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Register a new userStability criteria for the 2D $alpha$-Euler equations
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We derive analogues of the classical Rayleigh, Fjortoft and Arnold stability and instability theorems in the context of the 2D $alpha$-Euler equations.
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In this paper, we study desingularization of vortices for the two-dimensional incompressible Euler equations in the full plane. We construct a family of steady vortex pairs for the Euler equations with a general vorticity function, which constitutes a desingularization of a pair of point vortices with equal magnitude and opposite signs. The results are obtained by using an improved vorticity method.
We study stability of unidirectional flows for the linearized 2D $alpha$-Euler equations on the torus. The unidirectional flows are steady states whose vorticity is given by Fourier modes corresponding to a vector $mathbf p in mathbb Z^{2}$. We linearize the $alpha$-Euler equation and write the linearized operator $L_{B} $ in $ell^{2}(mathbb Z^{2})$ as a direct sum of one-dimensional difference operators $L_{B,mathbf q}$ in $ell^{2}(mathbb Z)$ parametrized by some vectors $mathbf qinmathbb Z^2$ such that the set ${mathbf q +n mathbf p:n in mathbb Z}$ covers the entire grid $mathbb Z^{2}$. The set ${mathbf q +n mathbf p:n in mathbb Z}$ can have zero, one, or two points inside the disk of radius $|mathbf p|$. We consider the case where the set ${mathbf q +n mathbf p:n in mathbb Z}$ has exactly one point in the open disc of radius $mathbf p$. We show that unidirectional flows that satisfy this condition are linearly unstable. Our main result is an instability theorem that provides a necessary and sufficient condition for the existence of a positive eigenvalue to the operator $L_{B,mathbf q}$ in terms of equations involving certain continued fractions. Moreover, we are also able to provide a complete characterization of the corresponding eigenvector. The proof is based on the use of continued fractions techniques expanding upon the ideas of Friedlander and Howard.
In this paper, we numerically study a class of solutions with spiraling singularities in vorticity for two-dimensional, inviscid, compressible Euler systems, where the initial data have an algebraic singularity in vorticity at the origin. These are different from the multi-dimensional Riemann problems widely studied in the literature. Our computations provide numerical evidence of the existence of initial value problems with multiple solutions, thus revealing a fundamental obstruction toward the well-posedness of the governing equations. The compressible Euler equations are solved using the positivity-preserving discontinuous Galerkin method.
In this paper, we investigate the well-posedness theory of compressible jet flows for two dimensional steady Euler system with non-zero vorticity. One of the key observations is that the stream function formulation for two dimensional compressible steady Euler system with non-zero vorticity enjoys a variational structure, so that the jet problem can be reformulated as a domain variation problem. This allows us to adapt the framework developed by Alt, Caffarelli and Friedman for the one-phase free boundary problems to obtain the existence and uniqueness of smooth solutions to the subsonic jet problem with non-zero vorticity. We also show that there is a critical mass flux, such that as long as the incoming mass flux does not exceed the critical value, the well-posedness theory holds true.
In the spirit of making high-order discontinuous Galerkin (DG) methods more competitive, researchers have developed the hybridized DG methods, a class of discontinuous Galerkin methods that generalizes the Hybridizable DG (HDG), the Embedded DG (EDG) and the Interior Embedded DG (IEDG) methods. These methods are amenable to hybridization (static condensation) and thus to more computationally efficient implementations. Like other high-order DG methods, however, they may suffer from numerical stability issues in under-resolved fluid flow simulations. In this spirit, we introduce the hybridized DG methods for the compressible Euler and Navier-Stokes equations in entropy variables. Under a suitable choice of the stabilization matrix, the scheme can be shown to be entropy stable and satisfy the Second Law of Thermodynamics in an integral sense. The performance and robustness of the proposed family of schemes are illustrated through a series of steady and unsteady flow problems in subsonic, transonic, and supersonic regimes. The hybridized DG methods in entropy variables show the optimal accuracy order given by the polynomial approximation space, and are significantly superior to their counterparts in conservation variables in terms of stability and robustness, particularly for under-resolved and shock flows.
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