Heat-transfer analysis of a transitional boundary layer over a concave surface with Gortler vortices by means of direct numerical simulations

This work studies the development of a thermal boundary layer during the laminar-to-turbulent transition process over a concave surface. Direct numerical simulations are performed where the temperature variable is treated as a passive scalar. The laminar flow is perturbed with wall-roughness elements that are able to produce centrifugal instabilities in the form of Gortler vortices with a maximum growth rate. It is found that Gortler vortices are able to greatly modify the surface heat-transfer by generating a spanwise periodic distribution of temperature. Similar to the Gortler momentum boundary layer, elongated mushroom-like structures of low-temperature are generated in the upwash region, whereas in the downwash region, the thermal boundary layer is compressed. Consequently, temperature gradients are increased and decreased in the downwash and upwash regions, respectively, thereby generating an overall enhancement of the heat-transfer rate of similar to 400% for the investigated Prandtl numbers (Pr = 0.72, Pr = 1, and Pr = 7.07). This enhancement surpasses the turbulent heat-transfer values during the transitional region, characterized by the development of secondary instabilities. However, downstream, the heat-transfer rate decays to the typical turbulent values. Streamwise evolution of several thermal quantities such as temperature wall-normal distribution, thermal boundary layer thickness, and Stanton number is reported in different regions encountered in the transition process, namely, linear, nonlinear, transition, and fully turbulent regions. These quantities are reported locally at upwash and downwash regions, where they present minima and maxima, as well as globally as spanwise-averaged quantities. Furthermore, it is found that the Reynolds analogy between streamwise-momentum and heat-transfer holds true throughout the whole transition process for the Pr = 1 case. Moreover, the turbulent thermal boundary layer over a concave surface is analyzed in detail for the first time. The viscous sub-layer and the log-law region are described for each investigated Pr. Besides, the root-mean-squared temperature fluctuations are computed, finding that its wall-normal distribution exhibits a higher peak when Pr is increased.