Continuous Adjoint Methods for Turbulent Flows, Applied to Shape and Topology Optimization: Industrial Applications

This article focuses on the formulation, validation and application of the continuous adjoint method for turbulent flows in aero/hydrodynamic optimization. Though discrete adjoint has been extensively used in the past to compute objective function gradients with respect to (w.r.t.) the design variables under turbulent flow conditions, the development of the continuous adjoint variant for these flows is not widespread in the literature, hindering, to an extend, the computation of exact sensitivity derivatives. The article initially presents a general formulation of the continuous adjoint method for incompressible flows, under the commonly used assumption of “frozen turbulence”. Then, the necessary addenda are presented in order to deal with the differentiation of both low- and high-Reynolds (with wall functions) number turbulence models; the latter requires the introduction of the so-called “adjoint wall functions”. An approach to dealing with distance variations is also presented. The developed methods are initially validated in 2D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2D$$\end{document} cases and then applied to industrial shape and topology optimization problems, originating from the automotive and hydraulic turbomachinery industries.