Validation of Higher-Order Discontinuous Galerkin Method for the Linearized Euler Equations

Abstract

A high-order implementation of the Discontinuous Galerkin (DG) method is presented for solving the three-dimensional Linearized Euler Equations on an unstructured hexahedral grid. The method is based on a quadrature free implementation and the high-order accuracy is obtained by employing higher-degree polynomials as basis functions. In the DG method the solution domain is divided into a set of non-overlapping elements and the approximate solution is expressed as a linear combination of linearly independent basis functions. The degree of the polynomials that constitute the basis functions determines the order of the accuracy of the method and, if desired, the degree can vary from element to element. The weak formulation of the problem is approximated by replacing the solution space by the space spanned by the basis functions. Hence the approximate solution, represented as an expansion in terms of basis functions is discontinuous at the interfaces between neighboring elements. To provide the crucial coupling and to handle the discontinuity at element interfaces the boundary-normal flux is replaced by the approximate Riemann flux which is the only means by which neighboring elements communicate, regardless of the order of the method. In the present work the Riemann flux is approximated by a Lax-Friedrichs flux. For the time discretization a four-stage Runge-Kutta scheme is used which for the present linear problem possesses fourth-order accuracy in time. The present DG method is implemented for an unstructured hexahedral grid and is up to fourth-order accurate in space. Non-reflecting boundary conditions are implemented at the boundaries of the computational domain. The method has been verified within previous studies and has been applied to the acoustic radiation from a vibrating wall segment in an infinite rectangular duct. In this study the method is applied to a problem in which an infinite long duct is split longitudinally through a thin wall that has a gap small compared to the duct height. An incoming wave is introduced at the open boundaries of the upper duct and the time histories of the perturbations of the mean flow variables are recorded at various microphone positions throughout the duct. The analytical solution is found by using a modal expansion of the pressure field in the three different regions of the duct (upper duct boundaries, lower duct boundaries and gap region). The pressure and velocities are matched at the interfaces between these regions. Reflection and transmission coefficients of the plane waves are calculated. Below the cut-off frequency we find an excellent agreement between numerical and analytical results. Note that since plane waves are considered, below the cut-off frequency energy conservation holds. Above the cut-off frequency this relationship does not hold because energy is transferred to a higher non-planar mode.