Three dimensional frequency-domain solution method for unsteady turbomachinery flows
The three-dimensional calculation of unsteady flows is increasingly gaining importance in the prediction of turbomachinery flow problems. A three-dimensional Euler/Navier-Stokes solver incorporating the time-linearized method and the nonlinear harmonic method in the frequency domain has been developed for predicting unsteady turbomachinery flows. In the time-linearized method, the flow is decomposed into a steady part and a harmonic perturbation part. Linearization results in a steady flow equation and a time-linearized perturbation equation. A pseudo-time time-marching technique is introduced to time-march them. A cell centred finite volume scheme is employed for spatial discretization and the time integration involves a four stage Runge Kutta scheme. Nonreflecting boundary conditions are applied for far field boundaries and a slip wall boundary condition is used for Navier-Stokes calculations. In the nonlinear harmonic method, the flow is assumed to be composed of a time-averaged part and an unsteady perturbation part. Due to the nonlinearity of the unsteady equations, time-averaging produces extra unsteady stress terms in the time-averaged equation which are evaluated from unsteady perturbations. While the unsteady perturbations are obtained from solving the harmonic perturbation equation, the coefficients of perturbation equations come from the solution of time-averaged equation and this interaction is achieved through a strong coupling procedure. In order to handle flows with strong nonlinearity, a cross coupling of higher order harmonics through a harmonic balancing technique is also employed. The numerical solution method is similar to that used in the time-linearized method. The numerical validation includes several test cases involving linear and nonlinear unsteady flows with specific attention to flows around oscillating blades. The results have been compared with other well developed linear methods, nonlinear time-marching method and experimental data. The nonlinear harmonic method is able to predict strong nonlinearities associated with shock oscillations well but some limitations have also been observed. A three-dimensional prediction of unsteady viscous flows through a linear compressor cascade with 3D blade oscillation, probably the first of its kind, has shown that unsteady flow calculation in the frequency domain is able to predict three-dimensional blade oscillations reasonably well.