Vorticity-stream function formulation of compressible and incompressible turbulent flows

Ducted turbulent flows with varying wall shapes are formulated with the Navier-Stokes equations written in terms of vorticity and stream function. Coordinate transformation and stretching functions are used to provide adequate resolution throughout the whole flow field. The turbulent characteristics of the flow are modeled by a low Reynolds number, two-equation k-epsilon model which is applicable in the wall sublayer as well as in the region away from the wall. In addition to the conventional Poisson-type pressure equation, a new pressure equation with both first- and second-order terms was derived from the momentum equations. A vector path function is included in the pressure equation to improve convergence and accuracy. Applications to a model wind tunnel diffuser and a sudden expansion diffuser are discussed. Good agreement with available experimental data is observed in terms of the pressure coefficient and the length of the recirculation region.