AN INTEGRAL-EQUATION JUSTIFICATION OF THE BOUNDARY-CONDITIONS OF THE DRIVEN-CAVITY PROBLEM

The driven-cavity problem, a renowned bench-mark problem of computatio nal, incompressible fluid dynamics, is physically unrealistic insofar as the inherent boundary singularities (where the moving lid meets the stationary walls) imply the necessity of an infinite force to drive t he flow: this follows from G. I. Taylor's analysis of the so-called sc raper problem. Using a boundary integral equation (BIE) formulation em ploying a suitable Green's function, we investigate herein, in the Sto kes approximation, the effect of introducing small ''leaks'' to replac e the singularities, thus rendering the problem physically realizable. The BIE approach used here incorporates functional forms of both the asymptotic far-field and singular near-field solution behaviours, in o rder to improve the accuracy of the numerical solution. Surprisingly, we find that the introduction of the leaks affects notably the global flow field a distance of the order of 100 leak widths away from the le aks. However, we observe that, as the leak width tends to zero, there is excellent agreement between our results and Taylor's, thus justifyi ng the use of the seemingly unrealizable boundary conditions in the dr iven-cavity problem. We also discover that the far-field, asymptotic, closed-form solution mentioned above is a remarkably accurate represen tation of the flow even in the near-field. Several streamline plots, o ver a range of spatial scales, are presented.