Development of immersed boundary methods for isothermal and thermal flows 9

  Chapter Conclusions and Recommendations     Chapter Conclusions and Recommendations 9.1 Conclusions In this thesis, several novel IBM solvers have been developed for fluid and thermal flows involving complex geometries or moving objects/boundaries: one for two dimensional incompressible viscous flows and two for thermal flows respectively with Dirichlet and Neumann temperature conditions. Additionally, a new version of boundary condition-enforced IBM in the framework of NS solver is presented. Their abilities for dealing with fluid and thermal problems have been well demonstrated by applications to various fluid and thermal flows with geometric and dynamic complexities. Note that the four methods proposed employ Dirac delta function in modeling the immersed boundaries. In other words, the immersed boundaries are assumed to be have finite thickness. Therefore they are truly immersed boundary methods. The critical issue of the IBM lies in the accurate evaluation of body forces. The conventional IBM generally calculates body forces explicitly, so the flow penetration to the surface of the immersed objects happens. On the other hand, 308      Chapter Conclusions and Recommendations the velocity correction-based IBM, which was originally developed within the framework of LBM solver, has proven to be effective in exactly enforcing the no-slip condition on the immersed boundary. In this regard, the present thesis firstly extends the method to a Navier-Stokes (NS) solver-based version where the popular primitive variable formulation is solved. Combined with the fractional step technique, the body force in the modified momentum equation is implicitly determined so that the no-slip condition on the immersed boundary is accurately satisfied. The good performance of the new IBM is extensively verified, first through the classical problems of flow over a single stationary circular cylinder and flow interference between two side-by-side circular cylinders, and then by the moving boundary cases such as vortex-structure interaction around a transversely oscillating cylinder and vortex-induced-vibration of an elastically mounted circular cylinder. Most IBM solvers are established under the framework of pressure-velocity formulation. For incompressible flows, the pressure-velocity formulation suffers from some difficulties and is usually rather expensive to solve. In contrast, the stream function-vorticity formulation has been recognized to be more efficient for two dimensional incompressible viscous flows. Inspired by the merits of the stream function-vorticity formulation and IBM techniques, a stream function-vorticity formulation-based IBM solver is proposed, in which the immersed object is modeled as localized vorticity sources and plays the 309      Chapter Conclusions and Recommendations role of vorticity corrections in the stream function equation. The main idea of the method is to accurately satisfy the governing equation and boundary condition, which is realized through velocity correction and vorticity correction procedures. The velocity corrections, in the present model, are evaluated in the principle of guaranteeing the no-slip condition on the immersed boundary, i.e., the velocity at the boundary (Lagrangian) point interpolated from the physical velocity at the Eulerian points equals to the given boundary velocity. The vorticity corrections, on the other hand, are directly evaluated from the first-order derivatives of velocity corrections by two proposed approximation methods. The whole fluid solver and the two vorticity correction methods were first validated by their applications to a stationary boundary problem of flow over a circular cylinder. The obtained flow characteristics agree very well with the published data in the literature both qualitatively and quantitatively. Additionally, no streamline penetration appears on the cylinder surface, implying that the boundary condition is accurately satisfied. Then the proposed solver was applied to examples of flow over a left moving circular cylinder and flow over an inline oscillating circular cylinder, to further examine its capability in handling moving boundary problems. Furthermore, a fluid-structure interaction problem, sedimentation of a circular particle between two parallel walls, was examined. All the obtained numerical results for the considered moving boundary cases were in very good agreement with the previous reported ones in the literature, indicating the 310      Chapter Conclusions and Recommendations promising potential of the stream function-vorticity formulation-based IBM for solving two dimensional incompressible viscous flows involving complex geometries and moving boundaries. To extend the IBM for heat transfer problems which are suffering from complex or moving/deforming boundaries, two thermal IBM solvers were developed. Firstly, a boundary condition-enforced IBM solver was suggested for heat transfer problems with Dirichlet-type boundary condition, where the heated immersed boundary is modeled as a set of heat sources added to the energy equation as the source term. All the previous IBM solvers treated the heat source term explicitly and no mechanism was implemented to enforce the boundary condition for temperature. Thus the prescribed temperature on the immersed boundary was not accurately satisfied and the accuracy of numerical results was negatively affected. In the present boundary condition-enforced IBM, the heat source/sink term is considered as unknown and determined implicitly such that the energy equation and the corresponding thermal boundary condition can be accurately satisfied. Furthermore, the critical issue of how to effectively calculate the average Nusselt number in IBM was properly addressed, as an additional contribution. Numerical experiment on accuracy analysis showed a second-order accuracy of the proposed IBM solver. The present method and proposed techniques to compute the Nusselt number were then validated by both forced and natural convections where the obtained 311      Chapter Conclusions and Recommendations numerical results compared considerably well with available data in the literature, implying that the suggested thermal solver and Nusselt number evaluation techniques provide a wonderful tool for thermal problems with Dirichlet condition. Secondly, a heat flux correction-based IBM was proposed for thermal flows with prescribed heat flux condition. Note that this is the first time IBM is applied to solve thermal flow problems with Neumann-type boundary conditions. In the proposed solver, the heated immersed boundary was regarded as localized heat sources whose evaluations and contributions were considered through a heat flux correction procedure. To compensate the differences between the calculated heat fluxes and the prescribed ones, boundary heat sources arising from the heat flux differences were introduced which were then distributed to the surrounding fluid as volumetric heat sources to correct the temperature field. The whole process was intelligible and easy to implement. The present solver, through a numerical analysis on a model problem of heat conduction, was recognized to be of second order in accuracy. Then its capability and efficiency were validated by well-established examples like forced convection over a stationary heated circular cylinder and natural convection in a horizontal concentric and eccentric annulus between two circular cylinders. The numerical results obtained matched very well with the reported ones, showing the high potential of the proposed IBM solver for problems with Neumann conditions. 312      Chapter Conclusions and Recommendations Subsequently, the developed methods were applied to various two- and three-dimensional fluid and thermal flows to further check their capabilities for handling geometric complexity and moving boundaries. In two dimensions, insect hovering flight where the insect wing undergoes a prescribed harmonic translational and rotational motion near or away from the ground was firstly simulated, which was followed by particulate flows where the motion of the involved particle is not given prior but coupled and mutually determined with the surrounding fluid. Obtained results showed that both types of moving boundary problems are correctly predicted and good agreements with available experimental and numerical results were observed. With the confidence achieved from the above validations, the methods were utilized to study the forced convective heat transfer from a transversely oscillating circular cylinder in the wake of a stationary one, where it was found that the flow behaviors and heat transfer characteristics were greatly affected by the mutual influences of spacing and downstream cylinder excitation. After successfully verifying their exciting performance in two-dimensional applications, the proposed temperature correction-based and heat flux correction-based IBM solvers are examined for their capabilities in dealing with three-dimensional thermal flows involving complex or moving boundaries. Examples related to forced convective heat and mass transfer from stationary or streamwise rotating spheres in uniform cross flow were simulated. 313      Chapter Conclusions and Recommendations The good agreement between the obtained results and the published ones in the literature clearly shows the easy implementation and accurate natures of the proposed IBM solvers for three-dimensional heat transfer problems. Finally, the simulation of complex moving boundary flows was conducted, where they were either with complex geometry or in complex movement, or both. One is the solid finite-span foil heaving and pitching in the air, and the other is the flexible-body fish swimming in the water. Both problems were accurately predicted as compared to the available numerical results, showing the great potential of the immersed boundary method for three-dimensional practical flow problems with complex solid or flexible moving boundaries. 9.2 Recommendations for the Future Work The temperature correction-based IBM is proposed to improve the accuracy of the existing IBM solvers for heat transfer problems subject to Dirichlet-type boundary conditions and the developed heat flux correction-based IBM is considered as an initial attempt to extend the IBM for solving problems with Neumann-type boundary conditions. While validation tests indicate their precious features such as accurate, efficient, simple and easy implementation, they can be extensively adopted to solve a large number of thermal flows with complex or moving geometries. However, both thermal solvers are limited to the cases where the temperature condition on the immersed boundary, either in Dirichlet or Neumann type, is prescribed in advance. From the viewpoint of 314      Chapter Conclusions and Recommendations methodology development, it may be desirable to generalize the solvers to more general cases where the temperature conditions change with time and are to be determined during the solution process. From the viewpoint of practical and engineering applications, the present study conducted simulations for three-dimensional forced and natural convective flows with complex geometries (Chapter 7) and three-dimensional moving boundary isothermal problems with prescribed motion (Chapter 8). Fluid and thermal problems with high dynamic complexity, such as mix convections, three-dimensional moving boundary thermal flows or fully fluid-structure interactions, are not included in the present thesis. All these would motivate the future investigations. 315      Reference     Reference Achenbach E., Vortex shedding from spheres, J. Fluid Mech., 62 (1974) 209-221. Ahmad R.A., Qureshi Z.H., Laminar mixed convection from a uniform heat flux horizontal cylinder in a crossflow, J. Thermophys Heat Transf., (1992) 277-287. Ahmed G.R., Yovanovich M.M., Approximate analytical solution of forced convection heat transfer from isothermal spheres for all Prandtl numbers, ASME, J. Heat Transf., 116 (1994) 838–843. Anagnostopoulos P., Numerical study of the flow past a cylinder excited transversely to the incident stream. Part 1: lock-in zone, hydrodynamic forces and wake geometry, J. Fluid Struct., 14 (2000) 819-851. Anderson J.M., Streitlien K., Barrett D.S., Truantafyllou M.S., Oscillating foils of high propulsive efficiency, J. Fluid Mech., 360 (1998) 41–72. Arthurs K.M., Moore L.C., Peskin C.S., Pitman E.B., Layton H.E., Modeling Arteriolar Flow and Mass Transport Using the Immersed Boundary Method, J. Comp. Phys., 147 (1998) 402-440. Barrett D.S., Triantafyllon M.S., Yue D.K.P., Grosenbaugh M.A., Wolfgang M., Drag Reduction in fish-like locomotion. J. Fluid Mech., 392 (1999) 316      Reference 183-212. Beyer Jr. R.P., A computational model of the cochlea using the immersed boundary method, J. Comp. Phys., 98 (1992) 145-162. Bharti R.P., Chahabra R.P., Eswaran V., A numerical study of the steady forced convection heat transfer from an unconfined circular cylinder, Heat Mass Transf., 43 (2007) 639–648. Birch J.M., Dickinson M.H., Spanwise flow and the attachment of the leading-edge vortex on insect wings, Nature, 412 (2001) 729-733. Bishop R.E.D., Hassan A.Y., The lift and drag forces on a circular cylinder oscillating in a flowing fluid, Proceedings of the Royal Society of London, Series A. 277 (1964) 51-75. Blackburn H.M., Karniadakis G.E., Two- and three-dimensional simulations of vortex-induced vibration of a circular cylinder, In 3rd Intl. Offshore & Polar Engng Conf., Singapore, 7150-720. Blondeaux P., Fornarelli F., Guglielmini L., Triantafyllou M.S., Verzicco R., Numerical experiments on flapping foils mimicking fish-like locomotion, Phys. Fluids, 17 (2005) 113601. Borazjani I., Sotiropoulos F., Numerical investigation of the hydrodynamics of carangiform swimming in the transitional and inertial flow regimes. J. Exp. Biol., 211 (2008) 1541-1558. Borazjani I., Sotiropoulos F., Numerical investigation of the hydrodynamics of anguiliform swimming in the transitional and inertial flow regimes. J. Exp. 317      Reference Kim Y.S., Peskin C.S., 2-D parachute simulation by the immersed boundary method, SIAM J. SCI. Comput., 28 (2006) 2294-2312. Kim Y.S., Peskin C.S., Penalty immersed boundary method for an elastic boundary with Mass, Phys. Fluids., 19 (2007). Kim Y.S., Peskin C.S., 3-D parachute simulation by the immersed boundary method, Computers & Fluids, 38 (2009) 1080-1090. Kim Y.S., Seol Y.C., Numerical simulations of two-dimensional wet foam by the immersed boundary method, Computers & Structures, 122 (2013) 259-269. Koopmann G.H., The vortex wakes of vibrating cylinders at low Reynolds number, J. Fluid Mech., 27 (1967) 501-512. Kostic Z.G., Oka S.N., Fluid flow and heat transfer with two cylinders in cross flow, Int. J. Heat Mass Transfer, 15 (1972) 279-299. Kuehn T.H., Goldstein R.J., An experimental and theoretical study of natural convection in the annulus between horizontal concentric cylinders, J. Fluid Mech., 74 (1976) 695–719. Lai M.C., Peskin C.S., An immersed boundary method with formal second-order accuracy and reduced numerical viscosity, J. Comp. Phys., 160 (2000) 705–719. Lange C.F., Durst F., Breuer M., Momentum and heat transfer from cylinders in laminar crossflow at 10−4 ≤ Re ≤ 200 , Int. J. Heat Mass Transf., 41(1998) 3409–3430. 328      Reference Le D.V., Khoo B.C., Lim K.M., An implicit-forcing immersed boundary method for simulating viscous flows in irregular domains, Comput. Methods Appl. Mech. Engrg., 197 (2008) 2119-2130. Lee D.S., Ha M.Y., Yoon H.S., Balachandar S., A numerical study on the flow patterns of two oscillating cylinders, J. Fluids Struct., 25 (2009) 263-283. Lee J., Kim J., Choi H., Yang K., Sources of spurious force oscillations from an immersed boundary method for moving-body problems, J. Comput. Phys., 230 (2011) 2677-2695. Lee K.C., Aerodynamic interaction between two spheres at Reynolds numbers around 10 , Aerosp. Q., 30 (1979) 371-385. Le D.V., Khoo B.C., Peraire J., An immersed interface method for viscous incompressible flows involving rigid and flexible boundaries, J. Comput. Phys., 220 (2006) 109-138. LeVeque R.J., Li Z.L., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal, 31 (1994) 1019-1044. LeVeque R.J., Lin Z.L., Immersed interface methods for Stokes flow with elastic boundaries or surface tension, SIAM J. Sci. Comput., 18 (1997) 709-735. Li J., Chambarel A., Donneaud M., Martin R., Numerical study of laminar flow past one and two circular cylinders, Comput. Fluids, 19 (1991) 155-170. Li Y., Jung E., Lee W., Lee H.G., Kim J., Volume preserving immersed 329      Reference boundary methods for two-phase fluid flows, Int. J. Numer. Meth. Fluids, 69 (2012) 842-858. Liao C.C., Lin C.A., Simulations of natural and forced convection flows with moving embedded object using immersed boundary method, Comput. Methods Appl. Mech. Engrg., 213 (2012) 58-70. Lighthill M.J., Hydrodynamics of aquatic animal propulsion, Annu. Rev. Fluid Mech., (1969) 413-445. Lima E Silva L.F., Silveira-Neto A., Damasceno J.J.R., Numerical simulation of two-dimensional flows over a circular cylinder using the immersed boundary method, J. Comput. Phys., 189 (2003) 351-370. Lin Z.L., Lai M.C., The immersed interface method for the Navier-Stokes equations with singular forces, J. Comput. Phys., 171 (2001) 822-842. Lindsey C.C., Form, function and locomotory habits in fish, Fish Physiology, New York: Academic Press, (1978) 1-100. Liu C., Zheng X., Sung C.H., Preconditioned multigrid methods for unsteady incompressible flows, J. Comput. Phys., 139 (1998) 35-57. Liu H., A numerical study on jet-stream propulsion of oscillating bodies, J. Soc. Naval Architects Jpn., 178 (1995) 101. Liu H., Ellington C.P., Kawachi K., Van Den Berg C., Willmott A.P., A computational fluid dynamic study of hawkmoth hovering, J. Exp. Biol., 201 (1998) 461-477. Liu H., Kawachi K., A numerical study of undulatory swimming, J. Comp. 330      Reference Phys., 155 (1999) 223-247. Liu H., Wassersug R., The three-dimensional hydrodynamics of tadpole locomotion, J. Exp. Biol., 200 (1997) 2807-2819. Liu H., Wassersug R.J., Kawachi K., A computational fluid dynamics study of tadpole swimming, J. Exp. Biol., 199 (1996) 1245-1260. Ljachowski D.N., J. techn. Phys., Leningrad, 999 (1940). Luo K., Wang Z., Fan J., A modified immersed boundary method for simulations of fluid-particle interactions, Comput. Meth. Appl. Mech. Eng., 197 (2007) 36-46. Luthander S., Rydberg A., Experimental investigation of the drag of a rotating body with the wind direction parallel to the axis, Phys. Z., 36 (1935) 552-558. Luo K., Wang Z.L., Fan J.R., A modified immersed boundary method for simulations of fluid-particle interactions, Comput. Methods Appl. Mech. Engrg., 197 (2007) 36-46. Magarvey R.H., Bishop, R.L., Transition ranges for three-dimensional wakes, Can. J. Phys., 39 (1961) 1418-1422. Magnaudet J., Rivero M., Fabre J., Accelerated flows past a rigid sphere or a spherical bubble. Part 1. Steady straining flow, J. Fluid Mech., 284 (1995) 97-135. Mahir N., Altac Z., Numerical investigation of convective heat transfer in unsteady flow past two cylinders in tandem arrangement, Int. J. Heat Fluid Flows, 29 (2008) 1309-1318. 331      Reference Majumdar S., Laccarino G., Durbin P., RANS solvers with adaptive structured boundary non-conforming grids, Center for Turbulence Research Annual Research Briefs, (2001). Maury B., Direct simulations of 2D fluid-particle flows in biperiodic domains, J. Comput. Phys., 156 (1999) 325-351. McQueen D.M., Peskin C.S., Heart simulation by an immersed boundary method with formal second-order accuracy and reduced numerical viscosity, Proceedings of the International Conference on Theoretical and Applied Mechanics, (2000). Miller L.A., Peskin C.S., A computational fluid dynamics study of ‘clap and fling’ in the smallest insects, J. Exp. Biol., 208 (2005) 195-212. Mittal R., Dong H., Bozkurttas M., Najjar F.M., Vargas A., Von Loebbecke A., A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries, J. Comput. Phys., 227 (2008) 4825-4852. Mittal R., Iaccarino G., Immersed Boundary Methods, Ann. Rev. Fluid Mech., 37 (2005) 239-261. Mizushima J., Ino Y., Stability of flows past a pair of circular cylinders in a side-by-side arrangement, J. Fluid Mech., 595 (2008) 491-507. Mohd-Yusof J., Combined immersed boundaries/B-Splines methods for simulations of flows in complex geometries, CTR Annual Research Briefs, NASA Ames/Stanford University, 1997. Morse T.L., Williamson C.H.K., Fluid forcing, wake modes, and transitions 332      Reference for a cylinder undergoing controlled oscillations, J. Fluids Struct., 25 (2009) 697-712. Mori Y., Peskin C.S., Implicit second-order immersed boundary methods with boundary mass, Comput. Methods Appl. Mech. Engrg., 197 (2008) 2049-2067. Moukalled F., Acharya S., Natural convection in the annulus between concentric horizontal circular and square cylinders, Jo. Thermop. Heat Transfer, 10 (1996) 524 –531. Muller U.K., Van Den Heuvel B.L.E., Stamhuls J.J., Vldeler J.J., Fish foot prints: morphology and energetic of the wake behind a continuously swimming mullet, J. Exp. Biol., 200 (1997) 2893-2906. Nachtigall W., Insects in flight: A glimpse behind the scene in biophysical research, McGraw-Hill, New York, 1974. Natarajan R., Acrivos A., The instability of the steady flow past spheres and disks, J. Fluid Mech., 254 (1993) 323-344. Nauen J.C., Lauder G. V., Hydrodynamics of caudal fin locomotion by chub mackerel, Scomber japonicas, J. Exp. Biol., 205 (2002) 1709-1724. Niu X.D., Shu C., Chew Y.T., Peng Y., A momentum exchange-based immersed boundary-lattice Boltzmann method for simulating incompressible viscous flows, Phys. Lett. A., 354 (2006) 173-182. Noor D.Z., Chern M.J., Horng T.L., An immersed boundary method to solve fluid-solid interaction problems, Comput. Mech., 44 (2009) 447-453. 333      Reference Ongoren A., Rockwell D., Flow structure from an oscillating cylinder, Part 1: Mechanisms of phase shift and recovery in the near wake, J. Fluid Mech., 191 (1988) 197-223. Pan D., Shen T.T., Computation of incompressible flows with immersed bodies by a simple ghost cell method, Int. J. Numer. Meth. Fluids, 60 (2009) 1378-1401. Paravento F., Pourquie M.J., Boersma B.J., An immersed boundary method for complex flow and heat transfer, Flow Turbulence Combust., 80 (2008) 187-206. Park H.G., Gharib M., Experimental study of heat convection from stationary and oscillating circular cylinder in cross flow, J. Heat Transf., 123 (2001) 51-62. 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 (2000) 1509-1524. Peskin C.S., Flow patterns around heart valves: a numerical method, J. Comp. Phys., 10 (1972) 252–271. Peskin C.S., The immersed boundary method, Acta Numerica (2002) 479-517. Polyanin A.D., Kutepov A.M., Vyazmin A.V., Kazenin D.A., Hydrodynamics, mass and heat transfer in chemical engineering, Taylor and Francis, London, (2002). Pottebaum T. S., Gharib M., Using oscillations to enhance heat transfer for a 334      Reference circular cylinder, Int. J. Heat Mass Transf., 49 (2006) 3190-3210. Prahl L., Holzer A., Arlov D., Revstedt J., Sommerfeld M., Fuchs L., On the interaction between two fixed spherical particles, Int. J. Multiphase Flow, 33 (2007) 707-725. Prahl L. Jadoon A., Revstedt J., Interaction between two spheres placed in tandem arrangement in steady and pulsating flow, Int. J. Multiphase Flow, 35 (2009) 963-969. Ravoux J.F., Nadim A., Haji-Hariri H., An embedding method for bluff body flows: interactions of two side-by-side cylinder wakes, Theoret. Comput. Fluid Dynamics, 16 (2003) 433-466. Rogers S.E. and Kwak D., Upwind differencing scheme for the time-accurate incompressible Navier–Stokes equations, AIAA J., 28 (1990) 253-262. Roshko A., Experiments on the flow pass a circular cylinder at very high Reynolds number, J. Fluid Mech., 10 (1961) 345-356. Rowe, P.N., Henwood, G.A., Drag forces in a hydraulic model of a fluidised bed–part 1, Trans. Inst. Chem. Eng., 39 (1961) 43–54. Russell D., Wang ZJ, A Cartesian grid method for modeling multiple moving objects in 2D incompressible viscous flow, J. Comput. Phys., 191(2003) 177-205. Saiki E.M., Biringen S., Numerical simulation of a cylinder in uniform flow: application of a virtual boundary method, J. Comp. Phys., 123(1996) 450–465. Sane S.P., Dickinson M.H., The control of flight force by a flapping wing: lift 335      Reference and drag production, J. Exp. Biol., 204 (2001) 2607-2626. Sane S.P., Dickinson M.H., The aerodynamic effects of wing rotation and a revised quasi-steady model of flapping flight, J. Exp. Biol., 205 (2002) 1087-1096 Sakamoto H., Haniu H., A study on vortex shedding from spheres in a uniform flow, Trans. ASME: J. Fluids Engng., 112 (1990) 386-392. Schlichting H., Boundary Layer Theory, 7th Edn. McGraw-Hill, (1979). Shao X.M., Pan D.Y., Deng J., Yu Z.S., Numerical studies on the propulsion and wake structures of finite-span flapping wings with different aspect ratios, J. Hydrodynamics, Ser. B, 22 (2010) 147-154. Shirayama S., Flow past a sphere: topological transitions of the vorticity field, AIAA J., 30 (1992) 349-358. Shu C. KHA W., Numerical simulation of natural convection in a square cavity by SIMPLE-generalized differential quadrature method, Computational Fluids., 31 (2002) 209-226. Shu C., Liu N.Y., Chew Y.T., A novel immersed boundary velocity correction-lattice Boltzmann method and its application to simulate flow past a circular cylinder, J. Comp. Phys., 226 (2007) 1607–1622 Shu C., Zhu Y.D., Efficient computation of natural convection in a concentric annulus between an outer square cylinder and an inner circular cylinder, Int. J. Numer. Methods Fluids, 38 (2002) 429-445. Shukla R.K., Tatineni M., Zhong X., Very high-order compact finite difference 336      Reference schemes on non-uniform grids for incompressible Navier-Stokes equations, J. Comput. Phys., 224 (2007) 1064-1094. Shun T., Taku N., Kota F., A numerical scheme based on an immersed boundary method for compressible turbulent flows with shocks: application to two-dimensional flows around cylinders, J. Appl. Math., (2014). Soares A.A., Ferreira J.M., Chhabra R.P., Flow and forced convection heat transfer in crossflow of non-Newtonian fluids over a circular cylinder, Ind. Eng. Chem. Research, 44 (2005) 5815–5827. Sotiropoulos F., Yang X.L., Immersed boundary methods for simulating fluid-structure interaction, Progress in Aerospace sciences, 65 (2014) 1-21. Sparrow E.M., Abraham J.P., Tong J.C.K., Archival correlations for average heat transfer coefficients for non-circular and circular cylinders and for spheres in crossflow, Int. J. Heat Mass Transf., 47 (2004) 5285–5296. Sudhakar Y., Vengadesan S., Flight force production by flapping insect wings in inclined stroke plane kinematics, Computers & Fluids, 39 (2010) 683-695. Sumner D., Two circular cylinders in cross-flow: a review, J. Fluids Struct., 26 (2010) 849-899. Sun M., Tang J., Unsteady aerodynamic force generation by a model fruit fly wing in flapping motion, J. Exp. Biol., 205 (2002a) 55-70. Tal R., Lee D.N., Sirignano W.A., Heat and momentum transfer around a pair of spheres in viscous flow, Int. J. Heat Mass Transf., 27 (1984) 1953-1962. Taneda S., Experimental investigation of the wake behind a sphere at low 337      Reference Reynolds numbers, J. Phys. Soc. Japan, 11 (1956) 1104-1108. Temam R., Sur l’approximation de la solution des equations de Navier-Stokes par la method des fractionnarires II, Arch. Rational Mech. Anal., 33 (1969) 377-385. Tomboulides A.G., Direct and large-eddy simulation of wake flows: flow past a sphere. PhD Thesis, Princeton University, (1993). Triantafyllou M.S., Triantafyllou G.S., An efficient swimming machine, Sci. Am., 272 (1995) 64-70. Tritton D.J., Experiments on the flow past a circular cylinder at low Reynolds numbers, J. Fluid Mech., (1959) 547-567. Tseng Y.H., Ferziger J.H., A ghost-cell immersed boundary method for flow in complex geometry, J. Comput. Phys., 192 (2003) 593-623. Tsuji Y., Morikawa Y., Terashima K., Fluid-dynamic interaction between two spheres, Int. J. Multiphase Flow, (1982) 71-82. Tytell E. D., Lauder G.V., The hydrodynamics of eel swimming. 1. Wake structure, J. Exp. Biol., 207 (2004) 1825-1841. Udaykumar H.S., Kan H.C., Shyy W., Tran-Son-Tay R., Multiphase dynamics in arbitrary geometries on fixed cartesian grids, J. Comput. Phys., 137 (1997) 366–405. Uhlmann M., An immersed boundary method with direct forcing for the simulation of particulate flows, J. Comp. Phys., 209 (2005) 448–476. Videler J.J., Hess F., Fast continuous swimming of two pelagic predators, 338      Reference saithe (Pollachius virens) and mackerel (Scomber scombrus): A kinematic analysis, J. Exp. Biol., 109 (1984) 209-228. Videler J.J., fish swimming, London: Chapman and Hall, (1993). Von Ellenrieder K.D., Parker K., Soria J., Flow Structures behind a Heaving and Pitching Finite-Span Wing, J. Fluid Mech., 290 (2003) 129-138 Wan D., Turek S., Direct numerical simulation of particulate flow via multigrid FEM techniques and the fictitious boundary method, Int. J. Numer. Meth. Fluids, 51 (2006) 531-566. Wang W.Q., Yin R., Hao D.W., Yan Y., Modeling and simulation of fish-like swimming in a straight-line swimming state using immersed boundary method, Advances in mechanical engineering, (2012). Wang X.Y., Yeo K.S., Chew C.S., Khoo B.C., A SVD-GFD scheme for computing 3D incompressible viscous fluid flows, Computers & Fluids, 37 (2008) 733-746. Wang Z., Fan J., Kun L., Cen K., Immersed boundary method for the simulation of flows with heat transfer, Int. J. Heat Mass Transf., 52 (2009) 4510-4518. Wang Z., Fan J., Luo K., Combined multi-direct forcing and immersed boundary method for simulating flows with moving particles, Int. J. Multiphase Flow, 34(2008) 283–302. Wang Z.L., Fan J.R., Cen K.F., Immersed boundary method for the simulation of 2D viscous flow based on vorticity-velocity formulations, J. Comput. Phys., 339      Reference 228 (2009) 1504-1520. Wang Z.L., Fan J.R., Luo K., Combined multi-direct forcing and immersed boundary method for simulating flows with moving particles, Int. J. Multiphase Flow, 34 (2008) 283-302. Wang Z.J., Two dimensional mechanism of hovering, Phys. Rev. Lett., 85 (2000a) 2216-2219. Wang Z.J., The role of drag in insect hovering, J. Exp. Biol., 207 (2004) 4147-4155. Wang Z.J., Birch J.M., Dickinson M.H., Unsteady forces and flows in low Reynolds number hovering flight: two-dimensional computations vs robotic wing experiments, J. Exp. Biol., 207 (2004) 449-460. Weis-Fogh T., Quick estimates of flight fitness in hovering animals, including novel mechanisms for lift production, J. Exp Biol., 59 (1973) 169–230 Weis-Fogh T, Jensen M., Biology and physics of locust flight. I. basic principles in insect flight. a critical review, Proc. R. Soc. B., 239 (1956) 415–458. Whitaker S., Forced convection heat transfer correlations for flow in pipes, past flat plates, single cylinders, single spheres, and flow in packed beds and tube bundles, AIChE J., 18 (1972) 361–371. White F.W., Viscous Fluid Flow, McGraw-Hill, New York, (1974). Williamson C.H.K., Roshko A., Vortex formation in the wake of an oscillating cylinder, J. Fluids Struct., (1988) 355-381. 340      Reference Wolfgang M.J., Anderson J.M., Grosenbaugh M.A., Yue D.K., Triantafyllou M.S., Near-body flow dynamics in swimming fish, J. Exp. Biol., 202 (1999) 2303-2327. Wu C.J., Wang L., Numerical simulations of self-propelled swimming of 3D bionic fish school, Sicence in China Series E: Technol. Science, 52 (2009) 658-669. Wu J., Shu C., Implicit velocity correction-based immersed boundary-lattice Boltzmann method and its applications, J. Comp. Phys., 228 (2009) 1963-1979. Wu J., Shu C., Particulate flow simulation via a boundary condition-enforced immersed boundary-lattice Boltzmann scheme, Commun. Comput. Phys., (2010) 793-812. Xia Z.H., Connington K.W., Rapaka S., Yue P.T., Feng J.J., Chen S.Y., Flow patterns in the sedimentation of an elliptical particle, J. Fluid Mech., 625 (2009) 249-272. Xu S. and Wang Z.J., An immersed interface method for simulating the interaction of a fluid with moving boundaries, J. Comput. Phys., 216 (2006) 454-493. Xu S., Wang Z.J., Systematic Derivation of Jump Conditions for the Immersed Interface Method in Three-Dimensional Flow Simulation, SIAM J. Sci. Comput., 27 (2006) 1948-1980. Yang J., Balaras E., An embedded-boundary formulation for large-eddy 341      Reference simulation of turbulent flows interacting with moving boundaries, J. Comput. Phys., 215 (2006) 12–40. Yang J.M., Stern F., A simple and efficient direct forcing immersed boundary framework for fluid-structure interactions, J. Comp. Phys., 231 (2012) 5029-5061. Yang X.F., Zheng Charlie Z.Q., Nonlinear spacing and frequency effects of an oscillating cylinder in the wake of a stationary cylinder, Phys. Fluids, 22 (2010) 043601. Ye T., Mittal R., Udaykumar H.S., Shyy W., An accurate cartesian grid method for viscous incompressible flows with complex immersed boundaries, J. Comput. Phys., 156 (1999) 209-240. Yoo J.S., Dual free-convective flows in a horizontal annulus with a constant heat flux wall, Int. J. Heat Mass Transf., 46 (2003) 2499-2503. Yoon D.H., Yang K.S., Flow-induced forces on two near-by spheres, Phys. Fluids, 19 (2007) 098103. Young D.L., Jan Y.J., Chiu C.L., A novel immersed boundary procedure for flow and heat simulations with moving boundary, Computers & Fluids, 38 (2009) 1145-1159. Yu Z.S., Shao X.M., Wachs A., A fictitious domain method for particulate flows with heat transfer, J. Comput. Phys., 217 (2006) 424-452. Zdravkovich M.M., Modification of vortex shedding in the synchronization range, ASME J. Fluids Eng., 104 (1982) 513-517. 342      Reference Zhang N., Zheng Z.C., An improved direct-forcing immersed-boundary method for finite difference applications, J. Comput. Phys., 221 (2007) 250-268. Zhang N., Zheng Z.C., Eckels S., Study of heat-transfer on the surface on a circular cylinder in flow using an immersed-boundary method, Int. J. Heat Fluid Flow, 29 (2008) 1558-1566. Zhou C.H., Shu C., A local domain-free discretization method for simulation of incompressible flows over moving bodies, Int. J. Numer. Meth. Fluids, 66 (2011) 162-182. Zhou C.H., Shu C., simulation of self-propelled anguiliform swimming by local domain-free discretization method, Int. J. Numer. Meth. Fluids, 69 (2012) 1891-1906. Zhu C., Liang S.C., Fan L.S., Particle wake effects on the drag force of an interactive particle, Int. J. Multiphase Flow, 20 (1994) 117-129. Zhu L.D., Peskin C.S., Simulation of a flapping flexible filament in a flowing soap film by the immersed boundary method, J. Comput. Phys., 179 (2002) 452-468. Zou J.F., Ren A.L., Deng J., Study on flow past two spheres in tandem arrangement using a local mesh refinement virtual boundary method, Int. J. Numer. Meth. Fluids, 49 (2005) 465-488. 343    [...]... Phys., 155 ( 199 9) 223-247 Liu H., Wassersug R., The three-dimensional hydrodynamics of tadpole locomotion, J Exp Biol., 200 ( 199 7) 2807-28 19 Liu H., Wassersug R.J., Kawachi K., A computational fluid dynamics study of tadpole swimming, J Exp Biol., 199 ( 199 6) 1245-1260 Ljachowski D.N., J techn Phys., Leningrad, 99 9 ( 194 0) Luo K., Wang Z., Fan J., A modified immersed boundary method for simulations of fluid-particle... Iaccarino G., Immersed Boundary Methods, Ann Rev Fluid Mech., 37 (2005) 2 39- 261 Mizushima J., Ino Y., Stability of flows past a pair of circular cylinders in a side-by-side arrangement, J Fluid Mech., 595 (2008) 491 -507 Mohd-Yusof J., Combined immersed boundaries/B-Splines methods for simulations of flows in complex geometries, CTR Annual Research Briefs, NASA Ames/Stanford University, 199 7 Morse T.L.,... multi-direct forcing and immersed boundary method for simulating flows with moving particles, Int J Multiphase Flow, 34(2008) 283–302 Wang Z.L., Fan J.R., Cen K.F., Immersed boundary method for the simulation of 2D viscous flow based on vorticity-velocity formulations, J Comput Phys., 3 39     Reference 228 (20 09) 1504-1520 Wang Z.L., Fan J.R., Luo K., Combined multi-direct forcing and immersed boundary. .. ( 199 9) 2303-2327 Wu C.J., Wang L., Numerical simulations of self-propelled swimming of 3D bionic fish school, Sicence in China Series E: Technol Science, 52 (20 09) 658-6 69 Wu J., Shu C., Implicit velocity correction-based immersed boundary- lattice Boltzmann method and its applications, J Comp Phys., 228 (20 09) 196 3- 197 9 Wu J., Shu C., Particulate flow simulation via a boundary condition-enforced immersed. .. (20 09) 576- 592 Braza M., Chassaing P., Minh H.H., Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder, J Fluid Mech., 165 ( 198 6) 79- 130 Breugem W.P., A second-order accurate immersed boundary method for fully resolved simulations of particle-laden flows, J Comput Phys., 231 (2012) 44 69- 4 498 Buchholz J.H.J., Smits A.J., On the evolution of. .. Numer Heat Transfer, 49 (2006) 717–731 Dhole S.D., Chhabra R.P., Eswaran V., A numerical study on forced convection heat transfer from an isothermal and isoflux sphere in the steady symmetric flow regime, Int J Heat Mass Transf., 49 (2006) 98 4 -99 4 Dickinson M.H., The effects of wing rotation on unsteady aerodynamic performance at low Reynolds number, J Exp Biol., 192 ( 199 4) 1 79- 206 Dickinson M.H.,... 156 ( 199 9) 2 09- 240 Yoo J.S., Dual free-convective flows in a horizontal annulus with a constant heat flux wall, Int J Heat Mass Transf., 46 (2003) 2 499 -2503 Yoon D.H., Yang K.S., Flow-induced forces on two near-by spheres, Phys Fluids, 19 (2007) 098 103 Young D.L., Jan Y.J., Chiu C.L., A novel immersed boundary procedure for flow and heat simulations with moving boundary, Computers & Fluids, 38 (20 09) ... rotation and the aerodynamic basis of insect flight, Science, 284 ( 199 9) 195 4- 196 0 Dickinson M.H., Gätz K.G., Unsteady aerodynamic performance of model wings at low Reynolds numbers, J Exp Biol., 174 ( 199 3) 45-64 Ding H., Shu C., Yeo K.S., Xu D., Simulation of incompressible viscous flows past a circular cylinder by hybrid FD scheme and meshless least square-based finite difference method, Comput Methods. .. 384 ( 199 6) 626-6 29 Fadlun E., Verzicco R., Orlandi P., Mohd-Yusof J., Combined immersed- boundary finite-difference methods for three dimensional complex flow simulations, J Comp Phys., 161 (2000) 35–60 Feng J., Hu H.H., Joseph D.D., Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid Part 1 Sedimentation, J Fluid 321      Reference Mech., 261 ( 199 4) 95 –134... ( 199 7) 366–405 Uhlmann M., An immersed boundary method with direct forcing for the simulation of particulate flows, J Comp Phys., 2 09 (2005) 448–476 Videler J.J., Hess F., Fast continuous swimming of two pelagic predators, 338      Reference saithe (Pollachius virens) and mackerel (Scomber scombrus): A kinematic analysis, J Exp Biol., 1 09 ( 198 4) 2 09- 228 Videler J.J., fish swimming, London: Chapman and . performance at low Reynolds number, J. Exp. Biol., 192 ( 199 4) 1 79- 206. Dickinson M.H., Lehmann F.O., Sane S.P., Wing rotation and the aerodynamic basis of insect flight, Science, 284 ( 199 9) 195 4- 196 0 on forced convection heat transfer from an isothermal and isoflux sphere in the steady symmetric flow regime, Int. J. Heat Mass Transf., 49 (2006) 98 4 -99 4. Dickinson M.H., The effects of wing. locomotion. J. Fluid Mech., 392 ( 199 9)  Reference 317  183-212. Beyer Jr. R.P., A computational model of the cochlea using the immersed boundary method, J. Comp. Phys., 98 ( 199 2) 145-162. Bharti