Two fluid space-time discontinuous Galerkin finite element method. Part II: Applications

The numerical method for two fluid flow computations presented in Sollie, Bokhove \& van der Vegt, Two Fluid Space-Time Discontinuous Galerkin Finite Element Method. Part I: Numerical Algorithm is applied to a number of one and two dimensional single and two fluid test problems, including a magma - ideal gas shocktube and a helium cylinder - shock wave interaction problem

Extension of a discontinuous Galerkin finite element method to viscous rotor flow simulations

Heavy vibratory loading of rotorcraft is relevant for many operational aspects of helicopters, such as the structural life span of (rotating) components, operational availability, the pilot's comfort, and the effectiveness of weapon targeting systems. A precise understanding of the source of these vibrational loads has important consequences in these application areas. Moreover, in order to exploit the full potential offered by new vibration reduction technologies, current analysis tools need to be improved with respect to the level of physical modeling of flow phenomena which contribute to the vibratory loads. In this paper, a computational fluid dynamics tool for rotorcraft simulations based on first-principles flow physics is extended to enable the simulation of viscous flows. Viscous effects play a significant role in the aerodynamics of helicopter rotors in high-speed flight. The new model is applied to three-dimensional vortex flow and laminar dynamic stall. The applications clearly demonstrate the capability of the new model to perform on deforming and adaptive meshes. This capability is essential for rotor simulations to accomodate the blade motions and to enhance vortex resolution

A space-time discontinuous Galerkin method for the incompressible Navier-Stokes equations

We introduce a space–time discontinuous Galerkin (DG) finite element method for the incompressible Navier–Stokes equations. Our formulation can be made arbitrarily high order accurate in both space and time and can be directly applied to deforming domains. Different stabilizing approaches are discussed which ensure stability of the method. A numerical study is performed to compare the effect of the stabilizing approaches, to show the method’s robustness on deforming domains and to investigate the behavior of the convergence rates of the solution. Recently we introduced a space–time hybridizable DG (HDG) method for incompressible flows [S. Rhebergen, B. Cockburn, A space–time hybridizable discontinuous Galerkin method for incompressible flows on deforming domains, J. Comput. Phys. 231 (2012) 4185–4204]. We will compare numerical results of the space–time DG and space–time HDG methods. This constitutes the first comparison between DG and HDG methods

Hamiltonian discontinuous Galerkin FEM for linear, rotating incompressible Euler equations: inertial waves

A discontinuous Galerkin finite element method (DGFEM) has been developed and tested for the linear, three-dimensional, rotating incompressible Euler equations. These equations admit complicated wave solutions, which poses numerical challenges. These challenges concern: (i) discretisation of a divergence-free velocity field; (ii) discretisation of geostrophic boundary conditions combined with no-normal flow at solid walls; (iii) discretisation of the conserved, Hamiltonian dynamics of the inertial-waves; and, (iv) large-scale computational demands owing to the three-dimensional nature of inertial-wave dynamics and possibly its narrow zones of chaotic attraction. These issues have been resolved, for example: (i) by employing Dirac’s method of constrained Hamiltonian dynamics to our DGFEM for linear, compressible flows, thus enforcing the incompressibility constraints; (ii) by enforcing no-normal flow at solid walls in a weak form and geostrophic tangential flow along the wall; and, (iii) by applying a symplectic time discretisation. We compared our simulations with exact solutions of three-dimensional incompressible flows, in (non) rotating periodic and partly periodic cuboids (Poincaré waves). Additional verifications concerned semi-analytical eigenmode solutions in rotating cuboids with solid walls. Finally, a simulation in a tilted rotating tank, yielding more complicated wave dynamics, demonstrates the potential of our new method

Variational space-time (dis)continuous Galerkin method for nonlinear free surface waves

A new variational finite element method is developed for nonlinear free surface gravity water waves. This method also handles waves generated by a wave maker. Its formulation stems from Miles' variational principle for water waves together with a space-time finite element discretization that is continuous in space and discontinuous in time. The key features of this formulation are: (i) a discrete variational approach that gives rise to conservation of discrete energy and phase space and preservation of variational structure; and (ii) a space-time approach that guarantees satisfaction of the geometric conservation law which is crucial in handling the deforming flow domain due to the wave maker and free surface motion. The numerical discretization is a combination of a second order finite element discretization in space and a second order symplectic Stormer-Verlet discretization in time. The resulting numerical scheme is verified against nonlinear analytical solutions and discrete energy conservation is demonstrated for long time simulations. We also validated the scheme with experimental data of waves generated in a wave basin of the Maritime Research Institute Netherlands

A Runge-Kutta discontinuous Galerkin method for linear free-surface gravity waves using high order velocity recovery

We present a higher order accurate discontinuous Galerkin finite element method for the simulation of linear free-surface gravity waves. The method uses the classical Runge–Kutta method for the time-discretization of the free-surface equations and the discontinuous Galerkin method for the space-discretization. In order to circumvent numerical instabilities arising from an asymmetric mesh a stabilization term is added to the free-surface equations. In combination with a higher order velocity recovery technique this stabilizes the numerical discretization with minimal effect on the accuracy of the wave computations. A stability analysis of the semi and fully-discrete scheme is presented, which suggests that for a suitable choice of the stabilization constant a relatively large time step can be chosen for accurate simulations over a long period of time. Numerical examples of a number of problems are also presented

A Hamiltonian vorticity–dilatation formulation of the compressible Euler equations

Using the Hodge decomposition on bounded domains the compressible Euler equations of gas dynamics are reformulated using a density weighted vorticity and dilatation as primary variables, together with the entropy and density. This formulation is an extension to compressible flows of the well-known vorticity–stream function formulation of the incompressible Euler equations. The Hamiltonian and associated Poisson bracket for this new formulation of the compressible Euler equations are derived and extensive use is made of differential forms to highlight the mathematical structure of the equations. In order to deal with domains with boundaries also the Stokes–Dirac structure and the port-Hamiltonian formulation of the Euler equations in density weighted vorticity and dilatation variables are obtained

A discontinuous Galerkin finite element discretization of the Euler equations for compressible and incompressible fluids

Using the generalized variable formulation of the Euler equations of fluid dynamics, we develop a numerical method that is capable of simulating the flow of fluids with widely differing thermodynamic behavior: ideal and real gases can be treated with the same method as an incompressible fluid. The well-defined incompressible limit relies on using pressure primitive or entropy variables. In particular entropy variables can provide numerical methods with attractive properties, e.g. fulfillment of the second law of thermodynamics. We show how a discontinuous Galerkin finite element discretization previously used for compressible flow with an ideal gas equation of state can be extended for general fluids. We also examine which components of the numerical method have to be changed or adapted. Especially, we investigate different possibilities of solving the nonlinear algebraic system with a pseudo-time iteration. Numerical results highlight the applicability of the method for various fluids