The influence of turbulence on surface heat transfer in boundary layers using a moment of temperature integral equation

Transition to turbulence dramatically increases the skin friction and heat transfer coefficients of boundary layer flows. The angular momentum integral (AMI) equation has previously shown that the skin friction coefficient is equal to the sum of the laminar skin friction (a function of Reynolds number only) and other terms representing skin friction enhancement by turbulent stresses ($-\overline{u^\prime v^\prime}$), freestream pressure gradients, and other physical effects. In this presentation, we extend this analysis to include heat transfer. A moment of temperature integral (MTI) equation is introduced to quantify the Stanton number as the sum of the laminar Stanton number (a function of Reynolds and Prandtl numbers) and other terms representing turbulent heat fluxes ($\overline{v^\prime T^\prime}$), streamwise growth, and other flow effects. The MTI equation is evaluated using direct numerical simulation results for transitional and turbulent boundary layers. The enhanced Stanton number in turbulent boundary layers is quantitatively tied to the integral of the turbulent heat flux across the boundary layer. The rapid increase in turbulent heat flux during transition creates a Stanton number peak that is partially mitigated by a reversal of the wall-normal mean velocity.