We verify the immersed boundary-lattice kinetic scheme (IB-LKS) by simulating different cases; a flow over a circular cylinder, a neutrally buoyant particle migration in a linear shear flow, and sedimentation of circular particles in a viscous fluid. Comparison between the numerical results and those of previous studies demonstrates the validity of the IBLKS for calculation of vicious incompressible flow with suspended solid particles. The IB-LKS is one solution approach for the problem of the velocity slip that is observed in the cylindrical Couette flow simulation with the Immersed boundary-lattice Boltzmann method

瀬田  剛

金森 拓哉

application/pdfWe verify the immersed boundary-lattice kinetic scheme (IB-LKS) by simulating different cases; a flow over a circular cylinder, a neutrally buoyant particle migration in a linear shear flow, and sedimentation of circular particles in a viscous fluid. Comparison between the numerical results and those of previous studies demonstrates the validity of the IBLKS for calculation of vicious incompressible flow with suspended solid particles. The IB-LKS is one solution approach for the problem of the velocity slip that is observed in the cylindrical Couette flow simulation with the Immersed boundary-lattice Boltzmann method.Article計算数理工学論文集 10, 17-22. (2010)journal articl

瀬田,  剛

金森, 拓哉

University of Toyama Repository

????????? Vol. 10 (2010? 12?), ??No. 04-101210 JASCOME???????????Lattice Kinetic???????????????????LATTICE KINETIC SCHEME FOR SIMULATION OF INCOMPRESSIBLE PARTICLE-FLUIDTWO PHASE FLOWS WITH AN IMMERSED BOUNDARY METHOD???? 1)?????? 2)Takeshi SETA and Takuya KANAMORI1)????????????? (??) (? 930-8555 ????? 3190, E-mail: seta@eng.u-toyama.ac.jp)2)?????????????????? (? 930-8555 ????? 3190, E-mail: s0770423@ems.u-toyama.ac.jp)We verify the immersed boundary-lattice kinetic scheme (IB-LKS) by simulating di®erentcases; a °ow over a circular cylinder, a neutrally buoyant particle migration in a linearshear °ow, and sedimentation of circular particles in a viscous °uid. Comparison betweenthe numerical results and those of previous studies demonstrates the validity of the IB-LKS for calculation of vicious incompressible °ow with suspended solid particles. TheIB-LKS is one solution approach for the problem of the velocity slip that is observed inthe cylindrical Couette °ow simulation with the Immersed boundary-lattice Boltzmannmethod.Key Words : Computational Fluid Dynamics, Lattice Kinetic Scheme, ImmersedBoundary Method, Particulate Flows1. ?????????????Lattice Boltzmann Method, LBM?(1)?????????????????????????????????Navier Stokes (NS)???????????????????????????????????????????? (Immersed Boundary Method, IBM)(2) ? LBM????? Immersed Boundary-Lattice Boltzmann Method(IB-LBM)?????????????? (3, 4, 5)?????IB-LBM?????????????????????????????????????????????????Le???????? (6)????LBM?????????????????????????????? Lattice KineticScheme (LKS)??? (7) ?????????????????????????????????????????????????? (8) ???????????????????LKS?????????????LBM????????¿ = 1?????????????????????IB-LBM?????????????????????? ¿ < 2????????????????????????? LKS????????????????????????????2010 ? 9 ? 22 ????2010 ? 10 ? 20 ???{Dedicated to the memory of Prof. Masataka TANAKA?? Direct forcing method(9) ????????????????2. IB-LKSIBM???Fig.1??????????????????????????????????????????? ~F????????????NS???????????????Direct forcing method???NS????@~u@t+ (~u ¢ r)~u = ¡1½rp+ ºr2~u+ ~g + ~F ; (1)???????????????½ ????~u ????p ????~g ????º ??????????? (1)??????~F =@~u@t+ ~u ¢ r~u+ 1½rp¡ ºr2~u¡ ~g=~u(n+1) ¡ ~u(n)±t¡ rhs; (2)??????????????rhs??Fig.1 A schematic of the immersed boundary-lattice kineticscheme.rhs = ¡¡~u ¢ r~u+ 1½rp¡ ºr2~u¡ ~g¢; (3)????Fig.1????????????? ~xs = (xs; ys)??????? (2)??????????~Fs =~u(n+1)s ¡ ~u(n)s±t¡ rhss=~u(n+1)s ¡ ~~us±t+~~us ¡ ~u(n)s±t¡ rhss = ~u(n+1)s ¡ ~~us±t; (4)??????????? s? ~xs ???????~~us ????~F = 0 ??? NS ??? (1) ?????????????n+ 1???? ~u(n+1) ??????? ~ud ???????????~Fs =~u(n+1)s ¡ ~~us±t=~ud ¡ ~~us±t; (5)??????IBM???? (5)??????? ~xs ????~Fs ?????Fig.1 ?????????????????~xi;j = (xi;j ; yi;j) ????? ~Fi;j ???????????~Fi;j ? NS??? (1)???????????????????????Fig.1?????????????? ~xi;j ?????~Fi;j =NXs=1~FsD(~xi;j ¡ ~xs) ¢¢Vs; (6)?????????????D(~x)?????????????????????N ?????????????¢Vs ??????????2?????????????????R????? ¢Vs = 2¼R±x=N ?????????????D(~x)?????????D(~xi;j ¡ ~xs) = 1±2x±hµxi;j ¡ xs±x¶±hµyi;j ¡ ys±x¶; (7)±h(z) =8>><>>:18¡3¡ 2jzj+p1 + 4jzj ¡ 4z2¢ 0 · jzj < 1;18¡5¡ 2jzj ¡p¡7 + 12jzj ¡ 4z2¢ 1 · jzj < 2;0 2 · jzj:(8)? (7)?(8)?????????????? ~xs ??????Fig.1???????? 2±x£2±x???????????????? ~Fi;j ??????? (5)????????????????????? ~xi;j ???? ~~ui;j ???????? ~~us?????????????????????? (7)?(8)??? D(~x)??????????????~~us =Xi;j~~ui;jD(~xi;j ¡ ~xs) ¢ ±2x; (9)?????????? (9)?????? ~~us ??? (5)????????????????????????????????? ~Fp ???? ~Tp ??~Fp = ¡NXs=1~Fs ¢¢Vs; (10)~Tp = ¡NXs=1(~xs ¡ ~xp)£ ~Fs ¢¢Vs; (11)????????~xp ????????????????????? up????????? !p ???????????msd~updt= ~Fp + ~Ff + (½s ¡ ½f )Vs~g (12)Isd~!pdt= ~Ts + ~Tf ; (13)???????????½f ???????½s ???????Vs ???????????ms ???????Is ????????????~Ff?~Tf ????????????????????????????????????R??????????? Vs = ¼R2???????ms = ½s¼R2????????? Is = 12½s¼R4 ????Ladd(10) ????????????????????½s½f> 1 +10R; (14)??????? (14) ????????????????????????????????????????????????????????????????????????????????????? Feng(12) ??~Ff = ½fVsd~updt= ½fVs~unp ¡ ~un¡1p±t; (15)~Tf = ½fVsd~!pdt= ½fVs~!np ¡ ~!n¡1p±t; (16)????~un+1p =µ1 +½f½s¶~unp ¡ ½f½s~un¡1p +¡~Fp + (ms ¡mf )~g±t¢ms;(17)~!n+1p =µ1 +½f½s¶~!np ¡ ½f½s~!n¡1p +~TpIs±t; (18)???????????????????????????????mf ??? Vs ?????????????LKS???Fig.1?????????????? ~xi;j ?????????? fk ????? ½??? ~u??½(~x; t) =8Xk=0fk(~x¡ ~ck±x; t¡ ±t); (19)~u(~x; t) =1½08Xk=0~ckfk(~x¡ ~ck±x; t¡ ±t); (20)??????????????½0????????He?????????????????????????? (11) ????? D2Q9??????? fk ??fk = !k·½+ ½0µ3ck®u® +92ck®ck¯u®u¯ ¡ 32u®u®+A±x³@u¯@x®+@u®@x¯´ck®ck¯¶¸; (21)??????????? fk(~x; t) ???? ~x??? t ???????? ~ck ????????????????±t ??????????????????????????????? (8)?!k ?????????!0 = 4=9?!1;2;3;4 = 1=9 ?!5;6;7;8 = 1=36 ??????????????p = ½=3; º =µ16¡ 29A¶±x; (22)?????????? Fk ???? ~g ????Fk = ½3!k~ck ¢ (~g + ~F ); (23)????????????Fk??P8k=0 Fk = 0?P8k=0 Fk~ck =25 30 35 40 45 50 55 6000.20.40.60.81 x u/u0 × ° • : Analytical solution : Numerical solution, τ = 1 : Numerical solution, τ = 2 : Numerical solution, τ = 5Fig.2 Pro¯les of the velocity at the horizontal central plane.½(~g + ~F )?????????IB-LBM????? (23)??????????Feng ?????????????????????? (4)????????????? LKS??????????????????Step 1. ???? ½(1)i;j?~u(1)i;j ?? (21)?????????????? f (1)k ?????Step 2. ??????????????????? f (n)k ?????f (n+1)k ??????? (19)?(20)?? ½(n+1)i;j ?~~u(n+1)i;j ?????Step 3. ? (9)????~~u(n+1)i;j ???????? ~~u(n+1)s ?????Step 4. ? (5)????????? ~F (n+1)s ?????Step 5. ? (10)?(11)?????????? ~Fp ???? ~Tp?????Step 6. ? (17)?(18)???~un+1p ?~!n+1p ?????????????? n+ 1??????????????????????Step 7. ? (6)??????????? ~F (n+1)i;j ?????Step 8. Step 7???? ~F (n+1)i;j ?~g ?? (23)?????f(n+1)k = f(n+1)k + ±tF(n+1)k ; (24)?????????f (n+1)k ??????Step 9. 1??????? ±t ?? Step 2????3. ?????????????????????????????????????????????????????? u0 = 0:01?????????????????? IB-LBM?????????????? Ri = 45??????? Ro = 70?????? 200£ 200?y = 100???????????? Fig.2???????? ¿= 1?2?5????????? £?±?²????????????????? Fig.2????(a) Re =20(b) Re =40Fig. 3 Flow past a circular cylinder.u^(R) = u0R=Ro ¡Ro=RRi=Ro ¡Ro=Ri : (25)???????????????????maxj(jujn+1 ¡ jujn)j · 10¡8; (26)????????????????????????????????????? Fig.2 ??????Le ?????????IB-LBM????????????????????????????? ¿ < 2????? (6)?????¿ = 5???? IB-LBM ?????????????????????LBM ? ¿ > 1 ????????????IB-LBM ??????????????? ¿ ????????????????????????? IBM ?????????????? ¿ = 1 ?????? LKS?????????????IB-LBM?????????????????????? ¿ < 2? LKS??????IB-LKS ??????????? Re = 20?Re= 40?????????????????? Fig.3????????? 1601 £ 1601??????? D = 40????????? (641,801)?????????? u0 = 0:1??????? º = Du0=Re ???????? A ?? (22) ???Table 1 Comparison of drag coe±cients and circulatinglength (2L=D) with previous studies.Re Method CD LwPresent 2.167 1.9020 Dennis (FDM)(13) 2.045 1.88He (LBM)(14) 2.152 1.92Niu (IB-LBM)(5) 2.144 1.89Present 1.607 4.8040 Dennis (FDM)(13) 1.522 4.69He (LBM)(14) 1.499 4.49Niu (IB-LBM)(5) 1.589 4.690 20 40 60 80 100 1200.20.30.40.50.6 t* y/ H ° • : Present IB-LKS : Present IB-LBM : IB-LBM by Z.G. Feng : FEM by J. FengFig. 4 The lateral migration of the particle.A = 9(1=6¡ º=±x)=2?????????? (21)? @u®=@x¯???????????????????????????????????????????????????? 1600???????????? CD???? Lw ????????? Table 1???????????????????????? IB-LKS????????????????????????????????????????W = 2000???H = 80?????????? R=10???? 0?????? 20?????????????????????360????????????????? ½f = ½s = 1?????????????? u0=2 = 1=120?????????????????????????????????? º=1/30?Re = u0H=º=40?????????????????? ±t = H=u0?????????????????????? Fig.4???????????????????????????????????? Penalty method ???? IB-LBM(3) ???? ±??FEM(15) ???? ²??????? Direct forcing method ???? IB-LKS ? IB-LBM??????????????????Fig.4?? IB-LKS????????????????????????????????????? IB-LKS???????????????????????? g = 981 [cm/s2]??????????????0 0.2 0.4 0.6 0.8 1-6-5-4-3-2-101 t uy  • : ∆t=0.5 ×10 -4 [s] : ∆t=1.0 ×10 -4 [s] : ∆t=2.0 ×10 -4 [s] : FEM by D. WanFig. 5 Translational velocity of a particle in a channel.0 0.2 0.4 0.6 0.8 101234 t y • : ∆t=0.5 ×10 -4 [s] : ∆t=1.0 ×10 -4 [s] : ∆t=2.0 ×10 -4 [s] : FEM by D. WanFig. 6 Longitudinal coordinates of a particle in a channel.?????????????????????????????Glowinski???????????????? (16)??????? 2 [cm]???? 6 [cm]?????? ºr? 0.1 [cm2/s]??????? 1.0 [g/cm3]??????? 1.25 [g/cm3]???? 0.125 [cm]?????? 4 [cm]?????????????? 201£ 601????????????? 360?????? 25???????????? ±t ? LKS??????????? º = (1=6¡ 2A=9)±x ????±t = ±2xº=ºr ?????????????? ±x = 0.01 [cm]???????? ±t =0:5£10¡4 [s] (A = 0.525) ?±t = 1:0£10¡4 [s] (A = 0.3)?±t= 2:0 £ 10¡4 [s] (A = ¡0:15) ?????????????????????????????????Fig.5? Fig.6??????????Fig.5?Fig.6???? ²?????Wan?????? (17) ???????????????????????????????????????????????????????Wan ?????????????????????????????????? Glowinski?????????? (12)?(13)??????????????????????????????? ~Ff?~Tf ???????????????????????????????????Wan??????????????????????????????????????????????????????? 2 [cm]???? 8 [cm]???????0 1 2 3 4 500.20.40.60.811.21.41.61.82 t (sec) x (cm) • ° : Present IB-LKS result : Particle 1 (X.D. Niu) : Particle 2 (X.D. Niu)Fig. 7 Transverse coordinates of the centers of the two par-ticles.0 1 2 3 4 5012345678 t (sec) y (cm) • ° : Present IB-LKS result : Particle 1 (X.D. Niu) : Particle 2 (X.D. Niu)Fig. 8 Longitudinal coordinates of the centers of the twoparticles.0.01 [cm2/s]??????? 1.0 [g/cm3]??????? 1.01[g/cm3]???? 0.1 [cm]????? 1 (Particle 1)? (0.999[cm],7.2[cm]) ???? 2 (Particle 2) ? (1[cm], 6.8[cm]) ??????????? 5:0£ 10¡4[s]????Fig.7?????????????????Fig.8?????????????????Fig.9? t = 0.5[s]?1.5[s]?2.5[s]?3.0[s]?4.5[s]?????????????Fig.7?Fig.8???IB-LKS??????????Niu????? (5) ?????????????????????Fig.9????????????????????????????? 1??? 2???????????????????????????????????????? DKT (Drafting, Kissing, Tumbling) ?? (5) ?????????????????Niu??????????????????????????????????????????????? Lennard-Jones potential ?????????????????????????????????????????????????????4. ????Direct forcing method ???? IB-LBM ??????????????????????IB-LBM???????????????????????????????????t =0.5[s] t =1.0[s] t =1.5[s] t =3.0[s] t =4.5[s]Fig. 9 Sedimentation of two particles in a channel.???????????IB-LBM????? ¿ = 1????? IB-LKS?????????????????????????????????????????????????????????????????????? DKT?????????????????????????????????????????????????????????????????????????????????????(1) Chen, S.?Doolen, G. D.? Lattice Boltzmann methodfor °uid °ows? Annu. Rev. Fluid Mech.? 30(1998)?pp. 329{364?(2) Peskin, C. S.?Flow patterns around heart valves: A nu-merical method?J. Comput. Phys.?10(1972)?pp. 252{271?(3) Feng, Z-G.?Michaelides, E. E.? The immersedboundary-lattice Boltzmann method for solving °uid-particles interaction problems? J. Comput. Phys.?195(2004)?pp. 602{628?(4) Feng, Z-G.?Michaelides, E. E.?Proteus: A direct forc-ing method in the simulations of particulate °ows? J.Comput. Phys.?202(2005)?pp. 20{51?(5) Niu, X. D.?Shu, C.?Chew, Y. 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Lattice kinetic scheme for simulation of incompressible particle-fluid two phase flows with an immersed boundary method

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Lattice kinetic scheme for simulation of incompressible particle-fluid two phase flows with an immersed boundary method

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