A Variable Turbulent Schmidt Number Formulation for Scramjet Application

In high speed engines, thorough turbulent mixing of fuel and air is required to obtain high performance and high efficiency. Thus, the ability to predict turbulent mixing is crucial in obtaining accurate numerical simulation of an engine and its performance. Current state of the art in CFD simulation is to assume both turbulent Prandtl number and Schmidt numbers to be constants. However, since the mixing of fuel and air is inversely proportional to the Schmidt number, a value of 0.45 for the Schmidt number will produce twice as much diffusion as that with a value of 0.9. Because of this, current CFD tools and models have not been able to provide the needed guidance required for the efficient design of a scramjet engine. The goal of this investigation is to develop the framework needed to calculate turbulent Prandtl and Schmidt numbers as part of the solution. This requires four additional equations: two for the temperature variance and its dissipation rate and two for the concentration variance and its dissipation rate. In the current investigation emphasis will be place on studying mixing without reactions. For such flows, variable Prandtl number does not play a major role in determining the flow. This, however, will have to be addressed when combustion is present. The approach to be used is similar to that used to develop the k–ζ model. In this approach, relevant equations are derived from the exact Navier-Stokes equations and each individual correlation is modeled. This ensures that relevant physics is incorporated into the model equations. This task has been accomplished. The final set of equations have no wall or damping functions. Moreover, they are tensorially consistent and Galilean invariant. The derivation of the model equations is rather lengthy and thus will not be incorporated into this abstract, but will be included in the final paper. As a preliminary to formulating the proposed model, the original k–ζ model with constant turbulent Prandtl and Schmidt numbers is used to model the supersonic coaxial jet mixing experiments involving He, O2 and air of Refs. 1 and 2. This step is important in order to evaluate the underlying turbulence model especially because of some discrepancies noted in comparing theory and experiment in Refs. 1 and 2. Comparisons are made with velocity, concentration, stagnation pressure and temperature at various stations in the jet. The results indicate sensitivity to Schmidt number. Moreover, the discrepancies noted in Ref. 1 and 2 are a result of the underlying k–ω model used and are absent in the current model. ∗Post-Doctoral Research Associate, Member AIAA. †Associate Professor, Senior Member AIAA. ‡Professor, Associate Fellow AIAA. §Associate Professor, Senior Member AIAA.