In order to complement associated experimental studies, the development of a theoretical approach to the evaluation of gas turbine combustors is extremely attractive. For this purpose a computer program is needed which starts from hypotheses about the fundamental processes and predicts the distributions of velocity, concentrations, and temperature. A new primitive pressure-velocity variable, finite-difference computer code has been developed to allow the computation of inert and reacting turbulent swirling flows in an axisymmetric combustor. The method and program involve a staggered grid system for axial and radial velocities, and a line relaxation technique for efficient solution of the equations. Turbulence simulation is by way of a two-equation k-e model combustion via a simple one-step chemical reaction model based on Arrhenius and eddy-breakup concepts for diffusion or premixed situations. Computational results show the interesting effects of combustor design parameters (such as: degree of swirl, existence of a recirculation zone amplifier, i.e., trip, and effect of laterally induced secondary air supply) on subsequent flowfield development and combustor performance. Available experimental data provide confirmatory comparisons.