A finite boundary method for fluid flow field computations

A powerful new finite boundary concept of seeking field solution, at few selected regions in the solution domain, is introduced. Also a highly economical finite boundary method (FBM) which would greatly reduce the size of the coefficient matrix of the resulting system of simultaneous algebraic equations, requiring lesser computer memory and lesser computing time, is developed. Fluid flow fields governed by the basic elliptic partial differential equations—the Laplace's and the Poisson's equations—in two independent variables are mainly considered. The computational merits of the FBM are shown by solving, as an example, a simple representative flow problem, and the relevant computational finite boundary formulae are given in tabular form. The formulae are numerically derived based on a generalized method presented here. The added feature of the FBM is that it proves to be equally economical even when the solution is sought in the entire flow domain. The problem of steady‐state viscous flows governed by the Navier‐Stokes equations, the system of two simultaneous partial differential equations—the Poisson's equation and the vorticity transport equation—makes the FBM doubly economical. The possibility of developing an efficient hybrid computational algorithm, for curved problem boundaries, in conjunction with the finite element method, is discussed. The extension of FBM to transient, non‐elliptic problems and to three‐dimensional problem fields is also indicated. The FBM has been discussed in a more detailed manner so as to clearly bring out the advantages of the new finite boundary concept.