Numerical solution of two-dimensional navier stokes equations
Gupta, Murli Manohar
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An attempt has been made to develop a convergent iterative method to obtain two-dimensional steady state solutions of the Navier-Stokes equations. In the last few years, various successful attempts have been made to apply numerical methods to obtain both the time dependent and steady state solutions of the Navier-Stokes equations for different types of boundary configurations. The non-linearity of the differential equations makes the convergence of the iterative method difficult. The limitations on the storage capacity of the available computing machines makes it impracticable to obtain solutions having any predetermined accuracy. In addition, the imposition of the natural boundary conditions which prescribe the velocities on the boundaries, also contributes to the complexity of the problem. A brief survey of some representative work done on the numerical solution of Navier-Stokes equations has been given here. The Navier-Stokes equations have been reduced to a system of two second order coupled differential equations which are then solved numerically. Some finite difference schemes have recently been suggested which guarantee the convergence of the iterations for solving the two difference equations. However, these schemes require a certain number of parameters which have to be determined by numerical experimentation. Our aim here is to develop a procedure which would be economical and would require a minimum of numerical experimentation. We have succeeded in showing that only one parameter besides the optimum over-relaxation factors is sufficient for convergence. Different criteria for convergence are studied and the one, which appears to be most efficient, has been used to obtain numerical solutions for the steady flow of a viscous fluid in a two-dimensional cavity flow. Another problem for which the numerical solutions have been obtained is the separating flow due to a thin plate of finite length placed at the point of symmetry of an otherwise two-dimensional stagnation point flow. A non-uniform mesh has been used for this problem and the numerical results for different values of Reynolds number have been obtained. These are given in the form of stream lines and equi-vorticity curves. We have also extended the difference schemes to problem of axisymmetric flows although no numerical results have been obtained. The results obtained in this thesis may be considered as qualitatively satisfactory in the sense that they exhibit the characteristics of the fluid flow. These results can not be considered quantitatively reliable unless some insight is gained about the convergence of the numerical solution of the discretized problem to that of the differential equations. This can partly be done by repeating the calculations using smaller mesh sizes. However, this would require a considerable amount of computing time. It would be interesting to investigate the possibilities of improving the results using some higher order stable difference schemes or finer mesh sizes in the boundary layer only without unduly increasing the computing time.