Equilibrium states of turbulent homogeneous buoyant flows

Abstract

The equilibrium states of homogeneous turbulent buoyant flows are investigated through a fixed-point analysis of the evolution equations for the Reynolds stress anisotropy tensor and the scaled heat flux vector. The mean velocity and thermal fields are assumed to be two-dimensional. Scalar invariants formed from the Reynolds stress anisotropy tensor, the scaled heat flux vector, and the strain rate and rotation rate tensors are governed by a closed set of algebraic equations derived for the stress anisotropy and scaled heat flux under a (weak) equilibrium assumption. Six equilibrium state variables are identified for the buoyant case and contrasted with the corresponding two state variables obtained for the non-buoyant homogeneous turbulence case. These results, while dependent on the functional forms of the models for the pressure–strain rate correlation tensor and the pressure–scalar-gradient correlation and viscous dissipation vector, can be used as in the non-buoyant case to either calibrate new closure models or validate the performance of existing models. In addition, since the analysis only involves the turbulent time scales (both velocity and thermal) and their ratio, the results of the analysis are independent of the specific models for the dissipation rates of the turbulent kinetic energy and the temperature variance. The analytical results are compared with model predictions as well as recent direct numerical simulation (DNS) data for buoyant shear flows. Good agreement with DNS data is obtained.