Application of finite differences in conjunction with the method of local potential in hydrodynamic stability
Abstract
The finite difference method was applied in conjunction with the theory of the local potential in order to solve numerically the OrrSommerfeld equation for Poiseuille flow between two parallel and infinite plates. The apparent advantages of this approach are that one does not require the use of trial functions, which in previous attempts have led to lowaccuracy results on the critical Reynolds number, and instead of integration only summation is needed. On the other hand, the price paid for these advantages is a considerable increase in the matrix size. The variational formulation of the OrrSommerfeld equation, constructed according to local potential theory, is discretized, using various schemes for treating the boundary conditions. It was found that applying finite differences to the potential formula leads to worse results numerically than direct application to the original differential equation, and that no gain in size of calculations, time, or accuracy is obtained by joint application of local potential theory and finite differencing.
 Publication:

Bulletin de l'Academie Royale de Belgique
 Pub Date:
 1975
 Bibcode:
 1975BARB...61..991V
 Keywords:

 Finite Difference Theory;
 Flow Stability;
 Laminar Flow;
 OrrSommerfeld Equations;
 Parallel Plates;
 Potential Flow;
 Boundary Conditions;
 Boundary Value Problems;
 Convergence;
 Hydrodynamics;
 Matrix Theory;
 Reynolds Number;
 Fluid Mechanics and Heat Transfer