Title Page
Contents
국문초록 12
Abstract 13
1. Introduction 14
2. Axial Green function formulation of the steady Stokes flow in unbounded domain 17
2.1. Separation of Operators for the axial Green function method 17
2.2. Preliminaries for the general solution of the representative ODE 19
2.3. AGM for one-dimensional elliptic problem in a half-infinite interval 23
2.4. AGM for Stokes flow in unbounded domain 26
3. Numerical Results for Stokes flows 30
3.1. Potential flow passing through a circular cylinder 30
3.2. Consistency of the informative boundary condition by an external Stokes flow 32
3.3. Half-infinite cavity flow 35
3.4. Infinite channel flow passing around a circular cylinder 39
4. A projection method using Axial Green function Method for the Navier-Stokes equations 44
4.1. A projection scheme for the incompressible Navier-Stokes flow 44
4.2. Separation of operators for the axial Green function method 45
4.3. Axial Green functions 47
4.4. Axial Green function formulation 52
4.5. Pseudo-pressure with the vanishing Neumann boundary condition 55
4.6. Numerical Implementation of the axial Green function formulation 58
5. Numerical Results for Navier-Stokes flows 59
5.1. Convergence 59
5.2. Lid-driven cavity flow 62
5.3. Backward-Facing Step flow 64
5.4. External flow 68
5.5. Flow in Tesla valve 74
5.6. 3D Lid-driven cavity flow 87
6. Conclusion 90
Appendices 93
Appendix A. Detailed calculation of the Green functions 93
Green function for the elliptic equation 93
Green function for the elliptic equation in the half-infinite interval 95
Green function for the reaction-diffusion equation 97
Appendix B. Proof of Theorems 101
proof of Theorem 1 101
proof of Theorem 2 101
Appendix C. M₁,₂ (x) and N₁,₂ (x) in (94) and (95) 103
Appendix D. Non-matching axis-parallel lines in the axial Green function method 104
Appendix E. Review of projection method 106
References 108
Table 1. Representative ODE for the separated equations from the two-dimensional steady Stokes flow. 19
Table 2. The external Stokes flow in the unbounded domain Ωext is compared between an *informative boundary condition and Dirichlet, utilizing (uext,₁∞, vext,₁∞) for velocity and pressure, respectively. All...[이미지참조] 37
Table 3. Half-infinite cavity flows are observed in two computational regions, Ω₋₂comp = Ωcavity ∩ {-2 〈 y 〈 0} and Ω₋₂comp = Ωcavity ∩ {-2.5 〈 y 〈 0}, which arise from informative boundary con-...[이미지참조] 38
Table 4. Drag coefficients exerted on the circular cylinder under the uniform channel flow for the case where R = 0.5. 43
Table 5. Primary, secondary, and ternary vortices in the lid-driven cavity flow are analyzed for Reynolds numbers Re = 1000, 5000, 7500, with an emphasis on vorticity strength, stream function, and location. 63
Table 6. Comparison of the wake length (Lsep), the separation angle (θsep), the drag coefficient (CD) when Re = 5, 10, 20, 40.[이미지참조] 74
Table 7. Comparison of the Strouhal number (St), lift (CL) and drag coefficient (CD)) for Reynolds numbers of 100 and 200.[이미지참조] 74
Figure 1. Unbounded domain Ω of an external flow is decomposed into Ωcomp and Ωinf. Axis-parallel lines are constructed at each A and B in Ωcomp: x-axis-parallel line (blue line) and y-axis-parallel line...[이미지참조] 18
Figure 2. One-dimensional configuration for the informative boundary condition in a half-infinite domain Ω∞ = (a, ∞): the computational region (──) is denoted by IR = (a, tR) in which (tk₋₁, tk)...[이미지참조] 24
Figure 3. The external flow passing around a circular cylinder with a radius of R: a square region except for the inner circular domain of radius R 〉 0 is the computational region Ωcomp.[이미지참조] 31
Figure 4. Informative and Dirichlet boundary conditions for external potential flow are outlined as follows: (a) the blue and red lines represent x- and y-axis-parallel lines, respectively, and have identical... 32
Figure 5. Potential flows of circulation γ passing around a circular cylinder with a radius R = 0.5 are analyzed in the case where U = 1: Numerical solutions for (a), (b), and (c) are calculated in... 33
Figure 6. Potential flows pass around cylinders in circular, elliptic, and star shapes, each with an area of πR². These cylinders are denoted by (a) Γcircle, (b) Γellipse, and (c) Γstar.. For cylinders with the...[이미지참조] 34
Figure 7. Convergence features are presented for the velocity and pressure of the external Stokes flow in the unbounded domain Ωext under an informative boundary condition. (a) The velocity is given in...[이미지참조] 36
Figure 8. Contour plots depicting the stream function and vorticity of a half-infinite cavity flow within Ωcavity = {-0.5 〈 x 〈 0.5, -∞ 〈 y 〈 0} are presented. The computational regions utilized are (a)...[이미지참조] 40
Figure 9. Stream values of the half-infinite cavity flow are observed along the section with x = 0 in two computational regions, namely Ω₋₂comp (the right zoomed-in box) and Ω₋₂.₅comp (the left zoomed-in box).[이미지참조] 41
Figure 10. Vorticities of the half-infinite cavity flow are observed along the x = 0 section for two computational regions, namely Ω₋₂comp (the right zoomed-in box) and Ω₋₂.₅comp (the left zoomed-in box).[이미지참조] 42
Figure 11. Unbounded uniform channel flow passing around a fixed circular cylinder with a radius R > 0: The computational region Ωcomp has limits of -L < x < L and -H < y < H. The walls...[이미지참조] 42
Figure 12. axis-parallel lines: (A) Axis-alined lines generate a set A comprising all cross points x inside Ω. (B) The minimal open line segments, X (blue line) and Y (red line) in Ω, that intersect x, are...[이미지참조] 53
Figure 13. Neumann Boundary Condition Implementation: (A) It is assumed that a line parallel to the axis touches the point x = (x, y) ∈ ∂Ω, and Ab represents the set of all such x's. (B) The x-axis-parallel...[이미지참조] 56
Figure 14. Two sets of axis-parallel lines are present: (A) uniform and (B) patterned in a 1h2h configuration. 60
Figure 15. Convergence orders of velocity, pressure, and vorticity are analyzed when only the initial conditions of velocities are given on two types of axis-parallel lines: (A) uniformly spaced lines, and... 61
Figure 16. Convergence orders of velocity, pressure, and vorticity are analyzed when the initial conditions of velocities and their derivatives are given on two types of axis-parallel lines: (A) uniformly... 61
Figure 17. Configuration of two-dimensional lid-driven cavity flow in Ωcav.[이미지참조] 64
Figure 18. u-velocity along the vertical centerline x = 0.5 (A, B, C) and v-velocity along the horizontal centerline y = 0.5 (D, E, F) are determined using uniformly distributed axis-parallel lines when Re=... 65
Figure 19. Streamlines and vorticity contours are presented for the lid-driven cavity flow at Reynolds numbers of 1000 in (A) and (D), 5000 in (B) and (E), and 7500 in (C) and (F). 66
Figure 20. (A) 127 x 127 randomly generated axis-parallel lines, parallel to the axis, are used to compare the stream function and vorticity of the lid-driven cavity flow at Re = 400. The results are compared... 67
Figure 21. Configuration of the backward-facing step flow problem 68
Figure 22. Velocity profiles were measured at the inlet channel for various Reynolds numbers, including Re = 100 (A), 500 (B), 1000 (C), and 1600 (D). 69
Figure 23. (A) velocity profiles for u and (B) vorticity profiles are presented across the channel for Re = 800, at x = 14h and x = 30h. 70
Figure 24. The lengths of (A) Normalized reattachment (X0/h) and (B) the first recirculating region on the upper wall (X1/h). 71
Figure 25. Pressure distribution and Streamlines for various Reynolds numbers (Re = 100 for (A), 500 for (B), 1000 for (C), and 1600 for (D)) are presented. 72
Figure 26. External flow domain ΩExt past a circular cylinder (A): stepwise axis-parallel lines (B) and zoomed-in axis-parallel lines around the cylinder body (C).[이미지참조] 73
Figure 27. (A) Vorticity and (B) pressure coefficient on the cylinder body for each Reynolds numbers. (Re = 5, 10, 20, 40) 75
Figure 28. Vorticity snapshots were captured at time t = 70, 80, 141, 250 for Reynolds numbers Re = 100. 76
Figure 29. Vorticity snapshots were captured at time t = 50, 60, 141, 250 for Reynolds numbers Re = 200. 77
Figure 30. (A) Configuration of the external flow past around an airfoil body with the angle of attack 30°, (B) axis-parallel line distribution for the external flow, and (C) the axis-parallel lines around the... 78
Figure 31. (A) vorticity and (B) pressure of the flow around the airfoil at t = 16 when Re = 1000. 79
Figure 32. Time evolution of drag and lift coefficients of the airfoil at the angle of attack 30° for Re = 1000. 80
Figure 33. (A) T45-R Tesla valve with forward/reverse flow direction and pressure measurement locations a and b, (B) Configuration of T45-R Tesla valve. All units in millimeters. 81
Figure 34. Diodicity versus Reynolds number was determined for the T45-R Tesla valve. 82
Figure 35. (A) Two-stage Tesla valve configuration (B) Four-state Tesla valve geometry (C) Diodicity as a function of Reynolds number for multi-stage Tesla valves. 83
Figure 36. The pressure within the two-stage Tesla valve at a Reynolds number of 300 for both forward (A) and reverse (B) flows. 84
Figure 37. Pressure along the centerline of the two-stage Tesla valve is analyzed for forward (A) and reverse (B) flow at Re = 300. 85
Figure 38. Velocity magnitude and streamlines in a four-stage Tesla valve at a Reynolds number of 300 for both forward (A) and reverse (B) flow directions. 86
Figure 39. (A) Configuration of 3D lid-driven cavity flow and (B) gradually accumulated axis-parallel lines used for computation. 87
Figure 40. u-velocity profiles are presented for the vertical center line of the cubic domain Ω₃DCV for Reynolds numbers of Re = 100(A), 400(B), and 1000(C).[이미지참조] 88
Figure 41. Iso-surfaces for different velocity magnitudes at Reynolds number 1000 are presented: (A) 0.3, (B) 0.25, (C) 0.2, and (D) 0.15. 89
Figure 42. Iso-surface for pressure and the y-component of vorticity (ωy) are presented: (A) pressure, (B) ωy[이미지참조] 90
Figure 43. Streamlines and magnitudes of u-velocities at x - y slices in (A), x - z slices in (B), and y - z slices in (C) are depicted for a Reynolds number of 400. 91
Figure 44. Non-matching axis-parallel lines along an interface Γ (black dashed line): virtual axis-parallel lines and its adjacent points at an interfacial point x.[이미지참조] 104