Forced response freeform optimization of a radial turbine using the adjoint method

Authors

Nicolas Lachenmaier  - Rolls-Royce Power Systems AG, Germany
Johannes Kech - Rolls-Royce Power Systems AG, Germany

Abstract

Radial turbines equipped with inlet guide vanes are subject to a forced excitation: The guide vanes distort the pressure field upstream of the turbine wheel and induce a wake, that excites the blades as they pass. This is problematic as soon as the excitation frequency and one of the blade's eigenfrequencies match, i.e. if resonance occurs. Hence, one is interesented in studying whether and how one may reduce the induced forced response of the blades to avoid high cycle fatigue. For the given turbine, around 10 resonances occur within the operating range. The forced response is evaluated using a custom finite element code. Herein, we take into account that the excitation amplitude scales with increasing resonance frequency or rotation rate, respectively. The viscous damping is modelled using an aero-acoustically motivated approach based on the averaged acoustic impedance of the surrounding flow. Aiming at a reduction of the resulting forced response, a gradient-based freeform optimization is conducted for the studied radial turbine: Sensitivities are evaluated by means of the adjoint method. Herein, the complex step method is exploited to gather accurate derivates of both dynamic stiffness matrix and excitation force with respect to the mesh nodes. The resulting mesh sensitivities are smoothed by applying the Vertex Morphing method before the mesh is actually deformed. This process is repeated several times. Few iterations are necessary to successfully reduce the blades' most prominent forced response. Hence, the method delivers a promising approach to make radial turbines more robust against guide vane induced excitations. The paper concludes with a comparison of the eigenmodes of both baseline and optimized design to show how those have been altered by the optimizer.