Some New Directions in Hoc Methodology:Tackling circular Geometries

Abstract

The present work is mainly deals with the development of a class of higher-order com- pact (HOC) Dnite di Derence formulations to tackle the circular geometries both for the continuous and discontinuous cases. Depending upon this, the contains of the present work can be divided into two parts. The Drst part concerned with the development of HOC schemes for convection-diDusion equations in general and incompressible viscous Dows in particular on nonuniform polar coordinate system. The basic diDerence be- tween the proposed scheme and the earlier HOC schemes is that the proposed schemes are able to handle variable coeDcients of the second order derivatives while the previ- ous schemes could deal only with unit diDusion coeDcients on cartesian or cylindrical polar coordinate on uniform grid. A fourth order accurate HOC scheme for the steady state convection-diDusion equations on non-uniform polar grid has been developed Drst. The scheme produces highly accurate results even in coarser grids for diDerent Duid Dow problems. An HOC treatment for the streamfunction-vorticity (D-!) formulation of the two-dimensional unsteady, incompressible, viscous Navier-Stokes equations on polar grid has been developed next, speciDcally designed for the motion past circular cylinder prob- lems. The scheme is second order accurate in time and at least third order accurate in space. The HOC treatment is also used to discretize the Neumann boundary conditions. The scheme is then used to solve the Dow past an impulsively started circular cylinder problem for a wild range of Reynolds numbers (Re) and to solve the Dow past rotating cylinder problems for wild range of both Re and rotation parameter (D). Present numer- ical results are then compared with the existing experimental and standard numerical results. In every case an excellent agreement has been found. In this process, some new properties have been found and some extended works have been carried out which have not been studied earlier. The second part of the present work deals with the development of Dnite diDerence algorithms which are obtained by clubbing the existing HOC methodology with a special treatment to tackle the immersed interfaces for problems having discontinuities along the circular interfaces. Firstly, a new methodology for numerically solving one-dimensional (1D) elliptic equations with discontinuous coeDcients, Duxes and singular source terms and the corresponding unsteady parabolic equations on nonuniform space grids have been developed. Stability and convergence analysis of the newly developed scheme have been carried out next. Then, this 1D idea has been extended for the 2D elliptic problems with same type of discontinuities. For both the 1D and 2D cases, numerous numerical studies on a number of problems have been done and compared present results with those obtained by well known methods. In all cases, our formulation is found to produce better results on relatively coarser grids..