Scalar Diffusion Equation-Based Model to Predict 2-Dimensional Film Cooling Effectiveness of a Shaped Hole

One-dimensional laterally averaged adiabatic film cooling effectiveness $η¯lat$-based correlations have been widely employed in the cooling design of the modern gas turbine and aero-engine; however, the flow field of the discrete film cooling is fully three dimensional, and thus, the cooling effectiveness distribution on the solid surface is two dimensional. Accurate prediction of the cooling effectiveness distribution in the lateral direction would help to optimize the film cooling design, but few paid attention to this issue in the literature. In this study, a simple yet accurate scalar diffusion equation based model is proposed to extend the one-dimensional correlation into two dimensional. The model is proved to be accurate and efficient. According to the accuracy analysis, the R² value is larger than 0.95 for the two-dimensional prediction and over 0.93 along the centerline. With given input parameters, the calculation cost for solving a certain case is in the magnitude of 1 × 10⁻³s in time using the space-marching method. There is only the effective diffusion coefficient left to be modeled in the control equation. It represents the balance between the diffusion and the passive transportation by the main flow. Analyses conducted within the typical experimental range show that $κ~eff$ is only dependent on the velocity ratio and the main-flow turbulence.