Development of immersed boundary methods for isothermal and thermal flows preface

Table 2.4 Comparison of drag coefficient C and recirculation length D/ w L D for flow over a stationary circular cylinder 58 Table 2.5 Comparison of drag coefficient C , lift coefficient

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

This thesis has also not been submitted for any

degree in any university previously

Ren Weiwei

2014

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

Chapter 2 Governing Equations and Boundary Condition-Enforced Immersed Boundary Method 292.1 Governing equations 30

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

84 3.2.4 Sedimentation of a single circular particle between two parallel

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 115

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

Chapter 6 Applications of Developed IBM Solvers to Simulate Two-Dimensional Fluid and Thermal Flows 1536.1 Unsteady insect hovering flight at low Reynolds numbers 153

VII

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 163 6.2 Particulate flow 165 6.2.1 Sedimentation of an elliptical particle between two closely

6.2.2 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 Vortex structure 176 6.3.1.1 In the “VS” regime, G=2 177 6.3.1.2 At the critical spacing, G=4 179 6.3.1.3 In the “VF” regime, G=7 180 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

Chapter 7 Applications of Developed IBM Solvers to Simulate

Chapter 8 Applications of Developed IBM Solver to Simulate Three Dimensional Moving Boundary Flows 2768.1 Incompressible flow over a heaving and pitching finite span foil 276 8.2 Hydrodynamics of flow over a fish-like body in carangiform

in primitive variable form is firstly presented, where the critical issue of how

to evaluate body forces is realized by an implicit velocity correction procedure such that the velocity on the immersed boundary interpolated from the surrounding fluid velocity through Dirac delta function interpolation equals the given boundary velocity, i.e., the velocity condition on the immersed boundary is exactly enforced

For two-dimensional incompressible flows, the stream function-vorticity formulation-based NS solver is more efficient and it is worthwhile to combine the IBM with the stream function-vorticity formulation-based fluid solver While the previous attempt incorporated a very complicated source term into

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the vorticity transport equation which brought extraordinary complexity into the computational process, a novel and efficient stream function-vorticity formulation-based IBM solver is proposed in the thesis In the present solver,

no source term is required in the vorticity transport equation Through an accurate velocity correction and efficient vorticity correction procedure, the present method can accurately satisfy both the governing equation and boundary condition

Heat transfer problems are frequently featured with complex configurations and moving boundaries In the present thesis, the IBM is creatively extended

to the heat transfer field and two novel IBMs are developed, one for thermal problems with Dirichlet conditions and the other for problems with Neumann conditions In both methods, the presence of the heated immersed boundary is replaced by a set of heat sources which are added to the energy equation as a source term Particular attentions are paid to the essential issue of how to properly determine the heat sources Through the proposed temperature correction procedure and heat flux correction procedure respectively, they are carefully evaluated in the two 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

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ones, it is confident to conclude that the proposed methods provide useful tools for fluid and thermal flows with complex geometries and moving boundaries

Table 2.4 Comparison of drag coefficient C and recirculation length D

/

w

L D for flow over a stationary circular cylinder 58

Table 2.5 Comparison of drag coefficient C , lift coefficient D C L and Strouhal number St for flow over a stationary circular cylinder 59

Table 2.6 Comparison of drag coefficient C , lift coefficient D C L and Strouhal number St for two side-by-side circular cylinders 59

Table 3.1 Comparison of drag coefficient C D and recirculation length

/

w

Table 3.2 Comparison of CPU time 91

Table 3.3 Comparison of drag coefficient C D, lift coefficient C , and L

Table 4.1 Comparison of average Nusselt number obtained by the two proposed methods (Re=10) 121 Table 4.2 Comparison of average Nusselt numbers 121 Table 4.3 Comparison of computed average Nusselt numbers 122

Table 5.1 Comparison of average Nusselt number Nu for Re=10,20,40

Table 7.1 Comparison of drag coefficient C for an isolated sphere D

immersed in a free stream at Re=100 and 200 250

Table 7.2 Comparison of surface-averaged Nusselt number Nu from an isolated hot sphere immersed in a cold free stream 250

Table 7.3 Comparison of drag coefficient C for a for an isolated sphere D

immersed in a free stream at Re=250 250

Table 7.4 Comparison of drag coefficient C for an isolated sphere D

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=40 251

Table 7.6 Comparison of the mean drag and lift coefficients for tandem-sphere

Table 7.9 Average Nusselt number Nu at different Rayleigh numbers 253

Table 7.10 Average Nusselt number Nu at different vertical eccentricities

253 Table 8.1 The variation range of AR and St 295

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List of Figures

Fig 2.1 A two-dimensional domain Ω containing an immersed object in the

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

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 40

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 F in one period 97 x

Fig 3.10 Schematic view of sedimentation of a single particle between two

Fig 3.12 Time evolution of translational kinetic energy E T 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 L -norm of relative error of the temperature versus the mesh spacing 1

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=1×104 125

Fig 4.6 Streamlines (left) and isotherms (right) for Ra=1×105 126

Fig 4.7 Streamlines (left) and isotherms (right) for Ra=1×106 127

Fig 5.1 the L -norm of relative error of the temperature versus the mesh 1

spacing for the model problem 144 Fig 5.2 Isotherms for flow over a heated stationary cylinder at different Re

145

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 5×104 150

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

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Fig 6.7 The vorticity field evolution in the first-half cycle at φ =0 196

Fig 6.8 The vorticity field evolution in the first-half cycle at φ = −π / 4 196

Fig 6.9 Comparison of time histories of drag and lift coefficients in one

198 Fig 6.13 The development of vortex structure in the forth stroke at G c = 5

199 Fig 6.14 Comparison of time-dependent drag and lift coefficient for dragonfly

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

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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

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 Instantaneous vorticity contours for G=2 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 / 1.0

Fig 6.34 Instantaneous isotherms for G=4 at different excitation conditions

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

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

Fig 7.9 Time evolution of drag coefficient and surface-averaged Nusselt

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/ =2.0 263

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