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Research Papers

Forced Response Sensitivity Analysis Using an Adjoint Harmonic Balance Solver

[+] Author and Article Information
Anna Engels-Putzka

Institute of Propulsion Technology,
German Aerospace Center (DLR),
Linder Höhe,
Cologne 51147, Germany
e-mail: anna.engels-putzka@dlr.de

Jan Backhaus

Institute of Propulsion Technology,
German Aerospace Center (DLR),
Linder Höhe,
Cologne 51147, Germany
e-mail: jan.backhaus@dlr.de

Christian Frey

Institute of Propulsion Technology,
German Aerospace Center (DLR),
Linder Höhe,
Cologne 51147, Germany
e-mail: christian.frey@dlr.de

1Corresponding author.

Contributed by the International Gas Turbine Institute (IGTI) of ASME for publication in the JOURNAL OF TURBOMACHINERY. Manuscript received September 20, 2018; final manuscript received October 4, 2018; published online January 21, 2019. Editor: Kenneth Hall.

J. Turbomach 141(3), 031014 (Jan 21, 2019) (8 pages) Paper No: TURBO-18-1260; doi: 10.1115/1.4041700 History: Received September 20, 2018; Revised October 04, 2018

This paper describes the development and initial application of an adjoint harmonic balance (HB) solver. The HB method is a numerical method formulated in the frequency domain which is particularly suitable for the simulation of periodic unsteady flow phenomena in turbomachinery. Successful applications of this method include unsteady aerodynamics as well as aeroacoustics and aeroelasticity. Here, we focus on forced response due to the interaction of neighboring blade rows. In the simulation-based design and optimization of turbomachinery components, it is often helpful to be able to compute not only the objective values—e.g., performance data of a component—themselves but also their sensitivities with respect to variations of the geometry. An efficient way to compute such sensitivities for a large number of geometric changes is the application of the adjoint method. While this is frequently used in the context of steady computational fluid dynamics (CFD), it becomes prohibitively expensive for unsteady simulations in the time domain. For unsteady methods in the frequency domain, the use of adjoint solvers is feasible but still challenging. The present approach employs the reverse mode of algorithmic differentiation (AD) to construct a discrete adjoint of an existing HB solver in the framework of an industrially applied CFD code. The paper discusses implementational issues as well as the performance of the adjoint solver, in particular regarding memory requirements. The presented method is applied to compute the sensitivities of aeroelastic objectives with respect to geometric changes in a turbine stage.

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Figures

Grahic Jump Location
Fig. 1

Midspan section of (non-split) computational grid

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Fig. 2

Residuals of the primal and adjoint harmonic balance solvers with one higher harmonic per blade row included. The adjoint residuals have been normalized so that the first residual is equal to one.

Grahic Jump Location
Fig. 3

Absolute values of surface sensitivities of Re[Fmod] for mode 2 on the pressure side of the TMTF

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Fig. 4

Original (gray) and deformed (orange) geometry of the TMTF for the linear (left) and quadratic (right) type of deformation

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Fig. 5

Modal force values of mode 1 for different geometry variations. All values have been normalized by that of the reference geometry.

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Fig. 6

Modal force values of mode 2 for different geometry variations. All values have been normalized by that of the reference geometry.

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Fig. 7

Modal force values of mode 3 for different geometry variations. All values have been normalized by that of the reference geometry.

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Fig. 8

Modal force values for different HB setups. The first number denotes the number of higher harmonics in the TMTF row, the second that of higher harmonics in the rotor row. All values are normalized with the result from the HB 1_1 computation for the respective mode.

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Fig. 9

Modal force sensitivities of mode 2 for different HB setups. The first number denotes the number of higher harmonics in the TMTF row, the second that of higher harmonics in the rotor row.

Grahic Jump Location
Fig. 10

Computational grid for the adTurb turbine stage

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