FOURIER TRANSFORMATION BASED ROTOR-STATOR COUPLING METHOD FOR TIME DOMAIN ANALYSIS OF TURBOMACHINERY UNSTEADY FLOWS

The paper presents an investigation of the Fourier transformation based method for analyzing the unsteady flow field within turbomachines due to blade row interaction. With the Fourier transformation method, the flow field in the circumferential direction at a given radius on a rotor-stator interface is represented using spatial Fourier series. During a flow solution process, the coupling method aims to match these Fourier coefficients. It does not involve interpolation of flow field in the circumferential direction if the grid distribution is uniform. Furthermore it does not require searching of matching points across an interface. Its implementation is much easier and more straightforward than the classical sliding plane method. This advantage is particularly useful for distributed memory parallel computing. In comparison, the sliding plane method requires point to point matching across an interface to be established at every time step and the matching of grid points and computing threads also varies with time, which needs extraordinary effort for a parallel computing. With the Fourier transformation based method, there is no need to seek the point matching as Fourier coefficients for all passages of one row are the same. However this method involves Fourier transformation and inverse Fourier transformation, which can be quite time consuming. The time cost, convergence rate, conservation and its effect on local and overall flow field are examined in this investigation using a transonic compressor stage test case. INTRODUCTION Blade row interaction is the major cause to the inherent unsteadiness of the flow field within multi-stage turbomachines. To resolve the unsteadiness, the unsteady Reynolds averaged Navier-Stokes (URANS) equations can be solved. Due to the relative rotation of rotors and stators, different reference frames are employed for stators and rotors with an interface separating rotor domains and stator domains. Subsequently an interface treatment enabling solution information exchange is needed to facilitate the solution of the URANS equations. The sliding plane method[1][2][3][4] is the wellestablished and widely used method for treating a rotor-stator interface for solving the URANS equations. There have been various ways to implement the sliding plane method. One way is to generate overlapped/patched grid[1] at an interface with the solution at the dummy cells next to the interface of one domain being interpolated from the solution at interior cells next to the interface of the other domain. The overlapped/patched grid generation is quite often accomplished by a flow solver. It can be easy to be implemented if the grids are of being structured (quadrilateral in 2D and hexahedral in 3D), but it can be nontrivial at all if the grid is unstructured. For this approach, its implementation is the same at all grids of a geometric multigrid solution. Another approach to implement the sliding plane method is to search the matching grid points from two domains at an interface. Then the flow field variables at those matching grid points or cells are forced to match by using their difference in solution between the matching points to define incoming flow characteristic variables with the outgoing characteristic variables being extrapolated. For this approach, and the previous one, the bilinear interpolation is often utilized. The bilinear interpolation quite often is not conservative. To ensure conservative interpolation, extra measure is needed when performing the interpolation. To avoid loss of conservation, Zha et al proposed a conservative sliding plane method[5]. This method requires uniformly, structured and matching grid point distribution along the circumferential direction. When the grid is rotating, the rotation is performed by one cell only so that the grids are always matched. For 3D blades, the requirement is very difficult, if not impossible, to achieve, as the blades can be highly twisted and grid radial distribution can be different across an interface. Very recently a Fourier transformation based approach was proposed to filter high reduced frequency unsteadiness to speed up analysis of low reduced frequency unsteadiness[6]. This approach is quite flexible, as it can