Boundary Elements and Viscous Flows

A Galerkin Implementation of the Generalized Helmholtz Decomposition for Vorticity Formulations

“…For two-dimensional problems, there are two components of the GHD but only one component of unknown vorticity. Wu [32] states that the normal and tangential component of the GHD are equivalent and either can be used to determine the boundary vorticity. In the following, it will be shown that, for Galerkin implementations, the normal component of the GHD leads to rank deficiency of the discretized linear system of equations.…”

“…Many investigators indicate that an overspecified set of equations must be solved to determine vorticity generation on the boundary including an integral constraint, although the precise mathematical justification for such constraints is not clear. For example, Wu [32] indicates that the linear system of equations based on a Helmholtz decomposition is rank deficient. For closure, Wu specifies that the volume integral of the vorticity field must be zero.…”

“…Approaches relying on kinematics include streamfunction-vorticity methods [1,13,22,[24][25][26], velocity-vorticity Cauchy methods [7], vorticity-velocity Poisson equation methods [5], Biot-Savart methods [4], and generalized Helmholtz decomposition (GHD) methods [18,19,[28][29][30][31][32]. Other approaches are based on dynamics (Navier-Stokes equations) on the boundary [12,33].…”

Parallelization of a vorticity formulation for the analysis of incompressible viscous fluid flows

“…Several approaches have been developed over the years to derive vorticity boundary conditions from the prescribed velocity boundary conditions and the vorticity within the domain. Some of these approaches include streamfunction-vorticity methods [1][2][3][4][5][6], velocity-vorticity Cauchy methods [7], vorticity-velocity Poisson equation methods [8], Biot-Savart methods [9], and generalized Helmholtz decomposition methods [10][11][12][13][14][15].…”

“…The generalized Helmholtz decomposition (GHD) is a kinematic statement which relates the velocity at a point within the domain to the domain vorticity and the imposed velocity boundary conditions. Wu [13] uses the GHD to determine values of the boundary vorticity which then yields Dirichlet boundary conditions for his solution of the vorticity equation. Wu states that, for a two-dimensional problem, either component of the GHD can be used to determine the boundary vorticity, since the tangential and normal components are equivalent.…”