Discontinuous Galerkin method for nonlinear standing waves in acoustic resonators

Based on the pseudo-one-dimensional unsteady compressible Navier-Stokes equations, state equation for an ideal gas, the three-order Runge-Kutta equation and slope limiter, discontinuous Galerkin method for solving the nonlinear standing waves in resonators is proposed. The standing waves in cylindrical resonators, exponential resonators and conical resonators are simulated by the discontinuous Galerkin method, respectively. The results obtained with the proposed discontinuous Galerkin method are in agreement with the results obtained with other numerical methods. The second peaks in the pressure waveforms, which were found in the existed experiments, are simulated by the discontinuous Galerkin method. In cylindrical resonators and exponential resonators, the computational accuracy, the ability for removing the numerical oscillations and exactly capturing the shock waves with two kinds of methods for increasing the number of grids are evaluated. How to choose the CFL number for removing the numerical oscillations is also discussed. Comparing the simulation results and CPU time with those of the finite volume method, we find that timesaving, high efficiency and high computational accuracy are the advantages of the proposed discontinuous Galerkin method. In conical resonators, we research the pressure and velocity waveforms under different flow velocities. As increasing of the flow velocity, the amplitudes of pressure at the small end and velocity at the large end increase. The shock waves emerge in the velocity waveforms. Finally, based on the properties of standing waves in resonators studied by the discontinuous Galerkin method, the important references for optimizing the shapes of resonators and engineering applications of nonlinear standing waves are built. © 2016 Journal of Mechanical Engineering.