59 



to mention the various methods and the problems they are designed to solve. • 

 The classical theory of two-dimensional flows about airfoils and strut shapes 

 and axisymmetric flows (ellipsoids, etc.) is given in Lamb 52 and Milne- 

 Thompson. 60 A method of computing the pressure distributions on arbitrary air- 

 foil shapes has been outlined by Theodorsen and Garrick 90 in which conformal 

 mapping methods are used to compute the pressure distribution on prescribed 

 shapes; various improvements in the methods for the successive approximations 

 have since been reported but without essential change in the theory. 



The inverse problem of specifying the pressure distribution and then 

 finding the corresponding shape has been treated for the two -dimensional case 

 by Mangier 91 and Peebles. 92 These methods should be particularly useful to 

 the designer since the pressure distribution can be specified for the lift dis- 

 tribution and the allowable pressure gradients for drag as well as for the min- 

 imum pressures for cavitation conditions, depending, of course, on whether the 

 conditions specified are compatible and lead to a closed surface. 



For bodies of revolution, if they cannot be replaced by ellipsoids 

 with sufficient accuracy, the pressures can be determined from distributions 

 of singularities on the axis which give the body outline as a closed stream- 

 line. One procedure for such computations has been outlined by von Karman. 93 

 Another procedure in which the flow field is obtained by approximations using 

 the Legendre functions is outlined by Kaplan; 94 other results are given in 

 Reference 95. These and other available methods are adequate for bodies of 

 rather large length-diameter ratios, but may lose accuracy for bodies with 

 rapid changes in curvature near the nose and afterbody. For such cases, the 

 body may be approximated by the flow about singularities distributed over a 

 ring or a disc placed perpendicular to the stream direction. Discussions of 

 such flows will be found in References 96 and 97. Estimates of the pressure 

 reduction due to wall proximity may also be obtained by the methods of poten- 

 tial theory (see, e.g., Reference 98). 



For those problems which are not conveniently handled by analytic 

 methods recourse may be had to analogy methods of solution. A most useful one 

 is that of the analogy of the potential-flow field to the electrostatic field — 

 the velocity potential and electrostatic potential both satisfying Laplace's 

 equation with corresponding boundary conditions (see, e.g. , References 99 and 

 TOO). 



The methods outlined above will be useful in only a limited number 

 of cases so that recourse must often be had to tests of models in the variable- 

 pressure water tunnels using the modelling laws discussed previously. However, 

 it will often be possible to use cavitation, itself, as an analogue method 



