Abstract
The possibility of taking into account unsteady flow effects if performing turbomachinery shape optimization is attractive to accurately address inherently time dependent design problems. The harmonic balance method is an efficient solution for computational fluid dynamics problems of turbomachinery characterized by quasi-periodic flows. If applied in combination with adjoint methods, it enables the possibility to deal with unsteady fluid-dynamic design in a cost effective manner, opening the way towards multi-disciplinary applications. This paper presents the development of a novel fully-turbulent discrete adjoint based on the time domain harmonic balance method and its application to the constrained fluid dynamic optimization of an axial turbine stage. As opposed to previous works, the proposed method does not require any assumption on the turbulent eddy viscosity and on the set of input frequencies. The results show that the method provides accurate gradients, if compared with second order finite differences, and significant deviation with respect to the sensitivity computed with the constant eddy viscosity approximation. The application of the method to the fluid-dynamic shape optimization of the exemplary stage leads to improve the total-to-static efficiency of 0.8%. The efficiency increase is found to be higher than that obtained by means of a steady state optimization method.
Highlights
Adjoint-based shape optimization methods are increasingly becoming essential for automated design
The main reasons for this difference are: i) a steady-state model with the MP interface cannot simulate the unsteady potential effects generated by the stator-rotor interaction; ii) the stator wake is not transported to the rotor when using the mixing plane interface; iii) for transonic calculations, the harmonic balance (HB) method is able to model the unsteady non-linear effects deriving from the shock waves appearing due to the imposed flow conditions and to the time-dependent mutual position of the blade rows
There are two main observations that can be drawn by analyzing Fig. 7b: i) the value of the gradients computed with the HB-based adjoint is comparatively the same for more than two frequencies; ii) the largest discrepancy between HB and MP simulation results occurs in the proximity of the rotor leading edge
Summary
Adjoint-based shape optimization methods are increasingly becoming essential for automated design. Due to the high computational cost and memory storage requirements associated with unsteady adjoints [16], several methods have been proposed to improve the efficiency of obtaining time-accurate design sensitivities. A design algorithm based on a fully-turbulent HB adjoint has been recently developed and applied to the optimization of problems characterized by quasi-periodic flows [22] This algorithm is restricted to a single computational domain, only suited for the automated design of isolated turbomachinery cascades. The baseline stage shape is optimized by means of both the proposed HB-based unsteady method and of a steady state method based on the mixing plane (MP) row interface The objective of this comparison is to assess whether the HB-based automated design provides a gain in computed fluid dynamic performance over the MP-based one, and if these are dependent on the operating conditions. The computational performance of the method is analyzed in detail in terms of computational cost, memory, and storage requirements
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