Hydrodynamic Problems Solved by Rheoelectric Analogies 



■ -±= i(x.,y.,z.) - \ gradn^nqdcT. " (17) 



Computation Procedure .- 



The analog simulation of the above boundary condition is very simple. 

 Nevertheless, two different rheoelectric tanks are necessary, one to represent 

 the actual flow field outside the body, and the other to represent the field inside 

 the body. 



The computation procedure is as follows: 



1. For the initial iteration it is supposed that B-^^/^n = i (x^.y^.z^), i.e., 

 that the regular part s^v oi 'i> is neglected. The computation corresponds to the 

 solution of the external field problem for a zero Froude number. The potential 

 distribution ^^Cx^.y^.z;) on each point of 2 is then obtained. 



2. These values are then introduced on the corresponding points of the 

 internal domain. The measure of the normal derivatives B0;/Bn gives a first 

 plausible distribution of q(Xi,yi,Zi) = (a^^/Bn - B^^/Bn). With these values it 

 is possible to compute numerically, for a given Froude number, the normal 

 derivatives over Z due to the regular part, i.e., the normal derivatives induced 

 by the free surface 



I 



grad n^n qda 



S 



3. Introducing this integral into Eq. (17) gives a new corrected distribution 

 of B(;6+/Bn, which is then imposed on the surface 2 of the external flow field. 

 Hence, a new distribution of the ^^ potential is obtained and permits us to con- 

 tinue the procedure by step 2. This iteration procedure is repeated until the 

 convergent values of ?^^ are obtained. 



Application of the Method 



To test the validity of this method the above computation procedure was 

 applied to the case of an immersed circular cylinder beneath the free surface. 

 The results obtained were in good agreement with those computed from the 

 analytical solution by Havelock (46) (Fig. 27). 



At the present time, the work carried on at the Centre de Calcul Analogique 

 concerns the study of three-dimensional flow fields with free surface. The stud- 

 ies will permit us to obtain the pressure distribution over a thick hull and the 

 wave drag attached to it, over a wide range of Froude numbers. The proposed 

 hull is represented in the rheoelectric tank by 240 electrodes, i.e., the velocity 

 tangential condition is satisfied on 240 control points over its surface. To com- 

 pute this problem numerically it was necessary to solve a 240 x 240 matrix at 



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