Recent Progress in the Calculation of Potential Flows 



A SHORT DESCRIPTION OF THE BASIC METHOD 



General Remarks ' 



In this section a general description of the basic method will be given, using 

 a minimum of mathematical formulas. All the basic equations and formulas can 

 be found in Ref. 1. If a maximum amount of detail is desired the reader is re- 

 ferred to the various reports and papers listed in (1). Three classes of shapes 

 are treated: two-dimensional bodies, axisymmetric bodies, and truly three- 

 dimensional bodies that may or may not have planes of symmetry. 



Mathematical Statement of the Problem 



The problem considered is the irrotational flow of an inviscid, incompres- 

 sible fluid about an arbitrary body surface or surfaces on which the normal ve- 

 locity of the fluid is either zero or a known quantity. Furthermore, the geometry 

 itself may vary with time. Except at the known body surface, the fluid is un- 

 bounded, and the onset flow, i.e., the velocity field that would exist in the fluid if 

 the body were removed, is prescribed. This is a so-called Neumann problem for 

 Laplace's equation and can be formulated mathematically in the following way. 



Let the surface of the body be denoted by s, and let the velocity field that 

 would exist in the fluid if the body were removed be denoted by V^. In most 

 cases the onset flow is a uniform stream, and hence V^ is a constant vector. 

 The situation is sketched in Fig. 2 for the case of a fully three-dimensional flow 

 about a single body surface s. For more than one body surface, the situation is 

 not essentially different. The disturbance velocity field due to the presence of 

 the body surface is assumed to be irrotational, and thus it may be expressed as 

 the negative gradient of a potential function qj. This function must satisfy three 

 conditions: It must satisfy Laplace's equation in the region R' exterior to s, 

 must approach zero at infinity, and must have a normal derivative on the surface 

 s equal but opposite to the normal component of the onset flow. (The last condi- 

 tion is where the total normal velocity on the body surface is prescribed as zero. 

 11 it is prescribed as nonzero, there is no essential change.) These three condi- 

 tions may be expressed symbolically as 



V^ (p = in region R' , (1) 



Igrad (p| -» for ( x^ + y ^ + z 2) -» co , (2) 





+ nV„ 



(3) 



where n is the unit outward normal vector on the surface as shown in Fig. 2 and 

 n denotes distance along this normal. The Laplacian operator is denoted by v^. 

 The plus sign in Eq. (3) is because the normal velocity due to the body is - 3(p/3n. 



319 



