Complex Standard Eigenvalue Problem Derivative Computation for Laminar-Turbulent Transition Prediction

As a high-fidelity approach to transition prediction, the coupled Reynolds-averaged Navier-Stokes (RANS) and linear stability theory (LST)-based e(N) method is widely used in engineering applications and is the preferred method for laminar flow optimization. However, the further development of gradient-based laminar flow wing optimization schemes is hindered by a lack of efficient and accurate derivative computation methods for LST-based eigenvalue problems with a large number of design variables. To address this deficiency and to compute the derivatives in the LST-based solution solver, we apply the adjoint method and analytical reverse algorithm differentiation (RAD), which scale well with the number of inputs. The core of this paper is the computation of the standard eigenvalue and eigenvector derivatives for the LST problem, which involves a complex matrix. We develop an adjoint method to compute these derivatives, and we couple this method with RAD to reduce computational costs. In addition, we incorporate the LST-based partial derivatives into the laminar-turbulent transition prediction framework for the computation of total derivatives. We verify our proposed method with reference to finite difference (FD) results for an infinite swept wing. Both the intermediate derivatives from the transition module and total derivatives agree with the FD reference results to at least three digits, demonstrating the accuracy of our proposed approach. The fully adjoint and the coupled adjoint-RAD methods both have considerable advantages in terms of computational efficiency compared with iterative RAD and FD methods. The LST-based transition method and the proposed method for efficient and accurate derivative computations have prospects for wide application to laminar flow optimization in aerodynamic design.