Assessment of One- and Two-Equation Turbulence Models for Hypersonic Transitional Flows

Many Navier-Stokes codes require that the governing equations be written in conservation form with a source term. The Spalart-Allmaras one-equation model was originally developed in substantial derivative form and when rewritten in conservation form, a density gradient term appears in the source term. This density gradient term causes numerical problems and has a small influence on the numerical predictions. Further work has been performed to understand and to justify the neglect of this term. The transition trip term has been included in the one-equation eddy viscosity model of Spalart-Allmaras. Several problems with this model have been discovered when applied to high-speed flows. For the Mach 8 flat plate boundary layer flow with the standard transition method, the Baldwin-Barth and both k-{omega} models gave transition at the specified location. The Spalart-Allmaras and low Reynolds number k-{var_epsilon} models required an increase in the freestream turbulence levels in order to give transition at the desired location. All models predicted the correct skin friction levels in both the laminar and turbulent flow regions. For Mach 8 flat plate case, the transition location could not be controlled with the trip terms as given in the Spalart-Allmaras model. Several other approaches have been investigated to allow the specification of the transition location. The approach that appears most appropriate is to vary the coefficient that multiplies the turbulent production term in the governing partial differential equation for the eddy viscosity (Method 2). When this coefficient is zero, the flow remains laminar. The coefficient is increased to its normal value over a specified distance to crudely model the transition region and obtain fully turbulent flow. While this approach provides a reasonable interim solution, a separate effort should be initiated to address the proper transition procedure associated with the turbulent production term. Also, the transition process might be better modeled with the Spalart-Allmaras turbulence model with modification of the damping function f{sub v1}. The damping function could be set to zero in the laminar flow region and then turned on through the transition flow region