Assessment of Temperature Fluctuation Models for RANS Simulations

The boundary layers that form on aerospace vehicles can be turbulent and chemically reacting.  The chemical reactions alter the temperature distribution in the boundary layer, modifying the heating rates. To aid the design of these vehicles, an accurate prediction of the chemical composition of the gas is needed.

k-epsilon turbulence models are widely used to simulate hypersonic flows. These models predict high speed perfect gas flows accurately. However, there are some model uncertainties when simulating chemically reacting flows. With the very high energies present in these flows, the temperature fluctuations will be very large. The reaction rate depends exponentially on temperature, and temperature fluctuations result in large increases in the reaction rates.  Also, the chemical source term can either damp or amplify turbulent fluctuations. Using direct numerical simulations (DNS) of
hypersonic boundary layers, Martin and Candler (2000) show that endothermic reactions damp the turbulent fluctuations whereas exothermic reactions strengthen the turbulence. Thus, there is a two-way interaction between turbulence and chemistry in these flows. The effect of this interaction must be accounted for while computing the average reaction rates in the Reynolds-averaged equations.

A general method for the closure of a non-linear chemical source term in the RANS approach is to use a probability density function (PDF) in which the unclosed species production term is represented by a PDF in terms of the independent variables. We use DNS of homogeneous isotropic turbulence to assess the accuracy of two turbulence- chemistry models, namely, the Martin and Candler model and the Gaffney etal model. We use the standard k-epsilon model to simulate the turbulence field. The k-epsilon model is found to work well in this flow field except for the initial transient regime of the flow. The turbulence-chemistry model of Gaffney etal solves a modeled transport equation for the internal energy variance. Comparison with the DNS shows that the modeling of the terms in this transport equation are inadequate in reproducing the observed trends. Also, correlations including species concentration fluctuations are neglected while obtaining the temperature variance from the internal energy variance. This leads to additional errors as a result of which the Gaffney etal model overpredicts the temperature fluctuations in the flow. On the other hand, the Martin and Candler model is found to predict the temperature fluctuations as well as the average reaction rate constants accurately. This is because the model is calibrated using the Taylor microscale which the k-epsilon model reproduces correctly.

Variation of RMS temperature fluctuations with simulation time: Martin and Candler model (purple symbols) and Gaffney etal model (blue symbols) are compared with the DNS data (line).