BEM SOLUTION OF THE 3D INTERNAL NEUMANN PROBLEM AND A REGULARIZED FORMULATION FOR THE POTENTIAL VELOCITY-GRADIENTS

The direct boundary element method is an excellent candidate for impos ing the normal flux boundary condition in vortex simulation of the thr ee-dimensional Navier-Stokes equations. For internal flows, the Neuman n problem governing the velocity potential that imposes the correct no rmal flux is ill-posed and, in the discrete form, yields a singular ma trix. Current approaches for removing the singularity yield unacceptab le results for the velocity and its gradients. A new approach is sugge sted based on the introduction of a pseudo-Lagrange multiplier, which redistributes localized discretization errors-endemic to collocation t echniques-over the entire domain surface, and is shown to yield excell ent results. Additionally, a regularized integral formulation for the velocity gradients is developed which reduces the order of the integra nd singularity from four to two. This new formulation is necessary for the accurate evaluation of vorticity stretch, especially as the evalu ation points approach the boundaries. Moreover, to guarantee second-or der differentiability of the boundary potential distribution, a piecew ise quadratic variation in the potential is assumed over triangular bo undary elements. Two independent node-numbering systems are assigned t o the potential and normal flux distributions on the boundary to accou nt for the single- and multi-valuedness of these variables, respective ly. As a result, higher accuracy as well as significantly reduced memo ry and computational cost is achieved for the solution of the Neumann problem.