Smith 



Once the entire procedure for determining the influence function for an ele- 

 ment covered by sources of this type has been established, it should be a rela- 

 tively minor problem to modify the existing three-dimensional computer pro- 

 gram. Exactly the same kind of modification procedure has already been 

 accomplished for hydrofoils. It appears that the present work is producing a 

 practical method of evaluating this special source, although whether or not it 

 is the best possible procedure is open to question. A full report on evaluation 

 of this source function is expected in late 1968. 



DIRICHLET AND OTHER PROBLEMS 



Problems of fluid mechanics are normally Neumann-type problems. There- 

 fore, in our work, we have been concerned with first boundary- value problems. 

 Although the computing programs have the inherent capability of solving a wide 

 variety of these problems, only a few of these have been run, so that we are 

 unable to make such definite statements about their accuracy and ability to ob- 

 tain solutions as those we can make for fluid problems. A few studies have 

 been made of a temperature distribution in solids for which analytic solutions 

 exist, and the accuracy was found to be good. Our principal reason for men- 

 tioning Dirichlet problems is to remind the reader that the basic procedure 

 encompasses that capability. To know this may be useful to someone who finds 

 himself faced with a problem that falls into the Dirichlet class, as was the case 

 with a missiles engineer who was studying the problem of cooling reentry bodies. 

 The basic type of problem will be described to show the Dirichlet capability. 

 John Hess conducted preliminary studies to ascertain our capability, and the 

 following is taken chiefly from his memo which summarizes the work. 



As part of a reentry study program, we were asked to perform 

 certain calculations with our axisymmetric -potential -flow program. 

 The problem of interest is the cooling of a reentry body by forcing 

 liquid from a reservoir in the interior of the body through a porous 

 medium to the surface. See Fig. 17. As is well known, the flow of 

 liquid through a porous medium is governed approximately by Laplace's 

 equation in the pressure for incompressible flow or in the square of the 

 pressure for compressible flow. The boundary conditions are that the 

 pressure equal the constant reservoir pressure on the interior surface 

 of the porous medium and that the pressure equal the surface pressure 

 of the exterior flow (as obtained from hypersonic theory) on the exte- 

 rior surface of the porous medium. This is thus a Dirichlet problem 

 in the "thick shell" region between the reservoir and the exterior. 



As part of a study of added-nnass effects sponsored by the Naval 

 Ordnance Test Station in Pasadena, program SOD had previously been 

 modified to handle axisymmetric Dirichlet problems. This capability 

 had been verified by comparison with analytic solutions for exterior 

 problems. Past experience had indicated that an interior problem 

 often leads to considerably more calculational difficulties than the 

 corresponding exterior problem, especially when, as in the present 

 case, a surface source distribution is used to obtain the solution. Ac- 

 cordingly, a test case was set up and run for several boundary condi- 

 tions for which simple analytic solutions are available. The configura- 

 tion is shown in Fig. 17. It consists of two concentric spherical shells. 

 The outer one has a radius of unity and the inner one has a radius of 0.8. 

 Four boundary conditions were considered for the potential <P. They are: 



358 



