This paper is a further study of the authors’ previous work on the continuous adjoint method based on the variation in grid node coordinates and Jacobi Matrices of the flow fluxes. This method simplifies the derivation and expression of the adjoint system, and reduces the computation cost. In this paper, the differences between the presented and the traditional methods are analyzed in details by comparing the derivation processes and the adjoint systems. In order to demonstrate the reliability and accuracy of the adjoint system deduced by the authors, the presented method is applied to different optimal problems, which include two inverse designs and two shape optimizations in both 2D and 3D cascades. The inverse designs are implemented by giving the isentropic Mach number distributions along the blade wall for 2D inviscid flow and 3D laminar flow. The shape optimizations are implemented with the objective function of the entropy generation in flow passage for 2D and 3D laminar flows. In the 3D optimal case, this method is validated by supersonic turbine design case with and without mass flow rate constraint. The numerical results testify the accuracy of this adjoint method, which includes only the boundary integrals, and furthermore, the universality and portability of this adjoint system for inverse designs and shape optimizations are demonstrated.

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