The adjoint method is attractive because of its low computational cost and high efficiency. Although it has been one of the hot issues in aerodynamic design, it is not so widely used in turbomachinery applications as it is in the aeronautical field. The purpose of this work is to apply the adjoint method to three-dimensional (3D) aerodynamic inverse design of axial turbine blades for inviscid compressible flow. The 3D continuous adjoint system of Euler equations is formulated for turbine internal flow. The 3D blade profile is parameterized with Non-uniform B-Spline patch, and the coordinates of the B-Spline control points are selected as the design variables. Characteristic analysis of adjoint equations is taken to set inlet/outlet boundary conditions. To avoid the discontinuity of boundary conditions of adjoint equations in the spanwise direction, a method for solving an ordinary differential equation is developed to smooth the residual distribution of aerodynamic parameter on blade surface. 3D adjoint equations are numerically solved by using time-marching method and finite volume method. Finally, combining the grid perturbation technique, CFD technique and quasi-Newton algorithm, the aerodynamic design approach for 3D axial turbine blades is presented and several numerical examples are demonstrated to validate this approach.

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