It is well-known that Navier-Stokes equations play an important role in several sci- entific and engineering fields. A small number of exact solutions of these equations are found. So, the required information by engineers/scientists can be provided only through numerical computations of these equations. In this study, the computational techniques
proposed are finite difference methods, the most widely used technique for engineering design and analysis. Despite significant progress in computational sciences, challenges persist in the accurate numerical simulation of a broad spectrum of dynamics such as fluid-structure interactions and technical applications such as solidification of castings, crude oil productions, geothermal reservoirs, grain storage etc. To resolve these complex flow phenomena more accurately, higher order accuracy of schemes are recommended. In addition to this, the fluid domain may not only contain several high gradient regions and boundary layers but also curvilinear boundaries. To resolve such flow phenomena accu- rately, non-uniform and non-orthogonal curvilinear grids could be of great help. This is where the motivation to develop HOC schemes on non-orthogonal curvilinear grids comes from, that can be used in irregular geometries with non-orthogonal curvilinear grid settings.
The present work is mainly concerned with the development of HOC finite differ- ence schemes (time independent and time dependent) and application on non-uniform and non-orthogonal curvilinear grids. In addition, the robustness of these schemes are checked when applied in complex geometries.
The first work proposes a time independent HOC scheme that works equally efficiently on problems described on both rectangular as well as curvilinear domains. The scheme can be applied to steady-state convection-diffusion, reaction-diffusion, and convection- reaction-diffusion equations having mixed derivative as well as non-homogeneous source terms, and it can also be used to solve equations of N-S type with slight adjustment of the convection coefficients. It is fourth order accurate in space and can handle Dirichlet, Neumann and Robin type boundary conditions with ease. To validate the proposed scheme, it is first applied to problems having analytical solutions on non-uniform grids.
Subsequently, it is applied to the lid driven cavity problem with bottom wavy wall on non-orthogonal grids and finally to the lid driven vase-shaped cavity problem using non-orthogonal curvilinear grids.
In the second work, an extension of the proposed HOC scheme to a transient one is made. It is implicit and of second order accurate in time and fourth order accurate in space. It handles Dirichlet, Neumann and Robin type boundary conditions with ease.
Numerical results for the transient solutions of the fluid flow in the lid-driven cavity with the same geometries used as in the first work and their temporal variations are presented and discussed.
The third work deals with the numerical solutions of a natural convection problem in a both sided (left and right) wavy wall cavity with and without porous medium in presence of a magnetic field. For the problem without porous medium, two different
thermal boundary conditions (as two cases) are used. Case I: the bottom wall is heated while the two vertical walls are cold and the top wall is adiabatic. Case II: the bottom wall is heated, the top wall is cold and the left and right vertical walls are adiabatic.
In the event of heated bottom wall, both uniform and non-uniform heating situations are considered. Numerical results are presented for variation of Prandtl number (P r), Rayleigh number (Ra) and Hartmann number(Ha). In the case of porous medium the thermal boundary conditions are assumed to be the same as those of Case I, the computed results are discussed by varying the values of different parametersP r, Haand Darcy number (Da).
In the fourth work, a mixed convection problem in trapezoidal cavity is considered with the intuition that this type of geometry enhances the heat transfer rate. In this problem the bottom wall is heated, the two inclined sidewalls are cold while the extended moving top wall is adiabatic. First, the lid driven problem in a trapezoidal cavity has been solved for comparison with the results obtained by Mcquain [54]. Then numerical results are presented for mixed convection flow for different Grashof numbers (Gr) and Richardson numbers (Ri).
In the fifth work, flow in a rectangular vented cavity is studied. First, the results of a rectangular vented cavity problem are compared with those studied in [81] where the inlet is at the bottom portion of the left wall and the outlet is at the top portion of the right wall. Then the mixed convection results for the rectangular cavity with bottom wavy wall with the variation of Richardson number (Ri) and Reynolds number (Re) are computed and discussed.
In the final work, magnetohydrodynamic flow in a dilated channel is studied. A Hart- mann profile is imposed at the inlet of the dilated channel, no-slip boundary conditions are used at bottom and top walls whereas at the outlet, a fully developed boundary condition is adopted. Streamline contours are presented for various Reto describe the flow phenomenon in the channel.
In case of the discretization of the Neumann boundary condition, one sided second order accurate finite difference approximation inside the flow region is considered.
The computed numerical results on a coarse grid are compared with both analytical and the established numerical results available in the literature, and excellent agreement is found in all the cases. It is to be noted that all of the computations were carried out on a Core i5 processor based PC.