Application of the Harmonic Balance Method for Large Spread Multiple Frequency Scales

Abstract This paper presents the application of the harmonic balance method to periodic turbomachinery flow problems containing multiple fundamental frequencies. It is well known that the solution convergence of a transient turbomachinery flow using classical integration methods is computationally intensive and requires the integration for multiple periods to achieve a converged periodic solution. The computational effort is even higher when multiple fundamental frequencies (vastly different time scales) are modeled since the time stepping required is limited by the higher frequency in the flow. Unlike the classical time integration methods, the multifrequency harmonic balance method is very well suited for these applications since it allows for much faster calculations than the standard time marching algorithms. In addition, to resolve all the fundamental frequencies and higher harmonics, the current implementation of the harmonic balance method allows for solutions on unequal time interval distributions of the time planes. Given a user-defined list of frequencies that govern the flow problem, this method utilizes an optimization strategy to compute the time planes. There are two simulation cases of interest: aerodynamic performance and forced response analyses, both with different accuracy requirements. The number of required frequencies is dependent on the goal of the simulation. For aerodynamic performance analysis, global quantities such efficiency, pressure ratio, etc. can be predicted with fewer fundamental frequencies than forced response analysis where accurate local flow details demand a higher number of fundamental frequencies. The advantages of the multifrequency harmonic balance are illustrated by modeling two radial turbine configurations subjected to an inlet pulse from a reciprocating engine. Not only the expected trends such as higher-modes modeling granularity and time transient accuracy are shown, but also the calculations agree well with experimental data. The computational effort can be up to tenfold lower than the standard time-marching simulations. Machine aerodynamic performance predictions from the harmonic balance method are compared to accurate time-marching solutions and experimental data measurements. The efficiency of the computation is also discussed. The second example will compare the predicted surface excitation from the harmonic balance method vs. the time-marching solution.