Aerodynamic Shape Optimization Using “Turbulent” Adjoint And Robust Design in Fluid Mechanics

This article presents adjoint methods for the computation of the first- and higher-order derivatives of objective functions \(F\) used in optimization problems governed by the Navier–Stokes equations in aero/hydrodynamics. The first part of the chapter summarizes developments and findings related to the application of the continuous adjoint method to turbulence models, such as the Spalart-Allmaras and k-\(\varepsilon \) ones, in either their low- or high-Reynolds number (with wall functions) variants. Differentiating the turbulence model, over and above to the differentiation of the mean–flow equations, leads to the computation of the exact gradient of \(F\), by overcoming the frequently made assumption of neglecting turbulence variations. The second part deals with higher-order sensitivity analysis based on the combined use of the adjoint approach and the direct differentiation of the governing PDEs. In robust design problems, the so-called second-moment approach requires the computation of second-order derivatives of \(F\) with respect to (w.r.t.) the environmental or uncertain variables; in addition, any gradient-based optimization algorithm requires third-order mixed derivatives w.r.t. both the environmental and design variables; various ways to compute them are discussed and the most efficient is adopted. The equivalence of the continuous and discrete adjoint for this type of computations is demonstrated. In the last part, some other relevant recent achievements regarding the adjoint approach are discussed. Finally, using the aforementioned adjoint methods, industrial geometries are optimized. The application domain includes both incompressible or compressible fluid flow applications.

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