A new approach is presented for a flow simulation system using generalized grids. In a generalized grid, the physical domain of interest is decomposed into cells with an arbitrary number of edges or faces. The grid can be of structured, unstructured, or hanging node type or an arbitrary combination of the types. A cell-face-based data structure is used to store the grid information. A flow simulation system is developed for generalized grids that can handle static and dynamic grids. The full Navier-Stokes equations, in the integral form, are taken as the relations that govern the fluid flow. A cell-centered finite volume scheme is developed for solving these governing equations. The numerical flux across the cell faces is evaluated by an upwind scheme based on Roe's approximate Riemann solver. A higher-order scheme is formulated by utilizing Taylor's series expansion and Green's theorem. Limiter functions are used to preserve monotonocity. The skin-friction coefficient is used to evaluate the accuracy of the limiter functions. The generalized minimal residual method is utilized to solve the sparse linear system of equations resulting from the linearization of the flux vectors. The Spalart-Allmaras one-equation turbulence model has been implemented for the generalized grids and is used to evaluate the turbulent viscosity. For dynamically moving bodies, the equations of classical mechanics are used to predict the trajectory based on the external aerodynamic and body forces. A variety of computational examples are presented to demonstrate the wide range of applications, and the results are compared with experimental data whenever available.