DEVELOPME T OF A OVEL IMMERSED BOU DARY-LATTICE BOLTZMA METHOD A D ITS APPLICATIO S WU JIE (B. Eng., M. Eng., anjing University of Aeronautics and Astronautics, China) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTME T OF MECHA ICAL E GI EERI G ATIO AL U IVERSITY OF SI GAPORE 2010 ACK OWLEDGEME TS I wish to express my deepest gratitude to my supervisor, Professor Shu Chang, for his invaluable guidance, supervision, patience and support throughout my Ph.D. study. In addition, I would like to express my appreciation to the National University of Singapore for giving me a research scholarship and an opportunity to this program at Department of Mechanical Engineering. It provides excellent conditions for me to complete my research work smoothly. I also wish to thank all the staff members in the Fluid Mechanics Laboratory for their valuable assistance. My sincere appreciation will go to my dear family. Their love, concern, support and continuous encouragement help me with tremendous confidence in solving the problems in my study and life. Finally, I would like to thank all my friends who have helped me in different ways during my whole period of study in NUS. WU JIE i TABLE OF CO TE TS ACKNOWLEDGEMENTS i TABLE OF CONTENTS ii SUMMARY viii LIST OF TABLES x LIST OF FIGURES xii NOMENCLATURE xix Chapter Introduction 1.1 Background 1.2 Non-Boundary Conforming Method 1.2.1 Sharp interface approach 1.2.2 Diffuse interface approach 1.2.2.1 Immersed boundary method 1.2.2.2 Force calculation in IBM 1.2.2.3 Advantages and disadvantages of IBM 10 1.3 Lattice Boltzmann Method 12 1.3.1 Features of LBM 12 1.3.2 Historic development of LBM 13 1.3.3 Implementation of boundary conditions in LBM 15 1.3.4 Some efforts on improvement of computational efficiency 17 1.3.5 Applications of LBM with curved boundary and/or 17 ii non-uniform mesh 1.3.6 Immersed boundary-lattice Boltzmann method (IB-LBM) 20 1.4 Objectives of The Thesis 21 1.5 Organization of The Thesis 23 Chapter Development of Efficient Lattice Boltzmann Method on Non-Uniform 25 Cartesian Mesh 2.1 Standard LBM 26 2.2 Taylor Series Expansion and Least Squares-base Lattice Boltzmann 30 Method 2.3 Efficient LBM on Non-Uniform Cartesian Mesh 33 2.4 Accuracy Analysis of Present LBM 39 2.5 Numerical Tests 41 2.6 Concluding Remarks 43 Chapter Boundary Condition-enforced Immersed Boundary-Lattice Boltzmann 51 Method 3.1 Immersed Boundary Method 52 3.2 Conventional IB-LBM 53 3.2.1 Combination of IBM and LBM 54 3.2.2 Penalty force model in IB-LBM 54 3.2.3 Direct forcing model in IB-LBM 56 3.2.4 Momentum exchange model in IB-LBM 58 iii 3.2.5 Drawback of conventional IB-LBM 59 3.3 Boundary Condition-enforced IB-LBM 60 3.4 Numerical Examples 68 3.4.1 Numerical test of overall accuracy 68 3.4.2 Flows over an array of circular cylinders placed at the middle of 70 straight channel 3.4.3 Flows over a circular cylinder 71 3.4.4 Flows over a pair of circular cylinders 75 3.4.4.1 Side-by side arrangement 76 3.4.4.2 Tandem arrangement 77 3.4.5 Flows over a NACA0012 airfoil 79 3.5 Concluding Remarks Chapter Application of New IB-LBM to Simulate Two-dimensional 81 105 Moving Boundary and Particulate Flow Problems 4.1 Brief Review of Boundary Condition-enforced IB-LBM 106 4.2 Simulation of Flows around Moving Boundaries 106 4.2.1 Brief review on moving boundary problems and computational 106 sequence 4.2.2 Numerical results and discussion 109 4.2.2.1 Flow over a moving circular cylinder 109 4.2.2.2 Flows over a rotationally oscillating cylinder 111 4.2.2.3 Unsteady flows at low Reynolds number flapping flight 115 iv 4.2.2.4 Flows over a flapping flexible airfoil 4.3 Simulation of Particulate Flows 117 120 4.3.1 Brief review on the study of particulate flow problems 120 4.3.2 Force, torque calculation on the particle and computational 124 sequence 4.3.3 Numerical results and discussion 126 4.3.3.1 A moving neutrally buoyant particle in linear shear flow 126 4.3.3.2 Particle sedimentation in viscous fluid 128 A. One particle sedimentation 128 B. Two particles sedimentation 129 4.3.3.3 Particle suspension in a 2D symmetric stenotic artery 131 A. One particle passes the stenosis throat with b = 1.75d 131 B. Two particles pass the stenosis throat with b = 1.75d 132 4.4 Concluding Remarks Chapter Application of New IB-LBM to Study Flows over a Stationary 133 158 Circular Cylinder with a Flapping Plate 5.1 Brief Review on Flow Wake behind a Bluff Body 158 5.2 Configuration of Problem and Numerical Validation 160 5.2.1 Problem definition 160 5.2.2 Numerical validation 161 5.3 Numerical Study of Flows over a Stationary Circular Cylinder with 162 a Flapping Plate v 5.3.1 Flow patterns due to flapping plate 162 5.3.1.1 Effect of flapping frequency 162 5.3.1.2 Effect of flapping amplitude 164 5.3.2 Drag force due to flapping plate 165 5.3.3 Near-wake structure 167 5.3.4 Vorticity control 170 5.4 Concluding Remarks Chapter Extension of New IB-LBM to Simulate Three-dimensional Flows 174 186 around Stationary/Moving Objects 6.1 Three-dimensional Boundary Condition-enforced IB-LBM 186 6.2 Efficient Three-dimensional LBM Solver on Non-Uniform Cartesian 189 Mesh 6.3 Computational Sequence 190 6.4 Numerical Simulation of Flows around Stationary Objects 191 6.4.1 Flows over a stationary sphere 191 6.4.2 Flows over a torus with small aspect ratio 195 6.5 Numerical Simulation of Flows around Moving Objects 198 6.5.1 Flows around a rotating sphere 199 6.5.2 Fish swimming 202 6.5.3 Dragonfly flight 206 6.6 Concluding Remarks 208 vi Chapter Conclusions and Recommendations 230 7.1 Conclusions 230 7.2 Recommendations 234 References 236 vii Summary In recent years, the immersed boundary method (IBM) has been developed into a popular numerical technique in the community of computational fluid dynamics (CFD). As a successful example of non-boundary conforming methods, the Cartesian mesh is utilized for resolving flow field in IBM. The effect of boundary is replaced by the body force density which influences flow phase around the boundary. The governing equations with this force density are solved on the whole computational domain including the exterior and interior of the boundary. On the other hand, as an alternative CFD tool, the lattice Boltzmann method (LBM) has gained wide range applications recently. Since the Cartesian mesh is also employed in LBM, an efficient solver can be generated by combining IBM with LBM, which is called IB-LBM. Some efforts have been made in this aspect and the achievement is obvious. However, there are still some shortcomings in this newly developed approach. In this work, two major improvements were made. Firstly, a new version of IB-LBM was proposed in order to strictly satisfy the non-slip boundary condition. In the conventional IB-LBM, the non-slip boundary condition is not enforced, and is only approximately satisfied at the converged state. As a consequence, the accuracy of solution is reduced, and the situation of streamline penetration to solid boundary is present. To overcome this drawback, a boundary condition-enforced IB-LBM was developed. Different from the conventional IB-LBM in which the body force is computed in advance, the unknown body force is employed viii in present method. Such force is resolved by enforcing the non-slip boundary condition. Applying the developed approach, the two-dimensional (2D) stationary and moving boundary flow problems, as well as particulate flow problems, were simulated. Since the non-slip boundary condition is enforced, no flow penetration happens and the accuracy of resolution is improved. All the obtained numerical results are compared well with previous experimental and numerical results. In the application of IB-LBM, the non-uniform mesh is usually employed in order to improve the computational efficiency. To apply LBM on the non-uniform mesh, many variants of LBM can be chosen. A simple way is to use Taylor series expansion and least squared-based LBM (TLLBM). Its final form is an algebraic formulation, in which the coefficients only depend on the coordinates of mesh points and lattice velocity. As compared to the standard LBM, the drawback of TLLBM is that additional memory is required to store the coefficients. Due to the limitation of virtual memory, it is not easy to apply TLLBM in three-dimensional (3D) simulations. To overcome this difficulty, an efficient LBM solver based on the one-dimensional interpolation was developed. As compared to TLLBM, much less coefficients are calculated. Combing with this efficient LBM solver, the new IB-LBM was easily extended to 3D simulation. The 3D flows around complex stationary and moving boundaries were simulated. The obtained numerical results are agreed well with the results and findings in the literature. ix References Fedkiw RP, Aslam T, Merriman B, and Osher S, A non-oscillatory Eulerian approach to interfaces in multimaterial flow flows (the ghost fluid method), J. Comput. Phys. 152, 457-492, 1999. Feng J, Hu HH, and Joseph DD, Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid. Part 2, Couette and Poiseuille flows, J. Fluid Mech. 277, 271-301, 1994. Feng Z and Michaelides E, The immersed boundary-lattice Boltzmann method for solving fluid-particles interaction problems, J. Comput. Phys. 195, 602-628, 2004. Feng Z and Michaelides E, Proteus: a direct forcing method in the simulations of particulate flows, J. Comput. Phys. 202, 20-51, 2005. Filippova O and Hänel D, Grid refinement for lattice-BGK models, J. Comput. Phys. 147, 219-228, 1998. Fogelson AL and Peskin CS, A fast numerical method for solving the three-dimensional Stokes equations in the presence of suspended particles, J. Comput. Phys. 79, 50-69, 1988. Fornberg B, A numerical study of steady viscous flow past a circular cylinder, J. Fluid Mech. 98, 819-855, 1980. Fortes A, Joseph DD, and Lundgren TS, Nonlinear mechanics of fluidization of beds of spherical particles, J. Fluid Mech. 177, 467-483, 1987. Frisch U, Hasslacher B, and Pomeau Y, Lattice-gas automata for the Navier-Stokes equations, Phys. Rev. Lett. 56, 1505-1508, 1986. 240 References Ghia U, Chia KN, and Shin CT, High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, J. Comput. Phys. 48, 387-411, 1982. Gilmanov A, Sotiropoulos F, and Balarasf E, A general reconstruction algorithm for simulating flows with complex 3D immersed boundaries on Cartesian grids, J. Comp. Phys. 191, 660-669, 2003. Glowinski R, Pan TW, Hesla TI, and Joseph DD, A distributed Lagrange multiplier/fictitious domain method for particulate flows, Int. J. Multiphase Flow, 25, 755-794, 1999. Goldstein D, Hadler R, and Sirovich L, Modeling a no-slip flow boundary with an external force field, J. Comput. Phys. 105, 354-366, 1993. Gopalkrishnan R, Triantafyllou MS, Triantafyllou GS, and Barrett D, Active vorticity control in a shear flow using a flapping foil, J. Fluid Mech. 274, 1-21, 1994. Gunstensen AK, Rothman DH, Zaleski S, and Zanetti G, Lattice Boltzmann model of immiscible fluids, Phys. Rev. A, 43, 4320-4327, 1991. Guo Z, Zheng C, and Shi B, Discrete lattice effects on the forcing term in the lattice Boltzmann method, Phys. Rev. E, 65, 046308, 2002. Hardy J, Pomeau Y, and de Pazzis O, Time evolution of a two-dimensional classical lattice system, Phys. Rev. Lett. 31, 276-279, 1973. 241 References He JW, Glowinski R, Metcalfe R, Nordlander A, and Periaux J, Active control and drag optimization for flow past a circular cylinder I. Oscillatory cylinder rotation, J. Comput. Phys. 163, 83-117, 2000. He X and Doolen GD, Lattice Boltzmann method on curvilinear coordinates system: flow around a circular cylinder, J. Comput. Phys. 134, 306-315, 1997. He X and Doolen GD, Lattice Boltzmann method on a curvilinear coordinate system: Vortex shedding behind a circular cylinder, Phys. Rev. E, 56, 434-440, 1997. He X, Luo LS, and Dembo M, Some progress in lattice Boltzmann method, Part I. Non-uniform mesh grids, J. Comput. Phys. 129, 357-363, 1996. Higuera F and Jimenez J, Boltzmann approach to lattice gas simulations, Europhys. Lett. 9, 663-668, 1989. Hirt CW, Amsden AA, and Cook JL, An arbitrary Lagrangian-Eulerian computing method for all flow speeds, J. Comput. Phys. 14, 227-253, 1974. Höfler K and 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. Hou S, Zou Q, Chen S, Doolen GD, and Cogley AC, Simulation of cavity flow by the lattice Boltzmann method, J. Comput. Phys. 118, 329-347, 1995. Hu HH, Direct simulation of flows of solid-liquid mixtures, Int. J. Multiphase Flow, 22, 335-352, 1996. 242 References Hu HH, Patankar NA, and Zhu MY, Direct numerical simulations of fluid-solid systems using arbitrary Lagrangian-Eulerian technique, J. Comput. Phys. 169, 427-462, 2001. Imamura I, Suzuki K, Nakamura T, and Yoshida M, Flow simulation around an airfoil using lattice Boltzmann method on generalized coordinates, AIAA, 2004-0244, 2004. Inamuro T, Maeha K, and Ogino F, Flow between parallel walls containing the lines of neutrally buoyant circular cylinder, Int. J. Multiphase Flow, 26, 1981-2004, 2000. Jeong J and Hussain J, On the identification of a vortex, J. Fluid Mech. 285, 69-94, 1995. Johnson AA and Tezduyar TE, Mesh update strategies in parallel finite element computations of flow problems with moving boundaries and interfaces, Comput. Meth. Appl. Mech. Eng. 119, 73-94, 1994. Johnson TA and Patel VC, Flow past a sphere up to a Reynolds number of 300, J. Fluid Mech. 378, 19-70, 1999. Kim D and Choi H, Laminar flow past a sphere rotating in the streamwise direction, J. Fluid Mech. 461, 365-386, 2002. Kim J, Kim D, and Choi H, An immersed-boundary finite-volume method for simulations of flow in complex geometries, J. Comput. Phys. 171, 132-150, 2001. 243 References Koelman JMVA, A simplified lattice Boltzmann scheme for Navier-Stokes fluid flow, Europhys. Lett. 15, 603-607, 1991. Kwon K and Choi H, Control of laminar vortex shedding behind a circular cylinder using splitter plates, Phys. Fluids, 8, 479-486, 1996. Ladd AJC, Short-time motion of colloidal particles: numerical simulation via a fluctuating lattice-Boltzmann equation, Phys. Rev. Lett. 70, 1339-1342, 1993. Ladd AJC, Numerical simulations of particulate suspensions via a discretized Boltzmann equation. I. Theoretical foundation, J. Fluid Mech. 271, 285-310, 1994a. Ladd AJC, Numerical simulations of particulate suspensions via a discretized Boltzmann equation. II. Numerical results, J. Fluid Mech. 271, 311-339, 1994b. Lai MC and Peskin CS, An immersed boundary method with formal second-order accuracy and reduced numerical viscosity, J. Comput. Phys. 160, 705-719, 2000. Le DV, Khoo BC, and Peraire J, An immersed interface method for viscous incompressible flows involving rigid and flexible boundaries, J. Comput. Phys. 220, 109-138, 2006. Lee L and LeVeque RJ, An immersed interface method for incompressible Navier-Stokes equations, SIAM J. Sci. Comput. 25, 832-856, 2003. Lehmann FO, Sane SP, and Dickinson MH, The aerodynamic effects of wing- wing interaction in flapping insect wings, J. Exp. Biol. 208, 3075-3092, 2005. 244 References LeVeque RJ and Li Z, The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. umer. Anal. 31, 1019-1044, 1994. LeVeque RJ and Li Z, Immersed interface method for Stokes flow with elastic boundaries or surface tension, SIAM J. Sci. Comput. 18, 709-735, 1997. Li H, Fang H, Lin Z, Xu S, and Chen S, Lattice Boltzmann simulation on particle suspensions in a two-dimensional symmetric stenotic artery, Phys. Rev. E, 69, 031919, 2004. Li J, Chambarel A, Donneaud M, Martin R, Numerical study of laminar flow past one and two circular cylinders, Comput. Fluids, 19, 155-170, 1991. Li J, Hesse M, Ziegler J, and Woods AW, An arbitrary Lagrangian Eulerian method for moving-boundary problems and its application to jumping over water, J. Comput. Phys. 208, 289–314, 2005. Li Z and Lai MC, The immersed interface method for the Navier-Stokes equations with singular forces, J. Comput. Phys. 171, 822-842, 2001. Li Z and Wang C, A fast finite difference method for solving Navier-Stokes equations on irregular domains, Comm. Math. Sci. 1, 180-196, 2003. Lighthill MJ, Note on swimming of slender fish, J. Fluid Mech. 9, 305-317, 1960. Lima E Silva LF, Silveira-Neto A, and Damasceno JJR, Numerical simulation of twodimensional flows over a circular cylinder using the immersed boundary method, J. Comput. Phys. 189, 351-370, 2003. 245 References Linnick MN and Fasel HF, A high-order immersed interface method for simulating unsteady incompressible flows on irregular domains, J. Comput. Phys. 204, 157-192, 2005. Liu C, Zheng X, and Sung CH, Preconditioned multigrid methods for unsteady incompressible flows, J. Comput. Phys. 139, 35-57, 1998. Liu XD, Fedkiw RP, and Kang MJ, A boundary condition capturing method for Poisson’s equation on irregular domains, J. Comput. Phys. 160, 151-178, 2000. Lockard DP, Luo LS, Milder SD, and Singer BA, Evaluation of powerflow for aerodynamic applications, J. Stat. Phys. 107, 423-478, 2002. Lu XY and Sato J, A numerical study of flow past a rotationally oscillating circular cylinder, J. Fluids Struct. 10, 829-849, 1996. Luo H and Bewley TR, On the contravariant form of the Navier-Stokes equations in time-dependent curvilinear coordinate systems, J. Comput. Phys. 199, 355-375, 2004. Luo K, Wang Z, and Fan J, A modified immersed boundary method for simulations of fluid-particle interactions, Comput. Meth. Appl. Mech. Eng. 197, 36-46, 2007. Marella S, Krishnan S, Liu H, and Udaykumar HS, Sharp interface Cartesian grid method I: An easily implemented technique for 3D moving boundary computations, J. Comput. Phys. 210, 1-31, 2005. Mateescu D, Mekanik A, and Paїdoussis MP, Analysis of 2-D and 3-D unsteady annular flows with oscillating boundaries, based on a time-dependent coordinate transformation, J. Fluids Struct. 10, 57-77, 1996. 246 References McNamara G and Zanetti G, Use of the Boltzmann equation to simulate lattice-gas automata, Phys. Rev. Lett. 61, 2332-2335, 1988. Maury B, Direct simulations of 2D fluid-particle flows in biperiodic domains, J. Comput. Phys. 156, 325-351, 1999. Mei R and Shyy W, On the finite difference-based lattice Boltzmann method in curvilinear coordinates, J. Comput. Phys. 143, 426-448, 1998. Miao JM and Ho MH, Effect of flexure on aerodynamic propulsive efficiency of flapping flexible airfoil, J. Fluids Struct. 22, 401-419, 2006. Mittal R and Iaccarino G, Immersed boundary methods, Annu. Rev. Fluid Mech. 37, 239-261, 2005. Mittal R, Dong H, Bozkurttas M, Najjar FM, Vargas A, and von Loebbecke A, A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries, J. Comput. Phys. 227, 4825-4852, 2008. Mittal S, Kumar V, and Raghuvanshi A, Unsteady incompressible flows past two cylinders in tandem and staggered arrangements, Int. J. umer. Meth. Fluids, 25, 1315-1344, 1997. Mittal S and Tezduyar TE, Massively parallel finite element computation of incompressible flows involving fluid-body interactions, Comput. Meth. Appl. Mech. Eng. 112, 253-282, 1994. Muller UK, Van Den Heuvel BLE, Stamhuis EJ, and Videler JJ, Fish foot prints: morphology and energetics of the wake behind a continuously swimming mullet (Chelon labrosus risso), J. Exp. Biol. 200, 2893-2906, 1997. 247 References Nieuwstadt F and Keller HB, Viscous flow past circular cylinders, Comput. Fluids, 1, 59-71, 1973. Niu XD, Chew YT, and Shu C, Simulation of flows around an impulsively started circular cylinder by Taylor series expansion- and least squares-based lattice Boltzmann method, J. Comput. Phys. 188, 176-193, 2003. Niu XD, Shu C, and Chew YT, A lattice Boltzmann BGK model for simulation of micro flows, Europhys. Lett. 67, 600-606, 2004. Niu XD, Shu C, Chew YT, and Peng Y, A momentum exchange-based immersed boundary-lattice Boltzmann method for simulating incompressible viscous flows, Phys. Lett. A, 354, 173-182, 2006. Noble DR, Chen S, Georgiadis JG, and Buckius RO, A consistent hydrodynamic boundary condition for the lattice Boltzmann method, Phys. Fluids, 7, 203-209, 1995. Noh WF, CEL: a time-dependent, two-space-dimensional, Coupled EulerianLagrange Code, Method in Computational Phys, Academic Press, New York, pp. 117-179, 1964. Peng G, Xi H, Duncan C, and Chou SH, Lattice Boltzmann method on irregular meshes, Phys. Rev. E, 58, R4124-R4127, 1998. Peng G, Xi H, Duncan C, and Chou SH, Finite volume scheme for the lattice Boltzmann method on unstructured meshes, Phys. Rev. E, 59, 4675-4682, 1999. 248 References Peng Y, Shu C, Chew YT, and Inamuro T, Lattice kinetic scheme for the incompressible viscous thermal flows on arbitrary meshes, Phys. Rev. E, 69, 016703, 2004. Peng Y, Shu C, Chew YT, Niu XD, and Lu XY, Application of multi-block approach in the immersed boundary-lattice Boltzmann method for viscous fluid flows, J. Comput. Phys. 218, 460-478, 2006. Peskin, CS, Numerical analysis of blood flow in the heart, J. Comput. Phys. 25, 220-252, 1977. Peskin CS, The immersed boundary method, Acta umer. 11, 479-517, 2002. Protas B and Wesfreid JE, Drag force in the open-loop control of the cylinder wake in the laminar regime, Phys. Fluids, 14, 810-826, 2002. Protas B and Wesfreid JE, On the relation between the global modes and the spectra of drag and lift in periodic wake flows, C. R. Mec. 331, 49-54, 2003. Qi D, Lattice-Boltzmann simulations of particles in non-zero-Reynolds-number flows, J. Fluid Mech. 385, 41-62, 1999. Qian YH, d’Humieres D, and Lallemand P, Lattice BGK model for Navier-Stokes equation, Europhys. Lett. 17, 479-484, 1992. Quirk JJ, An alternative to unstructured grids for computing gas dynamics flows around arbitrarily complex two-dimensional bodies, Comput. Fluids, 23, 125-142, 1994. Roma AM, Peskin CS, and Berger MJ, An adaptive version of the immersed boundary method, J. Comput. Phys. 153, 509-534, 1999. 249 References Rozhdestvensky KV and Ryzhov VA, Aerohydrodynamics of flapping-wing propulsors, Prog. Aerosp. Sci. 39, 585-633, 2003. Russell D and Wang ZJ, A Cartesian grid method for modeling multiple moving objects in 2D incompressible viscous flow, J. Comput. Phys. 191, 177-205, 2003. Saiki EM and Biringen S, Numerical simulation of a cylinder in uniform flow: Application of a virtual boundary method, J. Comput. Phys. 123, 450-465, 1996. Sheard GJ, Hourigan K, and Thompson MC, Computations of the drag coefficients for low-Reynolds-number flow past rings, J. Fluid Mech. 526, 257-275, 2005. Sheard GJ, Thompson MC, and Hourigan K, From spheres to circular cylinders: the stability and flow structures of bluff ring wakes, J. Fluid Mech. 492, 147-180, 2003. Sheard GJ, Thompson MC, and Hourigan K, From spheres to circular cylinders: nonaxisymmetric transitions in the flow past rings, J. Fluid Mech. 506, 45-78, 2004. Shu C, Liu NY, and Chew YT, A novel immersed boundary velocity correction-lattice Boltzmann method and its application to simulation flow past a circular cylinder, J. Comput. Phys. 226, 1607-1622, 2007. Shu C, Niu XD, and Chew YT, Taylor-series expansion and least-squares-based lattice Boltzmann method: Two-dimensional formulation and its applications, Phys. Rev. E, 65, 036708, 2002. Shu C, Niu XD, and Chew YT, Taylor series expansion and least squares-based lattice Boltzmann method: three-dimensional formulation and its applications, Int. J. Mod. Phys. C, 14, 925-944, 2003. 250 References Shu C, Niu XD, and Chew YT, A lattice Boltzmann kinetic model for microflow and heat transfer, J. Stat. Phys. 121, 239-255, 2005. Shukla RK, Tatineni M, and Zhong X, Very high-order compact finite difference schemes on non-uniform grids for incompressible Navier-Stokes equations, J. Comput. Phys. 224, 1064-1094, 2007. Singh P, Joseph DD, Hesla TI, Glowinski R, and Pan TW, A distributed Lagrange multiplier/fictitious domain method for viscoelastic particulate flows, J. on- ewtonian Fluid Mech. 91, 165-188, 2000. Succi S, Foti E, and Higuera F, Three dimensional flows in complex geometries with the lattice Boltzmann method, Europhys. Lett. 10, 433-438, 1989. Succi S, Amati G, and Benzi R, Challenges in lattice Boltzmann computing, J. Stat. Phys. 81, 5-16, 1995. Tezduyar TE, Liou J, and Behr M, A new strategy for finite element computations involving moving boundaries and interfaces—The DSD/ST procedure. I. The concept and the preliminary numerical tests, Comput. Meth. Appl. Mech. Eng. 94, 339-351, 1992a. Tezduyar TE, Liou J, Behr M, and Mittal S, A new strategy for finite element computations involving moving boundaries and interfaces—The DSD/ST procedure. II. Computation of free-surface flows, two-liquid flows, and flows with drafting cylinders, Comput. Meth. Appl. Mech. Eng. 94, 353-371, 1992b. Thiria B, Goujon-Durand S, and Wesfreid JE, The wake of a cylinder performing rotary oscillations, J. Fluid Mech. 560, 123-147, 2006. 251 References Tokumaru PT and Dimotakis PE, Rotary oscillation control of a cylinder wake, J. Fluid Mech. 224, 77-90, 1991. Tseng YH and Ferziger JH, A ghost-cell immersed boundary method for flow in complex geometry, J. Comput. Phys. 192, 593-623, 2003. Tsiveriotis K and Brown RA, Boundary-conforming mapping applied to computations of highly deformed solidification interfaces, Int. J. umer. Meth. Fluids, 14, 981-1003, 1992. Udaykumar HS, Kan HC, Shyy W, and Tran-Son-Tay R, Multiphase dynamics in arbitrary geometries on fixed Cartesian grids, J. Comput. Phys. 137, 366-405, 1997. Udaykumar HS, and Mal L, Sharp-interface simulation of dendritic solidification of solutions, Int. J. Heat Mass Transfer, 45, 4793-4808, 2002. Udaykumar HS, Mittal R, Rampunggoon P, and Khanna A, A sharp interface Cartesian grid method for simulating flows with complex moving boundaries, J. Comput. Phys. 174, 345-380, 2001. Uhlmann M, An immersed boundary method with direct forcing for the simulation of particulate flows, J. Comput. Phys. 209, 448-476, 2005. Wan D and Turek S, Direct numerical simulation of particulate flow via multigrid FEM techniques and the fictitious boundary method, Int. J. umer. Methods Fluids, 51, 531-566, 2006. Wang X, and Liu WK, Extend immersed boundary method using FEM and RKPM, Comput. Meth. Appl. Mech. Eng. 193, 1305-1321, 2004. 252 References Wang ZJ, Birch JM, and Dickinson MH, Unsteady forces and flows in low Reynolds number hovering flight: two-dimensional computations vs robotic wing experiments, J. Exp. Biol. 207, 449-460, 2004. White FW, Viscous Fluid Flow, McGraw-Hill, New York, 1974. Williamson CHK, Evolution of a single wake behind a pair of bluff bodies, J. Fluid Mech. 159, 1-18, 1895. Wolfgang MJ, Anderson JM, Grosenbaugh, MA, Yue DKP, and Triantafyllou, MS, Near-body flow dynamics in swimming fish, J. Exp. Biol. 202, 2303-2327, 1999. Wolfram S, Cellular automaton fluids. 1: Basic theory, J. Stat. Phys. 45, 471-526, 1986. Wu TYT, Hydromechanics of swimming propulsion. Part 3. Swimming and optimum movements of slender fish with side fins, J. Fluid Mech. 46, 545-568, 1971. Xi H, Peng G, and Chou SH, Finite-volume lattice Boltzmann method, Phys. Rev. E, 59, 6202-6205, 1999. Xu S and Wang ZJ, Systematic derivation of jump conditions for the immersed interface method in three-dimensional flow simulation, SIAM J. Sci. Comput. 27, 1948-1980, 2006a. Xu S and Wang ZJ, An immersed interface method for simulating the interaction of a fluid with moving boundaries, J. Comput. Phys. 216, 454-493, 2006b. Xu S and Wang ZJ, A 3D immersed interface method for fluid-solid interaction, Comput. Meth. Appl. Mech. Eng. 197, 2068-2086, 2008. 253 References Yang Y and Udaykumar HS, Sharp interface Cartesian grid method III: Solidification of pure materials and binary solution, J. Comput. Phys. 210, 55-74, 2005. Ye T, Mittal R, Udaykumar HS, and Shyy W, An accurate Cartesian grid method for viscous incompressible flows with complex immersed boundaries, J. Comput. Phys. 156, 209-240, 1999. Yu D, Mei R, and Shyy W, A multi-block lattice Boltzmann method for viscous fluid flows, Int. J. umer. Meth. Fluids, 39, 99-120, 2002. Yu Z, Phan-Thien N, Fan Y, and Tanner RI, Viscoelastic mobility of a system of particles, J. on- ewtonian Fluid Mech. 104, 87-124, 2002. Zdravkovich MM, Review of flow interference between two circular cylinders in various arrangements, ASME J. Fluids Eng. 99, 618-633, 1977. Zdravkovich MM, Flow induced oscillations of two interfering circular cylinders, J. Sound Vib. 101, 511-521, 1985. Zdravkovich MM, Flow around circular cylinder. Vol. 1: Fundamentals. Oxford University Press, 1997. Zdravkovich MM, Flow around circular cylinder. Vol. 2: Applications. Oxford University Press, 2002. Zhang L, Gerstenberger A, Wang X, and Liu WK, Immersed finite element method, Comput. Meth. Appl. Mech. Eng. 193, 2051-2067, 2004. Zhu L and Peskin CS, Simulation of a flapping flexible filament in a flowing soap film by the immersed boundary method, J. Comput. Phys. 179, 452-468, 2002. 254 References Zhu Q, Wolfgang MJ, Yue DKP, and Triantafyllou MS, Three-dimensional flow structures and vorticity control in fish-like swimming, J. Fluid Mech. 468, 1-28, 2002. Ziegler DP, Boundary conditions for lattice Boltzmann simulations, J. Stat. Phys. 71, 1171-1177, 1993. Zou Q and He X, On pressure and velocity boundary conditions for the lattice Boltzmann BGK model, Phys. Fluids, 9, 1591-1598, 1997. 255 [...]... interface method LBE Lattice Boltzmann equation LBM Lattice Boltzmann method N-S Navier-Stokes TLLBM Taylor series expansion- and least square-based LBM xxi Chapter 1 Introduction 1.1 Background As a branch of fluid mechanics, the computational fluid dynamics (CFD) plays an important role in the research and application of engineering problems The basic principle of CFD is to use numerical methods to get approximate... conventional IBM Practically, there is a demand to further develop a new version of IBM which can be implemented easily, and in the meantime, can satisfy the non-slip boundary condition This demand motivates the present work 1.3 Lattice Boltzmann Method In recent years, as an alternative and promising computational technique to the N-S solvers, the lattice Boltzmann method has achieved a great success... computational domain All the numerical computations are easily performed in the computational domain The most popular boundary conforming method is perhaps the arbitrary Lagrangian-Eulerian (ALE) approach (Hirt et al 1974; Hu et al 2001; Anderson et al 2004; Chew et al 2006) In many applications, the unstructured finite element (FE) mesh is employed in the ALE method In this category, a scheme named space-time... relaxation parameter φ Phase difference υ Kinematic viscosity Ω Computation domain Angular velocity vector ω , ωB Vorticity Abbreviations 2D Two-dimensional 3D Three-dimensional ALE Arbitrary Lagrangian-Eulerian DLM/FD Distributed Lagrange multiplier/fictitious domain DKT Drafting, kissing and tumbling IBM Immersed boundary method IB-LBM Immersed boundary- lattice Boltzmann method IIM Immersed interface... method (IFEM) was proposed by Wang and Liu (2004) and Zhang et al (2004) By applying the finite element technique to both fluid and object domains, the immersed body can be handled more appropriately and accurately On the other hand, the 10 Chapter 1 Introduction accuracy of IBM results can also be improved by incorporating the adaptive mesh refinement technique (Roma et al 1999) Another drawback of. .. boundary method The immersed boundary method was first proposed by Peskin (1977) to study the cardiac mechanics and associated blood flows In IBM, the N-S equations for flow field are discretized on the fixed Cartesian (Eulerian) mesh, and the boundary is represented by a set of Lagrangian points The basic idea of IBM is to treat the immersed boundary as deformable with high stiffness A small distortion of. .. Introduction point 1.2.2.3 Advantages and disadvantages of IBM The major advantage of IBM is its simplicity and easy implementation This is attributed to the decoupling of the solution of governing equation with the boundary In other words, the governing equation can always be solved on a regular domain without consideration of embedded boundary in the flow field The effect of boundary on the flow field... macroscopic limit, leads to the N-S equations when the underlying lattice guarantees isotropy However, the LGCA suffers from some drawbacks such as large statistical noise, non-Galilean invariance, unphysical velocity-dependent pressure and large numerical viscosities These shortcomings have greatly hampered its development as a good model in practical applications To overcome the drawbacks of LGCA,... the boundary along the horizontal and vertical mesh lines The approach is very simple However, it only has the first-order of accuracy and the computed forces at the boundary have some oscillations The reason may be that the linear relationship is applied along the horizontal/vertical mesh lines and the smooth Dirac delta function is not used Its implementation process is more complicated than 11 Chapter... computational approaches in the category of non -boundary conforming method for accurate simulation of flows over stationary and/ or moving objects In the present work, the flow field is obtained by the lattice Boltzmann method (LBM) In the following, we will firstly give a literature review on the non -boundary conforming method and the LBM, and then describe the objectives of this research and layout of . 25 Cartesian Mesh 2.1 Standard LBM 26 2.2 Taylor Series Expansion and Least Squares-base Lattice Boltzmann 30 Method 2.3 Efficient LBM on Non-Uniform Cartesian Mesh 33 2.4 Accuracy Analysis. Comparison of forces c f and w f for flows over an array of 83 circular cylinders placed at the middle of a straight channel Table 3.2 Comparison of drag coefficient, length of bubbles and separation. Schematic diagram of a neutrally buoyant particle in a linear 151 shear flow Figure 4.21 Comparison of lateral migration of particle with previous data 151 Figure 4.22 Comparison of particle