The coupled adjoint equation of laminar-turbulent transition considering the suction control effect is derived for the aerodynamic design of Hybrid Laminar Flow Control (HLFC) airfoils. The chain rule, automatic differentiation algorithm, and CK (Coupled Krylov) algorithms are used to accurately and efficiently solve the coupled adjoint equations. Finally, the gradient-based optimization framework, based on the discrete adjoint equations, for HLFC airfoil is constructed. The transition prediction model adopts the eN method based on the Amplification Factor Model (AFM). The laminar flight test results show that the transition prediction method based on the AFM can effectively capture the transition phenomenon induced by T-S (Tollmien-Schlichting) waves instability under the influence of suction control. The multi-point design of the laminar airfoil is carried out using the gradient-based optimization framework, and comparison of the results with those of the gradient-free optimization. The comparison of natural laminar flow design results show that the gradient-free optimization has a deformation trend similar to the gradient-free optimization. The results of multi-point design of the hybrid laminar flow airfoil show that the hybrid laminar flow optimization has stronger drag reduction ability than the natural laminar flow optimization. Compared with that of the natural laminar airfoil, the total drag of the hybrid laminar airfoil is reduced by 6.1%, 5.9%, 33.3%, 9.5%, respectively. In conclusion, the gradient-based optimization method based on the discrete adjoint equations can effectively improve the aerodynamic performance of the HLFC airfoil, thereby providing methodological support for the future drag reduction design of HLFC.
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