The time-averaged Navier-Stokes equations are solved numerically by a finite-volume method and applied to study flow around two-dimensional bluff bodies. The finite-volume equations are formulated in strong conservative form on a general, nonorthogonal grid system. The resulting equations are then solved by an implicit, time marching, pressure-correction based algorithm. If the flow problem has a steady state solution, then it is obtained by taking sufficient time steps until the flow field remains unchanged with time. As test cases for the developed methodology, two problems are selected; one has a steady state solution and the other has only a transient solution. Numerical predictions are obtained with the standard k-ϵ turbulence model for the steady state, turbulent flow problem. The k-ϵ model was able to predict the major, experimentally observed flow characteristics including the small separation bubble near the rear end of the body selected for the steady state test case. For the transient test case, the algorithm correctly captured the transient nature of the problem. However, agreement with the experimental results was only moderate because of the lower order differencing scheme employed in the method.