Solution of the three-dimensional Navier-Stokes equations for a steady laminar horseshoe vortex flow

A low Mach number formulation of the three-dimensional Navier-Stokes equations is solved for a steady laminar horseshoe vortex flow, using a time-iterative approach. A split linearized block implicit algorithm is used, with central spatial differences in a transformed coordinate system. The stability of this algorithm in three dimensions is examined for a scalar convection model problem, and results are obtained which suggest that the algorithm is both conditionally stable and rapidly convergent when nonperiodic inflow/outflow boundary conditions are used. A new form of artificial dissipation which acts along physical streamlines instead of coordinate grid lines is also tested and found to introduce less error when the local flow direction is not aligned with the computational grid. An accurate solution for a laminar horseshoe vortex flow is computed using an improved solution algorithm with small artificial dissipation. This solution does not change significantly when the mesh spacing is halved using (15 x 15 x 15) and (29 x 29 x 29) grids. Very good convergence rates were obtained, such that residuals were reduced by a factor of 1/100 in 30 and 60 iterations respectively, for 3,375 and 24,389 grid points.