Accounting for Buoyancy Effects in the Explicit Algebraic Stress Model: Homogeneous Turbulent Shear Flows

Most explicit algebraic stress models are formulated for turbulent shear flows without accounting for external body force effects, such as the buoyant force. These models yield fairly good predictions of the turbulence field generated by mean shear. As for thermal turbulence generated by the buoyant force, the models fail to give satisfactory results. The reason is that the models do not explicitly account for buoyancy effects, which interact with the mean shear to enhance or suppress turbulent mixing. Since applicable, coupled differential equations have been developed describing these thermal turbulent fields, it is possible to develop corresponding explicit algebraic stress models using tensor representation theory. While the procedure to be followed has been employed previously, unique challenges arise in extending the procedure for developing the algebraic representations to turbulent buoyant flows. In this paper the development of an explicit algebraic stress model (EASM) is confined to the homogeneous buoyant shear flow case to illustrate the methodology needed to develop the proper polynomial representations. The derivation is based on the implicit formulation of the Reynolds stress anisotropy at buoyant equilibrium. A five-term representation is found to be necessary to account properly for the effect of the thermal field. Thus derived, external buoyancy effects are represented in the scalar coefficients of the basis tensors, and structural buoyancy effects are accounted for in additional terms in the stress anisotropy tensor. These terms will not vanish even in the absence of mean shear. The performance of the new EASM, together with a two-equation (2-Eq) model, the non-buoyant EASM of Gatski and Speziale (1993) and a full second-order model, is assessed against direct numerical simulations of homogeneous, buoyant shear flows at two different Richardson numbers representing weak and strong buoyancy effects. The calculations show that this five-term representation yields better results than the 2-Eq model and the EASM of Gatski and Speziale where buoyancy effects are not explicitly accounted for.