A fictitious domain method for particulate flows with heat transfer. (English) Zbl 1160.76382

Yu, Zhaosheng; Shao, Xueming; Wachs, Anthony

J. Comput. Phys. 217, No. 2, 424-452 (2006).

Summary: The distributed-Lagrange-multiplier/fictitious-domain (DLM/FD) method of Glowinski et al. [R. Glowinski et al., Int. J. Multiphase Flow 25, No. 5, 755–794 (1999; Zbl 1137.76592)] is extended to deal with heat transfer in particulate flows in two dimensions. The Boussinesq approximation is employed for the coupling between the flow and temperature fields. The fluid-flow equations are solved with the finite-difference projection method on a half-staggered grid. In our operator splitting scheme, the Lagrange multipliers at the previous time level are kept in the fluid equations, and the new Lagrange multipliers for the rigid-body motion constraint and the Dirichlet temperature boundary condition are determined from the reduced saddle-point problem, whereas a very simple scheme based on the fully explicit computation of the Lagrange multiplier is proposed for the problem in which the solid heat conduction inside the particle boundary is also considered. Our code for the case of fixed temperature on the immersed boundary is verified by comparing favorably our results on the natural convection driven by a hot cylinder eccentrically placed in a square box and on the sedimentation of a cold circular particle in a vertical channel to the data in the literature. The code for the case of freely varying temperature on the boundaries of freely moving particles is applied to analyze the motion of a catalyst particle in a box and in particular the heat conductivities of nanofluids and sheared non-colloidal suspensions, respectively. Our preliminary computational results support the argument that the micro-heat-convection in the fluids is primarily responsible for the unusually high heat conductivity of nanofluids. It is shown that the Peclet number plays a negative role in the diffusion-related heat conductivity of a sheared non-colloidal suspension, whereas the Reynolds number does the opposite.

MSC:

76M20	Finite difference methods applied to problems in fluid mechanics
76T20	Suspensions
80A20	Heat and mass transfer, heat flow (MSC2010)

Keywords:

fictitious domain method; distributed Lagrange multiplier; particulate flows; heat transfer; heat conductivity; nanofluids

Citations:

Zbl 1137.76592

Software:

Proteus

References:

[1]	Hu, H.H.; Patankar, A.; Zhu, M.Y., Direct numerical simulations of fluid-solid systems using the arbitrary Lagrangian-Eulerian technique, J. comput. phys., 169, 427-462, (2001) · Zbl 1047.76571
[2]	Ladd, A.J.C.; Verberg, R., Lattice-Boltzmann simulations of particle-fluid suspensions, J. stat. phys., 104, 1191-1251, (2001) · Zbl 1046.76037
[3]	Höfler, K.; Schwarzer, S., Navier-Stokes simulation with constraint forces: finite-difference method for particle-laden flows and complex geometries, Phys. rev. E, 61, 7146-7160, (2000)
[4]	Glowinski, R.; Pan, T.-W.; Hesla, T.I.; Joseph, D.D., A distributed Lagrange multiplier/fictitious domain method for particulate flows, Int. J. multiphase flow, 25, 755-794, (1999) · Zbl 1137.76592
[5]	Sierou, A.; Brady, J.F., Accelerated Stokesian dynamics simulations, J. fluid mech., 448, 115-146, (2001) · Zbl 1045.76034
[6]	Lomholt, S.; Stenum, B.; Maxey, M.R., Experimental verification of the force coupling method for particulate flows, Int. J. multiphase flow, 28, 225-246, (2002) · Zbl 1136.76564
[7]	Lomholt, S.; Maxey, M.R., Force-coupling method for particulate two-phase flow: Stokes flow, J. comput. phys., 184, 381-405, (2003) · Zbl 1047.76100
[8]	Glowinski, R.; Pan, T.-W.; Périaux, J., A fictitious domain method for Dirichlet problems and applications, Comp. meth. appl. mech. eng., 111, 283-303, (1994) · Zbl 0845.73078
[9]	Glowinski, R.; Pan, T.-W.; Périaux, J., A fictitious domain method for external incompressible viscous flow modeled by Navier-Stokes equations, Comp. meth. appl. mech. eng., 112, 133-148, (1994) · Zbl 0845.76069
[10]	Glowinski, R.; Pan, T.-W.; Périaux, J., A Lagrange multiplier/fictitious domain method for the Dirichlet problem. generalization to some flow problems, Jpn. J. ind. appl. math., 12, 87-108, (1995) · Zbl 0835.76047
[11]	Glowinski, R.; Pan, T.-W.; Périaux, J., A Lagrange multiplier/fictitious domain method for the numerical simulation of incompressible viscous flow around moving rigid bodies (I): the case where the rigid body motions are known a priori, CR acad. sci. Paris, 324, 361-369, (1997) · Zbl 0885.76073
[12]	Bertrand, F.; Tanguy, P.A.; Thibault, F., A three-dimensional fictitious domain method for incompressible fluid flow problems, Int. J. numer. meth. fluids, 25, 719-736, (1997) · Zbl 0896.76033
[13]	Tanguy, P.A.; Bertrand, F.; Labie, R.; Brito-De La Fuente, E., Numerical modelling of the mixing of viscoplastic slurries in a twin-blade planetary mixer, Trans. icheme, 74, Part A, 499-504, (1996)
[14]	Glowinski, R.; Pan, T.-W.; Hesla, T.I.; Joseph, D.D.; Periaux, J., A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow, J. comput. phys., 169, 363-426, (2001) · Zbl 1047.76097
[15]	Yu, Z.; Phan-Thien, N.; Fan, Y.; Tanner, R.I., Viscoelastic mobility problem of a system of particles, J. non-Newtonian fluid mech., 104, 87-124, (2002) · Zbl 1058.76592
[16]	Yu, Z.; Phan-Thien, N.; Tanner, R.I., Dynamical simulation of sphere motion in a vertical tube, J. fluid mech., 518, 61-93, (2004) · Zbl 1131.76313
[17]	Hwang, W.R.; Hulsen, M.A.; Meijer, H.E.H., Direct simulations of particle suspensions in a viscoelastic fluid in sliding bi-periodic frames, J. non-Newtonian fluid mech., 121, 15-33, (2004) · Zbl 1143.76591
[18]	Peskin, C.S., Numerical analysis of blood flow in the heart, J. comput. phys., 25, 220-252, (1977) · Zbl 0403.76100
[19]	Yu, Z., A DLM/FD method for fluid/flexible-body interactions, J. comput. phys., 207, 1-27, (2005) · Zbl 1177.76304
[20]	Shi, X.; Phan-Thien, N., Distributed Lagrange multiplier/fictitious domain method in the framework of lattice Boltzmann method for fluid-structure interactions, J. comput. phys., 206, 81-94, (2005) · Zbl 1087.76543
[21]	Mittal, R.; Iaccarino, G., Immersed boundary methods, Annu. rev. fluid mech., 37, 239-261, (2005) · Zbl 1117.76049
[22]	Fadlun, E.A.; Verzicco, R.; Orlandi, P.; Mohd-Yusof, J., Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations, J. comput. phys., 161, 35-60, (2000) · Zbl 0972.76073
[23]	Patankar, N.A.; Singh, P.; Joseph, D.D.; Glowinski, R.; Pan, T.W., A new formulation of the distributed Lagrange multiplier/fictitious domain method for particulate flows, Int. J. multiphase flow, 26, 1509-1524, (2000) · Zbl 1137.76712
[24]	Dong, S.C.; Liu, D.; Maxey, M.R.; Karniadakis, G.E., Spectral distributed Lagrange multiplier method: algorithm and benchmark tests, J. comput. phys., 195, 695-717, (2004) · Zbl 1115.76349
[25]	Uhlmann, M., An immersed boundary method with direct forcing for the simulation of particulate flows, J. comput. phys., 209, 448-476, (2005) · Zbl 1138.76398
[26]	Sharma, N.; Patankar, N.A., A fast computation technique for the direct numerical simulation of rigid particulate flows, J. comput. phys., 205, 439-457, (2005) · Zbl 1087.76533
[27]	Feng, Z.; Michaelides, E.E., Proteus: a direct forcing method in the simulations of particulate flows, J. comput. phys., 202, 20-51, (2005) · Zbl 1076.76568
[28]	Kim, D.; Choi, H., Immersed boundary method for flow around an arbitrarily moving body, J. comput. phys., 212, 662-680, (2006) · Zbl 1161.76520
[29]	Baaijens, F.P.T., A fictitious domain/mortar element method for fluid-structure interaction, Int. J. numer. meth. fluids, 35, 743-761, (2001) · Zbl 0979.76044
[30]	McKenna, T.F.; Spitz, R.; Cokljat, D., Heat transfer from catalysts with computational fluid dynamics, Aiche j., 45, 2392-2410, (1999)
[31]	Keblinski, P.; Eastman, J.A.; Cahill, D.G., Nanofluids for thermal transport, Mater. today, 8, 36-44, (2005)
[32]	Maxwell, J.C., A treatise on electricity and magnetism, (1873), Clarendon Oxford · JFM 05.0556.01
[33]	Prasher, R.; Bhattacharya, P.; Phelan, P.E., Thermal conductivity of nanoscale colloidal solutions (nanofluids), Phys. rev. lett., 94, 025901, (2005)
[34]	Jang, S.P.; Choi, S.U.S., Role of Brownian motion in the enhanced thermal conductivity of nanofluids, Appl. phys. lett., 84, 4316-4318, (2004)
[35]	Nijemeisland, M.; Dixon, A.G., CFD study of fluid flow and wall heat transfer in a fixed bed of spheres, Aiche j., 50, 906-921, (2004)
[36]	Gan, H.; Chang, J.Z.; Feng, J.J.; Howard, H.H., Direct numerical simulation of the sedimentation of solid particles with thermal convection, J. fluid mech., 481, 385-411, (2003) · Zbl 1075.76059
[37]	Gan, H.; Feng, J.J.; Howard, H.H., Simulation of the sedimentation of melting solid particles, Int. J. multiphase flow, 29, 751-769, (2003) · Zbl 1136.76513
[38]	Kim, J.; Choi, H., An immersed-boundary finite-volume method for simulations of heat transfer in complex geometries, K.S.M.E. int. J., 18, 1026-1035, (2004)
[39]	Pacheco, J.R.; Pacheco-Vega, A.; Rodić, T.; Peck, R.E., Numerical simulations of heat transfer and fluid flow problems using an immersed-boundary finite-volume method on non-staggered grids, Numer. heat transf. part B: fundamentals, 48, 1-24, (2005)
[40]	Girault, V.; Raviart, P.-A., Finite element methods for Navier-Stokes equations: theory and algorithms, (1986), Springer-Verlag Berlin
[41]	Armfield, S.; Street, R., An analysis and comparison of the time accuracy of fractional-step methods for the Navier-Stokes equations on staggered grids, Int. J. numer. meth. fluids, 38, 255-282, (2002) · Zbl 1027.76033
[42]	Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P., Numerical recipes in FORTRAN: the art of scientific computing, (1992), Cambridge University Press Cambridge · Zbl 0778.65002
[43]	Crowe, C.; Sommerfeld, M.; Tsuji, Y., Multiphase flows with droplets and particles, (1998), CRC press Boca Raton, FL
[44]	Dance, S.L.; Climent, E.; Maxey, M.R., Collision barrier effects on the bulk flow in a random suspension, Phys. fluids, 16, 828-831, (2004) · Zbl 1186.76123
[45]	Nguyen, N.Q.; Ladd, A.J.C., Lubrication corrections for lattice-Boltzmann simulations of particle suspensions, Phys. rev. E, 66, 046708, (2002)
[46]	Jeffrey, D.J.; Onishi, Y., Calculation of the resistance and mobility functions for two unequal rigid spheres in low-Reynolds-number flow, J. fluid mech., 139, 261-290, (1984) · Zbl 0545.76037
[47]	Cichocki, B.; Jones, R.B., Image representation of a spherical particle near a hard wall, Physica A, 258, 273-302, (1998)
[48]	Kromkamp, J.; Van Den Ende, D.T.M.; Kandhai, D.; Van Der Sman, R.G.M.; Boom, R.M., Shear-induced self-diffusion and microstructure in non-Brownian suspensions at non-zero Reynolds numbers, J. fluid mech., 529, 253-278, (2005) · Zbl 1068.76085
[49]	Sierou, A.; Brady, J.F., Rheology and microstructure in concentrated noncolloidal suspensions, J. rheol., 46, 1031-1056, (2002)
[50]	Demirdžić, I.; Lilek, Ž.; Perić, M., Fluid flow and heat transfer test problems for non-orthogonal grids: bench-mark solutions, Int. J. numer. meth. fluids, 15, 329-354, (1992) · Zbl 0825.76631
[51]	Scardovelli, R.; Zaleski, S., Direct numerical simulation of free-surface and interfacial flow, Annu. rev. fluid mech., 31, 567-603, (1999)
[52]	Sethian, J.A.; Smereka, P., Level set methods for fluid interfaces, Annu. rev. fluid mech., 35, 341-372, (2003) · Zbl 1041.76057
[53]	Zhu, L.; Peskin, C.S., Simulation of a flapping flexible filament in a flowing soap film by the immersed boundary method, J. comput. phys., 179, 452-468, (2002) · Zbl 1130.76406
[54]	http:www.anter.com/6000.htm.
[55]	Batchelor, G.K., The stress in a suspension of force-free particles, J. fluid mech., 41, 545-570, (1970) · Zbl 0193.25702
[56]	Leal, L.G., Macroscopic transport properties of a sheared suspension, J. colloid interf. sci., 58, 296, (1977)
[57]	Brady, J.F.; Bossis, G., Stokesian dynamics, Annu. rev. fluid mech., 20, 111-157, (1988)
[58]	Lee, S.; Choi, S.U.S.; Li, S.; Eastman, J.A., Measuring thermal conductivity of fluids containing oxide nanoparticles, Trans. ASME, J. heat transf., 121, 280-289, (1999)
[59]	Perrins, W.T.; McPhedran, R.C.; McKenzie, D.R., Transport properties of regular arrays of cylinders, Proc. roy. soc., ser. A, 369, 207-225, (1979)
[60]	Banchio, A.J.; Brady, J.F., Accelerated stokesian dynamics: Brownian motion, J. comput. phys., 118, 10323-10332, (2003)
[61]	Ladd, A.J.C., Short time motion of colloidal particles: numerical simulation via a fluctuating lattice Boltzmann equation, Phys. rev. lett., 70, 1339-1342, (1993)
[62]	Sharma, N.; Patankar, N.A., Direct numerical simulation of the Brownian motion of particles by using fluctuating hydrodynamic equations, J. comput. phys., 201, 466-486, (2004) · Zbl 1061.76086