Methodology for determining coefficients of turbulent mixing model

The accurate prediction of turbulent mixing induced by Rayleigh-Taylor (R-T), Richtmyer-Meshkov (R-M) and Kelvin-Helmholtz (K-H) instabilities is very important in understanding natural phenomena and improving engineering applications. In applications, the prediction of mixing with the Reynolds-averaged Navier-Stokes (RANS) equation remains the most widely used method. The RANS method involves two aspects, i.e. physical modelling and model coefficients. Generally, the latter is determined empirically; thus, there is a lack of universality. In this paper, inspired by the well-known Reynolds decomposition, we propose a methodology to determine the model coefficients with the following three steps: (i) preset a set of analytical RANS solutions by fully using the knowledge of mixing evolutions; (ii) simplify the differential RANS equations to algebraic equations by imposing the preset solutions to RANS equations; (iii) solve the algebraic equations approximately to give the values of the entire model coefficients. The specific application of this methodology in the widely used K-L mixing model shows that, using the same set of model coefficients determined from the current methodology, the K-L model successfully predicts the mixing evolutions in terms of different physical quantities (e.g. temporal scalings and spatial profiles), density ratios and problems (e.g. R-T, R-M, K-H and reshocked R-M mixings). It is possible to extend this methodology to other turbulence models characterised with self-similar evolutions, such as K- mixing models.