Deliberate continuation of the combustion in the turbine passages of a gas turbine engine has the potential to increase the efficiency and the specific thrust or power of current gas-turbine engines. This concept, known as a turbine-burner, has challenges with the injection, mixing, ignition, and burning of fuel within a short residence time in a turbine passage characterized by large multidirectional accelerations. One method of increasing the residence time is to inject the fuel into a cavity adjacent to the turbine passage, creating a low-speed zone for mixing and combustion. This concept is simulated numerically, with the turbine passage modeled as a converging, turning channel flow of high-temperature, vitiated air adjacent to a cavity. Constant-area and converging channels with both straight and curving centerlines are modeled. Two-dimensional, unsteady calculations are performed, examining the effects of channel convergence and curvature, and injection configurations. These direct simulations address flows with Reynolds number values up to 5000. Calculations show that higher aspect ratio cavities improve the fluid interaction between the channel flow and the cavity, and that the cavity dimensions are important for enhancing the mixing. Results show that converging channels improve the combustion efficiency.