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Irrotional Flow of Frictionless Fluids , Mostly of Invariable Density

Engineering Applications and Design
Hydraulics Engineering and Design
Fluids Engineering and Design

Irrotional Flow of Frictionless Fluids, Mostly of Invariable Density

Irrotational Definition: 1 : not rotating or involving rotation. 2 : free of vortices irrotational flow.

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This report is a wide-ranging account of the fundamentals of the potential flow of frictionless fluids, and its value is greatly enhanced by the large number of actual examples included in the text. It will be of great value both to the practicing engineer concerned with fluid flows and to the student.

Dr. Earle H. Kennard was formerly Chief Scientist in the Hydromechanics Laboratory at the David Taylor Model Basin, and later head of its Structural Mechanics Laboratory. Through- out his service at the Model Basin he devoted his efforts to the advancement of knowledge in these fields and to the physics of underwater explosions. The value of his work in these areas and in the associated one of structural vibration is well attested by the many papers and reports which he has published.

He has also devoted much time to the education and training of the younger members of the staff. His educational work, indeed, began much earlier as a professor at Cornell University, and unnumbered students have profited from his well-known text book on physics.

His colleagues have learned to respect his judgments, to enjoy his friendship and to appreciate his wit, even though it is sometimes somewhat sharp!

This report is a typical example of Earle Kennard's clear, explanatory writing, combined nevertheless with an admirable economy of words. It is a great pleasure to his many friends and admirers to see it published while Dr. Kennard, though over 80 years of age, is still active and still continuing to work in the field of structural vibration. We look forward to the privilege of making more of his work available to the profession of naval architecture through the medium of Model Basin reports, for it is upon such people as Dr. Kennard and the results of their research that the reputation and image of the establishment depends.

TOC

INTRODUCTION xxiv

CHAPTER I - FUNDAMENTALS OF THE IRROTATIONAL FLOW OF FRICTIONLESS FLUIDS
1. Particle Velocity and Stream Lines 1
2. The Equation of Continuity 2
3. Euler's Equations of Motion 4
4. Boundary Conditions 6
5. Rotational Motion; the Circulation 6
6. The Velocity Potential for Irrotational Flow 8
7. The Laplace Equation 12
8. Some Properties of Irrotational Flow 13
9. The Pressure Equation for Irrotational Flow 15
10. The Bernoulli Equation for Steady Irrotational Flow 17
11. The Pressure Equation for Rotating Boundaries 18
12. Three-Dimensional Sources, Sinks, and Dipoles 19
13. Two-Dimensional Flow 20
14. Two-Dimensional Flow in Multiply Connected Spaces 24
15. Two-Dimensional or Line Sources, Sinks, Vortices, and Dipoles 26
16. Axisymmetric Three-Dimensional Flow 27
17. Kinetic Energy of the Fluid 29 18. Units of Measurement 32

CHAPTER II - THE USE OF COMPLEX FUNCTIONS IN HYDRODYNAMICS 19.
Complex Numbers 33
20. Some Common Functions of z 36
21. Powers of z 37
22. Regular Functions of a Complex Variable 38
23. Conformal Representation or Mapping 39
24. Examples of Conformal Transformations 42
25. Relation of Regular Functions to Two-Dimensional Irrotational Flow 46
26. The Transformation of Irrotational Motions 50
27. The Laurent Series 51
28. Complex Integration 52
29. The Cauchy Integral Theorem 53
30. Singular Points and Residues 56
31. The Schwarz-Christoffel Transformation 57
32. The Hyperbolic Functions 66
33. Some Series 69

CHAPTER III - CASES OF TWO-DIMENSIONAL FLOW 34.
Notation and Form of Presentation 70
SOME SIMPLE TYPES OF FLOW 74
35. Uniform Motion 74
36. Hyperbolic Flow 75
37. Line Dipole 75
38. Line Quadrupole 78
39. Flow in an Angle 79
40. Logarithmic Flow 82

LINE SINGULARITIES IN COMBINATION 86
41. Line Source and Sink; Line Vortex Pair 86
42. Circulating Flow: Cylinders, Vortices, a Wall 91
43. Line Source and Plane Wall 98
44. Row of Equal Sources or Vortices; Source Midway between Walls or on One Wall; Contracted Channel 100
45. Alternating Vortices or Sources; Vortex Midway between Walls 107
46. Row of Equal Line Dipoles on a Transverse Axis; Dipole Midway between Walls, with Parallel Axis; Flow Past Cylinder between Walls or through a Grating 108
47. Row of Equal Line Dipoles on a Longitudinal Axis; Flow Past a Grating 112
48. Alternating Line Dipoles; Dipole Midway between Walls, with Perpendicular Axis 114
49. Line Source, Vortex or Dipole Anywhere between Parallel Walls 115
50. Two Line Dipoles in Opposition; Dipole and a Wall 116
51. Line Source and Cylindrical Barrier 117
52. Line Dipole and Cylindrical Barrier 119
53. Line Source in Uniform Stream 121
54. Line Source and Sink in Uniform Stream 124
55. Vortex Pair in a Uniform Stream 127
56. Other Combinations Involving Line Sources or Dipoles 129
57. Sheets of Line Sources or Vortices 130
58. Source Sheet in a Uniform Stream 131
59. The Simpler Singularities and Their Transformation 133
60. Line Singularity in an Angle

134 TRANSFORMATIONS DEFINED INVERSELY 136
61. Ellipses and Hyperbolas 136
62. Straight Spout 140
63. Diverging Spout 142
64. Two-Dimensional Pitot Tube 143
65. Lamina between Walls 146
66. Laminas or Cylinders and Surfaces 148

CIRCULAR CYLINDERS 148
67. Symmetrical Flow Past a Circular Cylinder; Dipole in a Parallel Stream, or Inside a Coaxial Shell 148
68. Translation of a Circular Cylinder 151
69. Flow with Circulation Past a Circular Cylinder 152
70. Translation of a Circular Cylinder with Circulation 156
71. Cylinder and Vortices in a Stream 158

FORCES ON CYLINDERS 160
72. The Distant Motion 160
73. Lift on a Cylinder; the Kutta-Joukowski Theorem 162
74. The Blasius Theorem 164
75. The Lagally Theorems 168
76. Kinetic Energy in Translational Motion 170

AIRFOILS 173
77. The Joukowski Transformation 173
78. Circular Arcs by the Joukowski Transformation 177
79. The Joukowski Airfoils 183
80. Improved Airfoils 189

VARIOUS CYLINDERS 190
81. Circles Into Ellipses 190
82. Elliptic Coordinates 192
83. Flow Past an Elliptic Cylinder 196
84. Elliptic Cylinder in Translation 201
85. Flow Past a Plane Lamina 204
86. Plane Lamina in Translation 206
87. Parabolic Cylinders 208
88. The Circular-arc Transformation 210
89. Circular-arc Cylinder, Boss or Groove 213
90. Double Circular Cylinder, or Cylinder against a Wall 218
91. Cylinders of Other Forms 222
92. Two Equal Line Dipoles with Axes Longitudinal; Flow Past One or Two Similar Cylinders 225
93. Two Equal Line Dipoles with Axes Transverse; Flow Past One or between Two Similar Cylinders 228
94. Two Circular Cylinders in a Stream; Cylinder and Wall 232
95. Slender Circular Cylinders Moving Independently, or Near a Wall 235
96. Two or Three Laminas 241
97. Gratings 242
98. Vortices Near Cylinders or Walls 244

ROTATING BOUNDARIES 245
99. Moving Boundaries 245
100. Rotating Channel 247
101. Rotating Angle 248
102. Fluid within a Rotating Sector 250
103. Motion within a Rotating Triangular Prism 253
104. Two Coaxial Cylinders 255
105. Fluid within a Rotating Shell of Elliptic or Other Shape 255
106. Rotation of Elliptic Cylinder or Lamina 258

CHANNELS 262
107. Flow Past a Square End or an Offset 262
108. Straight Channel Varied in Width 266
109. Channels of Various Forms 270

FREE STREAMLINES
110. Nature of Free Streamlines 270
111. Efflux from a Two-Dimensional Orifice 272
112. Two-Dimensional Borda's Mouthpiece 278
113. Infinitely Wide Stream Incident Normally on a Plane Lamina 281
114. Infinite Stream Oblique to a Plane Lamina 285
115. Infinite Stream on a V-Shaped Lamina 288
116. Jet on a Wall 291
117. Other Free-Streamline Problems 293

CHAPTER IV - CASES OF THREE-DIMENSIONAL FLOW

118. Introduction 298
119. Potential and Stream Functions for a Uniform Stream, a Point Source or a Point Dipole 299
120. Variable Point Source, or Flow Near a Spherical Cavity 303
121. Point Source in a Uniform Stream 305
122. Point Source and Sink in a Uniform Stream; Rankine Solids 308
123. Line Distributions of Point Sources 310
124. Line of Point Sources in a Stream 312
125. Airship Forms 314
126. Space Distributions of Point Sources 318
127. Translation of a Sphere in Infinite Fluid 318
128. Streaming Flow Past a Sphere 320
129. Sphere within a Concentric Sphere 322
130. Sphere and a Wall; Two Spheres 323
131. Point Dipoles Near a Sphere 326
132. Line of Transverse Dipoles 330
133. Transverse Flow Past a Solid of Revolution 332
134. Point Source Near a Sphere 335
135. Boundary Conditions in Rotation 338
136. General Formulas for Orthogonal Curvilinear Coordinates 340
137. Ovary Ellipsoids (or Prolate Spheroids) 346
138. Planetary Ellipsoids (or Oblate Spheroids) and Circular Disks 356
139. Circular Aperture 369
140. Rotating Ellipsoidal Shell 371
141. Ellipsoid with Unequal Axes 373
142. Ellipsoid Changing Shape 374
143. Flow Past a Paraboloid 375
144. Axisymmetric Jets 378
145. Other Three-Dimensional Cases 379

CHAPTER V - COEFFICIENTS OF INERTIA

146. Effects of Fluid Inertia 380
147. Notes on Units 383
148. Table of Energies and Inertia Coefficients 384
A. Two-Dimensional Cases 385
B. Three-Dimensional Cases 389
REFERENCES 396


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