Fast Multipole Boundary Element Method (FMM/BEM) for the solution of the Navier-Stokes in primitive variables based on the Burton and Miller formulation in two-dimensions

Journal of Computational Physics, 471, 2022, 111615, (2022),
https://www.sciencedirect.com/science/article/pii/S0021999122006787

In the present work the fast multipole boundary element method (FMM/BEM) is presented for the solution of incompressible Navier-Stokes equations in 2 dimensions. This is a velocity-pressure primitive variable formulation abbreviated as uv-p. The Stokes-let fundamental solution is used as a weight function in the momentum equation eliminating the pressure from free variable. This way, the LBB conditions are bypassed removing any restriction in the type of grid, placement of nodes, or interpolation. Most of the numerical methods cannot use constant elements for all fields to solve the Navier-Stokes. BEM is the only known so far semi-analytical method capable of such an achievement. The continuity is enforced with the correction of velocity, using a scalar potential, the gradient of which is computed directly with hypersingular equations. The stress hypersingular equation is added to the boundary nodes rendering the well known Burton and Miller formulation and recommendation about the coupling constant is given. Accurate results for cavity Re = 1000 with constant elements and Re=10000 with linear quadrilaterals are obtained and presented. The reached degrees of freedom (dof) are beyond 1 million. Two more problems with mixed BCs are solved. The flow past cylinder is a kind of problem in which both singular and hypersingular equations must be used to ensure convergence. The main advantage of the method is the direct computation of the wall normal stresses or tractions, which is of cornerstone importance in many realistic applications. The tractions are obtained directly from the solution of the system of equations and can be integrated on the surface of a bluff body, to obtain forces and drag and lift coefficients. The pressure is computed in a post-processing phase with hypersingular equation derived in the framework of the present method. Smooth pressure field is obtained in all cases, even when the problem is solved with unstructured grids and constant elements. The method permits the use of non-conforming grids, and it can easily be combined with other numerical methods to obtain wall tractions and stresses via hypersingular equations. Moments for pressure and convective terms are derived.