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13.04

LECTURE

NOTES

HYDROFOILS

AND

PROPELLERS

Justin

E.

Kerwin

January2001

Contents

1 TWODIMENSIONAL FOILTHEORY 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Foil Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Conformal Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.2 Potential FlowAround A Circle . . . . . . . . . . . . . . . . . . . 9

1.3.3 Conformal Mapping forDummies . . . . . . . . . . . . . . . . . . 13

1.3.4 TheKarmanTrefftzMapping Function .. .. .. .. .. .. .. . 15

1.3.5 TheKuttaCondition . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.3.6 PressureDistributions . . . . . . . . . . . . . . . . . . . . . . . . 19

1.3.7 Lift and Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.4 LinearizedTheoryfor a2Dimensional FoilSection .. .. .. .. .. .. 26

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1.4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.4.2 Vortex andSourceDistributions. . . . . . . . . . . . . . . . . . . 27

1.4.3 Glauerts Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1.4.4 ExampleThe Flat Plate . . . . . . . . . . . . . . . . . . . . . . . 35

1.4.5 ExampleThe ParabolicMeanLine . . . . . . . . . . . . . . . . . 36

1.4.6 TheDesignofMeanLinesTheNACAaSeries. . . . . . . . . . . 37

1.4.7 LinearizedPressure Coefficient. . . . . . . . . . . . . . . . . . . . 40

1.4.8 ComparisonofPressureDistributions. .. .. .. . .. .. .. .. 41

1.4.9 SolutionoftheLinearizedThicknessProblem .. . .. .. .. .. 42

1.4.10 TheEllipticalThickness Form . . . . . . . . . . . . . . . . . . . . 43

1.4.11 The Parabolic Thickness Form . . . . . . . . . . . . . . . . . . . . 44

1.4.12 Superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

1.4.13 Lighthills Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

1.5 2D VortexLattice Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 52

1.5.1 Constant Spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . 531.5.2 Cosine Spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

1.5.3 Convertingfrom n to(x) . . . . . . . . . . . . . . . . . . . . . 55

1.5.4 Drag andLeading EdgeSuction . . . . . . . . . . . . . . . . . . . 56

1.5.5 Adding Foil Thickness to VLM . . . . . . . . . . . . . . . . . . . 62

1.5.6 The Cavitation BucketDiagram. . . . . . . . . . . . . . . . . . . 66

2 LIFTINGSURFACES 74

2.1 Introductory Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

2.2 TheStrengthofthe Free VortexSheetinthe Wake .. . . .. .. .. .. 78

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2.3 The velocity induced byathreedimensionalvortexline. . .. .. .. .. 81

2.4 Velocity Induced by aStraight VortexSegment . . . . . . . . . . . . . . 84

2.5 LinearizedLiftingSurfaceTheoryfor aPlanar Foil .. . . .. .. .. .. 87

2.5.1 FormulationoftheLinearizedProblem .. .. .. . .. .. .. .. 87

2.5.2 TheLinearizedBoundaryCondition . . . . . . . . . . . . . . . . . 89

2.5.3 Determining the Velocity. . . . . . . . . . . . . . . . . . . . . . . 90

2.5.4 RelatingtheBoundand Free Vorticity . . . . . . . . . . . . . . . 91

2.6 Lift and Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

2.7 Lifting Line Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

2.7.1 Glauerts Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

2.7.2 VortexLatticeSolutionforthePlanarLiftingLine . . . . . . . . 104

2.7.3 ThePrandtl Lifting Line Equation . . . . . . . . . . . . . . . . . 115

2.8 LiftingSurfaceResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

2.8.1 Exact Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

2.8.2 VortexLatticeSolutionoftheLinearizedPlanar Foil . . . . . . . 122

3 PROPELLERS 133

3.1 Inflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

3.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

3.3 Actuator Disk 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4 Propeller Lifting Line Theory . . . . . . . . . . . . . . . . . . . . . . . . 150

3.4.1 TheActuatorDiskas a ParticularLiftingLine . . . . . . . . . . . 157

3.5 OptimumCirculationDistributions . . . . . . . . . . . . . . . . . . . . . 161

3.5.1 AssigningThe WakePitch Anglew . . . . . . . . . . . . . . . . 165

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3.5.2 PropertiesofConstantPitch Helical VortexSheets . . . . . . . . 166

3.5.3 TheCirculationReduction Factor . . . . . . . . . . . . . . . . . . 169

3.5.4 Application of theGoldstein Factor . . . . . . . . . . . . . . . . . 172

3.6 LiftingLineTheoryforArbitraryCirculationDistributions. . . . . . . . 175

3.6.1 LerbsInduction FactorMethod . . . . . . . . . . . . . . . . . . . 175

3.7 Propeller VortexLatticeLiftingLineTheory . . . . . . . . . . . . . . . . 178

3.7.1 Hub effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

3.7.2 The VortexLatticeActuator Disk. . . . . . . . . . . . . . . . . . 185

3.7.3 Hub and TipUnloading . . . . . . . . . . . . . . . . . . . . . . . 185

4 COMPUTERCODELISTINGS 194

5 APPENDIX 218

5.1 Derivation ofGlauerts Integral . . . . . . . . . . . . . . . . . . . . . . . 219

ListofFigures

1 Illustration of notation for foil section geometry. . . . . . . . . . . . . . . 3

2 Sample of tabulated geometry and flow data for an NACA mean line andthickness form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 An example of a trailing edge modification used to reduce singing. Thisparticular procedure is frequently used for U.S. Navy and commercial ap-plications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

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4 An example of a complete geometrical description of a foil section (includ-ing antisinging trailing edge modifications) using a fourth order uniformBspline. The symbols connected with dashed lies represent the Bsplinecontrol polygon which completely defines the shape of the foil. The result-ing foil surface evaluated from the Bspline is shown as the continuous

curve. The upper curves show an enlargement of the leading and trailingedge regions. The complete foil is shown in the lower curve. . . . . . . . . 8

5 Flow around a circle with zero circulation. The center of the circle islocated at x= .3, y = 0.4. The circle passes through x= a= 1.0. Theflow angle of attack is 10 degrees. . . . . . . . . . . . . . . . . . . . . . . 10

6 Flow around a circle with circulation. The center of the circle is locatedat x= .3, y = 0.4. The circle passes through x=a= 1.0. Note that therear stagnation point has moved to x =a. . . . . . . . . . . . . . . . . . . 12

7 Flow around a KarmanTrefftz foil derived from the flow around a circleshown in figure 6 with a specified tail angle of = 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 the resultof adjusting the circulation to provide smooth flow at the trailing edge. . . 17

9 Early flow visualization photograph showing the development of a startingvortex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

10 Streamlines and pressure contours for a thin, highly cambered section atzero angle of attack. This section is symmetrical about midchord, andtherefore has sharp leading and trailing edges. As expected, the pressurecontours show low pressure on the upper surface (green) and high pressureon 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 hencelow 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 theu velocity at the midchord of a constant strengthvortex panel of strength = 1. . . . . . . . . . . . . . . . . . . . . . . . 28

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14 Illustration of the circulation path used to show that the jump in uvelocityis equal to the vortex sheet strength, . . . . . . . . . . . . . . . . . . . . 29

15 Horizontal distribution of the v velocity along a constant strength vortexpanel of strength= 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

16 Horizontal distribution of the v velocity along a constant strength sourcepanel of strength= 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

17 Horizontal distribution of the v velocity along a constant strength vortexpanel of strength= 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

18 Enlargement of figure showing the difference between an NACA a = 1.0and parabolic mean line near the leading edge. . . . . . . . . . . . . . . . 42

19 Shape and velocity distribution for elliptical and parabolic thickness forms

from linear theory. The thickness/chord ratio, to/c

= 0.1. The verticalscale of the thickness form plots has been enlarged for clarity. . . . . . . . 43

20 Comparison of surface velocity distributions for an elliptical thickness formwith to/c= 0.1 and to/c= 0.2 obtained from an exact solution and fromlinear theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

21 Local representation of the leading edge region of a foil by a parabola withmatching curvature at x= 0. This is sometimes referred to as an oscu-lating parabola. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

22 Surface velocity distribution near the leading edge of a semiinfinite parabola. 49

23 Vortex lattice approximation of the vortex sheets representing a marinepropeller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

24 Arrangement of vortex positions, xv, and control point positions, xc. Thevortices are plotted as filed circles, and the control points are shown asopen triangles. The number of panels, N= 8. . . . . . . . . . . . . . . . 54

25 Comparison of exact solution and vortex lattice method for a flat plateusing 10 and 20 panels. The vortex sheet strength and total lift coefficient

is exact. Increasing the number of panels improves the resolution in therepresentation of (x). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

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26 Comparison of exact solution and vortex lattice method for a parabolicmean line using 10 and 20 panels. The vortex sheet strength and total liftcoefficient is exact. Increasing the number of panels improves the resolu-tion in the representation of (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 strength and totallift coefficient is not exact, but very close to the analytic result. The errorin is visible near the leading edge, where VLM cannot deal with thelogarithmic singularity in slope of the mean line. . . . . . . . . . . . . . 58

28 Vector diagram of force components on a flat plate. For clarity, the angleof attack, , 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 vortexelements for unit angle of attack, . . . . . . . . . . . . . . . . . . . . . 61

30 Suction parameterC(x) for a flat plate, parabolic and NACAa =.8 meanline at unit lift coefficient, computed with 32 vortex elements . . . . . . . 62

31 Comparison of source lattice and exact conformal mapping calculations ofthe pressure distribution around a symmetrical KarmanTrefftz foil. Thefoil was generated with xc = 0.1, yc = 0.0 and = 5 degrees. Sourcelattice results are given for 20 panels (symbols) and 50 panels (continuouscurve). The Scherer/Riegels version of Lighthills leading edge correctionhas been applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

32 Vortex lattice approximation of the vortex sheets representing a marinepropeller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

33 Pressure distributions for a cambered Karman 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 Karman Trefftz section. The numbersindicate the approximate chorwdise locations, in percent of chord from theleading edge. The dashed curves are for the corresponding points on thelower surface. Since the foil is symmetrical, the curves for points on the

lower surface are the mirror image of the corresponding points on the uppersurface. The foil was generated with xc= 0.1, yc= 0.0, = 10 degrees. . 68

35 This is the same data as is shown in Figure 34, except that it is plottedfor 180 chordwise positions on the foil. Note that a well defined envelopecurve is now apparent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

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36 This is the same presentation of data as shown in Figure 35, but for thecambered Karman Trefftz foil computed in Figure 33. The envelope curveis no longer symmetrical, and shows two distinct knuckles at angles ofattack of approximately +3 and 2 degrees.The mapping parameters arexc=

.05, yc= 0.1, = 10 degrees. . . . . . . . . . . . . . . . . . . . . . 70

37 Here is the same type of data presentation, but for a nearly ogival foil sec-tion. The mapping parameters in this case are xc= 0.01, yc= 0.1, = 2 0degrees. The region within the envelope curve has now narrowed consider-ably in comparison to Figure 36. . . . . . . . . . . . . . . . . . . . . . . . 71

38 Sample family of bucket diagrams for NACA66 thickness forms with NACAa = .8 mean lines, all with a camber ratio of fo/c = 0.02. The sectionthickness/chord ratios (labeled as on the chart) range from to/c= 0.02to to/c= 0.20. The data is from T. Brockett, Minimum Pressure En-velopes for Modified NACA66 Sections with NACA a = .8 Camber and

BuShips Type I and Type II Sections, DTMB Report 1780, February 1966. 72

39 Design chart for optimum NACA66 sections from Brockett. The left handplot shows CP(min) along the xaxis versus to/calong the y axis. Thelines on the graph are for constant camber ratio, fo/c ranging from zeroto 0.06. The right hand graph shows the resulting width, in degrees, of thecavitation free range within the bucket. . . . . . . . . . . . . . . . . . . . 73

40 A lifting surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

41 Velocity diagram in the tangent plane. . . . . . . . . . . . . . . . . . . . . 76

42 Relating to Velocity Difference. . . . . . . . . . . . . . . . . . . . . . . 77

43 Velocity and vortex sheet strength for the special cases of twodimensionalflow and free vortex flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

44 General case Bound and free vorticity is present. . . . . . . . . . . . . 79

45 Circulation path used to determine the strength of the free vorticity in thewake. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

46 Notation for velocity, V

at point P

(x,y,z) induced by a volume distributionof vorticity (, , ) contained in volume V. . . . . . . . . . . . . . . . . 81

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47 Development of a vortex line. On the left is a volume distribution of vor-ticity . In the middle, the volume hs been put through a pasta machineto form a noodle with cross section area da. On the right, the noodle hasbeen turned into angels hair, with zero cross sectional area and infinitevorticity, but with the total circulation kept fixed. . . . . . . . . . . . . . . 83

with the xaxis coincident with the vortex, and the field point, P locatedon the yaxis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

48 Notation for a straight line vortex segment using a local coordinate system

49 Normalized velocity, 2yw(x, y)/ induced by a straight vortex segment. . 86

50 Notation for a planar hydrofoil. . . . . . . . . . . . . . . . . . . . . . . . 87

51 Cut through foil section at fixed spanwise location, y. . . . . . . . . . . . 88

52 Circulation contours to get free vorticity on the foil. . . . . . . . . . . . . 91

53 Control volume for momentum analysis for lift. . . . . . . . . . . . . . . 93

54 Control volume for kinetic energy far downstream. . . . . . . . . . . . . . 96

55 Concentration of bound vorticity along a lifting line. . . . . . . . . . . . . 97

56 Interpretation of lift and drag in terms of local flow at a lifting line. . . . 98

57 Plot of first four terms of Glauerts circulation series. . . . . . . . . . . . 101

58 Plot of velocity induced by first four terms of Glauerts circulation series. 101

59 Notation for a vortex lattice lifting line. In this case, there are 8 uniformlyspaced panels, with a 1/4 panel inset at each end. . . . . . . . . . . . . . 105

60 Spanwise distribution of velocity induced by a vortex lattice. The spacingis uniform, with ten panels and 25% tip inset. Due to symmetry, only halfthe span is shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

lifting line with a1= 1.0. The solution was obtained with 8 panels, using

uniform spacing with zero tip inset. . . . . . . . . . . . . . . . . . . . . . 109

61 Comparison of vortex lattice and exact results for an elliptically loaded

lifting line with a1= 1.0. The solution was obtained with 8 panels, usinguniform spacing with 25 % tip inset. . . . . . . . . . . . . . . . . . . . . 110


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63 Comparison of vortex lattice and exact results for an elliptically loadedlifting line with a1= 1.0. The solution was obtained with 8 panels, usingcosine spacing with central control points. . . . . . . . . . . . . . . . . . . 110


lifting line with a1= 1.0. The solution was obtained with 64 panels, usingcosine spacing with central control points. . . . . . . . . . . . . . . . . . . 111

65 Comparison of vortex lattice and exact results for an elliptically loadedlifting line with a1= 1.0. The solution was obtained with 8 panels, usingcosine spacing with cosine control points. . . . . . . . . . . . . . . . . . . 111

66 Comparison of vortex lattice and exact results for a tipunloaded lifting linewith a1 = 1.0 and a3 = 0.2. The solution was obtained with 8 panels,using cosine spacing with cosine control points. . . . . . . . . . . . . . . . 113

67 Comparison of vortex lattice and exact results for a tipunloaded lifting linewith a1= 1.0 and a3= 0.2. The solution was obtained with 32 panels,using cosine spacing with cosine control points. . . . . . . . . . . . . . . . 114

68 Lift slope, dCL/d, of an elliptic wing as a function of aspect ratio, A.(from Van Dyke 1975) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

69 Effect of planform shape on spanwise distribution of circulation obtainedfrom Prandtls lifting line equation. The foils all have an aspect ratio of= 4, and are at unit angle of attack. Equation 146 was used withM= 32,which is more than enough for a converged solution. . . . . . . . . . . . . 119

70 Notation for a vortex lattice solution for a rectangular foil. . . . . . . . . 122

71 Vortex lattice grid for a rectangular foil with aspect ratio A = 2. In thisexample, there are 32 spanwise and 16 chordwise panels. The plot on theupper right is an enlargement of the starboard tip near the trailing edge. . 124

72 Convergence of vortex lattice calculation for rectangular foil with aspectratio 1.0. Tabulated values of dCL/d. Each row shows convergence withnumber of chordwise vortices. Each column shows convergence with num-ber of spanwise panels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

73 Vortex lattice grid for a circular foil with an8 8 grid. . . . . . . . . . . 125

74 Vortex lattice grid for a circular foil with 64 spanwise and 32 chordwisepanels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

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75 Enlargement of the tip region of the vortex lattice grid for a circular foilwith 64 spanwise and 32 chordwise panels. . . . . . . . . . . . . . . . . . 128

76 Vortex lattice grid for a swept, tapered foil. The root chord is cr/s= 0.5and the tip chord is ct = 0.2. The leading edge is swept back 45 degrees.

The grid consists of 16 spanwise and 8 chordwise panels. One particularhorseshoe element is highlighted. . . . . . . . . . . . . . . . . . . . . . . . 129

77 Vortex lattice grid for a swept, untapered foil. The root chord iscr/s= 0.2and the tip chord is ct = 0.2. The leading edge is swept back 45 degrees.The grid consists of 16 spanwise and 8 chordwisae panels. . . . . . . . . . 130

78 Vortex lattice grid for a swept, untapered foil. The root chord iscr/s= 0.2and the tip chord is ct= 0.2. The leading edge is swept forward 45 degrees.The grid consists of 16 spanwise and 8 chordwisae panels. . . . . . . . . . 130

79 The effect of sweep on the spanwise circulation distribution. . . . . . . . . 131

80 Typical nominal axial wake field for a singlescrew container ship . . . . . 135

81 Propeller coordinate system and velocity notation. . . . . . . . . . . . . . 137

82 Control volume for actuator disk momentum calculation. The stream tubecontraction has been exaggerated for clarity. . . . . . . . . . . . . . . . . 143

83 Results of numerical calculation of slipstream radius and velocity field inthe plane of the disk and far downstream. The thrust coefficient is CT = 2.0.145

84 Ultimate slipstream radius as a function of thrust coefficient, CTfromEq. 183 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

85 Stream tubes near the tip of an actuator disk in static thrust, from Schmidtand Sparenberg. Note that the tip streamtube (labeled 1.0) initially goesupstream. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

86 Control volume for actuator disk energy balance . . . . . . . . . . . . . . 148

87 Efficiency as a function of thrust coefficient for the general case of anactuator disk with swirl. The curve for J= 0 corresponds to Eq. 190. . . 149

88 Illustration of the concept of a liftingline propeller as a limit of vanishingchord length. The radial distribution of blade circulation, (r) remains thesame, so that the strength of the trailing vortex sheet, f(r) is unchanged. 150

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89 Velocity and force diagram at a particular radial position on a lifting line. 151

90 Velocity induced on a lifting line at radius rc by a set of semiinfinitehelical vortices originating at rv= 1.0. The number of blades in this caseis Z = 5. Results are shown for pitch angles w = 10,20,30,40,50,60degrees. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

91 Effect of blade number on the velocity induced on a lifting line at radiusrc by a set of semiinfinite helical vortices originating at rv = 1.0. Thepitch angle is w = 30 degrees. Results are also shown for an infinitenumber of blades from Equations 206207. The total circulation, Z, iskept constant as the blade number is varied, and matches the value usedfor the five bladed propeller shown in Figure 90 . . . . . . . . . . . . . . . 156

92 Axial induction factors for a 5 bladed propeller derived from Figure 90.The enlarged plot shows the local behavior near rc/rv= 1. The analytical

limit of ia= cos w is plotted as square symbols on the graph. . . . . . . 158

93 Efficiency versus advance coefficient for a five bladed propeller with opti-mum radial distribution of circulation in uniform flow. Results are givenfor inviscid flow, and for viscous flow with sectional Lift/Drag ratios of 25and 50. The actuator disk result is shown as the symbol plotted at Js= 0.0.164

94 Induced velocities resolved into components normal to and along the helicalsurface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

95 Circulation path relating circulation around blades to circumferential mean

tangential velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

96 Prandtls simplified representation of the flow induced by helical vortex sheets.171

97 Example of Circulation Reduction Factors . . . . . . . . . . . . . . . . . 172

98 Kramer Diagram for Ideal Propeller Efficiency . . . . . . . . . . . . . . . 174

99 Geometric representation of the Glauert cosine transformation . . . . . . 176

100 Illustration of the image of a 2D point vortex in a circle of radius rh. The

vortex is at radius r, while the image is at radius ri = rh2

/r. If the twovortices have equal and opposite strengths, the normal (radial) componentof the velocity induced by the pair of vortices cancels at all points on thecircle of radius rh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

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101 Sample input data file for PVL. This file was used to generate the resultsplotted in figure 102 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

102 Lifting line results for a 5 bladed propeller obtained with the PVLcode.In this example, there is no hub, and a Lerbs optimum circulation distri-

bution has been selected. CT = 1.0 CP = 1.3432 KT = 0.2513 KQ =0.0430 Va/Vs= 0.8526 = 63.47%. . . . . . . . . . . . . . . . . . . . . . 186

103 Lifting line results for a 5 bladed propeller obtained with the PVLcode. Inthis example, there is an image hub, and a Lerbs optimum circulation dis-tribution has been selected. CT = 1.0 CP = 1.3744 KT = 0.2513 KQ=0.0440 Va/Vs = 0.8526 = 62.03%. The efficiency has been reducedslightly due to hub vortex drag. . . . . . . . . . . . . . . . . . . . . . . . . 187

104 Propeller operating alone. A substantial cavitating hub vortex is evident. . 188

105 Preswirl stator operating alone. A substantial hub vortex is again evident.The sign of this vortex is opposite from the one shown in figure 104. . . . 188

106 Propeller and stator operating together. The hub vortex has been canceled. 189

107 Lifting line results for a 5 bladed propeller obtained with the PVLcode.In this example, there is an image hub, and a Lerbs optimum circulationdistribution has been modified to unload the tip, using HT = 1.0. CT =1.0 CP = 1.4391 KT = 0.2513 KQ = 0.0461 Va/Vs = 0.8526 =59.24%. The efficiency has been further reduced due to tip unloading. Notethe very different shape of the axial induced velocity distribution. . . . . . 190

108 Lifting line results for a 5 bladed propeller obtained with the PVLcode.In this example, there is an image hub, and a Lerbs optimum circula-tion distribution has been modified to unload the hub, using HR = 1.0.CT = 1.0 CP = 1.3442 KT = 0.2513 KQ= 0.0431 Va/Vs= 0.8526 =63.43%. The efficiency has actually improved, since the reduced hub load-ing reduces the hub vortex drag. . . . . . . . . . . . . . . . . . . . . . . . 191

109 Lifting line results for a 5 bladed propeller obtained with the PVLcode.In this example, there is an image hub, and a Lerbs optimum circulationdistribution has been specified. In addition, the idealized counter rotating

propeller option has been selected, so that there are no tangential inducedvelocities. Note that the circulation near the hub has been greatly increased.CT = 1.0 CP = 1.2532 KT = 0.2513 KQ= 0.0401 Va/Vs= 0.8526 =68.03%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

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110 Lifting line results for a 25 bladed propeller obtained with the PVLcode.The induced velocities correspond to those of an infinite bladed propeller,and the tangential induced velocities have been canceled. Viscous drag hasbeen set to zero. This, therefore, corresponds to an actuator disk. CT =1.0 CP = 1.2071 KT = 0.2513 KQ = 0.0386 Va/Vs= 1.000 = 82.84%.

The circulation is a constant, with a value of G

= 0.005093and the axialinduced velocity is u/Va = 0.20711, which agrees exactly with actuatoradisk theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

.

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1 TWODIMENSIONALFOILTHEORY

1


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1.1 Introduction

Wewillbeginourexaminationofhydrofoilandpropellerflowsbylookingattheflowaroundtwodimensionalfoilsections. Itisimportanttorecognizeattheoutsetthatatwodimensionalflowisanidealization. Flowsaroundmarinepropellers,sailboat keelsorcontrolsurfacesareinherentlythreedimensional. Moreover,itisevenimpossibletocreateatrulytwodimensionalflowinawindorwatertunnel. Whilethefoilmodelmay be perfectly placed between the walls of thetunneltestsection, interaction betweenthetunnelwallboundarylayersandthefoilgeneratethreedimensional features thatdisturbthetwodimensionality oftheflowfield.Reliableexperimentalmeasurementsoftwodimensionalfoilsectionsthereforerequirecarefulattentiontotheissueofavoidingunwantedthreedimensionaleffects.

Ofcourse, twodimensional flowscan be modeledtheoretically, andare much easier todealwiththanthreedimensionalflows.Moreover,thefundamentalmechanismforcreat-

ingliftaswellasmuch ofthemethodologyfordesigningoptimumfoilsectionshapescanbeexplainedbytwodimensionalconcepts. Designmethodsforairplanewings,marinepropellers,andeverythingin betweenrelyheavilyontheuseofsystematicfoilsectiondata. But,itisimportanttorecognizethatonecannotsimplypiecetogetherathree-dimensional wing or propeller in astripwisemannerfrom asequence of twodimensionalfoilsectionsandexpecttogetanaccurateanswer.Wewillseelaterwhy thisistrue,andhow twoandthreedimensionalflowscan beproperlycombined.

A surprisingly largenumber ofmethods exist for predicting the flowaround foilsections,anditisimportanttounderstandtheiradvantagesanddisadvantages. Theycan be characterized inthefollowingthree ways,

1.AnalyticalorNumerical

2.PotentialFlow(inviscid)methods,FullyViscousMethodsorCoupledPotentialFlow/BoundaryLayermethods

3.Exact,Linearized,or PartiallyLinearizedmethods.

Notallcombinationsofthesethreecharacteristicsare possible. Forexample,fullyviscousflows(exceptinafewtrivialcases)mustbesolvednumerically. Perhapsone

couldconstructa threedimensionalgraphshowingallthe possiblecombinations, butthis will notbeattemptedhere! In this chapter, wewill startwith themethodofconformalmapping,whichcaneasily beidentifiedas beinganalytical, inviscid and exact.Wewillthenlookat inviscid,linear theory,which iscaneither beanalytical ornumerical.Theprincipalattributeoftheinviscid, linear, numerical methodisthatcan bereadilyextendedtothreedimensionalflows.

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Thiswill befollowed byabrief look atsomecorrections tolineartheory,afterwhichwewilllookatpanelmethods,which canbecategorized asnumerical, inviscid, andexact.Wewillthenlookatcoupled potential flow/boundary layer methods,whichcan becharacterizedasanumerical, exact method1Finally,wewilltakeabrieflookatresultsobtainedbyaReynolds Averaged NavierStokes (RANS) code,which isfully viscous,

numerical, and exact 2.

1.2 Foil Geometry

Figure1:Illustration of notation for foil section geometry.1Well,moreorless. Boundarylayertheoryinvolveslinearizingassumptionsthattheboundarylayer

isthin,butthecoupledmethodmakesnoassumptionsthatthefoil isthin.2Herewegoagain! Thefoilgeometryisexact,buttheturbulencemodelsemployed inRANScodes

areapproximations.

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Beforewestartwiththedevelopmentofmethodstoobtaintheflowaroundafoil,wewillfirstintroducetheterminologyusedtodefinefoilsectiongeometry.Asshowninfigure1,good foilsectionsaregenerallyslender,withasharp(ornearlysharp)trailingedge,andaroundedleadingedge.Thebaselineforfoilgeometryisalineconnectingthetrailingedgetothe pointofmaximumcurvatureattheleadingedge,andthisisshown

asthedashedlineinthefigure.Thisisknownasthenosetail line,anditslengthisthechord, cofthefoil.

Theparticularcoordinatesystemnotationusedtodescribeafoilvarieswidelyde-pendingonapplication,andone musttherefore becarefulwhenreadingdifferenttextsorresearch reports.Itisnaturaltousex, yasthecoordinateaxesfor a twodimensionalflow,particularlyifoneisusingthecomplexvariablez =x+iy. Thenosetail lineisgenerallyplacedonthe xaxis,butinsomeapplicationsthe xaxisistakento beinthedirectionoftheonsetflow,inwhichcasethenosetaillineisinclinedatanangle ofattack,withrespecttothexaxis. Positivexcan beeitherorientedintheupstreamordownstreamdirection,butweshallusethedownstreamconventionhere.

Forthreedimensionalplanarfoils,itiscommontoorient theycoordinate inthespanwise direction. Inthiscase,thefoilsectionordinateswill beinthezdirection.Finally,in the case of propeller blades, a specialcurvilinear coordinate system must be adopted,andwewillintroducethislater.

Asshown infigure 1, afoilsectioncan bethought ofasthecombination of ameanline,f(x) withmaximumvaluefoand asymmetricalthickness form,t(x), withmaximumvalueto.Thethicknessformisaddedatrightanglestothemeanline,sothat pointsontheupperandlowersurfacesofthefoilwillhavecoordinates,

xu = xt(x) sin2

yu = f(x) + t(x)

cos2

xl = x+t(x)

sin2

yl = f(x)t(x) cos (1)2

where

=arctan(df/dx) istheslopeofthemeanlineat point x.

Thequantityfo/ciscalledthecamber ratio, and in asimilarmanner,to/ciscalledthethickness ratio. Ithasbeencommonpracticetodevelopfoilshapesby scalinggenericmeanlineand thickness formstotheir desired values, and combining thenby usingequation 1toobtain thegeometry of thefoilsurface. A majorsourceofmeanline and

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thickness form data wascreated by the NACA (now NASA)in the1930s and1940s3Forexample,Figure2showssampletabulationsofthegeometryoftheNACA Mean Linea=0.8 andtheNACA65A010BasicThicknessForm. Notethatthetabulatedmeanlinehasacamberratio to/c = 0.0679,whilethethicknessformhasathicknessratioto/c= 0.10.Includedinthetablesissomecomputedvelocity andpressuredatathatwewill

.4

.8

1.2

1.6

0 .2 .4 .6 .8 1.0

x/c

V( )2

NACA 65A 010 NACA =0.8

mean line

.2 .4 .6 .8 1.00

.2

0

1.0

2.0

x/c

yec

PR

0c1= .10 Upper surface

10 Lower surface

0

x(per cent c)

y(per cent c) (/V)2 /V a/V

/V =PR/4PRdyc/dxYc

(per cent c)x

(per cent c)

cli=1.0 ; = 1.540 cmc/4= - 0.202

0.5

0.75

1.25

2.5

5.0

7.5

1015

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

100 0.021

0.604

1.188

1.771

2.352

2.912

3.432

3.899

4.304

4.632

4.863

4.983

4.995

4.912

4.742

4.483

4.127

3.6583.040

2.650

2.182

1.623

1.183

0.928

0.765 0.897

0.948

1.010

1.089

1.148

1.176

1.1941.218

1.234

1.247

1.257

1.265

1.272

1.277

1.271

1.241

1.208

1.172

1.133

1.091

1.047 1.023

1.045

1.064

1.083

1.099

1.114

1.127

1.130

1.128

1.125

1.121

1.117

1.111

1.1041.093

1.084

1.071

1.044

1.005

0.974

0.947

0.999 0.999

0.949 0.974

0.945 0.029

0.045

0.058

0.071

0.084

0.097

0.111

0.127

0.144

0.161

0.178

0.198

0.222

0.249

0.281

0.323

0.3820.478

0.559

0.679

0.936

1.303

1.619

1.878

2.987

0 0

0.5 0.287 0.48535

0.75 0.404 0.44925

1.25 0.616 0.40359

2.5 1.077 0.34104

5.0 1.841 0.277187.5 2.483 0.23868

10 3.043 0.21050

15 3.985 0.16892

20 4.748 0.13734

25 5.367 0.11101

30 5.863 0.08775

35 6.248 0.06634

40 6.528 0.04601

45 6.709 0.02613

50 6.790 0.00620

55 6.770 - 0.01433

60 6.644 - 0.03611

65 6.405 - 0.06010

70

75

80

85

90

95

100 0

1.163

2.435

3.683

4.771

5.514 - 0.12311

- 0.18412

- 0.23921

- 0.25583

- 0.24904

- 0.20385

Data for NACA Mean Line = 0.8NACA 65A010 Basic Thickness Form

L.E. radius: 0.639 per cent c

T.E. radius: 0.023 per cent c

0.833

1.111 0.278

0.556

0.278

0

0.208

0.139

0.069

0

6.037 - 0.08790

0.893

0 0 0

0 0 0

Sample of tabulated geometry and flow data for an NACA mean line and thickness form.

refer to later.

5

Figure by MIT OCW. Adapted from Abbott and von Doenhoff, 1959.


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Animportantgeometricalcharacteristicofafoilisitsleadingedgeradius, rL, as showninfigure1.Whilethisquantity is,inprinciple,containedinthethicknessfunctiont(x),extractinganaccuratevaluefromsparselytabulateddataisrisky. ItisthereforeprovidedexplicitlyintheNACAtablesforexample,theNACA65A010hasaleadingedgeradiusof0.639 percentofthechord. If youwishtoscalethisthicknessformto

anothervalue,alloftheordinatesaresimplyscaledlinearly.However,theleadingedgeradiusscaleswiththesquare ofthethicknessofthefoil,sothat afifteen percentthicksection of thesameform would have aleading edgeradius of 1.44 percent ofthe chord.Wecanshowwhythisistruebyconsideringanexamplewherewewishtogeneratethicknessform(2) by linearlyscalingalltheordinatesofthicknessform(1),

t(2)(x) =t(1)(x)[to/c]

(2)

(2)[to/c](1)

forallvaluesofx. Then,thederivativesdt/dxandd2t/dx2willalsoscalelinearlywith

thickness/chordratio.Now,attheleadingedge,theradiusofcurvature,rL is 23/21 + dt

dxrL=lim (3)

x0 d2t

dx2

evaluatedattheleadingedge,whichwewilllocateatx=0.Sincetheslopedt/dxgoestoinfinityat aroundedleadingedge,equation 3 becomes

3dt 2

rL=limdx

= (const)to

(4)x0 d

2t cdx2

which confirmstheresultstatedearlier.

Someattentionmustalso begiventothedetailsofthetrailingedgegeometry. Aswewillsee,theuniquesolutionfortheflowaroundafoilsectionoperatinginaninviscidfluidrequiresthatthetrailingedge besharp.However,practicalissuesofmanufacturingandstrengthmakesharptrailingedgesimpractical. Insomecases,foilsarebuiltwithasquare(butrelativelythin)trailingedge,asindicatedinfigure2,althoughthesearesometimesrounded.Anadditionalpracticalproblemfrequentlyarisesinthecaseoffoilsectionsformarinepropellers.Organized vortexsheddingfrombluntorroundedtrailing

edgesmayoccuratfrequencieswhich coincidewithvibratorymodesofthebladetrailingedgeregion. Whenthishappens,strongdiscreteacousticaltonesaregenerated,whicharecommonlyreferredtoassinging.Thisproblemcansometimes becured bymodifyingthetrailingedgegeometryinsuchawayastoforceflowseparationontheuppersurfaceofthefoilslightlyupstreamofthetrailingedge.

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Anexampleofanantisingingtrailingedgemodificationisshowninfigure3.Itisimportant tonotethatthenosetaillineofthemodifiedsectionnolongerpassesthroughthetrailing edge, sothat theconvenientdecomposition of thegeometry into ameanlineandthicknessformissomewhatdisrupted.

Figure3:An example of a trailing edge modification used to reduce singing. This partic-ular procedure is frequently used for U.S. Navy and commercial applications.

Courtesy of U.S. Navy. Used with permission.

Theprocedureforconstructingfoilgeometrydescribedsofarisbasedontraditionalmanualdraftingpracticeswhichdatebackatleasttotheearly1900s. Definingcurves

by sparse pointdata,withtheadditionalrequirement offairing into a specifiedradius

ofcurvature leaves a lot ofroom for interpretation and error. Inthe presentworldofCADsoftwareandnumerically controlledmachines, foil surfacesandultimatelythreedimensional propeller blades, hubsand filletsare best described intermsofstandardizedgeometricentitiessuch asNonUniform Rational BSplines (NURBS)curvesandsurfaces. Asanexample,figure4showsaBsplinerepresentationofafoilsection with proportions typical ofcurrent marine propeller. In this case, the foil, togetherwithitssurfacecurvatureandnormal vector, isuniquelydefined byaset of 16 (x, y)coordinatesrepresentingtheverticesoftheBsplinecontrol polygon. Thisisallthatisneededtointroducetheshapeintoacomputationalfluiddynamicscode,constructamodel,orconstructthefullsizeobject.

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Figure4: An example of a complete geometrical description of a foil section (includingantisinging trailing edge modifications) using a fourth order uniform Bspline. The sym-bols connected with dashed lies represent the Bspline control polygon which completelydefines the shape of the foil. The resulting foil surface evaluated from the Bspline isshown as the continuous curve. The upper curves show an enlargement of the leadingand trailing edge regions. The complete foil is shown in the lower curve.

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1.3 Conformal Mapping

1.3.1 History

Theinitialdevelopmentofthefieldofairfoiltheorytookplaceintheearly1900s,longbeforetheinventionofthecomputer.Obtaininganaccuratesolutionfortheflowaroundsuchacomplexshapeasafoilsection,evenintwodimensions,wasthereforeaformi-dabletask. Fortunately,oneanalyticaltechnique,knownasthemethodofconformalmapping,wasknownatthattime,andprovidedameansofdeterminingtheexact in-viscidflowaroundalimitedclassoffoilsectionshapes.Thistechniquewasfirstappliedby Joukowski [ ] in1914,andthesetoffoilgeometriescreated by themappingfunctionwhichhedevelopedbearshisname. Amoregeneralmappingfunction,whichincludestheJoukowskimapping as a specialcase, wasthen introduced byKarman and Trefftz[ ]. Whileotherseveralinvestigatorsintroduceddifferentmappingfunctions,thenextsignificantdevelopment was by Theodorsen [ ],whodevelopedanapproximateanalyti-

cal/numericaltechniqueforobtainingthemappingfunctionforafoilsectionofarbitraryshape.Theodorsens work was thebasis for the development of anextensivesystematicseriesof foilsectionspublishedbytheNationalAdvisoryCommissiononAeronautics(NACA) inthelate1930sand1940s [ ]. Theold NACAsectionresultsweredone, ofnecessity, byacombination ofgraphical andhandcomputation. An improvedconformalmappingmethod ofcomputingtheflowaroundarbitrarysections,suitableforimplemen-tationonadigitalcomputer,wasdevelopedby4 whofound,notsurprisingly,thatinaccuraciesexistedintheearlierNACAdata.Brockettsworkledtothedevelopmentoffoilsectiondesignchartswhichareusedforpropellerdesignatthepresenttime.

Thetheoreticalbasisforthemethod ofconformalmappingisgiveninmostadvancedcalculustexts [ Hildebrand?],soonlytheessentialhighlightswill bedevelopedhere.Onestartswiththeknownsolutiontoasimpleprobleminthiscasetheflowofauniformstream past a circle. The circle is then mapped into some geometry thatresembles a foilsection, and if you follow the rules carefully, the flow around the circle will betransformedinsuchawayastorepresentthecorrectsolutionforthemappedfoilsection.

1.3.2 PotentialFlowAround ACircle

Letus startwiththeflowaroundacircle. Weknowthat ina twodimensional idealflow, the superposition of auniform freestream and a dipole (whose axis isoriented inopposition to thedirection of thefreestream)willresult in adividingstreamlinewhoseformiscircular.Wealsoknowthatthisisnotthemostgeneralsolutiontotheproblem,because wecanadditionally superimpose the flowcreated by a pointvortex ofarbitrary

4DTMBReport1780,1966

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strengthlocatedatthecenterofthecircle.Thesolutionisthereforenotunique,butthisproblem will beaddressedlater when we look at theresulting flowaround afoil.

Tofacilitatethesubsequentmappingprocess,wewillwritedownthesolutionforacircleofradiusrcwhosecenter is located at anarbitrary point (xc, yc) inthexyplane,asshowninFigure5.Thecirclewill berequiredtointersectthe positivex

axisatthepointx =a, sothat theradius of thecircle must be,

rc= (xc+a)2+y2 (5)c

Figure5:Flow around a circle with zero circulation. The center of the circle is locatedat x=.3, y = 0.4. The circle passes through x=a= 1.0. The flow angle of attack is10 degrees.

Wewillseelaterthatinordertoobtain physicallyplausiblefoilshapes,the pointx=amusteither beinteriorofthecircleorlieonits boundary.Thissimplyrequires

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thatxc 0. Finally,theuniformfreestreamvelocitywillbeofspeedU andwillbeinclinedatananglewithrespecttothexaxis.

Withthesedefinitions,the velocity components (u, v) inthexandydirectionsare

2u(x, y) = U

cos() U rc cos(2

) sin()r 2r rc2

v(x, y) = Usin() U sin(2 ) + cos() (6)r 2r

whererandare polarcoordinateswithoriginatthe center of the circle,sothat

x = xc+rcos()

y = yc+rsin() (7)

Notethat wearefollowing astrictrighthandedcoordinatesystem,sothat positiveangles and positivetangential velocitiesare in acounter clockwisedirection. Avortex ofpositivestrength,,thereforeinducesavelocity whichisinthe negative xdirectiononthetopofthecircleandapositivexdirectionatthe bottom.

Figure 5 shows theresult in the specialcasewhere thecirculation, , has been settozero,andtheresultingflowpatternisclearlysymmetricalaboutalineinclinedattheangle ofattack which in thiscase wasselected to be tendegrees. If,instead, we set thecirculation equal to a value of =7.778695,theflowpatternshowninfigure 6 results.

Clearly,theflow isnolongersymmetrical,andthe twostagnation pointsonthecirclehave bothmoveddown. Theangularcoordinatesofthestagnation pointsonthecirclecan beobtaineddirectlyfromequation 6 by settingr=rcandsolvingforthetangentialcomponentofthevelocity,

ut = vcos() usin()

= 2Usin( ) +2rc

(8)

Ifwesetut= 0inequation 8anddenotetheangularcoordinatesofthestagnationpointsass,weobtain

sin(s ) =

4rcU(9)

11
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Figure6: Flow around a circle with circulation. The center of the circle is located atx=.3, y = 0.4. The circle passes through x=a= 1.0. Note that the rear stagnationpoint has moved to x=a.

Fortheexampleshowninfigure6,substituting rc = (1.32+ 0.42) = 1.3602,=7.778695,U= 1.0 and=10degreesintoequation9,weobtain

sin(s ) = 0.45510 : s=17.1deg,142.9deg (10)

Inthis specialcase, weseethat we havecarefullyselected insucha wayas to movetherearstagnation pointexactlytothe pointaonthexaxis,sinces=,where

=arcsinyc

(11)rc

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1.3.3 ConformalMappingforDummies

Conformalmapping is ausefultechniqueforsolving twodimensionalidealfluidproblemsbecause ofthe analogybetween the properties ofananalytic functionofacomplexvariableandthegoverningequationsofafluid.Weknowthattheflowofanidealfluidintwodimensionscanberepresentedeitherbyascalarfunction (x, y) knownasthevelocity potential,or byascalarfunction(x, y) knownasthe stream function. To be a legitimateidealfluidflow, both mustsatisfyLaplacesequation.Thefluid velocitiescanthen beobtainedfromeither,asfollows,

u = =

x y

v =y

=x

(12)

Nowletussupposethatthe physicalx, ycoordinates ofthefluidflowaretherealandimaginarypartsofacomplexvariablez=x+iy.Wecanconstructacomplex potential(z) by assigning thereal part to be the velocity potential and theimaginary part to bethestreamfunction,

(z) =(x, y) +i(x, y) (13)

Sincethe realand imaginary partsof each satisfyLaplaces equation, isananalytic function5.Inaddition,thederivativeof hastheconvenientproperty of beingtheconjugateoftherealfluidvelocity,u+ iv.Aneasyway toshowthisistocomputed/dzby takingtheincrementdz inthexdirection,

d = = +i

dz x x x = u iv (14)

wherethesecondlineofequation14followsdirectlyfromequation12. If youarenothappy withthisapproach,trytakingtheincrementdz intheiydirection,and youwillget theidenticalresult.This has to betrue,since isanalytic anditsderivative musttherefore beunique.

5Remember,ananalyticfunctionisonethatissinglevaluedandwhosederivativeisuniquelydefined,i.e. thevalueof itsderivativeisindependentofthepathtakentoobtainthelimitingvalueof/z

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Wenowintroduceamapping function (z),withrealpart andimaginarypart.Wecaninterpretthe zplaneandthe graphicallyastwodifferentmaps.Forexample,ifthez planeistherepresentationoftheflowaroundacircle(showninfigures5or6),theneachpairof x, y coordinatesonthesurfaceofthecircle,oronanyoneoftheflowstreamlines,willmaptoa corresponding point, inthe plane,dependingon

theparticularmappingfunction (z). Thisideamaymakemoresenseif youtakeanadvancedlookatfigure7. Thefancy lookingfoilshapewas, indeed,mappedfromacircle.

Figure 7:Flow around a KarmanTrefftz foil derived from the flow around a circle shownin figure 6 with a specified tail angle of = 2 5 degrees.

Whileitiseasytoconfirmthatthecirclehas beenmappedintoamoreusefulfoilshape,howdoweknowthatthefluidvelocitiesandstreamlinesintheplanearevalid?Theanswer isthat if (z) andthemappingfunction(z) are bothanalytic,then () isalsoanalytic. Itthereforerepresents a valid 2Dfluid flow, but it may notnecessarily beonethatwewant.However,ifthedividingstreamlineproducesashapethatweaccept,

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thentheonlyremainingflowproperty thatweneedtoverifyiswhetherornottheflowatlargedistancesfromthefoilapproachesauniformstreamofspeedU andangleofattack.Thelatterisensuredifthemappingfunctionisconstructedinsuchaway that=z inthelimitaszgoestoinfinity.

Finally,thecomplexvelocity inthe

planecansimply beobtainedfromthecomplexvelocity inthezplane,

dz z[u iv]=d

=d

=[u iv]

(15)d d d

dz dz

Even though we introduced theconcept of the complex potential, ,wedontactuallyneedit. Fromequation15,allweneedtogetthevelocityfieldaroundthefoilisthevelocity aroundthecircleandthederivativeofthemappingfunction.And,ofcourse,weneedthemappingfunctionitselftofindthelocationoftheactual pointinthe plane

wherethis velocityoccurs.

Performingcomplex operationshas beengreatlyfacilitated by the availability ofcom-puterlanguagesthatunderstandhowtodoit.Inparticular,complexarithmeticisbuiltintotheFortranlanguage.A listingofaFortran90computercodecalledMAPSL ispro-vided in the lastsection of thesenotes.This code performs all of the operationsdescribedin thissection, and mayserve as auseful guide inunderstanding the process.

1.3.4 TheKarman-TrefftzMappingFunction

TheKarmanTrefftz transformationmapsa pointz to a point using thefollowingrelationship

a (z+a)+ (z a)=

(z+a) (z a) (16)where anda aregivenrealconstants,whosepurposewewilldiscover shortly. Thederivativeofthemappingfunction,whichwewillneedtotransformthevelocitiesfromthezplanetotheplanecan beobtaineddirectlyfrom16

d 2 (z a)1(z+a)1

dz

= 42a

[(z

+a)

(z

a)

]2 (17)

Wecanseeimmediatelyfromequation16that= 1themappingfunctionreducesto=z,sothisproducesanexactphotocopy oftheoriginalflow!Notealso,thatwhenz=a,=a.Sincewewant tostretch outthecircle,usefulvaluesofwillthereforebegreaterthat 1.0.

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Finally, fromequation17,thederivativeofthemappingfunctioniszerowhenz=a.Thesearecalledcritical points inthemappingfunction,meaningthatstrangethingsarelikelytohappenthere. Mostdifficultconceptsofhighermathematicscanbestbeunderstood by observing the behavior ofsmall bugs. Suppose a bug is walking along theperimeterofthecircleinthe z plane,startingatsome pointz belowthe pointa. The

bugsfriendstarts walkingalongthe perimeter ofthefoil inthe

planestartingatthemapped point(z). Themagnitudeanddirectionofthemovementofthesecondbugisrelatedtothatofthefirstbug by thederivativeofthemappingfunction. Ifd/dz isnonzero,therelativeprogressof bothbugswill besmoothandcontinuous. Butwhenthefirstbuggetstothe pointa,thesecondbugstopsdeadinitstracks,whilethefirstbugcontinuessmoothly. After pointa,thederivativeofthemappingfunctionchangessign,sothesecondbugreversesitsdirection. Thus,asharpcornerisproduced,asisevidentfromfigure7.

Theincludedangleofthecorner(ortailangleinthiscase)dependsonthewayinwhichd/dzapproacheszero.Whilewewillnotproveithere,thetailangle(indegrees)andtheexponent

inthemappingfunctionaresimplyrelated,

= 2 =180(2 ) (18)

180

sothatthetailanglecorrespondingto = 1.86111is25degrees,whichisthevaluespecifiedforthefoilshown infigure 7.Notethat if= 2 inequation18theresultingtailangleiszero,i.e. acuspedtrailingedgeresults. Inthatcase,themappingfunctioninequation 16reduces to a much simpler form whichcan berecognized as themorefamiliarJoukowskitransformation,

a2=z+ (19)

z

Finally,if=1,thetailangleis=180degrees,orinotherwords,thesharpcornerhasdisappeared. Sincewesawearlierthat=1resultsinnochangetotheoriginalcircle,thisresultisexpected. Thusweseethatthe permissiblerangeof is between(1,2).Infact,sincepracticalfoilsectionshavetailanglesthataregenerallylessthan30degrees,thecorrespondingrangeofisroughlyfrom(1.8,2.0).

Ifthecirclepassesoutsideofz=a,thereisnosharpleadingedge. Ontheotherhand, we canconstruct a foil with a sharp leading and trailing edge by placing the centerofthecircleontheimaginaryaxis,sothatacirclepassingthroughz=awillalsopass

throughz

=a. Thisisshowninfigure10. Inthiscase,theupperandlowercontoursofthefoilcanbeshowntoconsistofcirculararcs. Inthelimitofsmallcamberandthickness,these becomethesameasparabolicarcs.

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1.3.5 TheKuttaCondition

Wecansee from equation 6that thesolution forthe potential flowaround acircleisnotunique,but containsanarbitraryvalueofthecirculation,. Ifwewereonlyinterestedinthisparticularflow,it would belogicaltoconclude,fromsymmetry,thattheonly physicallyrational valueforthecirculationwould bezero.Ontheotherhand,ifthecylinderwererotatingaboutitsaxis,viscousforcesactinginarealfluidmightbeexpectedtoinduceacirculationinthedirectionofrotation. Thisactuallyhappensinthecaseofexposedpropellershaftswhichareinclinedrelativetotheinflow. Inthiscase,atransverseforcecalledthemagnus effect willbepresent. Similarly, iffluid isejectedthroughjetsorientedtangenttothesurfaceofthecylinder,acirculationcanalsobeinduced. However,thesearenotofinterestinthepresentdiscussion,wheretheflowaroundacircleissimplyanartificialmeansofdevelopingtheflowaroundarealisticfoilshape.

Figure8:Flow near the trailing edge. The figure on the left is for zero circulation. Notethe flow around the sharp trailing edge and the presence of a stagnation point on theupper surface. The figure on the right shows the result of adjusting the circulation toprovide smooth flow at the trailing edge.

Figure 8 shows the local flownear thetrailing edge for theKarmanTrefftzfoil showninfigure5.Theflowintheleftfigureshowswhathappenswhenthecirculationaroundthecircleissettozero.Theflowontherightfigureshowsthecasewherethecirculationis adjusted toproduceastagnation pointat the pointa onthe x axis,asshowninfigure6.Intheformercase,thereisflowaroundasharpcorner,which fromequation15willresultininfinitevelocitiesatthat pointsinced/dz iszero. Ontheotherhand,

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theflowintherighthandfigureseemstoleavethetrailingedgesmoothly. Ifweagainexamineequation15,weseethattheexpressionforthevelocity isindeterminate,withbothnumeratoranddenominatorvanishingatz =a. Itcanbeshown fromalocalexpansionofthenumeratoranddenominatorintheneighborhood ofz=athatthereisactually astagnation pointthereprovidedthatthetailangle > 0.Ifthetrailingedge

iscusped (

=o),thevelocity isfinite,withavalueequaltothecomponentoftheinflowwhich istangenttothedirectionofthetrailingedge.

Kuttas hypothesiswasthat in arealfluid, theflowpatternshown in theleft offigure 8 is physically impossible,andthatthecirculationwilladjustitselfuntiltheflowleavesthetrailingedgesmoothly.Hisconclusionwasbased,inpart,onaverysimplebutclever experimentcarried out by L.Prandtl in theKaiserWilhelmInstitute in Gottingenaround1910. Amodelfoilsectionwassetupvertically, protrudingthroughthe freesurfaceofasmalltank. Finealuminumdustwassprinkledonthefreesurface,andthemodelwasstartedupfromrest. Theresultingflowpatternwasthenphotographed,asshowninfigure 9fromanearlytext6. Thephotographclearlyshowstheformationofavortexatthetrailingedgewhichisthenshedintotheflow. SinceKelvinstheoremstatesthatthetotalcirculationmustremainunchanged,avortexofequalbutoppositesigndevelopsaroundthefoil. Thus,theadjustmentofcirculationisnotarbitrary,butisdirectlyrelatedtotheinitialformationofvortexinthevicinity ofthesharptrailingedge. Whilethisprocessisinitiatedbyfluidviscosity,oncethevortexhasbeenshed,theflowaroundthefoilactsasthoughitisessentiallyinviscid.

ThisbasisforsettingthecirculationisknownastheKutta condition, andisuniversallyappliedwheninviscidflowtheoryisusedtosolve both twoandthreedimensionalliftingproblems. However, itis important tokeep inmindthattheKuttacondition isan

idealizationofanextremelycomplexrealfluidproblem. Itworksamazinglywellmuchofthetime,butitisnotanexactsolutiontotheproblem.Wewillseelaterhowgood itreallyis!

Inthe caseofthe present conformalmappingmethodofsolution, we simply setthe positionoftherearstagnation pointtos =. Therequiredcirculation, fromequation 9 is,

= 4rcUsin(+) (20)6L.Prandtland O.G. Tietjens,Applied Hydro and Aerodynamics, 1934. Dovereditionpublished in

18

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Figure9:Early flow visualization photograph showing the development of a starting vor-tex.

1.3.6 PressureDistributions

Thedistribution ofpressure on the upper and lowersurfaces of a hydrofoil is of interestinthedeterminationofliftanddragforces,cavitationinception,andinthestudyofboundarylayer behavior.Thepressurefieldintheneighborhood ofthefoilisofinterestinstudying the interaction between multiple foils, and in the interaction betweenfoils andadjacent boundaries.Thepressureatanarbitrary pointcan berelatedtothepressureat a pointfarupstreamfromBernoullisequation,

1 1p +

2U2=p+

2q2

where q is themagnitude of thetotalfluid velocityat the point inquestion,

q u2+v2

and (u, v) arethecomponentsoffluidvelocity obtainedfromequation15.Thequantityp is thepressure farupstream, taken at thesame hydrostatic level. A nondimensionalpressurecoefficientcan beformed by dividingthedifference betweenthelocalandup-streampressure by theupstreamdynamicpressure,

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pp q2CP 1 = 1 U2 U

2

Notethatata stagnation point,q = 0, so that thepressure coefficient becomes

CP = 1.0. Apressure coefficient ofzero indicates thatthe localvelocity isequal inmagnitudetothefreestreamvelocity,U,while anegativepressurecoefficientimplies alocal velocity which exceedsfreestream.Whilethis is theuniversallyacceptedconventionfordefiningthenondimensionalpressure,many authorsplotthenegative ofthepressurecoefficient. Inthatcase, astagnation pointwill beplottedwith a value ofCP =1.0.

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Figure10:Streamlines and pressure contours for a thin, highly cambered section at zeroangle of attack. This section is symmetrical about midchord, and therefore has sharpleading and trailing edges. As expected, the pressure contours show low pressure on theupper surface (green) and high pressure on the lower surface (blue).

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Figure11: 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.

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Figure12: Close up view of the flow near the leading edge at an angle of attack of 10degrees.

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1.3.7 LiftandDrag

Determining the overall lift and drag on a twodimensional foilsection in inviscid flow isincrediblysimple.Theforce(perunitofspan)directedatrightanglestotheoncomingflowofspeedU istermedlift andcan beshownto be

L=U (21)whiletheforceactinginthedirectionoftheoncomingflowistermeddrag iszero.Equa-tion21isknownasKuttaJoukowsks Law7.

Wecaneasily verifythatequation 21 iscorrectfortheflowaround acircle by inte-gratingthey andx componentsofthepressureactingonitssurface. Withoutlossofgenerality,letusassumethatthecircleiscenteredattheorigin,andthattheangleofattack iszero.In this case, the velocityon thesurface of the circle, from equation 8 is,

ut=2U

sin

+ 2rc (22)

As before, wecanwrite downthepressurefromBernoullisequation,

pp = 12

U2 u2t

(23)

andtheliftistheintegraloftheycomponentofthepressurearoundthecircle, 2

L= (pp)sin rcd (24)0

By substitutingequations22and23intoequation24,andrecognizingthatonlytheterm

containingsin2

survivestheintegration,onecanreadilyrecoverequation21.

Inasimilarway,wecanwritedowntheintegralfordrag, 2

D= (pp)cos rcd (25)0

andshowthatalltermsarezero.

Wecouldnowresorttofuzzymathandarguethatequation21mustapplytoanyfoilshape.Theargument isthatwecouldhavecalculatedtheliftforceonthecirclefromanapplication of themomentumtheoremaround acontrol volumeconsisting of acircular

pathatsomelargeradius r >>

rc. Theresultmustbethesameastheoneobtainedfrompressureintegrationaroundthefoil.Butifthisistrue,theresultmustalsoapplytoany foilshape,sincetheconformalmappingfunctionusedtocreateitrequiresthatthe flowfieldaround thecircle andaround thefoil become thesame at large values ofr.

7The negative sign in the equation is a consequence of choosing the positive direction forx to bedownstreamandusingarighthandedconventionforpositive

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Mapping Solutions for Foils of Arbitrary Shape

Closedformmappingfunctionsareobviouslylimitedinthetypesofshapeswhichtheycanproduce. WhilesomefurtherextensionstotheKarmanTrefftzmappingfunctionweredeveloped,thisapproachwaslargelyabandoned by the1930s. Then, in1931, T.Theodorsenpublishedamethodbywhichonecouldstart withthefoilgeometryanddevelop themappingfunctionthatwouldmap itback toa circle 8. Thiswasdonebyassumingaseriesexpansionforthemappingfunctionandsolvingnumericallyforafinitenumberofterms inthe series. Themethodwastherefore approximate, andextremelytimeconsumingintheprecomputerera. Nevertheless,extensiveapplicationofthismethodledtothedevelopmentoftheNACAseriesofwingsections,includingthesamplefoilsectionshowninfigure2.

An improved version of Theodorsens method,suitable for implementation on a digitalcomputer, wasdevelopedbyT.Brockettin 19669. Hefound,not surprisingly, that

inaccuraciesexistedinthetabulatedgeometryandpressuredistributionsforsomeoftheearlierNACAdata.BrocketsmodifiedNACA66thicknessformwasdevelopedatthattime,andhas beenusedextensivelyforpropellersections.

Bythemid1970s,conformalmappingsolutionshadgivenwaytopanelmethods,whichwewill bediscussinglater.Thishappenedforthreereasons,

1.Conformalmappingmethodscannot beextendedtothreedimensionalflow,whilepanelmethodscan.

2.Bothmethodsinvolvenumericalapproximationwhenappliedto foilsofa given

geometry,andimplementationandconvergencecheckingismorestraightforwardwithapanelmethod.

3. Panelmethodscan beextended toincludeviscous boundary layereffects.

8TheodoreTheodorsen,Theory of Wing Sections of Arbitrary Shape,NACARept. No. 383,19319TerryBrocket,Minimum Pressure Envelopes for Modified NACA66 Sections with NACA a=.8 Cam-

ber and Buships Type I and Type II Sections,DTMBReport1780,1966

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1.4 Linearized Theory for a 2Dimensional Foil Section

1.4.1 Problem Formulation

Inthissectionwewillreviewtheclassicallinearizedtheoryfor2Dfoilsininviscidflow.Theproblemwill besimplified by makingtheassumptionsthatthethicknessandcamberofthefoilsectionissmallandthattheangleofattackisalsosmall.Theflowfieldwillbeconsideredas the superposition of auniformoncoming flow of speedU andangleofattackand aperturbation velocity fieldcaused by thepresence ofthefoil. Wewillusethesymbolsu, vtodenotethe perturbation velocity,sothatthetotalfluid velocity inthexdirectionwill beUcos+u, whilethecomponentintheydirectionwill beUsin+v.

Theexactkinematic boundarycondition isthattheresultantfluid velocitymust betangenttothefoilon boththeupperandlowersurface,

dyu U

sin

+v

dx=

Ucos+uon y

=yu

dyl=

Usin+von u=yl (26)

dx Ucos+u

However, sincewearelookingforthelinearizedsolution,threesimplificationscanbemade. Firstofall,since issmall,cos 1andsin . Butifthecamberandthickness of thefoil isalsosmall, the perturbation velocitiescan be expected to besmallcomparedtotheinflow10. Finally,sincetheslopeofthemeanline,,isalsosmall,thecoordinatesoftheupperandlowersurfacesofthefoilshowninequation 1will beapproximately,

yu(x) f(x) + t(x)2

yl(x) f(x)t(x) (27)2

Introducingtheseapproximationsintoequation26,weobtainthefollowing,

dyu=

df(x)+

1 dt(x)=

U +v+(x)ony= 0

dx dx 2 dx Udyl

=df(x) 1 dt(x) =U +v

(x)ony= 0 (28)

dx dx 2 dx U

Notethatthe boundaryconditionisappliedontheliney= 0 ratherthanontheactualfoilsurface,which isconsistentwiththelinearizingassumptionsmadesofar.Thisresult

10Actually this assumption is not uniformly valid, since the perturbation velocity will not be smallin the case of the flow around a sharp leading edge, nor is it small close to the stagnation point at aroundedleadingedge. Wewillseelaterthat lineartheorywillbe locallyinvalidinthoseregions.

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can bederived in amoreformal way by carefullyexpanding thegeometry and flowfieldintermsofasmallparameter,butthisisalotofwork,andisunnecessarytoobtainthecorrectlinearresult.Thenotationv+andv meansthat the perturbation velocity is tobeevaluatedjustaboveandjust belowthexaxis.Now,ifwetakehalfofthesumandhalfofthedifferenceofthetwoequationsabove,weobtain,

df

(x)= +

[v+(x) +v(x)]

dx 2Udt(x)

=[v+(x) v(x)]

(29)dx U

Wenow seethat the linearized foilproblemhasbeenconvenientlydecomposed intotwoparts. Themeanvalueofthevertical perturbationvelocity alongthe x axisisdetermined by theslopeofthecamberdistributionf(x) andtheangleofattack,.The

jumpinverticalvelocityacrossthexaxisisdirectlyrelatedtotheslopeofthethicknessdistribution,t(x). This is the key to thesolution of theproblem,since wecangeneratethedesired even and odd behavior ofv(x) by distributingvorticesandsourcesalongthe

x

axisbetweentheleadingandtrailingedgeofthefoil,aswillbeshowninthenextsection.

1.4.2 VortexandSourceDistributions

The velocity field of a pointvortex ofstrength located at a pointonthexaxisis,

u(x, y) =

(x y

)2+y22

x

v(x, y) = 2

(x

)2+y2 (30)whilethecorresponding velocity fieldfor a pointsource ofstrengthS is,

S x u(x, y) =

2(x )2+y2

v(x, y) =2

S(x

y

)2+y2(31)

Wenextdefineavortex sheet asacontinuousdistributionofvorticeswithstrength perunitlength. Thevelocityfieldofavortexsheetdistributedbetweenx=c/2 to x= +c/2 will be,

c/2

u(x, y) = 2

1

c/2(x

(

)2)y

+y2d

v(x, y) =2

1

c/2(

x

(

)(

x

)2+

y

)2

d (32)

c/2

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Itisinstructivetolookatthevelocity fieldinthespecialcaseswherethevortexstrengthisconstant over the interval.Thisresultwillalso beusefullaterwhen we look at panelmethods. Inthiscase comesoutsidetheintegral,andequation32can beintegratedanalytically,givingtheresult, 11

u(x, y) = 2

tan

1x

y

c/2 tan

1x

+

y

c/2

(x c/2)2+y2v(x, y) =

4ln

(x+c/2)2+y2(33)

Figure13:Vertical distribution of the u

velocity at the midchord of a constant strengthvortex panel of strength =1.

Figure 13showsthe velocityfieldobtainedfromequation 33for pointsalongtheyaxis in thecasewhere the vortexsheetstrength has been set to=1.Notethat ajumpinhorizontalvelocity existsacrossthesheet,andthatthevalueofthevelocity jumpisequaltothestrengthofthesheet. Thisfundamentalproperty ofavortexsheetfollowsdirectlyfromanapplicationofStokestheoremtoasmallcirculationcontourspanningthesheet,asshowninfigure14.

dx = udx+ 0 + u+(dx) + 0 = u u+ (34)

11See, for example, J.Katz and A. Plotkin,LowSpeed Aerodynamics, From Wing Theory to PanelMethods,McGrawHill,1991

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Figure14:Illustration of the circulation path used to show that the jump in u

velocity isequal to the vortex sheet strength, .

Even though figure 13 wascomputed for a uniformdistribution of(x) betweenx1andx2,thelocal behavioroftheucomponent of velocity close to the vortexsheet would bethesameforanycontinouslyvaryingdistribution.Ontheotherhand,thevcomponentofvelocity dependson(x),butiscontinuousacrossthesheet. Figure15showsthevcomponentofvelocity alongthexaxis,againforthecasewhere=1.

Wecandevelopsimilarexpressionsforthevelocity fieldofauniformstrengthsource

sheet. If we let thestrength of thesourcesheet be

perunitlength,thevelocity fieldofasourcesheetextendingfromx

=c/2 to x

=c/2 will be,

c/2u(x, y) =

1 ()(x )d

2c/2

(x )2+y2

1c/2

()yd (35)v(x, y) =

2c/2

(x )2+y2

Again, if we specifythatthesourcestrength isconstant,equation 35can be integrated,

sogivetheresult,

ln

(x c/2)2+y2u(x, y) =

4 (x+c/2)2+y2

v(x, y) =2

tan1

xy

c/2 tan1

x+

y

c/2(36)

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Figure15:Horizontal distribution of the v

velocity along a constant strength vortex panelof strength =1.

Figure16showsthev componentofthevelocityobtainedfromequation36evaluatedjustaboveand justbelowthex axisforavalueof =1. Thejumpintheverticalvelocity isequaltothevalueofthesourcesheetstrength,whichfollowsdirectlyfromaconsiderationofmassconservation.

Returningtoequation29,wenowseethat,withintheassumptionsoflineartheory,a foilcan berepresented byadistribution ofsourcesand vortices along thexaxis.The

strengthofthesourcedistribution,(x) isknowndirectlyfromtheslopeofthethicknessdistribution,

dt(x) =U (37)

dx

whilethevortexsheetdistributionmustsatisfytherelationship,

c/2df(x) = 1 c () d (38)

dx 2U x c/2

Thesymbolcsuperimposedontheintegralsign isthereforareason. Thiswillbe

explainedinthenextsection.

Thisdecomposition offoilgeometry,velocity fieldsandsingularity distributionshasrevealed a very important result.According tolinear theory, the vortexsheetdistribution,andhencethetotalcirculation,isunaffected by foilthickness,sinceitdependsonlyonthemeanlineshapeandtheangleofattack.Thismeansthattheliftofafoilsectionis

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velocity along a constant strength source panelof strength = 1.

unaffected by itsthickness.Now,theexactconformalmappingproceduredevelopedintheprevioussectionshowsthatliftincreaseswithfoilthickness, but only slightly. So,there is no contradiction, sincelinear theory is only supposed to be valid for small valuesofthickness.Wewillseelaterthatviscouseffectstendtoreducetheamountofliftthatafoilproducesasthicknessisincreased.So,insomesense,lineartheoryismoreexactthanexacttheory!Wewillreturntothisfascinatingtalelater.

Tocomplete theformulationofthelinearproblem,wemust introducetheKuttacondition. Since the jump in velocity between the upper and lowersurface of thefoil isdirectlyrelatedtothevortexsheetstrength,itissufficient tospecifythatgamma(c/2) =0.Ifthis werenottrue,there would beflowaroundthesharptrailingedge.

1.4.3 GlauertsTheory

Inthissection,wewilldeveloptherelationshipbetweentheshapeofameanlineanditsboundvortexdistributionfollowingtheapproachofGlauert12. Adistributionof

boundcirculation(x) overthechordinducesavelocityfieldv(x) whichmustsatisfythelinearized boundaryconditiondevelopedearlierinequation38

12H.Glauert,The Elements of Aerofoil and Airscrew Theory,CambridgeUniversityPress,1926

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dfv(x) = U (39)

dx

Glauertassumedthattheunknowncirculation(x) could beapproximated by aseries

in atransformed xcoordinate,x,

cx= cos(x) (40)

2

Notethatattheleadingedge,x =c/2, x =0,whileatthetrailingedge,x =c/2, x =.Thevalueofxatthemidchordis/2.Theserieshasthefollowingform,

1 + cos(x)

(x) = 2U a0sin(x)

+ ansin(nx) (41)n=1

Alltermsinequation41vanishatthetrailingedgeinordertosatisfytheKuttacondition. Sincethesinetermsalso vanishattheleadingedge,theywillnot beabletogenerate aninfinite velocity whichmay bepresentthere.Thefirstterm in theseries hastherefore beenincludedtoprovideforthissingular behaviorattheleadingedge. Thisfirsttermisactuallythesolutionforaflatplateatunitangleofattack obtainedfromtheJoukowskitransformation,afterintroducingtheapproximationthatsin=. Itgoeswithoutsayingthatithelpstoknowtheanswer beforestartingtosolvetheproblem!

Withtheseriesforthecirculationdefined,wecannowcalculatethetotalliftforce

onthesectionfromKuttaJoukowskislaw,

L = Uc/2

= U (x)dxc/2

= 2U2

a01 + cos(x)

+

ansin(nx)

dxdx (42)

0sin(x) n=1 dx

Introducingtheexpressionforthederivative,

dx c= sin(x) (43)

dx 2

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andnotingthattheintegralofsin(nx)sin(x) overtheinterval (0, ) iszeroforn > 1, weobtainthefinalresult,

L

= cU2a0 (1 + cos(x))dx + an sin(nx) sin(x)dx

0 n=1 0

= cU2

a0+

a1

(44)2

Equation44 canbeexpressed innondimensional forminterms oftheusualliftcoefficient,

LCL= 1

2U2

c

= 2a0+a1 (45)

Wewillnextdevelopanexpressionforthedistributionofverticalvelocity,v , overthechordinduced by the bound vortices,

1c/2

()v(x) = c d (46)

2 x c/2

Notethattheintegralinequation46issingular,sincetheintegrandgoestoinfinity

whenx

=. ThisistermedaCauchy Principal Value integral,whichmeansitisnotone that youcan simply look up in thetables, and thecsymbolcenteredontheintegralsignisputtheretoserveasawarningsign.Thenextstepistorewriteequation46intermsofthetransformedxcoordinate,andtointroducetheseriesforthecirculation,

v(x) 1 a0(1 + cos ) + ansin(n)sin()

= c n=1 d (47)U cos x cos

0

Wewillnextintroducethefollowingtrigonometricidentities inordertoputequa-tion47intoaformsuitableforintegration,

1sin(n)sin() = [cos((n 1)) cos((n+ 1))]

21

cos(n)sin() = [sin((n+ 1)) sin((n 1))] (48)2

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which thengivestheresult,

v(x)

=1

c a0(1 + cos ) + 2

1 an cos((n 1)) cos((n+ 1))(49)n=1

U

cos x

cos

0

Withthesubstitutionofthefirstidentity above,theintegralofeach termoftheserieshastheform,

cosn)

In(x) = ccos x cosd

(50)0

A Cauchy principal value integral isobtained by takingthelimitasapproacheszero

of,

c f()d=lim

f()d+ f()d

(51)0

0 0 +

Asshownintheappendix,GlauertshowedthatI0= 0andI1=. Usingtrigono-metricidentities,hethendevelopedarecursionformulaexpressing In intermsofIn1andIn 2.Thesolutionofthisrecursionformulaproducedthegeneralresult,

In(x) = sin(nx) (52)sin x

Substitutingequation52intoequation50,andmakinguseofthesecondtrigonometricidentity inequation48,thefinalexpressionforthevelocitycomesoutinanamazinglysimpleform,

v(x)=a0+

ancos(nx) (53)U n=1

Solvingequation39 fordf/dx andsubstitutingequation53 forv, weobtain thedesiredrelationship betweentheshapeofthemeanlineandtheseriescoefficientsforthechordwisedistributionofthe boundcirculation,

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df = a0+ ancos(nx) (54)

dx n=1

Theresultingexpression looksline theFouriercosineseriesrepresentationforthe

functiondf/dx. Henceequation54canbeinvertedbytheusualmethodofharmonicanalysistogivetheresult

1

df a0 = dx

dx0

2

dfan = cos(nx)dx (55)

dx0

Aparticularlyimportantresultisobtainedby solvingequation55fortheangleofattack forwhich thea0 coefficientvanishes,

1

dfideal dx (56)

dx0

Thisisknownastheidealangleofattack,andisparticularlyimportantin hydrofoilandpropellerdesignsinceitrelatestocavitationinceptionattheleadingedge.Foranyshapeofmeanline,oneangleofattack existsforwhich thevelocity isfiniteattheleading

edge.Fromthesymmetryofequation56,weseethattheidealangleofattack iszeroforanymeanlinewhich issymmetricalaboutthemidchord.

1.4.4 ExampleTheFlatPlate

Foraflatplateatangleofattack wecanseeimmediatelyfromtheGlauertresultsthata0=andan= 0 for n > 0.Theliftcoefficientisthenfoundto beCL= 2andtheboundcirculationdistribution overthechordis

1+cosx

(x) =

2U sin x

Thisresult,togetherwithsomeothercasesthatwewilldealwithnext,areplottedinfigure 17. In this figure, all of themean lines have beenscaled to produce a lift coefficientofCL = 1.0. Inthecaseofaflatplate,theangleofattackhasthereforebeensetto= 1/(2) radians.

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velocity along a constant strength vortex panelof strength =1.

1.4.5 ExampleTheParabolicMeanLine

Theequationofaparabolicmeanlinewithmaximumcamberf0 is

f(x) =fo 1 (2x)2c

sothattheslopeis

df

8fox

=dx c2

butsincex=c/2 cos x,theslopecan bewrittenasdf fo

= 4 cos xdx c

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We cantherefore again solve for theGlauert coefficients of thecirculation very easily,

1

df a0= dx = 0

0 dx

2 4f0 2

f0a1= cos xdx = 4 c

0 c

an= 0 f or n > 1

Theliftcoefficientisthengiven by theexpression

f0CL= 2a0+a1= 2+ 4

c

andthecirculationdistribution becomes,

1+cosx

f0(x) =

2U

8U sin x

sin x c

Thesolution for the parabolic camber linethereforeconsists of the sum of two partsa lift and circulationdistribution proportionalto the angleof attack and a lift andcirculationdistributionproportionaltothecamberratio.Thisistrueforanymeanline,except that inthegeneralcase the liftdue toangleofattack isproportional tothedifferencebetweentheangleofattackandtheidealangleofattack. Thelatteriszerofortheparabolicmeanlineduetoitssymmetryaboutthemidchord.Theresultplottedinfigure 17 is for a parabolicmeanline operatingwith alift coefficient ofCL= 1.0 at itsdealangleofattackwhichiszero.

1.4.6 TheDesignofMeanLines-TheNACAa-Series

From a cavitation point of view, the ideal camber line is one which produces aconstantpressuredifferenceoverthechord. Inthisway,afixedamountofliftisgeneratedwiththeminimumreductioninlocalpressure. Sincethelocalpressurejumpisdirectlypro-portional to the bound vortexstrength,such acamberline has aconstant circulation overthe chord.Unfortunately,this type ofcamberline doesnot perform up to expectations,sincetheabruptchangeincirculationatthetrailingedgeproducesanadversepressuregradientwhichseparatestheboundarylayer. Onemustthereforebelessgreedy,andacceptaloaddistributionwhichisconstantuptosomepercentageofthechord,andthenallowthecirculationtodecreaselinearlytozeroatthetrailingedge. Aseriesofsuchmeanlines wasdeveloped by the NACA13,andisknownastheaseries,wheretheparameteradenotesthefractionofthechordoverwhichthecirculationisconstant.

13AbbottandVonDoenhoff,Theory of Wing Sections,Dover1959

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TheoriginalNACAdevelopmentofthesemeanlines,whichdatesbackto1939,wastoachievelaminar flow wingsections.Theuse ofthesemeanlines in hydrofoil and propellerapplicationstodelaycavitationinceptionwasalaterdevelopment.

Theseshapescould,inprinciple,bedevelopedfromtheformulasdevelopedinthe

precedingsection by expanding thedesiredcirculationdistribution in a sineseries. How-ever,thesineseriesapproximationtoasquarewaveconvergesveryslowly,sothatalargenumber ofterms would berequired. It istherefore better toderive the shapedirectly forthisspecialfamilyofshapes.

Wewilldothea= 1.0 camberlinehere,sincewealreadyhavetheexpressionforthevelocity induced by a constant strength vortexdistributionfromequation33.Thegeneralcaseinvolvesthecombinationofaconstantandalinearlyvaryingvortexdistribution,andwewillsimplyprovidethefinalresult.

Ifwesety= 0 inequation33,shifttheoriginsothatthefoilgoesfrom(0, c) rather

thanfromc/2, c/2 inordertomatch theoriginalNACAconvention,weobtain, 14

v(x) = [log(1 x/c) log(x/c)] (57)

2

Weknowthatthetotalcirculation isrelatedtotheliftcoefficient

2CL=

Uc

and in the specialcase ofuniform bound vortexstrength over the chord,

= c

sothat CL

=U 2

Thenondimensionalverticalinducedvelocityoverthechordistherefore

v(x)=

CL[log(1 x/c) log(x/c)] (58)

U 4

Thelinearized boundaryconditionis

df x(x)=

dx U14Toavoidtheembarrassmentofanegativelogarithm,Ireplaced(xc/2)2 with(c/2x)2 inderiving

theequation

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sothatf(x)

xv() d

=c 0 U c

f(x)=CL [(1 x/c) log(1 x/c) +x/clog(x/c)]

c 4

Forexample,themaximumcamber,which occursatthemid chord,x=c/2,is

f0 CL= (1/2 log(1/2 )+1/2 log(1/2))= 0.05516CL

c 4

Notethattheslopeofthemeanlineislogarithmicallyinfiniteattheleadingandtrailingedges,which isexpectedsincetheinducedverticalvelocities,fromequation57areinfinitethere.This is obviously anonphysicalsituationwhich needs to betreatedwithsomesuspicion! Asindicated at the beginning of thissection, maintaininguniform vortex

sheetstrengthrightuptothetrailingedgecertainlyviolatesthespiritoftheKuttacondition,andcan beexpectedtoresultinflowseparation.

Thecorrespondingequations for the shape of the generalseries ofmean lines are muchmorecomplicated,andwillnot bederivedhere. However,theyarestilllogarithmicinform,asindicated below,

f(x)=

CL

1

1(a x )2 log |ax | 1(1x )2 log(1x )

c 2(a+ 1) 1 a 2 c c 2 c c

1 x

1 x x x

x)2+ (1 (a

)2 log( ) +g

h4 c 4 c c c c

where

1 1 1 1log a ) +g=

1 a a2(

2 4 4

1

1 1

h= (1 a)2 log(1 a) (1 a)2 +g1 a 2 4

ExceptfortheNACAa

= 1.0 meanline,thisseriesofmeanlinesisnotsymmetricalaboutthemidchord. The idealanglesofattackarethereforenonzero,andmaybefoundfromthefollowingequation,

1

df CLhideal= dx =

0 dx 2(a+1)

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Experience has shown that the bestcompromise between maximumextent ofconstantcirculation, and avoidance of boundary layerseparationcorresponds to a choice ofa= 0.8.Thetabulatedcharacteristicsofthemeanline,takenfromTheory of Wing Sections aregiveninfigure2.

1.4.7 LinearizedPressureCoefficient

Thedistribution ofpressure on the upper and lowersurfaces of a hydrofoil is of interestbothinthedeterminationofcavitationinceptionandinthestudyofboundarylayerbehavior. Wesaw inthepreceding sectiononconformalmappingmethods thatthepressureatanarbitrary pointcan berelatedtothepressureat a pointfarupstreamfromBernoullisequation,

p +1

U2=p+1

q22 2

where q is themagnitude of thetotalfluid velocityat the point inquestion,

q (U+u)2+v2

andp is thepressure far upstream, taken at the same hydrostatic level. Anondimensionalpressurecoefficientcan beformed by dividingthedifference betweenthelocalandupstreampressure by theupstreamdynamicpressure,

pp q2CP = 1 1U2 U

2

Sincethedisturbancevelocities (u, v) areassumedto besmallcomparedwiththefreestreamvelocity inlineartheory,

2q= 1 + 2

u+ (

u)2+ (

v)2 1 + 2 u

U U U U U

sothatthepressurecoefficientcan beapproximated by

uCP 2

U

Thisisknownasthelinearized pressure coefficient,which isvalidonlywherethedistur-

bance velocitiesaresmallcomparedtofreestream. Inparticular,at astagnation pointwhereq = 0theexactpressurecoefficient becomesone,whilethe linearizedpressurecoefficient givesanerroneousvalue of two!

For alinearized twodimensional hydrofoilwithoutthickness,theucomponentofthedisturbance velocityat points just above and below thefoil isu =/2. Thus,the

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linearizedpressurecoefficientandthelocalvortexsheetstrengtharedirectlyrelated,with

CP =

Uontheuppersurface,and

CP = +U

onthelowersurface.

Cavitationinceptioncanbe investigatedbycomparing theminimumvalueofthepressure coefficienton the foilsurface to the value of the cavitation index,

p pv= 1U2

2

wherepv isthe vaporpressureofthefluidatthe operatingtemperature ofthe foil.Comparingthedefinitionsof andCP,itisevidentthatifCP > ,thenp < pv.Suppose that a foil is operating at a fixed angle ofattack at a value of the cavitation indexsufficientlyhightoinsurethatthepressureiswellabovethevaporpressureeverywhere.It istherefore safe toassume that no cavitation will be present at this stage. Nowreducethecavitationnumber,eitherby reducingp orincreasing U. The pointonthefoilsurfacewiththeminimumpressurecoefficient, (CP)min willreachvaporpressurefirst,andthiswilloccurwhen(CP)min =. Atthis pointequilibriumcanexist betweenliquidandvapor,sothatinprinciplefluidcanevaporatetoformacavity.

Thephysicsofthisprocessisactuallyverycomplicated,anditturnsoutthattheactualpressure at whicha cavity forms may be below the vaporpressure, and will dependon thepresence of cavitation nuclei in thefluid.These may be microscopicfree airbubbles

orimpuritiesinthefluidoronthesurfaceofthefoil.Ifthereisanabundanceoffreeairbubbles,asisgenerallythecaseneartheseasurface,cavitationwilloccuratapressureveryclosetovaporpressure. Ontheotherhand,

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