Sensitivity analysis of unsteady inviscid flow through turbomachinery cascades
We present a novel sensitivity analysis for predicting the effect of airfoil shape on the unsteady aerodynamic and aeroacoustic response of turbomachinery blading. The nominal steady and unsteady flow in a cascade of turbomachinery blades is modeled using the steady Euler equations and the time-linearized Euler equations, respectively. Both the steady and unsteady Euler equations are solved using a two-step finite-volume Lax-Wendroff discretization with multigrid acceleration. We compute the unsteady aerodynamic loads due to both incoming gusts and plunging motion of the cascade airfoils. Once the nominal steady and unsteady flows have been computed, a sensitivity analysis is performed using the discrete adjoint equations of the computational fluid dynamics scheme used to discretize the Euler equations. For each objective function (e.g., the amplitude peak of the aeroelastic blade motion), the resulting adjoint equations are solved using the adjoint Lax-Wendroff scheme, which is also accelerated using a multigrid technique. Once the adjoint equations have been solved, the computed adjoint variables may be used to compute rapidly the sensitivities of the aeroelastic and aeroacoustic objective functions due to arbitrary changes in geometry. The method is computationally efficient, with similar convergence rate histories for both the nominal and the adjoint solutions. To demonstrate the utility of the present method, we use the sensitivity analysis to redesign the shape of the airfoils of a cascade for increased aeroelastic stability. We also redesign the shape of the airfoils of an exit guide vane for reduced downstream radiated noise.
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