Numerical Modelling of Multi-dimensional Steady Ideal Gas and Fluid Flows

The thesis is devoted to the numerical modelling of two- and three-dimensional ideal fluid flows in channels of complicated geometries on the basis of the adaptive grid method. Iterative finite difference methods for the numerical modelling of ideal gas and fluid flows are developed using new dependent variables. For two-dimensional flows these variables are stream function and vorticity function, for three-dimensional flows these variables are: vector potential and vorticity vector. The set of equations for the stream function and vorticity, for which the appropriate boundary conditions are taken into account, are solved together with the energy equation and the relations of the marching method for the pressure calculation by a finite difference approximation on a curvilinear grid. Also the equations for the covariant components of vector potential, the boundary conditions for these components, and the equations for the contravariant components of the vorticity vector are constructed on curvilinear grids. For the construction of curvilinear grids - adapted either to the singularities of the solution or to the complicated geometry of domains - the equidistribution method is developed. The governing equations of the equidistribution method are solved using a finite difference approximation. The equations for the construction of curvilinear adaptive grids are obtained for two- and three-dimensional domains, for surfaces and curves in two- and three-dimensional spaces. The developed algorithms for calculating ideal gas flows are modified for the numerical solution of problems on steady ideal fluid flows with gravitational surface waves within the framework of the shallow water theory. For river channels of complicated configuration in the presence of islands the first approximation of this method is applied. For the realization of the developed algorithms the effective complexes of computer codes are created. The investigation of the problems on the fluid flows through channels and rivers of complicated form is carried out