Integral properties of temperature variance production in a turbulent channel flow with passive scalar transport

Abstract

This paper investigates the integral properties of the mean temperature dissipation α(∂Θ/∂y)2 and the temperature variance production Pθ=Rvθ(∂Θ/∂y) in fully developed turbulent channel flow, and in particular the effects of Prandtl number Pr=ν/α and Reynolds number Reτ=δuτ/ν (and/or Peclet number Peτ=ReτPr). Here Θ is the mean temperature and Rvθ is the wall-normal turbulent heat flux. y is the wall-normal direction and δ is the channel half-height. ν and α are the fluid viscosity and thermal diffusivity, respectively. uτ is the friction velocity. The main findings of the present work include: (1) Identities are derived from the mean thermal energy equation for the global integral of the mean temperature dissipation and temperature variance production for four heating conditions: constant heat source (CHS), constant wall heat flux (CHF), constant temperature difference (CTD), and constant wall temperature (CTD). Identities for the CHS and CHF conditions are consistent with those derived by Pirozzoli et al. (2016) and Abe and Antonia (2017). The identities are verified using publicly available direct numerical simulation (DNS) data under three heating conditions CHS, CHF, and CTD. (2) The global integral of the mean temperature dissipation is found to depend strongly on the fluid Prandtl number, but it is largely independent of the Reynolds number of the flow. In inner-scaling, this global integral can be approximated as(1/Pr)∫0δ+(∂Θ+/∂y+)2dy+≈9.13Pr1-n where Θ+=Θ/θτ is the inner-scaled mean temperature and θτ is the friction temperature. y+=yuτ/ν is the inner-scaled wall-normal distance. n≈1/2 for Pr≲1 and n≈1/3 for Pr≳1. The strong dependence of (1/Pr)∫0δ+(∂Θ+/∂y+)2dy+ on the Prandtl number is directly rooted in the molecular thermal diffusion sub-layer. (3) The global integral of temperature variance production is found to strongly depend on both Prandtl number and Reynolds number. (3.1) At sufficiently high Reynolds number and Peclet number, the integral of temperature variance production is found to exhibit a logarithmic-like layer similar to that of the mean temperature profile. (3.2) The effect of Prandtl number is mainly caused by the molecular thermal diffusion sub-layer and buffer layer. (3.3) Between the peak wall-normal turbulent heat flux Rvθ location, ymt, and the channel centerline, δ, the integral of temperature variance production increases with Peclet number in a logarithmic-like fashion. (3.4) At low Prandtl number Pr≲1, the integral of temperature variance production is found to be equally partitioned around ymt.