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BEM SOLUTION OF THE 3D INTERNAL NEUMANN PROBLEM AND A REGULARIZED FORMULATION FOR THE POTENTIAL VELOCITY GRADIENTS
Author(s) -
Gharakhani Adrin,
Ghoniem Ahmed F.
Publication year - 1997
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/(sici)1097-0363(19970115)24:1<81::aid-fld480>3.0.co;2-l
Subject(s) - mathematics , discretization , mathematical analysis , singularity , boundary (topology) , neumann boundary condition , boundary value problem , lagrange multiplier , boundary element method , finite element method , mathematical optimization , physics , thermodynamics
The direct boundary element method is an excellent candidate for imposing the normal flux boundary condition in vortex simulation of the three‐dimensional Navier–Stokes equations. For internal flows, the Neumann problem governing the velocity potential that imposes the correct normal flux is ill‐posed and, in the discrete form, yields a singular matrix. Current approaches for removing the singularity yield unacceptable results for the velocity and its gradients. A new approach is suggested based on the introduction of a pseudo‐Lagrange multiplier, which redistributes localized discretization errors—endemic to collocation techniques— over the entire domain surface, and is shown to yield excellent results. Additionally, a regularized integral formulation for the velocity gradients is developed which reduces the order of the integrand singularity from four to two. This new formulation is necessary for the accurate evaluation of vorticity stretch, especially as the evaluation points approach the boundaries. Moreover, to guarantee second‐order differentiability of the boundary potential distribution, a piecewise quadratic variation in the potential is assumed over triangular boundary elements. Two independent node‐numbering systems are assigned to the potential and normal flux distribu‐ tions on the boundary to account for the single‐ and multi‐valuedness of these variables, respectively. As a result, higher accuracy as well as significantly reduced memory and computational cost is achieved for the solution of the Neumann problem. © 1997 John Wiley & Sons, Ltd.