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.

Fast and accurate computation of thermo-physical properties is essential in computationally expensive simulations involving fluid flows that significantly depart from the ideal gas or ideal liquid behavior. A look-up table algorithm based on unstructured grids is proposed and applied to non-ideal compressible fluid dynamics simulations. The algorithm grants the possibility of a fully automated generation of the tabulated thermodynamic region for any boundary and to use mesh refinement. Results show that the proposed algorithm leads to a computational cost reduction up to one order of magnitude, while retaining the same accuracy level compared to simulations based on more complex equation of state. Furthermore, a comparison of the LuT algorithm with a uniformly spaced quadrilateral tabulation method resulted in similar performance and accuracy.

Shape optimization in unsteady flow problems enables the consideration of dynamic effects on design. The ability to treat unsteady effects is attractive, as it can provide performance gains when compared to steady-state design methods for a variety of applications in which time-varying flows are of paramount importance. This is the case, for example, in turbomachinery or rotorcraft design. Given the high computational cost involved in time-accurate design problems, adjoint-based shape optimization is a promising option. However, efficient sensitivity analysis should also be accompanied by a significant decrease in computational cost for the primal flow solution, as well. Reduced-order models, like those based on the harmonic balance concept, in combination with the calculation of gradients via adjoint methods, are proposed for the efficient solution of a certain class of aerodynamics optimization problems. The harmonic balance method is applicable if the flow is characterized by discrete finite dominant flow frequencies that do not need to be integer multiples of a fundamental harmonic. A fully-turbulent harmonic balance discrete adjoint formulation based on a duality-preserving approach is proposed. The method is implemented by leveraging algorithmic differentiation and is applied to two test cases: the constrained shape optimization of both a pitching airfoil and a turbine cascade. A key advantage of the current approach is the accurate computation of gradients as compared to second order finite differences without any approximation in the linearization of the turbulent viscosity. The shape optimization results show significant improvements for the selected time-dependent objective functions, demonstrating that design problems involving almost-periodic unsteady flows can be tackled with manageable computational effort.