10-5 τ=5x10-4 τ=5x10-3 τ=0.05 τ= Analytic Solution in the collisionless regime: 3 2 nt ρu2 = n LT L 4 [ ( ) z m 3erfc + z t 2T L t [ + n RT R 4 3erfc ] 2m e mz 2 /2t 2 TL πt L ( z ) m z t 2T R t ] 2m e mz 2 /2t 2 TR. πt R 5 Z. Guo, R. Wang, K. Xu, Phys. Rev. E 91 (2015) R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 11 / 44

12 Ultrarelativistic case: equation The advection part is 1D: t f + ξ z f = γ L(1 β L ξ) τ [f f (eq) M J ], f (eq) M J = n [ 8πT exp pγ ] L 3 T (1 β Lξ), where the Anderson-Witting SRT approximation was used for J[f]. 6 The momentum space is discretised using spherical coordinates: p x = p sin θ cos ϕ, p y = p sin θ sin ϕ, p z = p cos θ, while ξ = cos θ. Integrating over the ϕ degree of freedom, p and ξ are discretised following the Gauss-Laguerre and Gauss-Legendre prescriptions: 7 M s,r = d 3 p f ps+1 ξ r = 2π p 0 0 dp e p p dξ(e p f)p s ξ r = Q L Q ξ k=1 j=1 f jk p s k ξr j, where f jk = w L k wp j ep k f(p k, ξ j ), while L QL (p k ) = 0 and P Qξ (ξ j ) = 0. The quadrature weights w L k and wp j are given by: w L k = (Q L + 1)(Q L + 2)p k [(Q L + 1)L (2) Q L +1 (p k)] 2, wp j = 2(1 ξ 2 j ) [(Q ξ + 1)P Qξ +1(ξ j )] 2. For the planar case: Q L = 2 and Q ϕ = 1. 6 J. L. Anderson, H. R. Witting, Physica 74, (1974); (1974). 7 R. Blaga, V. E. Ambruș, arxiv: [physics.flu-dyn]. R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 12 / 44

13 Ultrarelativistic: n n (a) n Q ξ = 6 Q ξ = 20 Q ξ = 200 ballistic init. cond. η/s = 10 2 η/s = 10 3 η/s = 10 4 inviscid 0.25 (a) z z η/s ratio of shear viscosity to entropy density. Ballistic limit: n Eck = ( nl + n R 2 n L n R 2 ) z 2 t ( ) nl n 2 R (1 z2 4 t 2 ) 2 7 R. Blaga, V. E. Ambruș, arxiv: [physics.flu-dyn]. R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 13 / 44

14 Ultrarelativistic: P P (b) η/s = 10 2 η/s = 10 3 η/s = 10 4 inviscid P Q ξ = 6 Q ξ = 20 Q ξ = 200 ballistic (c) z z Ballistic limit: T 00 = 3 2 (P L + P R ) 3 2 (P L P R ) z t, T 0z = 3 ( ) 4 (P L P R ) 1 z2, t 2 T zz = 1 2 (P L + P R ) 1 ( ) z 3 2 (P L P R ), t R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 14 / 44 7 R. Blaga, V. E. Ambruș, arxiv: [physics.flu-dyn].

15 R-SLB vs BAMPS: n and β n BAMPS: η/s = 0.1 BAMPS: η/s = 0.01 R SLB: η/s = 0.1 R SLB: η/s = 0.01 β BAMPS: η/s = 0.2 BAMPS: η/s = 0.1 BAMPS: η/s = 0.01 R SLB: η/s = 0.2 R SLB: η/s = 0.1 R SLB: η/s = (a) (d) z z BAMPS Boltzmann approach to multiparton scattering. 8 8 I. Bouras, E. Molnár, H. Niemi, Z. Xu, A. El, O. Fochler, C. Greiner, D. H. Rischke, Phys. Rev. Lett. 103 (2009) R. Blaga, V. E. Ambruș, arxiv: [physics.flu-dyn]. R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 15 / 44

16 R-SLB vs BAMPS: P P BAMPS: η/s = 0.2 BAMPS: η/s = 0.1 BAMPS: η/s = 0.01 R SLB: η/s = 0.2 R SLB: η/s = 0.1 R SLB: η/s = 0.01 P η/s = 0.01 η/s = 0.05 η/s = 0.1 η/s = 0.2 η/s = (c) z z BAMPS Boltzmann approach to multiparton scattering. 8 I. Bouras, E. Molnár, H. Niemi, Z. Xu, A. El, O. Fochler, C. Greiner, D. H. Rischke, Phys. Rev. Lett. 103 (2009) R. Blaga, V. E. Ambruș, arxiv: [physics.flu-dyn]. R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 16 / 44

17 Section 3 Spherical shocks R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 17 / 44

18 Non-relativistic case: Equation Let us consider the Sod shock tube problem with spherical symmetry: f(r, t = 0) = { f (eq) L, r < r d (n L, P L, u L ) = (1, 1, 0) f (eq) R, r > r d (n R, P R, u R ) = (0.125, 0.1, 0). Spherical coordinates: r, ϑ, φ and pˆr, p ˆϑ, p ˆφ: t f + pξ mr 2 r(fr 2 ) + p mr ξ[(1 ξ 2 )f] = 1 τ (f f (eq) M B ), where pˆr = p cos θ, p ˆϑ = p sin θ cos ϕ and p ˆφ = p sin θ sin ϕ, while ξ = cos θ. f can be expanded w.r.t. ξ: 2s + 1 f = F s P s (ξ). 2 Thus, ξ [(1 ξ 2 )f] can be written as: s=0 ξ [(1 ξ 2 )f] = 1 1 dξ K(ξ, ξ )f(ξ ), where K(ξ, ξ ) = s=1 s(s + 1) P s (ξ)[p s+1 (ξ ) P s 1 (ξ )]. 2 R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 18 / 44

19 Non-relativistic case: Discretisation p, ξ and ϕ are discretised using the Gauss-Laguerre, Gauss-Legendre and Mysovskikh quadratures: 9 π d 3 pf p l x ps y pr z = dp p 2 dθ sin θ = 0 Qp Q ξ Qϕ k=1 j=1 i=1 0 2π 0 dϕ fp l+s+r (sin θ) l+s (cos θ) r (cos ϕ) l (sin ϕ) s f ijk p l+s+r k (sin θ j ) l+s (cos θ j ) r (cos ϕ i ) l (sin ϕ i ) s. The discrete magnitudes p k are obtained by solving L (1/2) Q L (p 2 k ) = 0. The elevations ξ j are obtained by solving P Qξ (ξ j ) = 0. ϕ i = 2π(i 1)/Q ϕ. The discrete populations f ijk are: f ijk = π w ξ 2 j Q ep k w L k f(p k, ξ j, ϕ i ). ϕ The derivative ξ [(1 ξ) 2 f] { ξ [(1 ξ) 2 f]} ijk = Q ξ j =1 Kξ j,j f i,j,k, where K ξ j,j = w ξ j Q ξ s=1 s(s + 1) P s (ξ j )[P s+1 (ξ 2 j ) P s 1 (ξ j )].. 9 V. E. Ambruș, V. Sofonea, Phys. Rev. E 86 (2012) R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 19 / 44

20 Ultrarelativistic case: equation The relativistic Boltzmann equation reads: t f + ξ r 2 r(fr 2 ) + ξ [(1 ξ 2 )f] = γ L τ (1 β Lξ)[f f (eq) M J ]. The derivative w.r.t. ξ is computed in exactly the same way as in the non-relativistic case. The momentum space is discretised in the same way as for the planar shock (including Q ϕ = 1). R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 20 / 44

21 Non-relativistic: Inviscid limit: Primary shock t=0 t=0.02 t=0.04 t=0.06 t=0.08 t=0.1 t=0.12 t=0.14 t=0.16 P r R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 21 / 44

22 Non-relativistic: Inviscid limit: Reverse shock 10 1 t=0.16 t=0.18 t=0.2 t=0.22 t=0.24 t=0.26 t=0.28 t= P r R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 22 / 44

23 Non-relativistic: Inviscid limit: Secondary shock P r t=0.3 t=0.33 t=0.36 t=0.39 t=0.42 t=0.45 t=0.48 t=0.51 t=0.54 R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 23 / 44

24 Ultra-relativistic: Inviscid limit 10 p z t = 0.00 t = 0.05 t = 0.13 t = 0.17 t = 0.19 t = 0.25 p z t = 0.41 t = 0.42 t = 0.50 t = 0.60 t = T. P. Downes, P. Duffy, S. S. Komissarov, Mon. Not. R. Astron. Soc. 332 (2002) R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 24 / 44

25 Non-relativistic: Collisionless limit: solution y r d x (n L, P L, u L ) r d r r d θ max p f(r, p, θ, t) = { f (eq) L, θ < θ max and p < p < p + f (eq) R, otherwise., θ max = { arcsin r d r r > r d, π r < r d while p ± = m t (r cos θ ± r 2 d r2 sin 2 θ) (p = 0 when r < r d ). Solution (ζ ±,L/R p ± / 2mT L/R ): n = n R + etc. θ max 0 dθ sin θ { nl π [(ζ,l e ζ2,l ) ] } π 2 erfζ,l (+ ) [L R], R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 25 / 44

26 Non-relativistic: Ballistic limit t=0.0 t=0.01 t=0.02 t=0.03 t=0.04 t=0.05 t= P r R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 26 / 44

27 Ultra-relativistic: Collisionless limit 11 r r d t θ m θ m t r d r r d r r d r f(r r d > t) = f R (causality: no L particles reached r); f(r d r > t) = f L (causality: no R particles reached r); f(r + r d < t) = f R (all L particles have flown away); 11 C. Greiner, D.-H. Rischke, Phys. Rev. C 54 (1996) R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 27 / 44

28 Ultra-relativistic: Collisionless limit: Solution r r d t θ m θ m t r d r r d r r d r When r > r d and r r d < t < r + r d : { f 1, 0 < θ < θ m, f(r, t; θ, ϕ) = f 2, otherwise, When r < r d and r d r < t: f(r, t; θ, ϕ) = { f 2, θ m < θ < π, f 1, otherwise. θ m =, θ m = { arccos r2 +t 2 r d 2 2rt, r r d < t < r + r d, 0, otherwise. π, r d r > t, arccos r2 +t 2 r 2 d 2rt, r d r < t < r d + r, 0, t > r d + r. R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 28 / 44

29 Ultra-relativistic: Ballistic limit n t = 0 t = 0.09 t = t = 0.1 t = t = n t = 0.11 t = 0.12 t = 0.13 t = 0.14 t = 0.15 t = 0.16 n max z z R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 29 / 44

30 Section 4 Cylindrical shocks R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 30 / 44

31 Non-relativistic case: Equation Let us consider the Sod shock tube problem with cylindrical symmetry: { f (eq) f(r, t = 0) = L, R < R d (n L, P L, u L ) = (1, 1, 0) f (eq) R, R > R d (n R, P R, u R ) = (0.125, 0.1, 0). Cylindrical coordinates: R, φ, z and p ˆR, p ˆφ, pẑ: t f + p cos ϕ mr where p ˆR = p cos ϕ and p ˆφ = p sin ϕ f can be expanded w.r.t. ϕ: f = 1 2π A π Thus, ϕ (f sin ϕ) can be written as: where 2πK(ϕ, ϕ ) = cos ϕ + m=1 R(fR) p mr ϕ(f sin ϕ) = 1 (eq) (f f M B τ ), [A j cos(jϕ) + B j sin(jϕ)]. j=1 ϕ (f sin ϕ) = 2π The pẑ degree of freedom can be integrated out: 0 dϕ K(ϕ, ϕ )f(ϕ ), { (m + 1) cos[mϕ (m + 1)ϕ] (m 1) cos[mϕ (m 1)ϕ] }. g = dp z f, h = dp z f p 2 z. R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 31 / 44

32 Non-relativistic case: Discretisation p and ϕ are discretised using the Gauss-Laguerre and Mysovskikh 12 quadrature methods: 2π dp x dp y g p s x pr y = dp p 0 0 dϕ gp s+r (cos ϕ)s (sin ϕ) r = Q Qϕ k=1 j=1 g jk p s+r,k (cos ϕ j) s (sin ϕ j ) r. The discrete magnitudes p,k are obtained by solving L QL (p 2,k ) = 0, while ϕ j = 2π(j 1)/Q ϕ. The discrete populations g jk are: g jk = π e p 2 k w L k Q g(p,k, ϕ j ), w L k = p 2 k ϕ [(Q + 1)L Q (p 2. k )]2 The derivative ϕ (f sin ϕ) [ ϕ (f sin ϕ)] jk = Qϕ j =1 Kϕ j,j f j,k, where K ϕ j,j = 1 Q φ /2 Qϕ/2 1 n cos[nϕ j (n 1)ϕ Q ϕ j ] n cos[nϕ j (n + 1)ϕ j ]. n=1 n=1 g (eq) jk = 2πn Qϕ F ke jk is implemented using: F k = w k 2π N l=0 (1 2mT ) l L l (p 2 k ), E jk = w j Nϕ s=0 12 I. P. Mysovskikh, Soviet Math. Dokl. 36 (1988) V. E. Ambruș, V. Sofonea, Phys. Rev. E 86 (2012) s! ( mu2 2T ) N ϕ 2s r=0 1 r! ( ) p u r. T R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 32 / 44

33 Ultrarelativistic case: equation Since f (eq) M J depends on p = p 2 x + p 2 y + p 2 z (instead of p 2 ), it is not convenient to switch to p. The relativistic Boltzmann equation reads: t f + sin θ cos ϕ mr R(fR) sin θ mr ϕ(f sin ϕ) = γ L τ (1 sin θ cos ϕβ L)[f f (eq) M J ]. The derivative w.r.t. ϕ is computed in exactly the same way as in the non-relativistic case. The momentum space is discretised in the same way as for the planar shock (now Q ϕ > 1!). R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 33 / 44

34 Ultra-relativistic: Inviscid limit: Primary shock P R t=0 t=0.03 t=0.06 t=0.09 t=0.12 t=0.15 t=0.18 t=0.21 t=0.24 t=0.27 t=0.3 R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 34 / 44

35 Ultra-relativistic: Inviscid limit: Reverse shock P R t=0.3 t=0.32 t=0.34 t=0.36 t=0.38 t=0.4 t=0.42 t=0.44 t=0.46 t=0.48 t=0.5 t=0.51 R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 35 / 44

36 Ultra-relativistic: Inviscid limit: Secondary shock t=0.54 t=0.57 t=0.6 t=0.63 t= P R R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 36 / 44

37 Non-relativistic: Collisionless limit: solution g(r, p, ϕ, t) = If R > R d : If R < R d : { g (eq) L, ϕ < ϕ max and p < p < p + g (eq) R, otherwise. ϕ max = arcsin(r d /R); p ± = m t (R ± R 2 1 R2 sin 2 ϕ); ϕ max = π; p = 0. Solution (ζ ±,L/R p ± / 2mT L/R ): n = n R + ρu = 3 ϕ max 0 ϕ max 0 dϕ dϕ cos ϕ [ nl π { nl 2 nt ρu2 = 3 2 n RT R + π y R d (n L, P L, u L ) ( ) ] e ζ2,l e ζ2 +,L (L R), ϕ max 0 h R d δ δ R R d ϕ max [ ] } p e ζ2,l p + e ζ2 πmtl +,L + (erfcζ,l erfcζ +,L ) (L R) 2 { [( ) ( ) ] } nl T L 3 dϕ π 2 + ζ2,l e ζ2,l ζ2 +,L e ζ2 +,L (L R) p x R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 37 / 44

38 Non-relativistic: Ballistic limit t=0.00 t=0.05 t=0.10 t=0.15 t= P R R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 38 / 44

39 Ultra-relativistic: Collisionless limit: solution (R > R d ) z R d R R d t y R d R R d t sin θ θ O n 1, P 1 n 2, P 2 v A R n 1, P 1 n 2, P 2 v ϕ x R + R d R + R d R t sin θ t t δ m ϕ m δ m A δ M A R R d R R d R d R R d R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 39 / 44

40 Ultra-relativistic: Collisionless limit: solution (R > R d ) where δ m = f(r > R d, t; θ, ϕ) = f 1, f 2, { arcsin R R d t R R d < t π/2 otherwise. ϕ m = θ (δ m, δ M ) (π δ M, π δ m ) and ϕ (0, ϕ m ) (2π ϕ m, 2π), otherwise, δ M = { arcsin R+R d t R + R d < t π/2 otherwise { arccos R2 +t 2 sin 2 θ R 2 d 2Rt sin θ, R R d < t sin θ < R + R d, 0, otherwise. R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 40 / 44

41 Ultra-relativistic: Collisionless limit: solution (R < R d ) R d + R R d R ϕ m δ m t sin θ t δ M { f 2 θ (δ m, π δ m ) and ϕ (ϕ m, 2π ϕ m ), f(r, t; θ, ϕ) = f 1 otherwise, { arcsin R R d δ m = t R R d < t π/2 otherwise. π, R d R > t sin θ, ϕ m = arccos R2 +t 2 sin 2 θ R d 2 2Rt sin θ, R d R < t sin θ < R d + R, 0, R d + R < t sin θ. R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 41 / 44

42 Ultra-relativistic: Ballistic limit P R t=0 t=0.01 t=0.02 t=0.03 t=0.04 t=0.05 t=0.06 t=0.07 t=0.08 t=0.09 t=0.1 t=0.11 t=0.12 R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 42 / 44

43 Ultra-relativistic: Ballistic limit P r t = 0.20 t = 0.25 t = 0.30 t = 0.40 t = 0.55 t = 0.70 t = 0.85 P max R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 43 / 44

44 Conclusion LB can be applied to study flows from the inviscid to the ballistic regime. The discretisation of the momentum space using Gauss quadratures ensures the exact recovery of the moments of f. Flows in the inviscid regime require small velocity sets (Q 6). For the ballistic regime, Q 100. Flows in arbitrary coordinate systems can be studied using the vielbein formalism. Aligning the momentum space directions along the coordinate unit vectors allows the flow symmetries to be preserved. The examples shown are 1D. However, the scheme is directly extendible to more complex geometries. In the inviscid regime, the rich phenomenology of cylindrical and spherical shocks was successfully captured. In the ballistic regime, the LB method was successfully validated against the analytic result. R. Blaga, S. Busuioc, V.E.Ambruș (WUT) LB models for rarefied flows 44 / 44