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 linear, three-dimensional, rotating incompressible Euler equations. These equations admit complicated wave solutions. The numerical 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: (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; (iii) by applying a symplectic time discretisation; and, (iv) by combining PETSc’s linear algebra routines with our high-level software. We compared our simulations with exact solutions of three-dimensional compressible and incompressible flows, in (non)rotating periodic and partly periodic cuboids (Poincar´e waves). Additional verifications concerned semi-analytical eigenmode solutions in rotating cuboids with solid walls

Selective decay by Casimir dissipation in fluids

The problem of parameterizing the interactions of larger scales and smaller scales in fluid flows is addressed by considering a property of two-dimensional incompressible turbulence. The property we consider is selective decay, in which a Casimir of the ideal formulation (enstrophy in 2D flows, helicity in 3D flows) decays in time, while the energy stays essentially constant. This paper introduces a mechanism that produces selective decay by enforcing Casimir dissipation in fluid dynamics. This mechanism turns out to be related in certain cases to the numerical method of anticipated vorticity discussed in \cite{SaBa1981,SaBa1985}. Several examples are given and a general theory of selective decay is developed that uses the Lie-Poisson structure of the ideal theory. A scale-selection operator allows the resulting modifications of the fluid motion equations to be interpreted in several examples as parameterizing the nonlinear, dynamical interactions between disparate scales. The type of modified fluid equation systems derived here may be useful in modelling turbulent geophysical flows where it is computationally prohibitive to rely on the slower, indirect effects of a realistic viscosity, such as in large-scale, coherent, oceanic flows interacting with much smaller eddies

Incompressible Limit of Solutions of Multidimensional Steady Compressible Euler Equations

A compactness framework is formulated for the incompressible limit of approximate solutions with weak uniform bounds with respect to the adiabatic exponent for the steady Euler equations for compressible fluids in any dimension. One of our main observations is that the compactness can be achieved by using only natural weak estimates for the mass conservation and the vorticity. Another observation is that the incompressibility of the limit for the homentropic Euler flow is directly from the continuity equation, while the incompresibility of the limit for the full Euler flow is from a combination of all the Euler equations. As direct applications of the compactness framework, we establish two incompressible limit theorems for multidimensional steady Euler flows through infinitely long nozzles, which lead to two new existence theorems for the corresponding problems for multidimensional steady incompressible Euler equations.Comment: 17 pages; 2 figures. arXiv admin note: text overlap with arXiv:1311.398

On the form of the viscous term for two dimensional Navier-Stokes flows

This is a pre-copyedited, author-produced PDF of an article accepted for publication in Quarterly Journal of Mechanics and Applied Mathematics following peer review. The version of record, Andrew D. Gilbert, Xavier Riedinger, and John Thuburn, On the form of the viscous term for two dimensional Navier–Stokes flows, Q J Mechanics Appl Math (2014) 67 (2): 205-228 first published online February 27, 2014 is available online at: http://qjmam.oxfordjournals.org/content/67/2/205The form of the viscous term is discussed for incompressible flow on a two-dimensional curved surface S and for the shallow water equations. In the case of flow on a surface three versions are considered. These correspond to taking curl twice, to applying the Laplacian defined in terms of a metric, and to taking the divergence of a symmetric stress tensor. These differ on a curved surface, for example a sphere. The three terms are related and their properties discussed, in particular energy and angular momentum conservation. In the case of the shallow water equations again three forms of dissipation are considered, the last of which involves the divergence of a stress tensor. Their properties are discussed, including energy conservation and whether the rotating bucket solution of the three-dimensional Navier–Stokes equation is reproduced. A derivation of the viscous term is also given based on shallow water equations as a truncation of the Navier–Stokes equation, with forces on a column determined by integration over the vertical. For both incompressible flow on a surface and for the shallow water equations, it is argued that a viscous term based on a symmetric stress tensor should be used as this leads to correct treatment of angular momentum