Efficient treatment of the nonlinear features in algebraic Reynolds-stress and heat-flux models for stratified and convective flows

Authors: Lazeroms, W.M.J., Brethouwer, G.B., Wallin, S.W., Johansson, A.V.J.
Document Type: Article
Pubstate: Published
Journal: Int. J. Heat and Fluid Flow
Volume: 53   15-28
Year: 2015


This work discusses a new and efficient method for treating the nonlinearity of algebraic turbulence models in the case of stratified and convective flows, for which the equations for the Reynolds stresses and turbulent heat flux are strongly coupled. In such cases, one finds a quasi-linear set of equations, which can be solved through an appropriate linear expansion in basis tensors and vectors, as discussed in earlier work. However, finding a consistent and truly explicit algebraic turbulence model requires solving an additional equation for the production-to-dissipation ratio $(\mat{P}+\mat{G})/\eps$ of turbulent kinetic energy. Due to the nonlinear nature of the problem, the equation for $(\mat{P}+\mat{G})/\eps$ is a higher-order polynomial equation for which no analytical solution can be found. Here we provide a new method to approximate the solution of this polynomial equation through an analysis of two special limits (shear-dominated and buoyancy-dominated), in which exact solutions are obtainable. The final result is a model that appropriately combines the two limits in more general cases. The method is tested for turbulent channel flow, both with stable and unstable stratification, and the atmospheric boundary layer with periodic and rapid changes between stable and unstable stratification. In all cases, the model is shown to give consistent results, close to the exact solution of $(\mat{P}+\mat{G})/\eps$. This new method greatly increases the range of applicability of explicit algebraic models, which otherwise would rely on the numerical solution of the polynomial equation.