Probability density function approach towards closure of turbulent chemistry in hypersonic boundary layers with dissociative chemical non-equilibrium effects

Fluid flows around hypersonic vehicles experience chemical non-equilibrium effects at extreme temperature conditions. Reynolds-averaged Navier–Stokes (RANS) equations are primarily used to simulate turbulent external flow at full vehicle scales. However, the turbulent closure of near-wall reactions related to gas dissociation is omitted in practice because it remains unknown how to close the associated mean reaction rate, despite research efforts in this direction for more than a decade. This paper aims to discover an appropriate turbulent closure strategy of the involved finite-rate dissociative reaction through direct numerical simulation of a hypersonic turbulent boundary at Mach 9.2 with an isothermal cold wall surface, computed using Park’s five-species air dissociation model. Three sets of calculations are conducted, including two sets with non-catalytic and catalytic wall surface conditions, and one set without chemical reaction. Results show that the involved endothermic reaction mainly affects the magnitude of mean temperature and its fluctuations, whereas it has a relatively slight influence on the velocity and wall surface statistics. Turbulence-chemistry interaction is analysed within the same probability density function (PDF) framework as Wang & Xu (2024 J. Fluid Mech. vol. 998, A1), which considers temperature and species compositions in sample space. We find that modelling only the PDF of temperature, with simple knowledge of the mean species concentrations, is sufficient to reasonably well close the turbulent reaction rates and heat absorption rates, except for quantitative errors in the reaction rate of atomic nitrogen. This finding avoids the need for a more complex multivariable PDF in closure and also eliminates the requirement to model species fluctuations in RANS. Assuming a log-normal distribution for temperature provides better results, owing to the strongly skewed temperature distribution near the wall surface. The dependence and sensitivity of the single model parameter, temperature skewness, are further investigated. It is shown that the accuracy of closure result is not highly sensitive to the exact skewness value, as long as a negative one within a relatively wide range is selected. The developed closure model is applied to a wall model with species balance equations, showing significant improvement over the laminar closure, while further closure modelling efforts in the atomic nitrogen are still needed to improve computation robustness.