Abstract:
The work is mainly concerned with the development of compact finite difference formulations for the biharmonic equation in irregular geometries. We specifically focus our attention towards the computation of solutions of complex flow problems by using biharmonic form of the Navier-Stokes (N-S) equations. When irregular physical domains are transformed onto computational domains that are expressible in terms of conformal mappings, the system of incompressible two dimensional NS equations reduces to a single biharmonic semi-linear equation. The formulation has the advantage that the entire flow field can be described in terms of only one equation with stream function as the dependent variable. Other flow field variables can easily be post processed from stream function. The work has been divided into two parts. In the first part we develop a new fourth order accurate essentially compact finite difference scheme for the steady N-S equations. The efficiency of the scheme is highlighted by performing numerical experiments on (i) a known constructed solution and is followed by its application on three different problems with varied complexities, viz. (ii) fluid flow in a constricted channel, (iii) driven polar cavity, and (iv) flow past an impulsively started circular cylinder. The computed solutions are then compared with the existing experimental and standard numerical results, and excellent agreement is found in all the cases. In the second part we propose a new compact implicit scheme for transient biharmonic form of the N-S equations. This scheme is second order accurate both temporally and spatially. Our main objective here is not only to document the versatility of biharmonic pure stream function formulation, but also the efficiency of the newly proposed scheme in simulating the dynamics of flow inside curved regions as well as fluid-embedded body interaction..
