Projection Methods For Numerically Solving Incompressible Flow

Among the numerical methods for solving the unsteady incompressible Navier-Stokes (N-S) equations in primitive variable form, the projection method has attracted increasing attention because of its high efficiency. However, most of the existing projection methods are only first order or incompletely second order accuracy in time. Therefore, the primary purpose of the present study is to construct the projection methods in higher order accuracy. Furthermore, the proposed projection methods will be applied to nonstaggered grids, curvilinear coordinates, and moving mesh cases.On the development of higher order accurate projection methods, the following works are completed in this dissertation: (1) The concept of continuous projection method is proposed, which overcomes some difficulties in the accuracy analysis of semi- and fully-discrete projection methods. (2) Using local truncation error analysis, the sufficient conditions for the continuous projection methods to be temporally second order as well as third order accurate are derived, and the fully second order and fully third order accurate discrete projection methods are proposed correspondingly. (3) These two sufficient conditions show clearly that the artificial boundary conditions are in close relative with the guessed pressure in order that the projection methods are higher order accurate, which is helpful to clarify some controversies on the relationship between the accuracy and the artificial boundary conditions in projection methods. (4) A heuristic stability analysis technique is applied to the second order as well as the third order accurate projection methods, which show both methods can be stable. It is also founded that the pressure updating formula has great impact on the stability of third order or higher order accurate projection methods: using the so called consistent pressure updating formula, the schemes can be stable, otherwise they can become unstable. These conclusions are verified by numerical results. On the application of the projection methods on nonstaggered grids, two techniques to cure the "checkerboard" phenomena in the pressure field are proposed (1) The geometrical grid Reynolds Number() and an adaptive factor that takes the degree of "checkerboard" phenomena into consideration are introduced and incorporated into the approximate projection method proposed by Armfield[71], which improves the applicability as well as the accuracy of the original Armfield method. (2) An exact projection method based on the pressure filter is proposed. This method shares some important features of the staggered grids, therefore, can be applied to incompressible flows with high Reynolds Number.Finally, the proposed fully second order accurate projection method on nonstaggered grids is extended successfully to curvilinear coordinates and moving mesh cases.