ON THE UNIFORMLY VALID APPROXIMATE SOLUTIONS 
OF LAPLACE EQUATION FOR AN INVISCID FLUID 
FLOW PAST A THREE-DIMENSIONAL THIN BODY 
J. S. Darrozes 
Ecole Nattonale Supérteure de Techntques Avancées 
Parts France 
ABSTRACT 
The classical solution of the Laplace equation for an 
inviscid incompressible fluid flow past a three-di- 
mensional thin body, is shown to be not uniformly 
valid in the vicinity of the planform edge (A). In order 
to find a solution which is valid in the vicinity of (A), 
the technique of ''matched asymptotic expansions" is 
used, The inner solution in the neighbourhood of a 
rounded leading edge brings a shift correction tothe 
classicalouter solution. The inner solution in the vi- 
cinity ofa sharptrailing edge givesthe starting shape 
of the vortex sheet. 
I. INTRODUCTION 
The study of three-dimensional flows past arbitrary bodies 
gives rise to many problems of greatinterestin Naval Hydrodynamics, 
Foraninviscidincompressible fluid, the velocity potential isa solution 
of Laplace equation, and in the few last years, most of basic works 
have dealt with the numerical methods used to solve this problem, The 
greatest difficulty comes from the fact that there is no rigourous ma- 
thematical theory available for such a problem and numericalattempts 
may be handled only with addition of physical assumptions, A unique 
solution could be obtained, only after the difficult analysis ofthe corres - 
ponding high Reynolds number flow, in the limiting case of an evanes- 
cent viscosity. As it is not possible to do so, it is necessary to guess 
[1 ]some results in order to define a problem which has a unique so- 
lution. For instance, if the physical flow takes place with separation, 
This work has been supported by the « Office National d’Etudes et de Recherches Aérospatiales » 
29, Av. de la Division Leclerc - 92320 CHATILLON 
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