Simulation of three rows of a multistage compressor with the harmonic balance method.
Essential physical phenomena occur in the simulation of flow in turbomachinery often with specific frequencies, e.g. at multiples of the rotation frequency or at the eigenfrequencies of blades. In these cases, it is sufficient to calculate the solution of the flow equations for some frequencies only. This is achieved by transforming the flow equations into the frequency domain and solving these equations for selected frequencies. Thus the computation time is reduced by one to two magnitudes compared to a nonlinear, unsteady calculation. In TRACE two types of frequency domain methods are implemented: in a classical time-linearized Navier-Stokes method and a harmonic-balance method which takes into account a set of frequency and their non-linear coupling
Time-linearized Navier-Stokes Solver
This method is based on the assumption that the solution of the flow equation can be decomposed into a time-independent part and a small time-dependent perturbation. This allows to derive linear equations for the perturbations. These equations are transferred, using a harmonic ansatz for the perturbation into the frequency domain. The spatial discretization then leads to a complex-valued linear system of equations which is solved with a pseudo time-stepping or GMRes method. The linearization and the formulation in the frequency domain makes it easy, compared to unsteady non-linear solvers, to implement some turbomachinery specific boundary conditions. These are non-reflecting boundary conditions on entry and exit surfaces and periodicity constraints, which allow for the simulation of configurations with multiple blade rows to calculate the solution in only one segment per row instead of the full wheel. Applications for this method are flutter and forced response analysis, as well as aeroacoustic studies.
Harmonic Balance Solver
The harmonic balance approach for the simulation of unsteady flows works by transforming the nonlinear flow equations in the frequency domain and solves them for selected frequencies. In contrast to the time-linearized approach the equations are not linearized which results in a coupling of the equations for the selected frequencies. By taking into account sufficiently many frequencies the solution of the nonlinear unsteady equations can be arbitrarily accurately approximated. For each frequency, the Fourier coefficients of the unsteady flow are computed by discrete Fourier and invers Fourier transforms of the flow variables and the nonlinear residuals. Here it proves particularly useful that an existing formulation of the flow residual, i.e. the spatial discretization of the flow equations, can be employed. The system of equations for the different frequencies is non-linear and must be solved with an appropriate iterative solution procedure. A pseudo time-step solver is used here which is already used in the steady and unsteady non-linear solver. Because the basic equations are formulated in the frequency domain, some turbomachinery specific boundary conditions can be used as in the time-linearized solvers which are very difficult to integrate in time domain solvers and cause significant run-time problems there. These are on the one hand non-reflecting boundary conditions at inlet and outlet and on the other hand the periodic boundary conditions with time shift which allows in multi-stage configurations to solve the equations in only one segment of every blade row instead of the full wheel. These two extensions of the solution method are prerequisites for an efficient and high-quality simulation of unsteady flows in turbomachines.