Contribution of viscous stress work to wall heat flux in compressible turbulent channel flows

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.

Skin-friction and heat-transfer decompositions in hypersonic transitional and turbulent boundary layers

“…According to the viewpoints of Zhang & Xia (2020) and Wenzel, Gibis & Kloker (2021 a ), the twofold repeated integration was better than the threefold repeated integration in the FIK identity. They showed that the decomposition based on the twofold repeated integration gets rid of the linearly weighted term on the Reynolds shear stress and has more intuitive physical interpretation of the underlying mechanisms of the generation of the skin-friction and heat-transfer coefficients.…”

“…They showed that the decomposition based on the twofold repeated integration gets rid of the linearly weighted term on the Reynolds shear stress and has more intuitive physical interpretation of the underlying mechanisms of the generation of the skin-friction and heat-transfer coefficients. Zhang & Xia (2020) used this new decomposition to investigate the heat-transfer coefficient in supersonic turbulent channel flows. They found that the viscous dissipation gives the dominant contribution to the heat-transfer coefficient.…”

A decomposition formula for the wall heat flux of a compressible boundary layer

“…Ghosh [11] proposed that the wall heat flux can be expressed by the sum of the integrations of different viscous terms in the incompressible turbulent channel and pipe flow. Zhang and Xia [12] further proposed a formula to assess the contributions of the viscous stresses to the heat transfer in a turbulent channel flow by the method of Fukagata et al [13], but the turbulent stresses were missing in their expression due to the simplification assumptions used in the channel flow.…”

“…As far as the authors' knowledge, this is the first decomposition formula proposed for the wall heat flux of the compressible boundary layer. The new formula is derived by the method of Renard et al [14], which is more physical and feasible than those [11,12] based on the method of Fukagata et al [13] It serves an insight into the complex transport processes of the wall heat flux, and helps us find the key factors.…”

“…Kim et al [7] proposed a direct approach for the time-dependent heat flux by assuming the temperature approximated as a third-order polynomial of position. Zhang and Xia [8] computed the contributions of the viscous stresses to the heat transfer in a turbulent channel flow by an exact expression, but the turbulent stresses were missing in their expression due to the simplification assumptions used in the channel flow.…”