Nowadays unsteady flows within turbomachines grow stronger with the trend of more compact and higher loading designs. Consequently, this asks for the impact of unsteady flows on blade aerodynamics and aeroelasticity to be assessed as early as possible in a design loop. In a design phase, particularly at the early stage, fast turnaround of tools/methods is vitally important to ensure that targets can be achieved in time. To fulfil this requirement, a few efficient reduced order methods have been developed for analyzing turbomachinery unsteady flows over the last four decades. One of them is the time spectral method, which has been widely used due to its convenience to be extended from an existing steady flow solver. The time spectral method resolves unsteady flows at some selected time instants, then temporal Fourier coefficients are obtained through Fourier transformation using solutions at those time instants. The temporal Fourier coefficients can be used to calculate time derivatives at all the considered time instants. The choice of the time instants is not arbitrary, as they determine the condition number of Fourier transform matrix. The principle of choosing time instants is to minimize the condition number of Fourier transform matrix. It has long reported that the condition number can affect solution stability and convergence of the time spectral equation system. Nevertheless, there is very limited study of the effect of the condition number. This paper aims at filling this knowledge gap by evaluating the effect of the condition number of the Fourier transform matrix on the solution stability and convergence of the time spectral method from two aspects. The first aspect is to assess the impact of condition number using a matrix stability analysis based upon the time spectral form of the scalar advection equation. The relationship between maximum allowable Courant number and the condition number will be derived. The second aspect is to assess the impact of condition number for real life applications. Two case studies will be provided: one is a flutter case and the other is a blade row interaction case. A series of numerical analyses will be performed for each case using different sets of time instants corresponding to different condition numbers. The conclusion drawn from the two real life case studies will corroborate the relationship derived from the matrix stability analysis. This study will also provide guidance for choosing time instants for solving the time spectral equation system.


Effect of the Condition Number of the Inverse Fourier Transform Matrix on the Solution Behavior of the Time Spectral Equation  System

Division / Sponsor:
International Gas Turbine Institute (IGTI)

Paper Proceedings: https://asmedigitalcollection.asme.org/GT/GT2020/volume/84058

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