ON IRROTATIONAL FLOWS EQUIVALENT TO THE 

 BOUNDARY LAYER AND WAKE 



INTRODUCTION 



One of the simplest and most useful results of boundary-layer theory 

 is that the flow exterior to the boundary layer, which will be assumed to 

 be irrotational, is pushed outwards by an amount called the displacement 

 thickness. This suggested, early in the development of boundary-layer 

 theory, that the accuracy of a boundary-layer calculation for a body could 

 be improved by adding the boundary-layer thickness to the body dimensions 

 and using the predicted pressure distribution on the so-thickened body in 

 a recalculation of the boundary layer. The irrotational field about the 

 thickened body, including the displacement thickness of the wake, is itself 

 of great interest, and numerous attempts have been made in this manner to 

 calculate the effect of the boundary layer and wake on the outer irrota- 

 tional flow. 



The concept of the thickened body gives an approximate model which is 

 usually justified by its consistency with the approximations of thin 

 boundary-layer theory. We shall review the basis of this model and suggest 

 ways of refining it. Such a development would be useful for several current 

 problems of ship hydrodynamics, among them the determination of Betz 

 sources in a method of calculating viscous drag by an analysis of wake 

 survey data, and the investigation of the effect of the boundary layer and 

 wake on wavemaking resistance. 



We shall assume that the boundary layer and wake (BLW) are known, 

 and seek an irrotational model which yields the actual outer flow. An 

 approximate solution of this problem has been given by Preston [1] and 

 Lighthill [2]. 



Solution of Preston and Lighthill - Two-Dimensional Case 



Let us consider a two-dimensional flow about a body of small curvature 

 in a uniform stream of velocity U , on which a thin boundary layer is 

 present. Coordinates parallel and normal to the surface will be denoted 

 by (s,n), and the corresponding velocity components by (u,v). We shall 



