Mean thermal energy balance analysis in differentially heated vertical channel flows

Direct numerical simulations of differentially heated vertical channel (DHVC) flows were performed for $Ra=105–109$ to investigate the characteristics of the streamwise mean momentum and mean thermal energy equations. The log law for mean temperature was observed for $Ra≥108$ at $y+>50$⁠, where y⁺ is the wall-normal distance normalized by the viscous wall unit. From the mean momentum equation, negligible viscous force and logarithmically increasing Reynolds shear stress were observed in the region where the log law for mean temperature occurred. The streamwise mean velocity did not exhibit a linear relationship with y⁺ close to the wall and did not show logarithmic development far from the wall due to the buoyancy force. In the mean thermal energy equation, a constant heat flux layer was observed, and the turbulent heat flux contribution was scaled by the inverse of wall-normal distance to satisfy the log law of mean temperature. For a high Rayleigh number (⁠$Ra=109$⁠), the turbulent heat flux spectra contained scale-separated inner and outer sites with linearly growing energetic structures along the wall-normal distance, which was not observed for a low Rayleigh number (⁠$Ra=106$⁠). The flow structures of turbulent heat flux originated from the upward wall-normal velocity fluctuations that triggered the non-directional structures of the temperature. These results suggest that the scale separation between the viscous and outer length scales with the wall-attached energetic structures resulted in the log law for mean temperature. These findings could serve as the basis of scaling formulations for the mean velocity and temperature in DHVC flows.