Multiscaling analysis of the mean thermal energy balance equation in fully developed turbulent channel flow

Tie Wei
Phys. Rev. Fluids 3, 094608 – Published 28 September 2018

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 Peτ=PrReτ in the MHB equation is the counterpart of Reτ in the MMB equation. Here Prν/α denotes the Prandtl number of the fluid, which is the ratio between the kinematic viscosity ν and the molecular thermal diffusivity α. Reτ=δuτ/ν is the Reynolds number of the flow defined with the channel half-height δ and the friction velocity uτ. 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 (Pr<1), the thickness of the molecular thermal diffusion sublayer is yIt=O(1Pr1/2νuτ) or yIt+=O1Pr1/2, a proper scaling for the wall-normal distance is Pr1/2y+ where y+=y/(ν/uτ) is the inner-scaled wall-normal distance, and a relevant Péclet number is Pr1/2Reτ. At very low Prandtl number (Pr1), the molecular thermal diffusion sublayer becomes much thicker than the viscous sublayer (molecular momentum diffusion sublayer). At large Prandtl number (Pr>1), the thickness of the molecular thermal diffusion sublayer is yIt+=O1Pr1/3, a proper scaling for the wall-normal distance is Pr1/3y+, and a relevant Péclet number is Pr1/3Reτ. At very high Prandtl number (Pr1), 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 y/(α/uτ)=Pry+. 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.

  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
9 More
  • Received 9 April 2018

DOI:https://doi.org/10.1103/PhysRevFluids.3.094608

©2018 American Physical Society

Physics Subject Headings (PhySH)

Fluid Dynamics

Authors & Affiliations

Tie Wei*

  • Department of Mechanical Engineering, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA

  • *tie.wei@nmt.edu

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand
Issue

Vol. 3, Iss. 9 — September 2018

Reuse & Permissions
Access Options
Author publication services for translation and copyediting assistance advertisement

Authorization Required


×
×

Images

×

Sign up to receive regular email alerts from Physical Review Fluids

Log In

Cancel
×

Search


Article Lookup

Paste a citation or DOI

Enter a citation
×