The exploration of superhydrophobic drag reduction has been and continues to be of significant interest to various industries. In the present work, direct numerical simulation (DNS) is utilized to investigate the effect of the parameters on the drag-reducing performance of superhydrophobic surfaces (SHS). Simulations with a friction Reynolds number of 180 were carried out at solid fraction values of ${\varphi }_s=\frac{1}{16},\frac{1}{11}$, and $\frac{1}{4}$, and three distribution shapes: aligned, staggered, and random. The top wall is the smooth one, and the bottom wall is a superhydrophobic surface (SHS). Drag reduction and Reynolds stress profiles are compared for all cases. The turbulent kinetic energy budget, including production, dissipation, and diffusion, is presented with respect to the solid fraction and type of distribution to investigate the drag reduction mechanism. The sizes of the longitudinal vortices and formation of hairpin vortices are investigated through the observation of coherent structures. The simulation of a post model is a useful method to study the drag reduction for different solid fraction values and distribution geometries. Our study demonstrates that the drag reduction could acquire 42% with the solid fraction value ${\varphi }_s=\frac{1}{16}$ and an aligned distribution shape for post superhydrophobic surface geometry. Our study also showed the relationship of the Reynolds stress component (R₁₁, R₂₂, and R₃₃) to the drag reduction with the differences in the solid fraction values and distribution geometry. In which, the R₁₁ component has the most change between an aligned distribution and a random one. The peak value of R₁₁ tends to shift away from the SHS wall. In addition, the analysis of the TKE budget over the superhydrophobic surface was performed, which can be adopted as a useful resource in turbulence modeling based on RANS methodology.

In the present work, a comparative study of two major non-intrusive polynomial chaos methods, Point-Collocation Non-Intrusive Polynomial Chaos (NIPC) and Non-Intrusive Spectral Projection (NISP), was conducted for the transitional $\gamma -R_{\theta }$ transitional model. Three multiple model coefficients, C_a2, C_e1, and C_e2 were considered with multiple random inputs with the assumption of uniform distributions with ±10% deviation. The target transitional flows were one around a flat plate and Aerospatiale A-airfoil. Deterministic solutions were obtained by employing the open source software OpenFOAM. The results of two methods were compared to the results of Monte Carlo simulation with 500 runs. The order convergence of the mean value and the standard deviation (STD) were compared in terms of the quantities of interest, drag and lift coefficients. Further, the most effective model coefficient for each transitional flow could be found through the calculation of the Sobol index.