An efficient discontinuous Galerkin method for aeroacoustic propagation

An efficient discontinuous Galerkin formulation is applied to the solution of the linearized Euler equations and the acoustic perturbation equations for the simulation of aeroacoustic propagation in two-dimensional and axisymmetric problems, with triangular and quadrilateral elements. To improve computational efficiency, a new strategy of variable interpolation order is proposed in addition to a quadrature-free approach and parallel implementation. Moreover, an accurate wall boundary condition is formulated on the basis of the solution of the Riemann problem for a reflective wall. Time discretization is based on a low dissipation formulation of a fourth-order, low storage Runge-Kutta scheme. Along the far-field boundaries a perfectly matched layer boundary condition is used. For the far-field computations, the integral formulation of Ffowcs Williams and Hawkings is coupled with the near-field solver. The efficiency and accuracy of the proposed variable order formulation is assessed for realistic geometries, namely sound propagation around a high-lift airfoil and the Munt problem

Validation of a Discontinuous Galerkin Implementation of the Time-Domain Linearized Navier–Stokes Equations for Aeroacoustics

The propagation of small perturbations in complex geometries can involve hydrodynamic-acoustic interactions, coupling acoustic waves and vortical modes. A propagation model, based on the linearized Navier–Stokes equations, is proposed. It includes the mechanism responsible for the generation of vorticity associated with the hydrodynamic modes. The linearized Navier–Stokes equations are discretized in space using a discontinuous Galerkin formulation for unstructured grids. Explicit time integration and non-reflecting boundary conditions are described. The linearized Navier–Stokes (LNS) model is applied to two test cases. The first one is the time-harmonic source line in an incompressible inviscid two-dimensional mean shear flow in an infinite domain. It is shown that the proposed model is able to capture the trailing vorticity field developing behind the mass source and to represent the redistribution of the vorticity. The second test case deals with the analysis of the acoustic propagation of an incoming perturbation inside a circular duct with a sudden area expansion in the presence of a mean flow and the evaluation of its scattering matrix. The computed coefficients of the scattering matrix are compared to experimental data for three different Mach numbers of the mean flow, M0 = 0.08, 0.19 and 0.29. The good agreement with the experimental data shows that the proposed method is suitable for characterizing the acoustic behavior of this kind of network

Arbitrary-Lagrangian-Eulerian discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes

We present a new family of high order accurate fully discrete one-step Discontinuous Galerkin (DG) finite element schemes on moving unstructured meshes for the solution of nonlinear hyperbolic PDE in multiple space dimensions, which may also include parabolic terms in order to model dissipative transport processes. High order piecewise polynomials are adopted to represent the discrete solution at each time level and within each spatial control volume of the computational grid, while high order of accuracy in time is achieved by the ADER approach. In our algorithm the spatial mesh configuration can be defined in two different ways: either by an isoparametric approach that generates curved control volumes, or by a piecewise linear decomposition of each spatial control volume into simplex sub-elements. Our numerical method belongs to the category of direct Arbitrary-Lagrangian-Eulerian (ALE) schemes, where a space-time conservation formulation of the governing PDE system is considered and which already takes into account the new grid geometry directly during the computation of the numerical fluxes. Our new Lagrangian-type DG scheme adopts the novel a posteriori sub-cell finite volume limiter method, in which the validity of the candidate solution produced in each cell by an unlimited ADER-DG scheme is verified against a set of physical and numerical detection criteria. Those cells which do not satisfy all of the above criteria are flagged as troubled cells and are recomputed with a second order TVD finite volume scheme. The numerical convergence rates of the new ALE ADER-DG schemes are studied up to fourth order in space and time and several test problems are simulated. Finally, an application inspired by Inertial Confinement Fusion (ICF) type flows is considered by solving the Euler equations and the PDE of viscous and resistive magnetohydrodynamics (VRMHD).Comment: 39 pages, 21 figure

High-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamics

The paper develops high-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamical (RHD) equations, built on the local Lax-Friedrich splitting, the WENO reconstruction, the physical-constraints-preserving flux limiter, and the high-order strong stability preserving time discretization. They are extensions of the positivity-preserving finite difference WENO schemes for the non-relativistic Euler equations. However, developing physical-constraints-preserving methods for the RHD system becomes much more difficult than the non-relativistic case because of the strongly coupling between the RHD equations, no explicit expressions of the primitive variables and the flux vectors, in terms of the conservative vector, and one more physical constraint for the fluid velocity in addition to the positivity of the rest-mass density and the pressure. The key is to prove the convexity and other properties of the admissible state set and discover a concave function with respect to the conservative vector replacing the pressure which is an important ingredient to enforce the positivity-preserving property for the non-relativistic case. Several one- and two-dimensional numerical examples are used to demonstrate accuracy, robustness, and effectiveness of the proposed physical-constraints-preserving schemes in solving RHD problems with large Lorentz factor, or strong discontinuities, or low rest-mass density or pressure etc.Comment: 39 pages, 13 figure

High Order Cell-Centered Lagrangian-Type Finite Volume Schemes with Time-Accurate Local Time Stepping on Unstructured Triangular Meshes

We present a novel cell-centered direct Arbitrary-Lagrangian-Eulerian (ALE) finite volume scheme on unstructured triangular meshes that is high order accurate in space and time and that also allows for time-accurate local time stepping (LTS). The new scheme uses the following basic ingredients: a high order WENO reconstruction in space on unstructured meshes, an element-local high-order accurate space-time Galerkin predictor that performs the time evolution of the reconstructed polynomials within each element, the computation of numerical ALE fluxes at the moving element interfaces through approximate Riemann solvers, and a one-step finite volume scheme for the time update which is directly based on the integral form of the conservation equations in space-time. The inclusion of the LTS algorithm requires a number of crucial extensions, such as a proper scheduling criterion for the time update of each element and for each node; a virtual projection of the elements contained in the reconstruction stencils of the element that has to perform the WENO reconstruction; and the proper computation of the fluxes through the space-time boundary surfaces that will inevitably contain hanging nodes in time due to the LTS algorithm. We have validated our new unstructured Lagrangian LTS approach over a wide sample of test cases solving the Euler equations of compressible gasdynamics in two space dimensions, including shock tube problems, cylindrical explosion problems, as well as specific tests typically adopted in Lagrangian calculations, such as the Kidder and the Saltzman problem. When compared to the traditional global time stepping (GTS) method, the newly proposed LTS algorithm allows to reduce the number of element updates in a given simulation by a factor that may depend on the complexity of the dynamics, but which can be as large as 4.7.Comment: 31 pages, 13 figure