Numerical Method

For the investigation, a time-domain aeroelastic method that incorporates fhid-structure coupling is used (DLR’s TRACE-code: Nurnberger et al, 2001; Schmitt et al, 2001). The method consists of a non-linear Navier-Stokes solver for the aerodynamic part, and a linear modal model for the structural dynamics part. Some details of this method working in the time-domain are given as follows.

The aerodynamic part solves the three-dimensional Reynolds-Averaged Navier-Stokes equations in the relative frame of reference. The Spalart-Allmaras

one-equation turbulence model (Spalart and Allmaras, 1992) in combination with wall functions is used here for turbulence closure. The system of equation is discretized around the centers of the cells of the structured mesh in a blended finite-volume/finite-difference approach. For the convective terms, Roe’s up­wind scheme is used (Roe, 1981) in combination with van Leer’s MUSCL – extrapolation (van Leer, 1979). The viscous terms are discretized by central differences. At entry and exit, non-refecting boundary conditions based on a harmonic approach are applied (Giles, 1990). The coupling of multiple blade rows is realized here using the ‘sheared cells” approach (Giles, 1991). The time-accurate integration of the aerodynamic conservation laws uses a dual­time-stepping algorithm where the solution at the pseudo-time-levels is com­puted by an implicit solver. The computations are carried out on structured multi-block meshes. The blade vibration is taken into account by an algebraic mesh deformation. For the computations here, the blade count ratio is taken into account by meshing the geometry segment under consideration (the so called multi-passage approach). Parallel computing based on domain decom­position is applied.

The structural part uses a linear modal model. The mode shapes are taken from a real eigenvalue analysis done beforehand and are interpolated on the aerodynamic mesh. The mode shapes are constant during the aeroelastic sim­ulation. Each mode shape of each blade is one degree of freedom in the linear structural model. The system of equations of motion for the structure is in­tegrated in time using the Newmark method (Newmark, 1959). The coupling between fliid and structure is done by exchanging the modal forces and the displacements. The fuid-structure coupling exchanges information as well as updating the structural solution at every pseudo-time-subiteration of the aero­dynamic solver. This iterative improvement of the coupled solution leads to a diminishing coupling error between fliid and structure. It gives a quasi­implicit character to the solution of the coupled problem.

Validation and prior applications of the aeroelastic method have been re­ported previously (Schmitt et al, 2001).