An Explicit Algebraic Model for Turbulent Buoyant Flows

Abstract

This paper presents a derivation of an explicit algebraic stress model (EASM) and an explicit algebraic heat flux model (EAHFM) for buoyant shear flows. The models are derived using a projection methodology. The derived EASM has a four-term representation and is applicable to 2-D and 3-D flows. It is an extension of the three-term EASM for incompressible flow and the fourth term is added to account for the effect of buoyancy. The projection methodology is further extended to treat the heat flux transport equation in the derivation of an EAHFM. Again, the weak equilibrium assumption is invoked for the scaled heat flux equation. The basis vector used to represent the scaled heat flux vector is formed with the mean temperature gradient vector and 3×3 tensors, not necessarily symmetric or traceless, deduced from the shear and rotation rate tensors and the stress anisotropy tensor. An explicit algebraic model for buoyant shear flows is then formed with the derived EASM and EAHFM. From the derived EAHFM, an expression for the thermal diffusivity tensor in buoyant shear flows can be deduced. Thus, a turbulent Prandtl number for each of the three heat flux directions can be determined. These Prandtl numbers are functions of the gradient Richardson number. Alternatively, a scalar turbulent Prandtl number can be derived; its value is compared with the directional turbulent Prandtl numbers. The EASM and EAHFM are specialized to calculate 2-D homogeneous buoyant shear flows and the results are compared with direct numerical simulation (DNS) data and other model predictions. Good agreement with DNS data and other model predictions is obtained.