Onset of large-scale convection in wall-bounded turbulent shear flows

Abstract

Large-scale coherent rolls are observed frequently in unstably stratified turbulent wall-bounded flows where they influence strongly the turbulent transport and the mean flow properties. Here we address the question of their genesis by means of a linear stability analysis of the turbulent mean flow where a model of turbulent Reynolds stresses is embedded in the linear stability operator. We use the unstably stratified turbulent channel flow as a testbed for the analysis. We show that the onset of large-scale convection is associated to the linear instability of the mean flow to large-scale streamwise-uniform coherent rolls of aspect ratio$A \approx 3\unicode{x2013}3.3$when the friction Richardson number exceeds$|Ri_{\tau,c}|=0.86$. This corresponds to critical Rayleigh numbers that increase with the Reynolds number approximately as$Ra_c \approx 0.04\,Re_b^{1.8}$. These results are consistent with those obtained in recent direct numerical simulations performed in the same setting. It is also found that if turbulent Reynolds stresses are not modelled in the linear operator used in the stability analysis, then predictions are not consistent with direct numerical simulations results.