Exact mathematical formulas for wall-heat flux in compressible turbulent channel flows

TL;DR

This paper derives exact formulas for wall heat flux in compressible turbulent channel flows using energy equations and verifies them with numerical data, highlighting their mathematical nature and limited physical interpretability.

Contribution

It introduces new exact mathematical expressions for wall heat flux in compressible turbulence, derived via different energy equations and verification methods.

Findings

Formulas verified with DNS data across various Reynolds and Mach numbers.

Most formulas are mathematically exact but lack clear physical interpretation.

The derived formulas are useful for estimation but not for understanding underlying physics.

Abstract

In this paper, several exact expressions for the mean heat flux at the wall ($q_w$) for the compressible turbulent channel flows are derived by using the internal energy equation or the total energy equation. Two different routes, including the FIK method and the RD method, can be applied. The direct numerical simulations data of compressible channel flows at different Reynolds and Mach numbers verify the correctness of the derived formulas. Discussions related to the different energy equations, and different routes are carried out, and we may arrive at the conclusion that most of the formulas derived in the present work are just mathematically ones which generally are lack of clear physical interpretation. They can be used to estimate $q_w$, but might not be suitable for explore the underlying physics.