| A new numerical method for solving the unsteady, incompressible Navier-Stokes equations has been developed. This new technique called the spectral-difference method is based on the multidimensional extension of the Jacobi polynomial collocation technique. This method has been tested using the cavity flow problem as the test case. Both the lid driven cavity flow problem for the cartesian coordinate system and the polar cavity flow problem for the cylindrical polar coordinate system has been solved. The lid driven cavity flow problem has been solved for both two- and three-dimensional cases using a square cavity and a rectangular cavity with an aspect ratio of two. Comparisons of the centerline velocity profiles for corresponding two- and three-dimensional problems have been presented. The results indicate that for moderate to high Reynolds number, the two-dimensional results are significantly different from results obtained using the full three dimensions. In addition the spectral-difference method, a new computational grid has been developed. This grid does not contain corner points for the two-dimensional case. A new iterative pressure solver has also been developed. This new pressure solver has been tested in both cartesian and polar coordinate systems and is applicable for both two- and three-dimensional situations. Finally, this new numerical method has been used to simulate the flow that occurs in the inlet section of the hydrocyclone. Results indicate that the flow is not symmetrical and is greatly influenced by the vortex finder radius, the Reynolds number, the entry slit width and the radial component of the entering fluid. The simulation of the header section was performed in two dimensions. |
