As a high-fidelity approach to transition prediction, the coupled Reynolds-averaged Navier–Stokes (RANS) and linear stability theory (LST)-based [Formula: see text] 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.

Based on the adjoint method, the afterbody of a military transport aircraft was optimized and designed to meet engineering constraints under real flight conditions. Guidance for the key design parameters of the afterbody of the military transport aircraft is given. The vortex dynamics and boundary layer extraction methods were used to analyze the optimization results of military transport aircraft. It was found that, upstream of the vortex shedding point, the circumferential accumulation process of the vorticity is weakened. The position of the vortex shedding and the appearance of the saddle line are delayed by reducing the circumferential inverse pressure gradient and the intensity of the crossflow. The afterbody vortex system of the optimized configuration is further away from the surface. Meanwhile, the distance between the counter-rotated vortex decreases, and the upwashing speed of the vortex core is smaller. Therefore, vortex-induced drag is reduced. Finally, compared with the initial configuration, the optimized configuration has a relative drag reduction of 23.2%.

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Reducing fuel consumption and improving the economy by effectively reducing cruising drag is the main objective of the aerodynamic design of supersonic civil aircraft. In this paper, the aerodynamic optimization design system based on the Reynolds-Averaged Navier–Stokes (RANS) equation and discrete adjoint theory is applied to supersonic wing design. Based on this system, a single-point optimization design study of aerodynamic drag reduction in cruise conditions was carried out for two typical supersonic wing layouts, subsonic leading edge and supersonic leading edge, and the drag reduction reached 3.78% and 4.53%, respectively. The aerodynamic design characteristics of different types of supersonic wings were explored from the perspectives of wing load, twist angle distribution, pressure distribution, airfoil shape characteristics, and flow field characteristics. The optimization results show that the drag reduction of the subsonic leading edge configuration is dominated by the induced drag, while the optimizer mainly focuses on reducing the shock wave drag for the supersonic leading edge configuration. By comparing the sensitivity analysis of lift and drag coefficients to airfoil deformation with the optimization results, the optimized dominant directions of two types of supersonic wings are qualitatively analyzed. The derivatives obtained from discrete adjoint equations are useful to elaborate the design tendency and the reason for the trade-off generation of supersonic wings under specific layouts and engineering constraints, which provides a reference for the design of supersonic wings in the future.