Investigation Of Application Of Harmonic Balance Method To Periodic Unsteady Flow Problems

In recent years,a novel nonlinear frequency domain harmonic balance method for the numerical simulation of unsteady flows has been developed,it can be considered as an alternative method to the classical time domain method.The basic principle of this method is to decompose the flow variables into a Fourier series,which transforms the unsteady-flow into several steady problems coupled by a HB source term,and then rebuild the whole history of the unsteady flow.According to the theory of the harmonic balance method,we obtained the harmonic balance equations for Reynolds Averaged Navier-Stokes Equations.The approach is applied to solve the one-dimensional wave equation and inviscid Burgers' equation with a variety of periodic boundary conditions to verify the reliability of harmonic balance method.The results show that boundary conditions,amplitude,fundamental frequency,and grid density have impact on the harmonic balance solution.On a given grid,a flow that is smoothly unsteady(i.e.,without moving discontinuities)will require fewer harmonics than a flow containing a moving shock to achieve the same level of fidelity.The unsteady flows around an enforced pitching HBS standard model with different parameters were simulated to demonstrate the computational accuracy,efficiency,and memory requirements of the harmonic balance method.A comparison was made with the time domain method,and the results show that the harmonic balance method is applicable to simulate middle and small amplitude pitching,and the frequency has less effect on its computing time.Therefore the harmonic balance method has obvious advantage for problems with a long period compared with the time domain method.Based on the unsteady Etkin pneumatics model,we applied the harmonic balance method to calculate the dynamic derivatives,and obtained the dynamic derivatives of small amplitude pitching HBS by harmonic balance method and time domain method.The results agreed well with each other,and both close to the results of the experiments,which further verified the correctness of the codes and the reliability of harmonic balance method.The unsteady flow around an enforced pitching X-51-Type hypersonic vehicle in different attack was simulated,and calculated the dynamic derivatives.A comparison was made with the time domain method to verify the ability of harmonic balance method for the numerical simulation of unsteady flows around complicated model and the ability for the simulation of dynamic derivative.