Unsteady aerodynamic analysis of flows in multistage turbomachinery using the time-linearized Navier-Stokes equations

An efficient and accurate computational method for predicting unsteady flows in multistage turbomachinery is presented. Particularly, a method is developed to solve the flutter and forced response problems accounting for blade row coupling. The aerodynamic blade row coupling is provided by propagation of waves in the working fluid. A set of travelling waves is represented by multiple unsteady solutions for each blade row. Each solution has a particular frequency and interblade phase angle defined by the scattering and frequency shifting mechanisms. The waves are allowed to propagate between the rows by exchanging the information between various unsteady solutions at the interface boundaries. Only a finite number of unsteady solutions must be retained in the model to compute accurately the unsteady aerodynamic response. The unsteady solutions are defined by solving a system of time-linearized Reynolds-averaged Navier-Stokes equations with the Spalart-Allmaras one-equation turbulence model. After linearization, the resulting system of equations consists of the non-linear steady equations and linear unsteady equations with variable coefficients. The equations are discretized on a deforming, multi-block grid and advanced in the pseudo-time domain using a finite-volume Lax-Wendroff integration technique. Using the circumferential periodicity only one blade-to-blade passage is required to model each blade row. Various acceleration techniques such as multi-grid, local time-stepping and smoothing are employed to improve the convergence rate. The steady flow solver is validated against available experimental data. Results from the unsteady flow solver are compared with a semi-analytical method for the flutter problem. The analysis of the flutter problem for a front one-and-half stage of a modern axial compressor demonstrated the capability of the method of predicting the unsteady loads on blades with complex realistic geometry. The unsteady flow calculations in the forced response problem revealed the importance of the first two blade passing frequencies for computing the unsteady load on blades in a downstream blade row. The present time-linearized method is several orders of magnitude faster than conventional non-linear time-marching methods.