From bluff bodies to optimal airfoils: Numerically stabilized RANS solvers for reliable shape optimization

The robustness of Reynolds-Averaged Navier-Stokes (RANS) solvers is a significant challenge in gradient-based aerodynamic shape optimization , especially when encountering shapes that produce highly separated flows . These cases often cause solvers to diverge or exhibit limit-cycle oscillations. Common practices rely on averaging non-converged solutions and often yield gradients that mislead optimizers, resulting in poor designs or failure. To address this, we extend the modified-BoostConv method to ensure full convergence of SIMPLE-based RANS solvers in such conditions, providing model-accurate gradients using the complex-step derivative technique. Building on previous work, we introduce a novel coupling approach that resolves stability issues in earlier implementations of the modified-Boostconv method by splitting the basis construction for complex residuals into its solution (real) and gradient (imaginary) components. The gradient accuracy of averaged solutions from non-converging solvers is compared to fully converged gradients using the chaotic logistic map equation and a RANS simulation of a bluff body undergoing massive flow separation . We then optimize a shape starting from a 80% thick bluff body, demonstrating the importance of accurate gradients. Using model-accurate gradients, we obtain an optimized 18% thick airfoil with a high lift-to-drag ratio, matching the results of optimizing a NACA 0018 airfoil and indicating the presence of a global optimum. Conversely, optimization with averaged gradients failed to improve the initial shape. This study highlights the limitations of gradient averaging for non-converging solvers. It demonstrates that fully converged gradients reliably guide the optimizer toward attached flow regions, where RANS models are more accurate and reliable.