This article describes a nonlinear frequency domain method for the simulation of unsteady blade row interaction problems across several blade rows in turbomachinery. The capability to efficiently simulate such interactions is crucial for the improvement of the prediction of blade vibrations, tonal noise, and the impact of unsteadiness on aerodynamic performance. The simulation technique presented here is based on the harmonic balance approach and has been integrated into an existing flow solver. A nontrivial issue in the application of harmonic balance methods to turbomachinery flows is the fact that various fundamental frequencies may occur simultaneously in one relative system, each one being due to the interaction of two blade rows. It is shown that, considering the disturbances corresponding to different fundamental frequencies as mutually uncoupled, one can develop an unsteady simulation method which from a practial view point turns out to be highly attractive. On the one hand, it is possible to take into account arbitrarily many nonlinear interaction terms. On the other, the computational efficiency can be increased considerably once it is known that the nonlinear coupling between certain subsets of the harmonics plays only a minor role. To validate the method and demonstrate its accuracy and efficiency a multistage compressor configuration is simulated using both the method described in this article and a conventional time-domain solver.

This paper compares various approaches to simulate unsteady blade row interactions in turbomachinery. Unsteady simulations of turbomachinery flows have gained importance over the last years since increasing computing power allows the user to consider 3D unsteady flows for industrially relevant configurations. Furthermore, for turbomachinery flows, the last two decades have seen considerable efforts in developing adequate CFD methods which exploit the rotational symmetries of blade rows and are therefore up to several orders of magnitude more efficient than the standard unsteady approach for full wheel configurations. This paper focusses on the harmonic balance method which has been developed recently by the authors. The system of equations as well as the iterative solver are formulated in the frequency domain. The aim of this paper is to compare the harmonic balance method with the time-linearized as well as the non-linear unsteady approach. For the latter the unsteady flow fields in a fan stage are compared to reference results obtained with a highly resolved unsteady simulation. Moreover the amplitudes of the acoustic modes which are due to the rotor stator interaction are compared to measurement data available for this fan stage. The harmonic balance results for different sets of harmonics in the blade rows are used to explain the minor discrepancies between the time-linearized and unsteady results published by the authors in previous publications. The results show that the differences are primarily due to the neglection of the two-way coupling in the time-linearized simulations.

This paper describes the implementation of a set of nonreflecting boundary conditions of increasing approximation quality for time-accurate and time-linearized 3D RANS solvers in the time and frequency domain. The implementations are based on the computation of eigenfunctions, either analytically or numerically, of the linearized Euler or Navier-Stokes equations for increasingly complex background flows. This results in a hierarchy of nonreflecting boundary conditions based on 1D characteristics, 2D circumferential mode decomposition, and 3D circumferential and radial mode decomposition, including viscous effects in the latter, for the frequency domain solver. By applying a Fourier transform in time at the boundaries the frequency domain implementations can be employed in the time domain solver as well. The limitations of each approximation are discussed and it is shown that increasing the precision of the boundary treatment the nonreflecting property of the boundary conditions is preserved for more complex flows without incurring an excessive increase in computing time. Results of a flutter analysis of a low pressure turbine blade obtained by time and frequency domain simulations are validated against each other and against reference results obtained with a 3D Euler frequency domain solver. The comparison of the results for different boundary conditions reveals the importance of using high quality boundary conditions.

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Since adjoint flow solvers allow for the computation of sensitivities of global flow parameters under geometric variations in an amount of time which is nearly independent of the number of geometric parameters, automatic shape optimization can be accelerated considerably by the use of an adjoint solver. In this article, a systematic approach for the development of an exact discrete adjoint of a turbomachinery flow solver is described. By using finite differences to differentiate the numerical fluxes, the problems associated with automatic and hand differentiation are circumvented. Moreover, a general treatment of the adjoint numerical boundary conditions is presented. As a result, an exact adjoint boundary condition for the conservative mixing planes is obtained. In combination with nonreflecting boundary conditions the latter are crucial for accurate flow simulations in turbomachinery. The adjoint is validated on the basis of a transonic compressor stage.