Least squares grid free kinetic upwind method

Least Squares Kinetic Upwind Method(LSKUM) is a node based scheme. This is a kinetic theory based upwind scheme derived from the Boltzmann equation. The method makes use of the fact that, Euler equations are suitable moments of the Boltzmann equation when the velocity distribution function is a Maxwellian. LSKUM has the potential to operate on any type of mesh structured,unstructured,Cartesian,Hybrid, Overlapping(Chimera) meshes, or even an arbitrary distribution of points. Given any distribution of points(which can be obtained from any type of grid generator), the LSKUM solver needs the data of the immediate surrounding nodes at every point. This data of the immediate neighbouring points is called as the connectivity data or simply the connectivity. A simple approach to build connectivity for the nodes, would be to search through all the points in the data set and then retain only those which lie within a small radius which defines the immediate neighbourhood of the nodes. When the number of points become more, a lot of time is unneccesarily spent in search operations. Therefore it is desirable to use suitable data structures to optimise the search operations. Towards this we have used quadtree data structures to build the connectivity data. Generation of connectivity is basically a geometric proximity problem. The quadtree data structures are well known for their applications to geometric proximity problems. In our present work we have developed a preprocessor which uses the quadtree data structures to generate the connectivity data. Given any arbitrary distribution of points, the preprocessor builds a background quadtree data structure around these points. Then using this quadtree structure the desired connectivity data is generated. The solver then operates on the given distribution of points with the connectivity data generated by the preprocessor. The main advantage of this approach is, for any given distribution of points the solver can operate on them without the need to tesselete the points to form lines along any coordinate directions or to form any edges defining cells. For complex configurations, with such an approach we can use any simple grid generator to generate grids over each individual component. The final distribution of points is then obtained by combining all the nodes from each component grid. The preprocessor will then operate on these points to generate the connectivity data, thus facilitating the solver to operate. With this approach, we can also use overlapping grids without any need to keep track of the overlapped regions. In the conventional methods for example using finite volume methods, the solver has to keep track of the various overlapped boundaries, also construct accurate interpolation schemes for transfer of solution across these various overlapped boundaries. In our present work using LSKUM in conjunction with the quadtree preprocessor, we do not face any of these difficulties. The solver only sees a distribution of points, at which the solution is directly updated. In fact in the overlapped regions we have more number of points at which the solution is directly updated. Hence we get a better resolution of the flow compared to the interpolation procedures which do not solve for any fluid dynamic equations in the overlapped regions. In the first chapter we present the details of formulation for LSKUM. This includes the extension of LSKUM for 3-D problems which is a recent addition from the existing 2-D LSKUM. Along with the extension of LSKUM to 3-D, we also give a new kinetic theory based boundary treatment for implementing the far field boundary condition. This is called as the Kinetic Outer Boundary Condition(KOBC). This treatment significantly differs in its approach from the Riemann treatment for the outer boundary condition. KOBC is very similar for both 2-D and 3-D problems and unless like the Riemann treatment does not assume a local 1-D flow model normal to the outer boundary. Another important feature of KOBC, which makes it different from the Riemann treatment is, KOBC is independent of the flow conditions at the outer boundary. That is the formulae for KOBC remain identically same for all subsonic inflow/outflow and supersonic inflow/outflow conditions at the outer boundary. We have successfully used KOBC in all our 2D and 3D computations. In the thesis we present results for 2-D LSKUM on a variety of point distributions conclusively establishing the grid free nature of the LSKUM solver. This has been made possible by the use of the quadtree preprocessor. Results are presented for flow past NACA0012 airfoil and NACA 0012 biplane configuration for various test conditions. The flow computations have been done using point distributions obtained from structured, unstructured, Cartesian and overlapping grids. In order to validate the 3D~LSKUM solver we have tested it on a variety of problems. The first problem considered is the simulation of an intense blast wave in air. This problem deals with extreme flow conditions involving very high temperature gradients. This a problem of spherical symmerty with the spherical blast wave propagating radially outwards in time. In our computations we have used points from a simple Cartesian grid. In spite of using points from rectangular grid the method captures the shock front very well maintaining the spherical symmetry. In the next test case, we consider supersonic flow past a hemisphere. Through these two test cases we first establish the proper working of the interior and the boundary schemes. In order to further prove the working of the method, we compute transonic flow past ONERA-M6 wing. This is a more realistic 3D configurartion. The computed results compare very favourably with the AGARD values. Having obtained good results for the various test cases, we finally compute high supersonic flow past a generic flight vehicle. This is a fairly complex configuration consisting of Bluntcone-cylinder-flare with lifting surfaces. Extensive computations for different Mach numbers and angle of attack have been carried out for this geometry. All the computations compare satisfactorily with the available experimental values. In the end, we present two further extensions to LSKUM. The first case consists of extension of LSKUM to moving grids. This is called as LSKUM-MG. In the second extension we give a new novel approach for obtaining higher order schemes for LSKUM. We call this as LSKUM-NH (LSKUM-Novel Higher order). Formulation for 1D LSKUM-MG is followed by results on the standard moving piston problem. The formulation for 2D LSKUM-MG is then given along with the kinetic theory based boundary conditions for the moving boundaries. Results for flow past an airfoil oscillating in pitch is presented using the 2D LSKUM-MG. Finally we give some preliminary results for 1-D LSKUM-NH for 1-D shock tube problem.