Hydrodynamic Problems Solved by Rheoelectric Analogies 



induced by the vortices, may be computed numerically by the composition of 

 known (41) elementary perturbations. 



The solution of the problem may be obtained for a given shape of the hydro- 

 foil by a series of operations, each consisting of two stages. First, for arbi- 

 trary values of i- in the linearized free surface, the vortex distribution over the 

 chord of the foil is computed, by rheoelectric analogy, which fulfills Joukowski's 

 condition on the trailing edge, without, however, complying with the constant 

 pressure condition at the free surface. Second, the ordinates of the free sur- 

 face, which would in reality induce the preceding vortex, are determined nu- 

 merically. This allows a new distribution of potentials on the z-axis and a new 

 analog computation of the connected vortex. The cycle of operations is continued 

 until the potentials on the free surface and the vortex distribution converge 

 simultaneously towards functions which represent the solution of this boundary 

 value problem. A few approximations are generally sufficient. Instead of intro- 

 ducing an arbitrary free surface into the first analogical approximation, it is 

 easy to introduce the boundary conditions corresponding to zero or infinite 

 Froude numbers. 



The accuracy of this method was verified by comparison of analog results 

 to those obtained by Isay (42) in the case of a flat plate with incidence (Fig. 4). 

 The application of the rule of reverse flows to free- surface flows and finite 

 Froude number (15) permits the useful exploitation of results obtained in the 

 case of the flat plate and the rapid determination of the influence of the free 

 surface on foils of arbitrary shapes (Fig. 5). An interesting example of the pos- 

 sibilities of the method is given in Fig. 6 which shows for different Froude num- 

 bers the distribution of perturbation velocities on the lower and upper surfaces 

 of a flat plate with flap slightly immersed. 



Design of Siibcavitating Foils Near the Free Surface— The same method 

 may be used to design hydrofoils with given load and thickness distributions. 

 Two effects must be then considered separately; the first corresponding to the 

 distribution of the connected vortex y (<f ), i.e., the lifting effect, and the second 

 to the equivalent distribution of sources and sinks, i.e., the thickness effect. 

 The boundary value problem is now completely defined and the rheoelectric 

 simulation is very simple. 



Figure 7 shows, for different Froude numbers, the mean lines obtained for 

 the NACA 65 pressure distribution. From the linearized theory results and in 

 order to verify them, a hydrofoil and the corresponding free surface were repre- 

 sented in a rheoelectric tank. By considering the streamlines of this flow as 

 shown in Fig. 8 it is possible to verify how the Joukowski condition on the trail- 

 ing edge and the free- entry shock condition at the leading edge are fulfilled. The 

 lift coefficient computed from the value of the circulation, corresponding to the 

 electric results, is 0.3% higher than that chosen to design the hydrofoil. 



Supercavitating Hydrofoils Near the Free Surface 



Small Froude Numbers— \n the case of small Froude numbers the gravity 

 field effects on the free surface and on the boundaries of the cavity may be con- 

 sidered. The rheoelectric method enables us to take them into account with 



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