Abstract
In this work we investigate properties of the mean thermal energy balance equation using a multiscaling analysis approach. The analysis of the mean thermal energy balance (MHB) equation and the mean momentum balance (MMB) equation are presented side by side to better demonstrate the similarities and differences between the two. The main findings of this work include, first, the multiscaling for the MHB equation are similar to those for the MMB equation in the outer layer, mesolayer, and log layer. Péclet number in the MHB equation is the counterpart of in the MMB equation. Here denotes the Prandtl number of the fluid, which is the ratio between the kinematic viscosity and the molecular thermal diffusivity . is the Reynolds number of the flow defined with the channel half-height and the friction velocity . In the outer layer, mesolayer, and log layer, the shapes of the mean temperature and the mean velocity are very similar, and the shapes of the wall-normal turbulent transport of heat and momentum are also similar. Second, the molecular thermal diffusion sublayer is strongly influenced by the Prandtl number. At low Prandtl number (), the thickness of the molecular thermal diffusion sublayer is or , a proper scaling for the wall-normal distance is where is the inner-scaled wall-normal distance, and a relevant Péclet number is . At very low Prandtl number (), the molecular thermal diffusion sublayer becomes much thicker than the viscous sublayer (molecular momentum diffusion sublayer). At large Prandtl number (), the thickness of the molecular thermal diffusion sublayer is , a proper scaling for the wall-normal distance is , and a relevant Péclet number is . At very high Prandtl number (), the molecular thermal diffusion sublayer becomes much thinner than the viscous sublayer. Third, to be consistent with the scaled MHB equation, the “log law” for the mean temperature is presented using an inner-scaled wall-normal distance . The main effect of the Prandtl number is the shifting of the additive constant in the log law, due to the thickness of the molecular thermal diffusion sublayer and buffer layer. Fourth and finally, Zagarola-Smits-style scaling is applied to the mean velocity and the mean temperature deficit in the outer layer. An interpretation of the Zagarola-Smits-style scaling is provided. At sufficiently high Reynolds number and Péclet number, the Zagarola-Smits-style scaling is shown to be equivalent to the traditional scaling.
9 More- Received 9 April 2018
DOI:https://doi.org/10.1103/PhysRevFluids.3.094608
©2018 American Physical Society