Boundary element solution of Poisson\u27s equations in axisymmetric laminar flows

The primitive variable Navier-Stokes equations may be replaced by two equations using the derived variable of vorticity. These equations model separately the kinematic and kinetic parts of the problem. Two boundary element solutions for the kinematic equations were developed for axisymmetric flow geometries. The first was based on the fluid mechanics analogy of the Biot and Savart formula for the magnetic effects of a current. The second was the solution of the vector Poisson\u27s velocity equation using the direct boundary element equation. Numerical integration algorithms were developed which were used for all integrals;Integral solutions for Poisson\u27s pressure equation and Poisson\u27s vector potential equation were derived using the direct boundary element equation. The equations were integrated using the algorithms developed for the velocity solutions;The axisymmetric laminar Navier-Stokes solution was completed by solving the kinetic vorticity transport equation with finite difference methods. Two finite difference methods developed for the complete 2 dimensional non-linear Burger\u27s equation were modified for use on the axisymmetric form of the vorticity transport equation;This complete Navier-Stokes solution was then used to verify the form of the six boundary element equations and the accuracy of the integration algorithm developed. This was done by solving three steady state flow problems and one time dependent flow problem which were designed to simulate flow in power hydraulic components;Flow problems were encountered which produced ill-conditioned kinematic systems with attendant unstable solutions and large errors. Solution algorithms were developed which stabilized the associated matrix operator and improved solution performance. The method is based on the theory and numerical methods of Tikhonov regularization as it applies to linear algebraic systems of equations