An explicit algebraic Reynolds stress and heat flux model for incompressible turbulence: Part II Buoyant flow

Abstract

This paper presents a derivation of an explicit algebraic model for two-dimensional (2-D) buoyant flows. It is an extension of the work reported in Part I (So et al. [27]). The tensor representation method of Jongen and Gatski [14] is extended to derive an explicit algebraic Reynolds stress model (EASM) for 2-D buoyant flow invoking the Boussinesq approximation. The projection methodology is further extended to treat the heat flux transport equation in the derivation of an explicit algebraic heat flux model (EAHFM) for buoyant flow. Again, the weak equilibrium assumption is invoked for the scaled Reynolds stress and scaled heat flux equation. An explicit algebraic model for buoyant flows is then formed with the EASM and EAHFM. From the derived EAHFM, an expression for the thermal diffusivity tensor in buoyant shear flows is deduced. Furthermore, a turbulent Prandtl number (PrT) for each of the three heat flux directions is determined. These directional PrT are found to be a function of the gradient Richardson number. Alternatively, a scalar PrT can be derived; its value is compared with the directional PrT. The EASM and EAHFM are used to calculate 2-D homogeneous buoyant shear flows and the results are compared with direct numerical simulation data and other model predictions, where good agreement is obtained.