Session: 34-05 Harmonic balance methods

Paper Number: 127820

127820 - Using Pseudotime Marching for the Solution of Harmonic Balance Problems 

The prediction of unsteady aerodynamic loads is a central problem during the design of turbomachinery.
Over the last 20 years, harmonic balance methods have been shown to be highly efficient for this task.
In contrast to linearized frequency-domain methods, harmonic balance takes the nonlinear interaction between
harmonics into account. Several researchers have presented approaches to generalize the pseudotime marching
technique, usually employed in steady solvers, to the harmonic balance system equations, be they formulated in the
frequency or the time domain.

The aim of this paper is to compare various approaches to implement implicit pseudotime marching for the harmonic
balance system in the frequency domain. We first give a motivation for using pseudotime marching as a solution technique. It turns out that, when the discretization errors of the pseudospectral time derivative and the pseudotime derivative are neglected, the harmonic balance solution converges to a stable periodic flow, provided the initial solution is sufficiently close to a stable periodic solution. This motivates the choice of a robust pseudotime marching approach, e.g., an implicit solver based on backward Euler integration. For this approach the Jacobian of the harmonic balance residual is needed. As for the steady problem, it is possible to approximate this Jacobian without changing the final solution as long as the solver converges. Therefore the question arises about what are appropriate simplifications with regard to the overall efficiency and robustness of the solver.

We show by means of a very simple model problem that the spectral time-derivative  operator should be taken into acount in the implicit system. Moreover, we show that, up to terms which scale with the amplitude of the disturbances, the linear system matrix is the sum of a scalar diagonal and a block diagonal matrix with identical blocks for each harmonic. The deviation from this structure is due to the nonlinearity of the unsteady flow problem. We show that when the unsteadiness is small, the nonlinear coupling terms can be neglected in the implicit solver and the resulting special matrix structure can be exploited to massively speed up the solver. In contrast, simple test cases with a strong linearity, such as the shock oscillation in a converging-diverging nozzle, show that this linear system approximation can lead to significant losses in robustness. To illustrate our findings we apply the implemented methods to predict aerodynamic damping and the flow response to a disturbance prescribed at the inlet for a transonic compressor at off-design conditions.

Presenting Author: Christian Frey German Aerospace Center (DLR)

Presenting Author Biography: 1995-2001: Study of Mathematics at Humboldt University Berlin
2001-2005: PhD Student at University of Cologne
2005: Doctoral Degree in Mathematics
2005-today: Scientist at the Institute of Propulsion Technology in Cologne, German Aerospace Center (DLR)
2010-today: Team lead of "New Methods" group within dept. "Numerical Methods"

Authors:

Christian Frey German Aerospace Center (DLR)
Jan Backhaus German Aerospace Center (DLR)
Graham Ashcroft German Aerospace Center (DLR)
Georg Geiser German Aerospace Center (DLR)
Benjamin Winhart MTU Aero Engines AG
Heinrich Stüer Siemens Energy