  In this work, a higher order compact (HOC) discretization is developed on the
nonuniform polar grid. The discretization conceptualized using the unsteady
convection-diffusion equation (CDE) is further extended to flow problems
governed by the Navies-Stokes (N-S) equations as well as the Boussinesq
equations. The scheme developed here combines the advantages of body-fitted
mesh with grid clustering, thereby making it efficient to capture flow
gradients on polar grids. The scheme carries a spatial convergence of order
three with temporal order of convergence being almost two. Diverse flow
problems are being investigated using the scheme. Apart from two verification
studies we validate the scheme by time marching simulation for the benchmark
problem of driven polar cavity and the problem of natural convection in the
horizontal concentric annulus. In the process, a one-sided approximation for
the Neumann boundary condition for vorticity is also presented. Finally, the
benchmark problem of forced convection around a circular cylinder is tackled.
The results obtained in this study are analyzed and compared with the
well-established numerical and experimental data wherever available in the
literature. The newly developed scheme is found to generate accurate solutions
in each case.
