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A harmonic balance technique for the analysis of nonlinear unsteady flows in turbomachinery arising from several excitation sources, possibly with frequencies with irrational ratios, is presented in this paper. This method uses a mixed time-domain/frequency-domain approach that allows one to model the blade row on a computational grid spanning just a single blade passage, no matter the interblade phase angles of the original disturbances. Using this approach, we compute several solutions, each one corresponding to one of several times spanning one period for periodic flows or over a representative time interval for aperiodic flows. These time levels are coupled through a spectral time-derivative operator in the interior of the computational domain and through the far-field and periodic boundary conditions around the boundary of the domain. In this paper, we apply the method to the two-dimensional Euler equations (although the method can be applied to three-dimensional and viscous flows) and examine the nonlinear interaction of wake passing with blade vibration.

A harmonic balance technique for the analysis of two-dimensional linear (small-disturbance) and nonlinear unsteady flows in multistage turbomachines is presented. The present method uses a mixed time-domain/frequency-domain approach that allows one to compute the unsteady aerodynamic response of multistage machines to both blade vibration (the flutter problem) and wake interaction (the forced response problem). In general, the flowfield may have multiple excitation frequencies that are not integer multiples of each other, so that the unsteady flow is (sometimes) aperiodic in time. Using our approach, we model each blade row using a computational grid spanning a single blade passage. In each blade row, we store several subtime level solutions. For flows that are periodic in time, these subtime levels span a single time period. For aperiodic flows, the temporal period spanned by these subtime level solutions is sufficiently long to sample the relevant discrete frequencies contained in the aperiodic flow. In both cases, these subtime level solutions are related to each other through the time-derivative terms in the Euler or Navier-Stokes equations and boundary conditions; complex periodicity conditions connect the subtime levels within a blade passage, and interrow boundary conditions connect the solutions among blade rows. The resulting discretized equations, which are mathematically steady because time derivatives have been replaced by a pseudospectral operator in which the excitation frequencies appear as parameters, can be solved very efficiently using multigrid acceleration techniques. In this paper, we apply the technique to both flutter and wake-interaction problems and illustrate the influence of neighboring blade rows on the unsteady aerodynamic response of a blade row. DOI: 10.2514/1.22888

We present a three-dimensional time-linearized unsteady Euler solver for computing unsteady flows in multistage turbomachines. Using this approach, each blade row is modeled on a computational grid spanning a single blade passage. Within each blade passage, several time-linearized unsteady solutions, each of which have different frequencies and interblade phase angles, are computed. These various solutions are coupled together at the interrow boundaries between the blade rows. The coupling allows one to pass the pressure, entropy, and vorticity waves traveling between neighboring blade rows resulting in an efficient multistage analysis of unsteady flows in turbomachinery. Results that demonstrate the accuracy and efficiency of the method, as well as the importance of multistage effects, are presented for several geometries. In particular, we show that multistage effects can be significant for the aerodynamic loads acting on a given blade row. Furthermore, the method presented is highly efficient. For example, a flutter calculation requires only about three times the CPU time of one steady flow computation for each interblade phase angle and frequency considered.

The modification of unsteady three-dimensional Navier-Stokes codes for application on massively parallel and distributed computing environments is investigated. Previously, the Euler mode of the Navier-Stokes code TURNS has been parallelized. For the efficient implementation of the Navier-Stokes mode of TURNS on parallel computing systems, several algorithmic changes should be made. The main modification is done on the implicit operator, lower-upper symmetric Gauss-Seidel. Two new implicit operators are used because of convergence problems of traditional operators with high cell aspect ratio grids needed for viscous calculations. Results for Navier-Stokes cases are presented for various operators. The message passing interface protocol is used because of its portability to various parallel architectures.