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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.

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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