On the realizability of second-order models for a passive scalar in homogeneous turbulence

Predictions of the equilibrium states of homogeneous turbulent flows are reported for several orientations of the mean scalar gradient relative to mean strain. Differential second-order turbulent Reynolds stresses and scalar flux vector models predictions are compared with direct numerical simulation data. The realizability and performance of the solutions obtained with four different pressure–scalar gradient correlation models are critically compared. The models considered are the truncated zeroth-order model (also known as ‘isotropization of production’) of Gibson and Launder, the complete zeroth-order model of Jones and Musonge, the linear model of Shih and Lumley and the quadratic model of Craft and Launder. A complete formulation of all the models tested is presented in a unified manner. The comparison allowed us to isolate the pressure–scalar influence on the turbulence parameters. The results include evaluation criteria for the model which have not been employed in practice, namely the alignment between the principal angle of Reynolds stress tensor and heat flux vector and the joint-realizability Cauchy–Schwarz condition. It was found that the quadratic and the isotropization of production models are fully realizable in the investigated flow conditions. The linear and the zeroth-order model of Jones and Musonge are not realizable for specific situations of the relative orientation of the scalar gradient and for these cases it is demonstrated that the joint realizability conditions are not verified.

Keywords:

• Homogeneous turbulence
• Passive scalar turbulence
• Reynolds-averaged Navier–Stokes equations
• Turbulence modelling
• Reynolds-averaged pressure correlation realizability

Notes

†Optimized values suggested by Shih et al. [18].