Malavard 



From these three elements— experience acquired in incompressible aero- 

 dynamics, the linearized theory of cavitations, and auxiliary analytical data — 

 naval hydrodynamic studies have been developed, as outlined below and illus- 

 trated in Fig. 1. 



Two-Dimensional Problems 



In 1958, Luu carried out studies on the solution of the direct problems of 

 supercavitating hydrofoils (8,9). These studies were the continuation of impor- 

 tant research devoted to the problem of thin jetstreams in aerodynamics (Refs. 

 8 and 10 through 12) and came within the framework of linearized free boundaries. 



In 1960, a research program was envisaged concerning the effects of the free 

 surface on slightly immersed sub- and supercavitating hydrofoils. In the case of 

 small Froude numbers, where there is a considerable influence of the gravity- 

 field effect, it was possible to proceed easily to hydrofoil design for imposed 

 pressure distribution (inverse problem) (13, 14). These studies took into account 

 the gravity effect on the free surface and on the finite cavity, which, to our 

 knowledge, had not yet been treated. The direct problem in the case of the im- 

 mersed flat plate was also solved and allowed a useful comparison with analytical 

 results (13,15). 



In the case of high Froude numbers and zero cavitation number, Luu and 

 Fruman published, in 1963, a rheoelectric method permitting the design of ven- 

 tilated hydrofoils with arbitrary local pressure distribution (16). The results 

 obtained agreed with those of Auslaender (17), published shortly before, and ex- 

 tended them by the definition of shapes with larger lift-drag ratios. It was 

 proved that the drag of supercavitating hydrofoils is related to the angle of the 

 spray far downstream, and it seems natural that these studies led to the design 

 of base-vented hydrofoils with zero drag (13). 



Subcavitating cascades had been thoroughly studied earlier by Malavard, 

 Siestrunck, and Germain, Refs. 18 through 22, within the framework of the foil 

 theory. The linearization used by Luu in the case of thin- jet flap on the trailing 

 edge of cascades (8) was easily extrapolated to supercavitating cascades (23) 

 which were liable to be used in certain types of pumps and turbines. 



Three -Dimensional Problems 



Hydrofoils— The two-dimensional studies on supercavitating hydrofoils led 

 Luu to carry out an analog simulation with finite- span wings (24). The experi- 

 ence gained in lifting- surface problems, published in the work of Malavard, 

 Duquenne, Granjean, and Enselme, Refs. 25 through 28, allowed a very rapid 

 implementation of the supercavitating problem in an unbounded flow field, by the 

 introduction of an ingenious decomposition of the potential. This will be ex- 

 amined in detail later in this paper. 



The method used also permitted the design of supercavitating wings at zero 

 cavitation number near the free surface (29) . The optimal vortex distribution 

 over the span was obtained by using the properties of the potential in the Trefftz 



368 



