List of Figures

• 4.1. Redistribution of the circulation of a vortex $\Gamma ^n_i$.
• 6.1. Vortex pair, $Re=0$: Growth in number of computational vortices for the Stokes flow starting from a pair of counter-rotating point vortices. The small circle indicates the size of the computational neighborhood of a typical vortex.
• 6.2. Vortex pair: Total number of computational vortices versus time.
• 6.3. Vortex pair, $Re=50$: Growth in number of computational vortices for a flow starting from a pair of counter-rotating point vortices at $Re=50$. The small circle indicates the size of the computational neighborhood of a typical vortex.
• 6.4. Smoothing functions in (a) Fourier space and (b) Physical space. Broken lines are nonconvergent smoothing. Solid lines are modified smoothing.
• 7.1. Point vortex, $Re=50$: Growth in mean square radius of a single diffusing vortex. The solid line is exact and circles are vorticity redistribution solutions.
• 7.2. Point vortex, $Re=50$: Vorticity distribution of a single diffusing point vortex along the horizontal symmetry axis at times $\tau =0.082$ & $0.202$. The solid lines are exact and circles are vorticity redistribution solutions.
• 7.3. Vortex pair, $Re=0$: Vorticity along the connecting line at times $\tau =0.082$ & $0.202$. The solid lines are exact and circles are vorticity redistribution solutions.
• 7.4. Vortex pair, $Re=0$: Isovorticity contours (a) at time $\tau =0.082$: $\omega = 5.00$, 3.85, 2.70, 1.55, & 0.40; (b) at time $\tau =0.202$: $\omega = 1.40$, 1.10, 0.80, 0.50, & 0.20. The solid lines are exact and circles are vorticity redistribution solutions.
• 7.5. Vortex pair, $Re=50$: Isovorticity contours for a counter-rotating vortex pair. (a) at time $\tau =0.01025$: $\omega = 40$, 24, & 8; (b) at time $\tau =0.02050$: $\omega = 20$, 12, & 4. The dashed and solid lines represent orders of approximation in the analytical solution. Circles are vorticity redistribution solutions.
• 7.6. Vortex pair, $Re=50$: The effect of using exponentially decaying core shapes instead of algebraically decaying ones.
• 7.7. Vortex pair: Isovorticity contours $\omega =$ 5, 3, & 1 at time $\tau =0.082$ for different Reynolds numbers.
• 7.8. Vortex pair: Isovorticity contours $\omega =$ 1.7, 1.5, ..., 0.1 at time $\tau =0.202$ for different Reynolds numbers.
• 7.9. Vortex pair: Maximum vorticity for different Reynolds numbers. Stokes represents the exact solution for $Re=0$.
• 7.10. Vortex pair: Distance of the point of maximum vorticity from the symmetry plane. Stokes represents the exact solution for $Re=0$.
• 7.11. Vortex pair: Circulation in a half plane for different Reynolds numbers. Stokes represents the exact solution for $Re=0$.
• 7.12. Vortex pair: Average velocity $\bar v$, vortex center velocity $v_c$, and asymptotic velocity $v_g$ for vanishing Reynolds number. Short dash curves and dot dash curves represent the small time and the long time analytical solutions respectively.
• 7.13. Vortex pair: Deviation in average velocity $\bar v$ from the inviscid drift velocity. Short dash curves represent the small time analytical solutions. Stokes represents the exact solution for $Re=0$ and the asymptotic solution for large time for any Reynolds number.
• 7.14. Vortex pair: Deviation in vortex center velocity $v_c$ from the inviscid drift velocity. Short dash curves represent the small time analytical solutions. Stokes represents the exact solution for $Re=0$ and the asymptotic solution for large time for any Reynolds number.
• 7.15. Vortex pair: Average velocity $\bar v$ for different Reynolds numbers. Short dash curves represent the small time analytical solutions. Stokes represents the exact solution for $Re=0$.
• 7.16. Vortex pair: Vortex center velocity $v_c$ for different Reynolds numbers. Short dash curves represent the small time analytical solutions. Stokes represents the exact solution for $Re=0$.
• 7.17. Vortex pair: Asymptotic velocity $v_g$ for different Reynolds numbers. Short dashed curves represent the small time analytical solutions. Stokes represents the exact solution for $Re=0$.
• 7.18. Vortex pair: Average velocity $\bar v$, vortex center velocity $v_c$ and asymptotic velocity $v_g$ for Reynolds number 100.
• 7.19. Vorticity for three-dimensional diffusion of a pair of vortex poles: (a) Along a line through the vortices; (b) Isovorticity contours $\omega$=0.5, 1.0, 1.5, 2.0, & 2.5 in the plane of the vortices. Solid lines are exact and symbols are redistribution solutions.
• 7.20. Diffusing vortex ring, $Re=0$: Vorticity fields at two different times.
• 8.1. Rotating cylinder, $Re=0$: Vorticity distribution along a radial line at times $\tau =0.30$ & $0.60$. Solid line is a finite difference solution. Symbols are vorticity redistribution solutions.
• 8.2. Oscillating cylinder, $Re=0$: Vorticity distribution along a radial line at different times. Solid lines are finite difference solutions. Dashed lines are redistribution solutions.
• 8.3. Oscillating cylinder, $Re=0$: Total circulation during one time period of oscillation. Solid lines are finite difference solutions. Symbols are redistribution solutions.
• 8.4. Impulsively translated cylinder, $Re=550$: Instantaneous streamlines from vorticity redistribution ( $\Delta t=0.01; \epsilon _\Gamma =10^{-5}$); see subsection 8.2.2 for streamline values.
• 8.5. Impulsively translated cylinder, $Re=550$: (a) Streamlines from experiment (Bouard & Coutanceau [29], used by permission); (b) Instantaneous streamlines from vorticity redistribution ( $\Delta t=0.01; \epsilon _\Gamma =10^{-5}$); see subsection 8.2.2 for streamline values.
• 8.6. Impulsively translated cylinder, $Re=3,000$: Instantaneous streamlines from vorticity redistribution ( $\Delta t=0.01; \epsilon _\Gamma =10^{-5}$); see subsection 8.2.2 for streamline values.
• 8.7. Impulsively translated cylinder,$Re=3,000$: (a) Streamlines from experiment (Bouard & Coutanceau [29], used by permission); (b) Instantaneous streamlines from vorticity redistribution ( $\Delta t=0.01; \epsilon _\Gamma =10^{-5}$); see subsection 8.2.2 for streamline values.
• 8.8. Impulsively translated cylinder, $Re=9,500$: (a) Streamlines from experiment (Bouard & Coutanceau [29], used by permission); (b) Instantaneous streamlines from vorticity redistribution ( $\Delta t=0.01; \epsilon _\Gamma =10^{-6}$); see subsection 8.2.2 for streamline values.
• 8.8. (continued.)
• 8.9. Impulsively translated cylinder, $Re=550$: Radial velocity along the rear symmetry axis. Solid lines are vorticity redistribution solutions. Symbols are experimental values of Bouard & Coutanceau [29].
• 8.10. Impulsively translated cylinder, $Re=550$: Radial velocity along the rear symmetry axis. Solid lines are vorticity redistribution solutions. Symbols are solutions computed by Pépin [170], and Loc [133].
• 8.11. Impulsively translated cylinder, $Re=3,000$: Radial velocity along the rear symmetry axis. Solid lines are vorticity redistribution solutions. Symbols are experimental values of Loc & Bouard [134].
• 8.12. Impulsively translated cylinder, $Re=3,000$: Radial velocity along the rear symmetry axis. Solid lines are vorticity redistribution solutions. Symbols are solutions computed by Hakizumwami [105], Chang & Chern [40], Pépin [170], Cheer [42], Smith & Stansby [215], and Loc & Bouard [134].
• 8.13. Impulsively translated cylinder, $Re=9,500$: Tangential velocity profiles at time $t=0.50$ at various angular distances from the front symmetry line. Dotted lines are standard boundary layer theory. Solid lines are second-order boundary layer theory. Symbols are vorticity redistribution solutions ($\Delta t=0.01$; $\epsilon _\Gamma =10^{-6}$).
• 8.14. Impulsively translated cylinder, $Re=9,500$: Radial velocity along the rear symmetry axis. Solid lines are vorticity redistribution solutions. Symbols are experimental values of Loc & Bouard [134].
• 8.15. Impulsively translated cylinder, $Re=9,500$, part 1: Radial velocity along the rear symmetry axis. Solid lines are vorticity redistribution solutions. Symbols are solution computed by Anderson & Reider [3], Kruse & Fischer [120], Hakizumwami [105], and Wu, Wu, Ma & Wu [249].
• 8.15. Impulsively translated cylinder, $Re=9,500$, part 2: Radial velocity along the rear symmetry axis. Solid lines are vorticity redistribution solutions. Symbols are solution computed by Chang & Chern [40], Pépin [170], Cheer [42], and Loc & Bouard [134].
• 8.16. Impulsively translated cylinder, $Re=550$: Vorticity fields at different times for $\Delta t=0.01$ and $\epsilon _\Gamma =10^{-5}$.
• 8.17. Impulsively translated cylinder, $Re=3,000$: Vorticity fields at different times for $\Delta t=0.01$ and $\epsilon _\Gamma =10^{-5}$.
• 8.18. Impulsively translated cylinder, $Re=9,500$: Vorticity fields at different times for $\Delta t=0.01$ and $\epsilon _\Gamma =10^{-6}$.
• 8.18. (continued.)
• 8.19. Impulsively translated cylinder, $Re=20,000$: Vorticity fields at different times for $\Delta t=0.01$ and $\epsilon _\Gamma =5 \times 10^{-7}$.
• 8.20. Impulsively translated cylinder, $Re=3,000$: Vorticity fields. (a) Experimental data from Shih, Lourenco & Ding [212]; (b) Vorticity redistribution method ($\Delta t=0.01$; $\epsilon _\Gamma =10^{-5}$).
• 8.21. Impulsively translated cylinder, $Re=9,500$: Vorticity fields. (a) Spectral element method (preliminary data of Kruse & Fischer [120], used by kind permission); (b) Vorticity redistribution method ($\Delta t=0.01$; $\epsilon _\Gamma =10^{-6}$).
• 8.22. Impulsively translated cylinder, $Re=9,500$: Vorticity fields. (a) Particle strength exchange method (Koumoutsakos & Leonard [117], used by permission); (b) Vorticity redistribution method ( $\Delta t=0.01; \epsilon _\Gamma =10^{-6}$).
• 8.23. Impulsively translated cylinder, $Re=10,000$: Vorticity fields obtained from random walk computations (Unpublished data of Van Dommelen, used by permission); (a) and (b) refer to two different runs of the computation.
• 8.23. (continued.)
• 8.24. Impulsively translated cylinder: Drag coefficient at small times for various Reynolds numbers. Long dashed lines are standard boundary layer theory. Solid lines are second-order boundary layer theory. Dot-dashed lines are the small time expansion of Collins & Dennis [60]. Symbols are vorticity redistribution solutions ($\Delta t=0.01$; $\epsilon _\Gamma =10^{-5}$ for $Re$ = 550, 1000, & 3,000; $\epsilon _\Gamma =10^{-6}$ for $Re$ = 9,500; $\epsilon _\Gamma =5 \times 10^{-7}$ for $Re$ = 20,000; and $\epsilon _\Gamma = 10^{-7}$ for $Re$ = 40,000).
• 8.25. Impulsively translated cylinder, $Re=550$: Drag coefficient. Solid line is our vorticity redistribution solution. Symbols are solutions computed by Koumoutsakos & Leonard [117], Chang & Chern [40], Pépin [170], Van Dommelen [237], and Loc [133].
• 8.26. Impulsively translated cylinder, $Re=550$: Drag and lift coefficients. Solid lines are vorticity redistribution solutions ($\Delta t=0.01$; $\epsilon _\Gamma =10^{-6}$). The short and long dashed lined are random walk results of Van Dommelen [237] at $\Delta t=0.0125$ and $\Delta t=0.025$ respectively.
• 8.27. Impulsively translated cylinder, $Re=3,000$: Drag coefficient. Solid line is our vorticity redistribution solution. Symbols are solutions computed by Anderson & Reider [3], Koumoutsakos & Leonard [117], Chang & Chern [40], and Pépin [170].
• 8.28. Impulsively translated cylinder, $Re=9,500$, part 1: Drag coefficient. Solid line is our vorticity redistribution solution. Symbols are solutions computed by Anderson & Reider [3], Kruse & Fischer [120], Koumoutsakos & Leonard [117], and Wu, Wu, Ma & Wu [249].
• 8.28. Impulsively translated cylinder, $Re=9,500$, part 2: Drag coefficient. Solid line is our vorticity redistribution solution. Symbols are solutions computed by Chang & Chern [40], Pépin [170], and Van Dommelen (unpublished).
• 8.29. Impulsively translated cylinder, $Re=9,500$: Drag coefficient. Long dashed line is $\Delta t=0.04$. Short dashed line is $\Delta t=0.02$. Solid line is $\Delta t=0.01$. For all three cases $\epsilon _\Gamma =10^{-6}$.
• 8.30. Impulsively translated cylinder, $Re=20,000$: Drag coefficient. Long dashed line is $\Delta t=0.04$. Short dashed line is $\Delta t=0.02$. Solid line is $\Delta t=0.01$. For all three cases $\epsilon _\Gamma =5 \times 10^{-7}$.
• 8.31. Impulsively translated cylinder, $Re=9,500$: Radial velocity along the rear symmetry line at time $t=0.50$. Dashed line is standard boundary layer theory. Solid line is second-order boundary layer theory. Solid symbols are vorticity redistribution solutions for $\Delta t=0.01$ and $\epsilon _\Gamma =10^{-6}$. Open symbols are vorticity redistribution solutions for $\Delta t=0.01$ and $\epsilon _\Gamma =10^{-5}$. Dash-dot line is the irrotational flow solution.
• 8.32. Impulsively translated cylinder, $Re=9,500$: Radial velocity along the rear symmetry axis. Short dashed lines are computed velocity using $\Delta t=0.01$ and $\epsilon _\Gamma =10^{-5}$. Solid lines are computed velocity using $\Delta t=0.01$ and $\epsilon _\Gamma =10^{-6}$.
• 8.33. Impulsively translated cylinder, $Re=9,500$: Vorticity fields at time $t=2.50$ for $\Delta t=0.04$, 0.02, & 0.01. (a) $\epsilon _\Gamma =10^{-5}$; (b) $\epsilon _\Gamma =10^{-6}$.
• 8.34. Impulsively translated cylinder, $Re=9,500$: Vorticity fields at time $t=3.00$ for $\Delta t=0.04$, 0.02, & 0.01. (a) $\epsilon _\Gamma =10^{-5}$; (b) $\epsilon _\Gamma =10^{-6}$.
• 8.35. Impulsively translated cylinder, $Re=9,500$: Vorticity fields obtained in particle strength exchange computation (preliminary data of Shiels [208], used by kind permission); $\Delta t=0.005$, cut-off vorticity = $10^{-4}$ and Gaussian kernel size = 1.1 times the average particle spacing.
• 8.36. Impulsively translated cylinder, $Re=9,500$: Drag. (a) Solid line is $\Delta t=0.01$ and $\epsilon _\Gamma =10^{-6}$. Short dashed line is $\Delta t=0.01$ and $\epsilon _\Gamma =10^{-5}$. (b) Solid line is $\Delta t=0.01$ and $\epsilon _\Gamma =10^{-6}$. Dot-dashed line is $\Delta t=0.02$ and $\epsilon _\Gamma =10^{-5}$. Long dashed line is $\Delta t=0.04$ and $\epsilon _\Gamma =10^{-5}$.
• 8.37. Impulsively translated cylinder: (a) Local vorticity contours obtained from the Van Dommelen & Shen singularity [241]; (b) and (c) local vorticity fields at $t=1.50$ for $Re=9,500$ and $Re=20,000$ obtained from the vorticity redistribution method.
• 8.38. Vortex-pair/cylinder interaction, $Re=500$: Vorticity fields at different times for $\Delta t=0.02$ and $\epsilon _\Gamma =10^{-5}$.
• 8.39. (continued.)
• 8.40. Vortex-pair/cylinder interaction, $Re=500$: Path of the vortex approaching the cylinder. Symbols indicate vortex positions at various times.
• 8.41. Vortex-pair/cylinder interaction, $Re=500$: Circulation in a half plane. Solid line is the total circulation of the same sign as the primary vortex; dashed line is for opposite sign.
• 8.42. Vortex-pair/cylinder interaction, $Re=500$: Computational vortices at time $t=0.00$ and $t=7.00$.