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1.3.5 the Kutta Condition ...17 Deift UniVersity ofTechnology Ship HydromechafliCS laboratory Library Mekelweg 2 26282 CD Deift Phone: +31 (0)15 2786873 E-mail: p.w.deheertUdelft.flI 13.04 LECTURE NOTES HYDROFOILS AND PROPELLERS* Justin E. Kerwin January 2OOl Contents i TWO DIMENSIONAL FOIL THEORY - i 1.1 Introduction 2 1.2Foil Geometry 3 1.3Conformal Mapping 9 1.3.1History 9 1.3.2Potential Flow Around A Circle 9 1.3.3Conformal Mapping for Dummies 13 1.3.4The Karman-Trefftz Mapping Function 15 1.3.5The Kutta Condition ....................17 1.3.6Pressure Distributions 19 1.3.7Lift and Drag 24 1.4Linearized Theory for a 2Dimensional Foil Section 26 *©Justjfl E. Kerwin 2001 t Web document updated March 9 i 1.41Problem Formulation 26 27 1.4.2Vortex and Source Distributions 1.4.3Glauert's Theory 31 1.4.4Example-The Flat Plate 35 1.45 Example-The Parabolic Mean Line 36 37 1.4.6The Design of Mean Lines-The NACA a-Series ......... 1.4.7Linearized Pressure Coefficient 40 1.4.8Comparison of Pressure Distributions 41 42 1.4.9Solution of the Linearized Thickness Problem 1.4.10 The Elliptical Thickness Form 43 1.4.11 The Parabolic Thickness Form 44 1.4.12 Superposition 45 47 1.4.13 Lighthill's Rule . 52 1.52-D Vortex Lattice Theory 53 1.5.1 Constant Spacing 53 1.5.2Cosine Spacing . 55 1.5.3Converting from ['to -y(x) 56 1.5.4Drag and Leading Edge Suction 62 1.5.5Adding Foil Thickness to VLM 66 1.5.6The Cavitation Bucket Diagram 74 2LIFTING SURFACES 75 2.1Introductory Concepts 78 2.2The Strength of the Free VortexSheet in the Wake 11 C, 81 2.3The velocity induced by a three-dimensionalvortex line. 84 2.4Velocity Induced by a Straight VortexSegment 87 2.5Linearized Lifting-Surface Theory for a PlanarFoil 87 2.5.1 Formulation of the Linearized Problem 89 2.5.2The Linearized Boundary Condition 90 2.5.3Determining the Velocity 91 2.5.4Relating the Bound and Free Vorticity 93 2.6Lift and Drag 99 2.7Lifting Line Theory 2.7.1 Glauert's Method 99 104 2.7.2Vortex Lattice Solution for the PlanarLifting Line 115 2.7.3The Prandtl Lifting Line Equation 121 2.8Lifting Surface Results 2.8.1 Exact Results 121 122 2.8.2Vortex Lattice Solution of theLinearized Planar Foil 3 PROPELLERS 133 134 3.1Inflow 136 3.2 Notation 140 3.3Actuator Disk 150 3.4Propeller Lifting Line Theory 157 3.4.1The Actuator Disk as a ParticularLifting Line 161 3.5Optimum Circulation Distributions 165 3.5.1 Assigning The Wake Pitch Angle /3 II 3.5.2Properties of Constant Pitch Helical Vortex Sheets 166 169 3.5.3The Circulation Reduction Factor . 3.5.4Application of the Goldstein Factor 172 3.6Lifting Line Theory for Arbitrary CirculationDistributions 175 3.6.1 Lerbs Induction Factor Method 175 3.7Propeller Vortex Lattice Lifting Line Theory 178 181 3.7.1Hub effects . 3.7.2The Vortex Lattice Actuator Disk 185 3.7.3Hub and Tip Unloading 185 4 COMPUTER CODELISTINGS 194 5 APPENDIX 218 5.1Derivation of Glauert's Integral 219 List of Figures 1 Illustration of notation for foil section geometry 2 Sample of tabulated geometry and flow datafor an NA CA mean line and thickness form. This 3 An example of a trailing edge modificationused to reduce singing. particular procedure is frequently used for U.S.Navy and commercial ap- plications..... iv 4 An example of a complete geometrical description of a foil section (includ- ing anti-singing trailing edge modifications) using a fourth order uniform B-spume. The symbols connected with dashed lies represent the B-spume control polygon which completely defines the shape of the foil. The result- ing foil surface evaluated from the .B-spline is shown asthe continuous curve. The upper 'curves show am enlargementof the leading and trailing edge regions. The complete foil is shown in the lower curve 8 5 Flow around a circle with zero circulation.The center of the circle is located at x = .3,y = 0.4. The circle passes through x a = 1.0. The flow angle of attack is 10 degrees. 10 6 Flow around a circle with circulation. The center of the circle is located at z = .3, y = 0.4. The circle passes through z = a = 1.0. Note thatthe rear stagnation point has moved to z = a 12 7 Flow around a Karman- Trefftz foil derived from the flow around a circle' shown in figure 6 with a specified tail angle of T = 25 degrees 14 8 Flow near the trailing edge. The figure on the left is for zero circulation Note the flow around the sharp trailing edge and the presence of a stagna- tion point on the upper surface. The figure on the right shows theresult of adjustïng the circulation to provide smooth flow at the trailing edge.. 17 9 Early flow visualization photograph showing the development of a starting vortex 19 10 Streamlines and pressure contours for a thin, highly cambered section at zero angle of attack.This section is symmetrical about mid-chord, and therefore has sharp leading and trailing edges. As expected, the pressure contours show low pressure on the upper surface (green) andhigh pressure on the lower surface (blue). 21 11 This is the same section as before, but at an angle of attack of 10 degrees. The flow pattern is no longer symmetrical, with high velocities and hence low pressures (red) near the leading edge 22 12 Close up view of the flow near the leading edge at an angle of attack of 10 degrees 23 13 Vertical distribution of the u velocity at the mid-chord of a constant strength vortex panel of strength 'y = 1 28 V 14 Illustration of the circulation path used to showthat the jump in u velocity is equal to the vortex sheet strength, -y. 29 15 Horizontal distribution of the y velocity along a constantstrength vortex panel of strength -y 1. 30 16 Horizontal distribution of the y velocity along a constantstrength source panel of strength= 1. 31 17 Horizontal distribution of the y velocity along a constantstrength vortex panel of strength -y = 1. 36 18 Enlargement of figure showing the differencebetween an NACA a = 1.0 and parabolic mean line near the leading edge. 42 19 Shape and velocity distribution forelliptical and parabolic thickness forms from linear theory.The thickness/chord ratio, t0/c = 0.1.The vertical scale of the thickness formplots has been enlarged for clarity 43 20 Comparison of surface velocity distributionsfor an elliptical thickness form with t0/c = 0.1 and t0/c = 0.2 obtainedfrom an exact solution and from linear theory 47 21 Local representation of the leading edge regionof a foil by a parabola with matching curvature at x 0. This is sometimes referred to as an'oscu- 48 lating parabola". 22 Surface velocity distribution nearthe leading edge of a semi-infinite parabola. 49 23 Vortex lattice approximation of the vortexsheets representing a marine propeller. 52 24 Arrangement of vortex positions, x, andcontrol point positions, x. The vortices are plotted as filed circles,and the control points are shown as open triangles. Thenumber of panels, N 8. 54 25 Comparison of exact solution and vortexlattice method for a fiat plate using 10 and 20 panels. The vortexsheet strength and total lift coefficient is exact. Increasing the numberof panels improves the resolution in the representation of 'y(x). 56 vi 26 Comparison of exact solution and vortex lattice method for a parabolic mean line using .10 and 20 panels. The vortex sheetstrength and total lift coefficient is exact. Increasing the number of paneLs improves the resolu- tion in the representation of y(x) 57 27 Comparison of exact solution and vortex lattice method for an NACA a = .8 mean line using 10 and 20 panels. The vortex sheet strengthand total lift coefficient is not exact, but very close to the analytic result. The error in is visible near the leading edge, where VLM cannot deal with the logarithmic singularity in slope of the mean line 58 28 Vector diagram of force components on a flat plate. For clarity, the angle of attack, a, has been drawn at an unrealistically high value of 30 degrees 60 29 Suction parameter C(x) for a flat plate computed with 8 and 64 vortex elements for unit angle of attack, a 61 30 Suction parameter C(x) for a flat plate, parabolic and NACA a = .8 mean line at unit lift coefficient, computed with 32 vortex elements 62 31 Comparison of source lattice and exact conformal mapping calculations of the pressure distribution around a symmetrical Karman- TreJftz foil. The foil was generated with x = 0.1, Yc = 0.0 and r = 5 degrees.Source lattice results are given for 20 panels (symbols) and 50 panels (continuous curve). The Scherer/Riegels version of Lighthill's leading edge correction has been applied. 63 32 Vortex lattice approximation of the vortex sheets representing a marine propeller 65 33 Pressure distributions for a cambered Karrnan Trefftz section at two dif- ferent angles of attack 66 34 Variation of pressure coefficient with angle of attack at several fixed chord- wise locations for a symmetrical Karrnan Trefftz section.The numbers indicate the approximate chorwdise locations, in percent of chord from the leading edge. The dashed curves are for the corresponding points on the lower surface. Since the foil is symmetrical, the curves for points on the lower surface are the mirror image of the corresponding points on the upper surface.
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