J. Fluid Mech. (2013), vol. 723, pp. 91–125. c  Cambridge University Press 2013 91 doi:10.1017/jfm.2013.116 An explicit algebraic Reynolds-stress and scalar-flux model for stably stratified flows W. M. J. Lazeroms 1, †, G. Brethouwer 1 , S. Wallin 1,2 and A. V. Johansson 1 1 Linn´ e FLOW Centre, Department of Mechanics, KTH Royal Institute of Technology, SE-100 44, Stockholm, Sweden 2 Swedish Defence Research Agency (FOI), SE-164 90, Stockholm, Sweden (Received 10 August 2012; revised 14 January 2013; accepted 21 February 2013) This work describes the derivation of an algebraic model for the Reynolds stresses and turbulent heat flux in stably stratified turbulent flows, which are mutually coupled for this type of flow. For general two-dimensional mean flows, we present a correct way of expressing the Reynolds-stress anisotropy and the (normalized) turbulent heat flux as tensorial combinations of the mean strain rate, the mean rotation rate, the mean temperature gradient and gravity. A system of linear equations is derived for the coefficients in these expansions, which can easily be solved with computer algebra software for a specific choice of the model constants. The general model is simplified in the case of parallel mean shear flows where the temperature gradient is aligned with gravity. For this case, fully explicit and coupled expressions for the Reynolds-stress tensor and heat-flux vector are given. A self-consistent derivation of this model would, however, require finding a root of a polynomial equation of sixth- order, for which no simple analytical expression exists. Therefore, the nonlinear part of the algebraic equations is modelled through an approximation that is close to the consistent formulation. By using the framework of a K–ω model (where K is turbulent kinetic energy and ω an inverse time scale) and, where needed, near-wall corrections, the model is applied to homogeneous shear flow and turbulent channel flow, both with stable stratification. For the case of homogeneous shear flow, the model predicts a critical Richardson number of 0.25 above which the turbulent kinetic energy decays to zero. The channel-flow results agree well with DNS data. Furthermore, the model is shown to be robust and approximately self-consistent. It also fulfils the requirements of realizability. Key words: stratified turbulence, turbulence modelling, turbulent flows 1. Introduction In many applications, within both engineering and the atmospheric sciences, models of the turbulent fluxes that appear in the Reynolds-averaged Navier–Stokes (RANS) equations have been important and relatively inexpensive tools for making predictions of turbulent flows. Traditionally, simple two-equation models that make use of eddy- viscosity and eddy-diffusivity concepts have prevailed. Such models, however, lack the correct description of some important physical aspects, and they give poor results in † Email address for correspondence: werner@mech.kth.se