Discretization of the Incompressible Navier-Stokes Equations in GeneralCoordinates Using Contravariant Velocity Components

摘要

The report may be considered as a continuation and extension of the report ofCuvelier (1987), where a formulation was given of the Navier-Stokes equations coupled with an equation of convection diffusion type in general coordinates. In chapter 2 the authors recall some basic results from tensor analysis in relation to coordinate transformations. Chapter 3 treats the discretization of the divergence theorem. Chapter 4 is devoted to the discretization of a general tensor equation, which contains the Navier-Stokes momentum equations as a special case. Several discretizations are considered, each with their own merits and drawbacks. In chapter 5 the space-discretization of the incompressible Navier-Stokes equations is derived using the finite volume method derived in chapter 4. Only a global derivation is made, the detailed molecules are treated in appendix 1. The effect of the boundary conditions to the molecules of chapter 5 is given in chapter 6. The complete derivation is postponed to appendix 2. Finally chapter 7 treats the time-discretization in relation to the continuity equation.