Scaling in Internally Heated Convection: A Unifying Theory

Qi Wang,Detlef Lohse,Olga Shishkina

doi:10.1029/2020gl091198

Abstract

AbstractWe offer a unifying theory for turbulent, purely internally heated convection, generalizing the unifying theories of Grossmann and Lohse (2000, https://doi.org/10.1017/S0022112099007545; 2001, https://doi.org/10.1103/PhysRevLett.86.3316) for Rayleigh‐Bénard turbulence and of Shishkina et al. (2016, https://doi.org/10.1002/2015GL067003) for turbulent horizontal convection, which are both based on the splitting of the kinetic and thermal dissipation rates in respective boundary and bulk contributions. We obtain the mean temperature of the system and the Reynolds number (which are the response parameters) as function of the control parameters, namely the internal thermal driving strength (called, when nondimensionalized, the Rayleigh‐Roberts number) and the Prandtl number. The results of the theory are consistent with our direct numerical simulations of the Boussinesq equations.

Highlights

Driven turbulence is omnipresent in nature and technology
The thermal driving can be thanks to the temperature boundary conditions such as in Rayleigh-Bénard convection (RBC)—a flow in a container heated from below and cooled from above (Ahlers et al, 2009; Chilla & Schumacher, 2012; Lohse & Xia, 2010)—or in horizontal convection (HC) (Hughes & Griffiths, 2008; Shishkina & Wagner, 2016; Shishkina et al, 2016) or vertical convection (Ng et al, 2015, 2017, 2018; Shishkina, 2016), where parts of the top, bottom, or sidewalls of the container are set at different temperatures
Comparison With Direct Numerical Simulations. To check these predictions of the Grossmann and Lohse (GL) theory generalized to internally heated convection (IHC), we have performed 2-D direct numerical simulations (DNS) according to Equations 1 and 2 with the corresponding BCs

Summary

Introduction

Driven turbulence is omnipresent in nature and technology. The thermal driving can be thanks to the temperature boundary conditions such as in Rayleigh-Bénard convection (RBC)—a flow in a container heated from below and cooled from above (Ahlers et al, 2009; Chilla & Schumacher, 2012; Lohse & Xia, 2010)—or in horizontal convection (HC) (Hughes & Griffiths, 2008; Shishkina & Wagner, 2016; Shishkina et al, 2016) or vertical convection (Ng et al, 2015, 2017, 2018; Shishkina, 2016), where parts of the top, bottom, or sidewalls of the container are set at different temperatures. The thermal driving can be thanks to internal heating, where the temperature field is driven by some forcing in the bulk. The DNS are conducted in two dimensions (2-D), as (i) the theory is based on Prandtl's equations, which are 2-D in spirit, as (ii) 2-D and 3-D thermally driven turbulence show very close analogies with respect to the integral quantities, in particular for large Prandtl numbers Pr ≥ 1 (van der Poel et al, 2013), and as (iii) otherwise, due to unavoidable limitations in available CPU time, we could explore only a much smaller portion of the parameter space

Control and Response Parameters and Governing Equations

Comparison With Direct Numerical Simulations

Conclusions