A two-domain analytical approach for laminar and turbulent forced convection in a transpiration-cooled channel

Abstract

A two-domain analytical approach has been proposed to attack the conjugate problem associated with forced convective flows in a channel with uniform porous walls subject to transpiration cooling. Both temperature fields in the porous wall and the channel with uniform porous walls are treated simultaneously and matched on the porous wall. General analytical expressions for the temperature fields and wall heat transfer rates, which involve multiple integrals of the velocity profile function, were derived considering the energy balance equation for the two domains in both laminar and turbulent cases. A local thermal non-equilibrium solution was obtained for the volume average temperature of coolant gas phase and that of solid phase within the porous wall, while a local similarity solution was found for the energy equation for the channel flows with transpiration cooling for the first time. For the turbulent flows in a channel, Stevenson's law of the wall was used along with the laminar sublayer velocity profile with transpiration, so as to reveal the effect of the blowing ratio on the temperature fields and heat transfer rates. The present analytical and numerical results agree well with available RANs and DNS results. The present results obtained for both laminar and turbulent flows reveal that Kays and Crawford's correlation for the heat transfer coefficient ratio of with and without blowing holds reasonably well for both laminar and turbulent flow cases. The damping of the heat transfer coefficient ratio, predicted in this analysis, however, is much more substantial than that estimated by Kays and Crawford.