kerwin_notes - 13.04 LECTURE NOTES HYDROFOILS AND...

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13.04 LECTURE NOTES HYDROFOILS AND PROPELLERS Justin E. Kerwin January 2001 Contents 1 TWO DIMENSIONAL FOIL THEORY 1 . 1 In t r odu c t i on. ... . ......... ........ ... 1 . 2 F o i lG e om e t ry .... 1 . 3 C on f o rm a lM app in g. 1 . 3 . 1 H i s t o ry. ....... 1.3.2 Potential Flow Around A Circle . . . . . . . . . . . . . . . . . . . 1.3.3 Conformal Mapping for Dummies . . . . . . . . . . . . . . . . . . 13± 1.3.4 The Karman-Trefftz Mapping Function . . . . . . . . . . . . . . . 15± 1.3.5 The Kutta Condition . . . . . . . . . . . . . . . . . . . . . . . . . 17± 1.3.6 Pressure Distributions . . . . . . . . . . . . . . . . . . . . . . . . 19± 1 . 3 . 7 L i f tandD r a g .............. 2 1.4 Linearized Theory for a 2–Dimensional Foil Section . . . . . . . . . . . . 26± c Justin E. Kerwin 2001 Web document updated March 9 i
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1.4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 26± 1.4.2 Vortex and Source Distributions . . . . . . . . . . . . . . . . . . . 27± 1 . 4 . 3 G l au e r t sTh e o ry . ....... .... ........ 3 1.4.4 Example–The Flat Plate . . . . . . . . . . . . . . . . . . . . . . . 35± 1.4.5 Example–The Parabolic Mean Line . . . . . . . . . . . . . . . . . 36± 1.4.6 The Design of Mean Lines-The NACA a-Series . . . . . . . . . . . 37± 1.4.7 Linearized Pressure Coefficient . . . . . . . . . . . . . . . . . . . . 40± 1.4.8 Comparison of Pressure Distributions . . . . . . . . . . . . . . . . 41± 1.4.9 Solution of the Linearized Thickness Problem . . . . . . . . . . . 42± 1.4.10 The Elliptical Thickness Form . . . . . . . . . . . . . . . . . . . . 43± 1.4.11 The Parabolic Thickness Form . . . . . . . . . . . . . . . . . . . . 44± 1 . 4 . 1 2Sup e rpo s i t i on ..... ......... 4 1 . 4 . 1 3L i gh th i l l sRu l e ............. 4 1.5 2-D Vortex Lattice Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 52± 1 . 5 . 1 C on s t an tSp a c in g 5 1 . 5 . 2 C o s eSp a c g . ............ 5 1.5.3 Converting from Γ n to γ ( x ) ........... .. 5 1.5.4 Drag and Leading Edge Suction . . . . . . . . . . . . . . . . . . . 56± 1.5.5 Adding Foil Thickness to VLM . . . . . . . . . . . . . . . . . . . 62± 1.5.6 The Cavitation Bucket Diagram . . . . . . . . . . . . . . . . . . . 66± 2 LIFTING SURFACES 74± 2 . 1 In t r odu c t o ryC c ep t s . ... 7 2.2 The Strength of the Free Vortex Sheet in the Wake . . . . . . . . . . . . 78± ii
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2.3 The velocity induced by a three-dimensional vortex line . . . . . . . . . . 81± 2.4 Velocity Induced by a Straight Vortex Segment . . . . . . . . . . . . . . 84± 2.5 Linearized Lifting-Surface Theory for a Planar Foil . . . . . . . . . . . . 87± 2.5.1 Formulation of the Linearized Problem . . . . . . . . . . . . . . . 87± 2.5.2 The Linearized Boundary Condition . . . . . . . . . . . . . . . . . 89± 2.5.3 Determining the Velocity . . . . . . . . . . . . . . . . . . . . . . . 90± 2.5.4 Relating the Bound and Free Vorticity . . . . . . . . . . . . . . . 91± 2 . 6 L i f tandD r a g .... ......... . ........ ... 9 2 . 7 L i f t in gL eTh e o ry 9 2 . 7 . 1 G l au e r t sM e th od ....... 9 2.7.2 Vortex Lattice Solution for the Planar Lifting Line . . . . . . . . 104± 2.7.3 The Prandtl Lifting Line Equation . . . . . . . . . . . . . . . . . 115± 2.8 Lifting Surface Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121± 2 . 8 . 1 Ex a c tR e su l t s ..... 1 2 2.8.2 Vortex Lattice Solution of the Linearized Planar Foil . . . . . . . 122± 3 PROPELLERS± 133 3 . 1 Infl ow . ...... ± ... 1 3 4 3 . 2 N o t a t i on . ....± 3 6 3 . 3 A c tu a t o rD i sk 1 4 0 ...± 3.4 Propeller Lifting Line Theory . . . . . . . . . . . . . . . . . . . . . . . . 150± 3.4.1 The Actuator Disk as a Particular Lifting Line . . . . . . . . . . . 157± 3.5 Optimum Circulation Distributions . . . . . . . . . . . . . . . . . . . . . 161± 3.5.1 Assigning The Wake Pitch Angle β w .. 1 6 iii
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3.5.2 Properties of Constant Pitch Helical Vortex Sheets . . . . . . . . 166± 3.5.3 The Circulation Reduction Factor . . . . . . . . . . . . . . . . . . 169± 3.5.4 Application of the Goldstein Factor . . . . . . . . . . . . . . . . . 172± 3.6² Lifting Line Theory for Arbitrary Circulation Distributions . . . . . . . . 175± 3.6.1 Lerbs Induction Factor Method . . . . . . . . . . . . . . . . . . . 175± 3.7² Propeller Vortex Lattice Lifting Line Theory . . . . . . . . . . . . . . . . 178± 3 . 7 . 1 Hubeff e c t s . .............. ........ .......
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kerwin_notes - 13.04 LECTURE NOTES HYDROFOILS AND...

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