DEVELOPMENT OF IMMERSED BOUNDARY METHODS FOR ISOTHERMAL AND THERMAL FLOWS REN WEIWEI (B. Eng., M. Eng., Nanjing University of Aeronautics and Astronautics, China) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2014 DECLARATION I hereby declare that the thesis is my original work and it has been written by me in its entirety. I have duly acknowledged all the sources of information which have been used in the thesis. This thesis has also not been submitted for any degree in any university previously. Ren Weiwei 2014 Acknowledgements First of all, I would like to express my deepest thanks to my supervisor Professor Shu Chang and Dr. Yang Wenming, for their continuous guidances, supervisions and enjoyable discussions throughout this work. In addition, I wish to thank the National University of Singapore for her supports of the research scholarship, the abundant library resources, and the advanced computing facilities which are essential to the completion of the work. The gratitude also goes to all the friends of the Fluid Dynamic Laboratory in NUS for their valuable assistances. Finally, I would like to thank all my family members for their endless love, support and encouragement. Ren Weiwei I   Table of Contents Acknowledgements I Table of Contents II Summary X List of Tables XIII List of Figures XV Nomenclature XXVI Chapter Introduction 1.1 Background of computational fluid dynamics 1.1.1 Limitations of traditional body-fitted method 1.1.2 The concept of non-body-conforming method 1.2 Non-body-conforming method 1.2.1 Sharp interface method 1.2.1.1 Ghost cell method 1.2.1.2 Cut-cell method 1.2.1.3 Immersed interface method 1.2.2 Diffuse interface method 10 1.2.2.1 Fictitious domain method 11 1.2.2.2 Immersed boundary method 12 II   1.3 Brief review of immersed boundary methods 1.3.1 Pressure-velocity formulation-based 14 immersed boundary method 1.3.2 Stream 14 function-vorticity formulation-based immersed boundary method 18 1.3.3 Applications of immersed boundary method 20 1.4 Objective of this thesis 24 1.5 Organization of this thesis 26 Chapter Governing Equations and Boundary Condition-Enforced Immersed Boundary Method 2.1 Governing equations 30 2.2 Solution procedure 31 2.3 Calculation of Predicted velocity field – Projection method 32 2.4 Evaluation of Body force 34 2.4.1 The Convectional IBM 34 2.4.1.1 Penalty force scheme 34 2.4.1.2 Feedback forcing scheme 35 2.4.1.3 Direct forcing scheme 36 2.4.2 Boundary condition-enforced IBM 37 2.5 Computational sequence 41 2.6 Results and Discussion 42 III   29 2.6.1 Flow over an isolated stationary circular cylinder 42 2.6.2 Flow interference between a pair of side-by-side circular cylinders 48 2.6.3 Flow around a transversely oscillating circular cylinder in a free-stream 51 2.6.4 Vortex-induced vibration of an elastically mounted circular cylinder in a free-stream 2.7 Conclusions 53 55 Chapter Stream Function-Vorticity Formulation-based Immersed Boundary Method 71 3.1 Methodology 72 3.1.1 Governing equations 72 3.1.2 Velocity correction procedure 75 3.1.3 Vorticity correction procedure 77 3.1.4 Computational sequence 80 3.2 Results and Discussion 80 3.2.1 Flow over a stationary circular cylinder 81 3.2.2 Flow over a left moving circular cylinder 83 3.2.3 Flow over an inline oscillating circular cylinder in a fluid at rest 84 3.2.4 Sedimentation of a single circular particle between two parallel IV   walls 86 3.3 Conclusions Chapter A Boundary 88 Condition-Enforced Immersed Boundary Method for Heat Transfer Problems with Dirichlet Conditions 100 4.1 Methodology 101 4.1.1 Governing equations 101 4.1.2 Temperature correction procedure 103 4.1.3 Computational sequence 106 4.2 Evaluation of Average Nusselt Number 107 4.2.1 Method 1: Direct evaluation of average Nusselt number from temperature correction at Eulerian points 109 4.2.2 Method 2: Direct evaluation of average Nusselt number from heat flux at Lagrangian points 4.3 Numerical Examples 110 110 4.3.1 Numerical analysis of spatial accuracy 111 4.3.2 Forced convection over a stationary isothermal circular cylinder 112 4.3.3 Natural convection in a concentric annulus between a square outer cylinder and a circular inner cylinder 4.4 Conclusions 119 V   115 Chapter An Efficient Immersed Boundary Method for Thermal Flow Problems with Heat Flux Boundary Conditions 128 5.1 Methodology 129 5.1.1 Governing equations 129 5.1.2 Heat Flux Correction Procedure 130 5.1.3 Computational Sequence 134 5.2 Numerical Examples 134 5.2.1 Numerical analysis of spatial accuracy 135 5.2.2 Forced convection over a stationary isoflux circular cylinder 135 5.2.3 Natural convection in a concentric horizontal cylindrical annulus between an outer isothermal cylinder and an inner isoflux cylinder 138 5.2.4 Natural convection in an eccentric horizontal cylindrical annulus between an outer isothermal cylinder and an inner isoflux cylinder 140 5.3 Conclusions 142 Chapter Applications of Developed IBM Solvers to Simulate Two-Dimensional Fluid and Thermal Flows 6.1 Unsteady insect hovering flight at low Reynolds numbers VI   153 153 6.1.1 Normal hovering mode 157 6.1.1.1 Normal hovering flight without ground effect 157 6.1.1.2 Normal hovering flight with ground effect 160 6.1.2 Dragonfly hovering mode 6.2 Particulate flow 6.2.1 6.2.2 163 165 Sedimentation of an elliptical particle between two closely spaced walls 166 Cold particle settling in an infinitely long channel 170 6.3 Forced Convective Heat Transfer from a Transverse Oscillating Cylinder in the Tandem Cylinder System 174 6.3.1 176 Vortex structure 6.3.1.1 In the “VS” regime, G = 177 6.3.1.2 At the critical spacing, G = 179 6.3.1.3 In the “VF” regime, G = 180 6.3.2 Temperature field 181 6.3.3 Forces and average Nusselt number 183 6.3.3.1 Average drag 184 6.3.3.2 R.M.S. of lift 185 6.3.3.3 Average Nusselt number 187 6.4 Conclusions 189 Chapter Applications of Developed IBM Solvers to Simulate VII   Three Dimensional Incompressible Thermal Flows 215 7.1 Forced convective heat and mass transfer around a stationary isolated sphere 216 7.1.1 Steady axisymmetric flow regime 220 7.1.2 Steady planar-symmetric flow regime 222 7.1.3 Unsteady periodic flow regime 224 7.2 Forced convective heat and mass transfer around a pair of tandem spheres 227 7.2.1 The case at Re = 40 229 7.2.2 The case at Re = 300 232 7.3 Laminar flow past a streamwise rotating isothermal sphere 236 7.4 Natural convective heat transfer between concentric and vertically eccentric spheres 241 7.5 Conclusions 247 Chapter Applications of Developed IBM Solver to Simulate Three Dimensional Moving Boundary Flows 276 8.1 Incompressible flow over a heaving and pitching finite span foil 276 8.2 Hydrodynamics of flow over a fish-like body in carangiform swimming 286 8.3 Conclusions 293 VIII   List of Figures Fig. 2.1 A two-dimensional domain Ω containing an immersed object in the form of a closed curve Γ 60 Fig. 2.2 Schematic view of flow over a stationary circular cylinder 60 Fig. 2.3 Steady-state streamlines and vorticity patterns for flow over a stationary circular cylinder at Re = 40 61 Fig. 2.4 Instantaneous streamlines and vorticity patterns for flow over a stationary circular cylinder at Re = 100 62 Fig. 2.5 Instantaneous streamlines for flow over an isolated stationary circular cylinder at Re = 100 obtained using the conventional immersed boundary method 62 Fig. 2.6 Time evolution of drag and lift coefficients for an isolated circular cylinder in a free-stream at Re = 100 63 Fig. 2.7 Schematic diagram of flow over a pair of side-by-side circular cylinders 63 Fig. 2.8 Instantaneous streamlines (left) and vorticity pattern (right) for flow around a pair of side-by-side circular cylinders at G / D = 64 Fig. 2.9 Time histories of drag and lift coefficients on two side-by-side arranged circular cylinders at G / D = 64 Fig. 2.10 Instantaneous streamlines (left column) and vorticity patterns (right column) for flow around a pair of side-by-side circular cylinders at different gap ratios 65 Fig. 2.11 Time histories of drag and lift coefficients on the two side-by-side circular cylinders at different gap ratios 66 Fig. 2.12 Time history of drag and lift coefficients for one transversely oscillating cylinder at Re = 185 67 XV   Fig. 2.13 Comparison of time-mean drag and r.m.s of drag and lift coefficients for one transversely oscillating cylinder at Re = 185 68 Fig. 2.14 Instantaneous vorticity patterns for flow around a transversely oscillating cylinder at various oscillation frequencies 68 Fig. 2.15 Schematic diagram of an elastically mounted circular cylinder in a free-stream 69 Fig. 2.16 Trajectory of the cylinder center during its free vibration 69 Fig. 2.17 Phase plots between the cylinder location and its velocity 70 Fig. 2.18 Instantaneous vorticity pattern for a freely vibrating circular cylinder in a free-stream 70 Fig. 3.1 Streamlines and vorticity patterns in the vicinity of circular cylinder at Re = 40 92 Fig. 3.2 Streamlines and vorticity patterns in the vicinity of circular cylinder at Re = 100 93 Fig. 3.3 Time evolution of drag and lift coefficients at Re = 100 94 Fig. 3.4 Adjusted streamlines for flow over a left moving circular cylinder at Re = 40 94 Fig. 3.5 Vorticity patterns for flow over a left moving/stationary circular cylinder at Re = 40 94 Fig. 3.6 Vorticity distribution on the surface of cylinder at Re = 40 for stationary and left moving cylinder cases 95 Fig. 3.7 Evolution of drag coefficient at Re = 40 for stationary and left moving cylinder cases 95 Fig. 3.8 Comparison of velocity profiles at four different x locations and three phase angles of φ = 2π ft = 180°, 210°,330° 96 Fig. 3.9 Comparison of time evolution of inline force Fx in one period 97 Fig. 3.10 Schematic view of sedimentation of a single particle between two XVI   parallel walls 97 Fig. 3.11 Instantaneous flow structures (vorticity) at different times of t = 0.2s , 0.4s , 0.6s , 0.8s 98 Fig. 3.12 Time evolution of translational kinetic energy ET 98 Fig. 3.13 Time evolution of longitudinal coordinate Y of particle center 99 Fig. 3.14 Time evolution of longitudinal velocity V of particle center 99 Fig. 3.15 Time evolution of Reynolds number Re pc 99 Fig.4.1 Configuration for the model problem 123 Fig. 4.2 L1 -norm of relative error of the temperature versus the mesh spacing for the model problem 123 Fig. 4.3 Isotherms for flow over a heated stationary cylinder at Re = 20, 40 124 Fig. 4.4 Schematic view of natural convection in a concentric annulus 124 Fig. 4.5 Streamlines (left) and isotherms (right) for Ra = 1× 10 125 Fig. 4.6 Streamlines (left) and isotherms (right) for Ra = 1× 10 126 Fig. 4.7 Streamlines (left) and isotherms (right) for Ra = 1× 10 127 Fig. 5.1 the L1 -norm of relative error of the temperature versus the mesh spacing for the model problem 144 Fig. 5.2 Isotherms for flow over a heated stationary cylinder at different Re 145 XVII   Fig. 5.3 Comparison of local Nusselt number distribution on the cylinder surface for Re = 10,20 146 Fig. 5.4 Time evolution of average Nusselt number on cylinder surface for Re = 100 147 Fig. 5.5 Schematic view of natural convection in a horizontal concentric cylindrical annulus 147 Fig. 5.6 Streamlines (left) and isotherms (right) for different Ra 148 Fig. 5.7 Effect of Rayleigh number on local temperature distribution along the inner cylinder surface 149 Fig. 5.8 Comparison of local temperature distribution on the inner cylinder surface for Ra = 5700 and 5× 10 150 Fig. 5.9 Configuration of natural convection in an eccentric horizontal cylindrical annulus 150 Fig. 5.10 Streamlines (left) and isotherms (right) for different Ra 151 Fig. 5.11 Comparison of temperature profile along the inner cylinder surface 152 Fig. 6.1 The schematic view of normal hovering mode 192 Fig. 6.2 The schematic view of dragonfly hovering mode 192 Fig. 6.3 The schematic clarification of kinematic parameters 192 Fig. 6.4 The drag coefficient evolution in the first four flapping cycles at different φ 193 Fig. 6.5 The lift coefficient evolution in the first four flapping cycles at different φ 194 Fig. 6.6 The vorticity field evolution in the first-half cycle at φ = π / 195 XVIII   Fig. 6.7 The vorticity field evolution in the first-half cycle at φ = 196 Fig. 6.8 The vorticity field evolution in the first-half cycle at φ = −π / 196 Fig. 6.9 Comparison of time histories of drag and lift coefficients in one flapping cycle 197 Fig. 6.10 The time-mean drag and lift coefficients at different ground clearances 197 Fig. 6.11 The development of vortex structure in the forth stroke at Gc = 198 Fig. 6.12 The development of vortex structure in the forth stroke at Gc = 2.5 198 Fig. 6.13 The development of vortex structure in the forth stroke at Gc = 199 Fig. 6.14 Comparison of time-dependent drag and lift coefficient for dragonfly hovering mode 200 Fig. 6.15 Vorticity field evolution during one stroke for dragonfly hovering 201 Fig. 6.16 Time-mean drag and lift coefficients versus inclined angle 201 Fig. 6.17 Time evolution of force coefficients during two strokes (a) horizontal force (b) vertical force 202 Fig. 6.18 Snapshots of particle sedimentation at blockage ratios: 12/13, 18/13, 20/13, 22/13, 32/13 203 Fig. 6.19 Trajectories of particle center at different blockage ratios 203 Fig. 6.20 Instantaneous vorticity field at different blockage ratios corresponding to Fig. 6.18 204 XIX   Fig. 6.21 Streamlines, the vorticity and temperature contours at different Gr 206 Fig. 6.22 Time histories of the lateral particle positions at different Gr 206 Fig. 6.23 The terminal-settling-velocity based Reynolds number Retmn versus the Grashof number Gr 207 Fig. 6.24 Configuration of tandem cylinder system 207 Fig. 6.25 Instantaneous vorticity contours for G = at different vibration frequencies and amplitudes 208 Fig. 6.26 Instantaneous vorticity contours for two consecutive cycles of excitation at f c / f st = 0.9 and A = 0.35 208 Fig. 6.27 Instantaneous vorticity contours of a stationary tandem cylinder system at G = and for Re = 100 209 Fig. 6.28 Instantaneous vorticity contours for G = at different vibration frequencies and amplitudes 209 Fig. 6.29 Instantaneous vorticity contours for two consecutive cycles of excitation at the locked-on frequency of f c / f st = 1.0 and A = 0.15 209 Fig. 6.30 Instantaneous vorticity contours at the locked-on frequency of f c / f st = 1.0 for G = and A = 0.35 209 Fig. 6.31 Instantaneous vorticity contours for G = at different vibration frequencies and amplitudes 210 Fig. 6.32 Instantaneous isotherms for G = and A = 0.15 at different excitation frequencies 210 Fig. 6.33 Instantaneous isotherms for G = and A = 0.35 at the excitation frequency f c / f st = 0.9 210 Fig. 6.34 Instantaneous isotherms for G = at different excitation conditions 211 XX   Fig. 6.35 Instantaneous isotherms for G = at the excitation condition of f c / f st = 1.7 and A = 0.35 211 Fig. 6.36 Time-averaged drag coefficient versus vibration frequency 212 Fig. 6.37 Time-averaged r.m.s of lift coefficient versus vibration frequency 213 Fig. 6.38 Time-averaged Nusselt number versus vibration frequency 214 Fig. 7.1 Three-dimensional vortex structures in their λ2 -definition at different Re 254 Fig. 7.2 Streamlines in ( x, y ) -plane at Re = 100 and 200 254 Fig. 7.3 Isotherms in ( x, y ) -plane at Re = 100 (left) and 200 (right) for isothermal condition 255 Fig. 7.4 Isotherms in ( x, y ) -plane at Re = 100 and 200 for isoflux condition 255 Fig. 7.5 Local Nusselt number distribution along the sphere surface in the circumferential direction 255 Fig. 7.6 Streamlines in the ( x, y ) -plane and ( x, z ) -plane at Re = 250 256 Fig. 7.7 Isotherms in the ( x, y ) -plane and ( x, z ) -plane at Re = 250 256 Fig. 7.8 Local Nusselt number distribution along the sphere surface in the circumferential direction: (a) comparison between Re = 100 ,200 and 250 on ( x, y ) -plane; (b) comparison between ( x, y ) -plane and ( x, z ) -plane at Re = 250 256 Fig. 7.9 Time evolution of drag coefficient and surface-averaged Nusselt number at Re = 300 257 XXI   Fig. 7.10 Streamline development in one vortex shedding cycle in ( x, z ) -plane at Re = 300 257 Fig. 7.11 Streamline development in one vortex shedding cycle in ( x, y ) -plane at Re = 300 258 Fig. 7.12 Isotherm development in one vortex shedding cycle in ( x, z ) -plane at Re = 300 258 Fig. 7.13 Isotherm development in one vortex shedding cycle in ( x, y ) -plane at Re = 300 259 Fig. 7.14 Local Nusselt number distribution on the sphere surface in one vortex shedding cycle at Re = 300 259 Fig. 7.15 Three-dimensional vortex structures for flow around a pair of tandem spheres at Re = 40 260 Fig. 7.16 Streamlines and isotherms at Re = 40 and G / D = 1.2 260 Fig. 7.17 Streamlines and isotherms at Re = 40 and G / D = 2.5 260 Fig. 7.18 Streamlines for an isolated sphere at Re = 40 261 Fig. 7.19 Local Nusselt number distribution on the sphere surface along the circumferential direction for G / D = 1.2 261 Fig. 7.20 Local Nusselt number distribution on the sphere surface along the circumferential direction for G / D = 2.5 261 Fig. 7.21 Three-dimensional vortex structures for flow around a pair of tandem spheres at Re = 300 262 Fig. 7.22 Streamlines and isotherms at Re = 300 and G / D = 1.5 262 Fig. 7.23 Streamlines and isotherms at Re = 300 and G / D = 2.0 262 Fig. 7.24 Local Nusselt number distribution on the sphere surface in ( x, z ) -plane: a comparison between G / D = 1.5 and G / D = 2.0 XXII   263 Fig. 7.25 Local Nusselt number distribution on the sphere surface at G / D = 2.0 263 Fig. 7.26 Vortex structure evolution in one vortex shedding cycle at G / D = 3.0 264 Fig. 7.27 Local Nusselt number distributions on the sphere surfaces at Re = 300 and G / D = 3.0 265 Fig. 7.28 Instantaneous isotherms corresponding to Fig. 7.26(a) for Re = 300 and G / D = 3.0 265 Fig. 7.29 Three-dimensional vortex structures induced by streamwise rotating sphere for different rotating speed at Re = 100 265 Fig. 7.30 Inplane streamlines and isotherms for Re = 100 and Ω = 0.3 266 Fig. 7.31 Inplane streamlines and isotherms for Re = 100 and Ω = 1.0 266 Fig. 7.32 Comparison of local Nusselt number distributions on the sphere surface at Re = 100 for different rotating speed 266 Fig. 7.33 Three-dimensional vortex structures induced by streamwise rotating sphere for different rotating speed at Re = 250 267 Fig. 7.34 Time evolutions of the drag and lift coefficients on a streamwise rotating sphere at Re = 250 for different rotating speed 268 Fig. 7.35 Time histories of surface-averaged Nusselt number from a streamwise rotating sphere for different rotating speed at Re = 250 269 Fig. 7.36 Three-dimensional vortex structures induced by streamwise rotating sphere for different rotating speed at Re = 300 269 Fig. 7.37 Time evolutions of the drag and lift coefficients on a streamwise rotating sphere at Re = 300 for different rotating speed 270 Fig. 7.38 Time histories of surface-averaged Nusselt number from a streamwise rotating sphere for different rotating speed at Re = 300 271 Fig. 7.39 Overall performance of flow behavior and heat transfer from the rotating sphere in terms of time-mean drag coefficient and Nusselt number 272 XXIII   Fig. 7.40 The geometric configuration of natural convection inside concentric or vertically eccentric spherical annulus 272 Fig. 7.41 Steady-state streamlines and isotherms at different Rayleigh numbers 273 Fig. 7.42 Local Nusselt number along the inner sphere for different Rayleigh numbers 273 Fig. 7.43 Steady-state streamlines and isotherms at different vertical eccentricities 274 Fig. 7.44 Local Nusselt number along the inner sphere at e = 274 Fig. 7.45 Local Nusselt number along the inner sphere at e = −0.625 and e = 0.625 275 Fig. 8.1 Schematic view of a rigid finite-span foil heaving and pitching in a free stream 296 Fig. 8.2 Schematic diagram for the definition of parameters 296 Fig. 8.3 Perspective and side views of vortex structure evolution in one flapping cycle 298 Fig. 8.4 Top view of the vortex structure corresponding to Fig. 8.3(d) 298 Fig. 8.5 Contours of the spanwise vorticity along the spanwise symmetry plane for AR = and St = 0.6 298 Fig. 8.6 Contours of the mean streamwise velocity along the spanwise symmetry plane for AR = and St = 0.6 299 Fig. 8.7 Perspective and side views of vortex structures for different aspect ratios at St = 0.6 300 Fig. 8.8 Contours of the mean streamwise velocity in the spanwise symmetry plane 301 Fig. 8.9 Top view of vortex sturctures for different Strouhal number at AR = 301 Fig. 8.10 Contours of the mean streamwise velocity in plane y = for XXIV   different Strouhal number at AR = 302 Fig. 8.11 Time-averaged thrust coefficient versus foil aspect-ratio for St = 0.6 302 Fig. 8.12 Propulsive efficiency versus foil aspect-ratio for St = 0.6 303 Fig. 8.13 Variations of the propulsive efficiency and time-averaged thrust coefficient with respect to Strouhal number at AR = 303 Fig. 8.14 Schematic view of the fish geometry 303 Fig. 8.15 The three-dimensional flow structures induced by the swimming fish 304 at Re=300 for some representative Strouhal numbers Fig. 8.16 The three-dimensional flow structures induced by the swimming fish at Re=4000 for some representative Strouhal numbers 305 Fig. 8.17 The instantaneous in-plane streamlines and vorticity field in the mid plane of y = at St = 0.3 for Re=300 and 4000 305 Fig. 8.18 The time-dependent hydrodynamic force coefficients for Re = 300 and 4000 306 Fig. 8.19 The mean force coefficient for Re = 300 and 4000 XXV   307 Nomenclature a Major axis for elliptical particle A Oscillation amplitude [ A F ] , [ A P ] , [CT ] Coefficient matrix AR   Aspect ratio b Minor axis for elliptical particle BR Blockage ratio c  Chord length of flapping wing cp   Specific heat at constant pressure CD , CL Drag and lift coefficients CD , CL Time-mean drag and lift coefficients CF , CF Instantaneous and time-mean force coefficient for swimming fish CH , CV Drag and lift coefficient for hovering insect CH , CV Time-mean drag and lift coefficients for hovering insect CLy , CLz Lift coefficients for sphere CT Thrust coefficient dp Diameter of particle D Diameter of circular cylinder or sphere Dh Discrete delta function e Eccentricity of annulus f Body force density on the Eulerian mesh F Surface force density on the Lagrangian points fc , f N Forced and natural vibration frequency fs Vortex shedding frequency of an isolated stationary cylinder f st Vortex shedding frequency of tandem stationary cylinders FD , FL Drag and lift force Frepulsive Repulsive force between particles FT , FT Instantaneous and time-mean thrust Fx   x − component of hydrodynamic force g  Gravitational acceleration G Gap distance between centers of circular cylinders or spheres Gc Ground clearance Gr   Grashof number h Heave displacement or lateral excursion hc   Convective heat transfer coefficient k  Thermal conductivity KC   Keulegan-Carpenter number L , Lref , Ltotal   Length Lw Recirculation length mp Mass of particle mr Mass ratio Nu , Nu , Nu Local, surface-averaged and time-mean Nusselt number p Pressure ΔP Virtual boundary flux ΔPx , ΔPy   Components of virtual boundary flux Pr   Prandtl number q  Heat source/sink on the Eulerian mesh ΔQ   Heat source/sink at the Lagrangian points QB   Prescribed heat flux at the Lagrangian points Ri , Ro   Radius of inner and outer cylinder or square of annulus Ra   Rayleigh number Re Reynolds number Retmn Terminal Reynolds number Δs Length of boundary element St Strouhal number t Time Δt Time step size T ,T* Temperature ΔT   Temperature correction on the Eulerian mesh TB   Prescribed temperature on the Lagrangian points T flap   Stoke period of flapping wing or foil Tin , Tout   Temperature of inner and outer surface of annulus Tshed Vortex shedding period T∞   Ambient or free stream temperature u , u* , u Velocity vector on the Eulerian mesh Δu Velocity correction vector on the Eulerian mesh u , v, w Velocity components on the Eulerian mesh Δu , Δ v , Δ w Components of velocity correction on the Eulerian mesh U , UB Velocity vector on the Lagrangian points U tmn Terminal velocity of particle U∞ Free stream velocity W Span width of foil x Eulerian coordinates x, y , z Coordinate components for the Eulerian mesh Δx, Δy, Δz , h Mesh spacing X Lagrangian coordinates X ,Y , Z Coordinate components for the Lagrangian point α Thermal diffusivity α AOA   Angle of attack α spring Spring constant β  Thermal expansion coefficient β damp Damp constant δ Dirac delta function δh Kernel distribution function ε Eccentricity of annulus η Propulsion efficiency θ Angular coordinate Θ Vorticity source κ Stiffness of spring μ ,ν Dynamic and kinematic viscosity of fluid ρ Density of fluid ρp Density of particle ϕ ,ϕ* Stream function ψ Inclined angle of stroke plane φ ,ϑ Phase angle ω , ω* Vorticity Ω Rotational speed Abbreviation ALE Arbitrary Lagrangian-Eulerian BCF Body/Caudal Fin CFD Computational Fluid Dynamics DLM/FD Distributed Lagrangian Multiplier-based Fictitious Domain Method DNS Direct Numerical Simulation FD Finite Difference Method FE Finite Element Method FV Finite Volume Method IBM Immersed Boundary Method IIM Immersed Interface Method LBM Lattice Boltzmann Method LEV Leading Edge Vortex NS Navier-Stokes RHS Right Hand Side TEV Trailing Edge Vortex VF Vortex Formation VS Vortex Suppression [...]... 6.9 Comparison of time histories of drag and lift coefficients in one flapping cycle 197 Fig 6.10 The time-mean drag and lift coefficients at different ground clearances 197 Fig 6.11 The development of vortex structure in the forth stroke at Gc = 1 198 Fig 6.12 The development of vortex structure in the forth stroke at Gc = 2.5 198 Fig 6.13 The development of vortex structure in the forth stroke at... number isolated hot sphere immersed in a cold free stream 250 Nu from an 250 Table 7.3 Comparison of drag coefficient CD for a for an isolated sphere immersed in a free stream at Re = 250 250 Table 7.4 Comparison of drag coefficient CD for an isolated sphere immersed in a free stream at Re = 300 251 Table 7.5 Comparison of the mean drag coefficients and Nusselt numbers for tandem-sphere system at Re... types of problems so that the contribution of the heated immersed boundary to its surrounding is concisely modeled The performances of all the developed IBM solvers are extensively studied While the obtained results compare considerably well with the benchmark XI   ones, it is confident to conclude that the proposed methods provide useful tools for fluid and thermal flows with complex geometries and. .. swimming fish at Re=4000 for some representative Strouhal numbers 305 Fig 8.17 The instantaneous in-plane streamlines and vorticity field in the mid plane of y = 0 at St = 0.3 for Re=300 and 4000 305 Fig 8.18 The time-dependent hydrodynamic force coefficients for Re = 300 and 4000 306 Fig 8.19 The mean force coefficient for Re = 300 and 4000 XXV   307 Nomenclature a Major axis for elliptical particle... 207 Fig 6.24 Configuration of tandem cylinder system 207 Fig 6.25 Instantaneous vorticity contours for G = 2 at different vibration frequencies and amplitudes 208 Fig 6.26 Instantaneous vorticity contours for two consecutive cycles of excitation at f c / f st = 0.9 and A = 0.35 208 Fig 6.27 Instantaneous vorticity contours of a stationary tandem cylinder system at G = 2 and 4 for Re = 100 209 Fig 6.28... variation range of AR and St XIV   295 List of Figures Fig 2.1 A two-dimensional domain Ω containing an immersed object in the form of a closed curve Γ 60 Fig 2.2 Schematic view of flow over a stationary circular cylinder 60 Fig 2.3 Steady-state streamlines and vorticity patterns for flow over a stationary circular cylinder at Re = 40 61 Fig 2.4 Instantaneous streamlines and vorticity patterns for flow over... ratio b Minor axis for elliptical particle BR Blockage ratio c  Chord length of flapping wing cp   Specific heat at constant pressure CD , CL Drag and lift coefficients CD , CL Time-mean drag and lift coefficients CF , CF Instantaneous and time-mean force coefficient for swimming fish CH , CV Drag and lift coefficient for hovering insect CH , CV Time-mean drag and lift coefficients for hovering insect... coefficients for sphere CT Thrust coefficient dp Diameter of particle D Diameter of circular cylinder or sphere Dh Discrete delta function e Eccentricity of annulus f Body force density on the Eulerian mesh F Surface force density on the Lagrangian points fc , f N Forced and natural vibration frequency fs Vortex shedding frequency of an isolated stationary cylinder f st Vortex shedding frequency of tandem... Fig 3.7 Evolution of drag coefficient at Re = 40 for stationary and left moving cylinder cases 95 Fig 3.8 Comparison of velocity profiles at four different x locations and three phase angles of φ = 2π ft = 180°, 210°,330° 96 Fig 3.9 Comparison of time evolution of inline force Fx in one period 97 Fig 3.10 Schematic view of sedimentation of a single particle between two XVI   parallel walls 97 Fig 3.11... different times of t = 0.2s , 0.4s , 0.6s , 0.8s 98 Fig 3.12 Time evolution of translational kinetic energy ET 98 Fig 3.13 Time evolution of longitudinal coordinate Y of particle center 99 Fig 3.14 Time evolution of longitudinal velocity V of particle center 99 Fig 3.15 Time evolution of Reynolds number Re pc 99 Fig.4.1 Configuration for the model problem 123 Fig 4.2 L1 -norm of relative error of the temperature . DEVELOPMENT OF IMMERSED BOUNDARY METHODS FOR ISOTHERMAL AND THERMAL FLOWS REN WEIWEI (B. Eng., M. Eng., Nanjing University of Aeronautics and Astronautics, China). Pressure-velocity formulation-based immersed boundary method 14 1.3.2 Stream function-vorticity formulation-based immersed boundary method 18 1.3.3 Applications of immersed boundary method. history of drag and lift coefficients for one transversely oscillating cylinder at Re 185= 67 XVI  Fig. 2.13 Comparison of time-mean drag and r.m.s of drag and lift coefficients for one