Passive scalar cascade in the intermediate layer of turbulent channel flow for $\boldsymbol{Pr}\boldsymbol{\leqslant} \boldsymbol{1}$

Similarities and differences between Kolmogorov scale-by-scale equilibria/non-equilibria for velocity and scalar fields are investigated in the intermediate layer of a fully developed turbulent channel flow with a passive scalar/temperature field driven by a uniform heat source. The analysis is based on intermediate asymptotics and direct numerical simulations at different Prandtl numbers lower than unity. Similarly to what happens to the velocity fluctuations, for the fluctuating scalar field Kolmogorov scale-by-scale equilibrium is achieved asymptotically around a length scale $r_{\textit{min}}$ , which is located below the inertial range. The length scale $r_{\textit{min}}$ and the ratio between the inter-scale transfer and dissipation rates at $r_{\textit{min}}$ vary following power laws of the Prandtl number, with exponents determined by matched asymptotics based on the hypothesis of homogeneous two-point physics in non-homogeneous turbulence. The inter-scale transfer rates of turbulent kinetic energy and passive scalar variance are globally similar but show evident differences when their aligned/anti-aligned contributions are considered.