| This thesis presents ghost cell methods on adaptive Cartesian grids to solve two dimensional steady compressible flows. According to the properties of Cartesian mesh, a quadtree-based data structure, which is ideally suited for Cartesian grid scheme, is implemented, and the integer variables stored per cell and the relationship between each cell are presented. For mesh refinement, we provide three kinds of mesh refinement strategies: geometry-based adaptation, curvature-based adaptation, and solution-based adaptation. Two kinds of criterions of solution-based adaptation, which called pressure gradient and curl and divergence of velocity, are presented. Also the adaptation criterions are compared on a test case.In this thesis the governing equation is two-dimensional Euler equation. In order to solve the multi-dimensional Euler equations, dimensional splitting method is applied. To satisfy the nonuniformed Cartesian grid, we constructed a MUSCL scheme on the nonuniformed Cartesian grid. A MUSCL-type extrapolation using a MINMOD slope limiter, with a formal second order accuracy in space, has been applied to extrapolate the conserved variables onto the left- and right-hand sides of each cell face. An approximate Riemann solver, which called HLLC scheme, is used to get face flux. Time discretization use the optimal second TVD Runge-Kutta method。To deal with wall boundary condition, we used the ghost cell methods. In this paper, we constructed three kinds of ghost cell methods: Symmetrical Technique, FGCM method and GBCM method. We use a flow past a circular cylinder to test each method described above.Finally, we use our methods to simulate some problems, which include flow past NACA0012 airfoil, RAE2822 airfoil, two-displaced NACA0012 airfoils and GA(W)-1 multi-element airfoil. From the numerical results, we validated the reliability and stability of our code, also validated the methods presented by this thesis are suited for two-dimensional inviscid compressible steady flow problem.
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