| The nodal integral method can achieve the same accuracy as many conventional numerical methods using coarser mesh and less CPU time. In early applications of the nodal integral method to the Navier-Stokes equations, the nonlinear convection terms were treated as part of the pseudo source terms. The transverse-averaged continuity equations were used to solve for two of the transverse-averaged velocities, and two of the transverse-averaged momentum equations were used to solve for transverse-averaged pressures. This led to a numerical scheme that was asymmetric in spatial directions.; A modified nodal integral method is developed in this dissertation, in which a Poisson equation is used and the nonlinear convection terms are kept on the left hand side of the transverse-averaged momentum equations. The numerical scheme thus developed has the following advantages: (1) The use of Poisson equations leads to a model symmetric in all spatial directions. (2) The local cell-interior solutions of the transverse averaged velocities have a component that varies exponentially in space. These exponential terms can capture steep spatial variation of velocities within each cell, thus, allowing the use of coarse meshes. (3) The appearance of the local Reynolds number in the exponential terms leads to inherent upwinding in the numerical scheme.; In this dissertation, the modified nodal integral method is first developed for two-dimensional, time-dependent, incompressible Navier-Stokes equations, then extended to three dimensions. Results from both the two-dimensional and three-dimensional codes are compared with reference solutions and results obtained using commercial software. Comparison of the numerical results shows that the modified nodal integral method can achieve the same accuracy as other numerical methods using coarse meshes.; A parallel version of the modified nodal integral method is also developed for the two-dimensional Navier-Stokes equations with Drichlet boundary conditions. Good speedup is obtained for up to four processors for the modified lid driven cavity problem with exact solution. |
