An extended iterative direct-forcing immersed boundary method in thermo-fluid problems with Dirichlet or Neumann boundary conditions

Abstract

An iterative direct-forcing immersed boundary method is extended and used to solve convection heat transfer problems. The pressure, momentum source, and heat source at immersed boundary points are calculated simultaneously to achieve the best coupling. Solutions of convection heat transfer problems with both Dirichlet and Neumann boundary conditions are presented. Two approaches for the implementation of Neumann boundary condition, i.e. direct and indirect methods, are introduced and compared in terms of accuracy and computational efficiency. Validation test cases include forced convection on a heated cylinder in an unbounded flow field and mixed convection around a circular body in a lid-driven cavity. Furthermore, the proposed method is applied to study the mixed convection around a heated rotating cylinder in a square enclosure with both iso-heat flux and iso-thermal boundary conditions. Computational results show that the order of accuracy of the indirect method is less than the direct method. However, the indirect method takes less computational time both in terms of the implementation of the boundary condition and the post processing time required to compute the heat transfer variables such as the Nusselt number. It is concluded that the iterative direct-forcing immersed boundary method is a powerful technique for the solution of convection heat transfer problems with stationary/moving boundaries and various boundary conditions.

Keywords

immersed boundary method / direct forcing / thermo-fluid problems / neumann boundary condition

References

[1] PeskinC S. Flow patterns around heart valves: A numerical method [J]. Journal of Computational Physics, 1972, 10: 252-271

[2] GoldsteinD, HadlerR, SirovichL. Modeling a no-slip flow boundary with an external force field [J]. Journal of Computational Physics, 1993, 105: 354-366

[3] SaikiE M, BiringenS. Numerical simulation of a cylinder in uniform flow: Application of a virtual boundary method [J]. Journal of Computational Physics, 1996, 123: 450-465

[4] Mohd-YusofJ. Combined immersed boundaries/B-splines methods for simulations of flows in complex geometries [M]//. CTR Annual Research Briefs, 1997317-327

[5] FadlunE, VerziccoR, OrlandiP, Mohd-YusofJ. Combined immersed-boundary finite-difference methods for threedimensional complex flow simulations [J]. Journal of Computational Physics, 2000, 161: 35-60

[6] BalarasE. Modeling complex boundaries using an external force field on fixed Cartesian grids in large-eddy simulations [J]. Computers & Fluids, 2004, 33: 375-404

[7] MittalR, DongH, BozkurttasM, NajjarF M, VargasA v, LoebbeckeA. A Versatile sharp interface immersed boundary method for incompressible flows with complex boundaries [J]. Journal of Computational Physics, 2008, 227: 4825-4852

[8] UhlmannM. An immersed boundary method with direct forcing for the simulation of particulate flows [J]. Journal of Computational Physics, 2005, 209: 448-476

[9] WangZ, FanJ, LuoK. Combined multi-direct forcing and immersed boundary method for simulating flows with moving particles [J]. International Journal of Multiphase Flow, 2008, 34: 283-302

[10] JiC, MunjizaA, WilliamsJ J R. A novel iterative direct-forcing immersed boundary method and its finite volume applications [J]. Journal of Computational Physics, 2012, 231: 1797-1821

[11] PanD. An Immersed boundary method on unstructured Cartesian meshes for incompressible flows with heat transfer [J]. Numerical Heat Transfer B, 2006, 49: 277-297

[12] YoonH S, LeeJ B, ChunH H. A numerical study on the fluid flow and heat transfer around a circular cylinder near a moving wall [J]. International Journal of Heat and Mass Transfer, 2007, 50: 3507-3520

[13] WangZ, FanJ, LuoK, CenK. Immersed boundary method for the simulation of flows with heat transfer [J]. International Journal of Heat and Mass Transfer, 2009, 52: 4510-4518

[14] KimB S, LeeD S, HaM Y, YoonH S. A numerical study of natural convection in a square enclosure with a circular cylinder at different vertical locations [J]. International Journal of Heat and Mass Transfer, 2008, 51: 1888-1906

[15] MarkA, SvenningE, EdelvikF. An immersed boundary method for simulation of flow with heat transfer [J]. International Journal of Heat and Mass Transfer, 2013, 56: 424-435

[16] PachecoJ R, Pacheco-VegaA, RodicT, PeckR E. Numerical simulations of heat transfer and fluid flow problems using an immersed-boundary finite volume method on nonstaggered grids [J]. Numerical Heat Transfer B, 2005, 48: 1-24

[17] ZhangN, ZhengZ C, EckelsS. Study of heat-transfer on the surface on a circular cylinder in flow using an immersed-boundary method [J]. International Journal of Heat and Fluid Flow, 2008, 29: 1558-1566

[18] PanD. A general boundary condition treatment in immersed boundary methods for Incompressible Navier-Stokes Eqs with heat transfer [J]. Numerical Heat Transfer B, 2012, 61: 279-297

[19] RenW, ShuC, YangW. An efficient immersed boundary method for thermal flow problems with heat flux boundary conditions [J]. International Journal of Heat and Mass Transfer, 2013, 64: 694-705

[20] PeskinC. The immersed boundary method [J]. Acta Numerica, 2002, 11: 479-517

[21] BhartiR P, ChhabraR P, EswaranV. A numerical study of the steady forced convection heat transfer from an unconfined circular cylinder [J]. International Journal of Heat and Mass Transfer, 2007, 43: 639-648

[22] AhmadR A, QureshiZ H. Laminar mixed convection from a uniform heat flux horizontal cylinder in a crossflow [J]. J Thermophys Heat Transfer, 1992, 6: 277-287

[23] KhanaferK, AithalS M. Laminar mixed convection flow and heat transfer characteristics in a lid-driven cavity with a circular cylinder [J]. International Journal of Heat and Mass Transfer, 2013, 66: 200-209

[24] LiaoC C, LinC A. Mixed convection of a heated rotating cylinder in a square enclosure [J]. International Journal of Heat and Mass Transfer, 2014, 72: 9-22