Two-dimensional Incompressible Euler Equations Research Articles

The stabilizing effect of temperature and magnetic field on a 2D magnetic Bénard fluids

In this paper we study the stability of a special magnetic Bénard system near equilibrium, where there exists Laplacian magnetic diffusion and temperature damping but the velocity equation involves no dissipation. Without any velocity dissipation, the fluid velocity is governed by the two-dimensional incompressible Euler equation, whose solution can grow rapidly in time. However, when the fluid is coupled with the magnetic field and temperature through the magnetic Bénard system, we show that the solution is stable. Our results mathematically illustrate that the magnetic field and temperature have the effect of enhancing dissipation and contribute to stabilize the fluid.

Maximum-principle-preserving high-order discontinuous Galerkin methods for incompressible Euler equations on overlapping meshes

In this paper, we construct a new local discontinuous Galerkin (LDG) algorithm to solve the incompressible Euler equation in two space dimensions on overlapping meshes. This method solves the vorticity, velocity field and potential function on different meshes. Different from the traditional LDG method, the overlapping meshes used in this paper make the velocity to be continuous along the interfaces of the primitive meshes. Therefore, the upwind fluxes can be applied. We derive two sufficient conditions to obtain the maximum principle of vorticity. The first one is the divergence-free numerical approximation of the velocity field. This condition further grants that the scheme of the vorticity equation keeps constant solutions. The second one is to preserve the positivity of the numerical vorticity. We select suitable time step sizes to construct positive numerical cell averages of the vorticity provided the vorticity in the previous time step is positive. Then a slope limiter can be applied to enforce the positivity of the numerical approximation of the vorticity. Thanks to the above two conditions, we can arbitrarily add constants to the vorticity function and construct high-order MPP LDG methods on overlapping meshes for the two-dimensional incompressible Euler equation in the vorticity stream function formulation. Numerical tests will be given to demonstrate the good performance of the proposed method.

Rotation-dominant three-scale limit of the Cauchy problem to the inviscid rotating stratified Boussinesq equations

In this paper, we investigate the rotation-dominant three-scale singular limit of the initial-value problem to the inviscid rotating stratified Boussinesq equations. We first prove a dispersive decay estimate for the linear propagator of that limit regime, which requires novel techniques involving an elaborate frequency cut-off since the phase function is degenerate on both horizontal and vertical directions in some areas. Using the dispersive decay estimate, we then investigate rigorously the rotation-dominant limit of local strong solutions to Boussinesq equations with ill-prepared initial data. The limiting system is shown to be the two-dimensional incompressible Euler equations coupled with a two-dimensional transport equation but in three spatial dimensions.

Statistical solutions of the two-dimensional incompressible Euler equations in spaces of unbounded vorticity

We study statistical solutions of the incompressible Euler equations in two dimensions with vorticity in Lp, 1≤p≤∞, and in the class of vortex-sheets with a distinguished sign. Our notion of statistical solution is based on the framework due to Bronzi, Mondaini and Rosa in [4]. Existence in this setting is shown by approximation with discrete measures, concentrated on deterministic solutions of the Euler equations. Additionally, we provide arguments to show that the statistical solutions of the Euler equations may be obtained in the inviscid limit of statistical solutions of the incompressible Navier-Stokes equations. Uniqueness of trajectory statistical solutions is shown in the Yudovich class.

Existence of weak solutions to the two-dimensional incompressible Euler equations in the presence of sources and sinks

A classical model for sources and sinks in a two-dimensional perfect incompressible fluid occupying a bounded domain dates back to Yudovich's paper [44] in 1966. In this model, on the one hand, the normal component of the fluid velocity is prescribed on the boundary and is nonzero on an open subset of the boundary, corresponding either to sources (where the flow is incoming) or to sinks (where the flow is outgoing). On the other hand the vorticity of the fluid which is entering into the domain from the sources is prescribed. In this paper, we investigate the existence of weak solutions to this system by relying on {\it a priori} bounds of the vorticity, which satisfies a transport equation associated with the fluid velocity vector field. Our results cover the case where the vorticity has a $L^p$ integrability in space, with $p $ in $[1,+\infty]$, and prove the existence of solutions obtained by compactness methods from viscous approximations. More precisely we prove the existence of solutions which satisfy the vorticity equation in the distributional sense in the case where $p > \frac43$, in the renormalized sense in the case where $p > 1$, and in a symmetrized sense in the case where $p =1$.

Faraday instability of non-Newtonian fluids under low-frequency vertical harmonic vibration

Resonance Acoustic Mixing® (RAM) technology applies an external low-frequency vertical harmonic vibration to convey and mix the non-Newtonian fluid across space. However, although this method is used for various applications, its mechanism is yet not well understood. In this paper, we investigate the Faraday instability of power-law non-Newtonian fluids in RAM utilizing theory and simulations. According to the Floquet analysis and the dimensionless Mathieu equation, the critical stable region besides the stable region and the unstable region is discovered. Based on the numerical solutions of the two-dimensional incompressible Euler equations for a prototype Faraday instability flow, the temporal evolution of the surface displacement and the mechanism of Faraday waves for two cases are explored physically. For the low forcing displacement, there are only stable and critical stable regions. The surface deformation increases linearly and then enters the steady-state in which the fluctuation frequency is twice the vertical harmonic vibration. For the large forcing displacement, there are only stable and unstable regions. Under the effect of the inertial force, both cases have a sudden variation after the brief stabilization period. Furthermore, a ligament structure is observed, which signals that the surface is destabilized. In addition, a band-like pressure minimum distribution below the interface is formed. The fluid flows from the bottom to the crest portion to balance the pressure difference, which raises the crest.

On the approximation of vorticity fronts by the Burgers–Hilbert equation

This paper proves that the motion of small-slope vorticity fronts in the two-dimensional incompressible Euler equations is approximated on cubically nonlinear timescales by a Burgers–Hilbert equation derived by Biello and Hunter (2010) using formal asymptotic expansions. The proof uses a modified energy method to show that the contour dynamics equations for vorticity fronts in the Euler equations and the Burgers–Hilbert equation are both approximated by the same cubically nonlinear asymptotic equation. The contour dynamics equations for Euler vorticity fronts are also derived.

Analytic adjoint solutions for the 2-D incompressible Euler equations using the Green's function approach

The Green's function approach of Giles and Pierce (J. Fluid Mech., vol. 426, 2001, pp. 327–345) is used to build the lift and drag based analytic adjoint solutions for the two-dimensional incompressible Euler equations around irrotational base flows. The drag-based adjoint solution turns out to have a very simple closed form in terms of the flow variables and is smooth throughout the flow domain, while the lift-based solution is singular at rear stagnation points and sharp trailing edges owing to the Kutta condition. This singularity is propagated to the whole dividing streamline (which includes the incoming stagnation streamline and the wall) upstream of the rear singularity (trailing edge or rear stagnation point) by the sensitivity of the Kutta condition to changes in the stagnation pressure.

Harmonic solutions and weak solutions of two-dimensional rotational incompressible Euler equations

In this paper, two families of exact solutions to two-dimensional incompressible rotational Euler equations are constructed by connecting the Euler equations with the Laplace equation via a stream function. The constituent solutions in the first family are smooth, orthogonal, and conjugate harmonic solutions, while their constituent velocities are nonlinear with respect to the spatial variables. The second family are weak solutions in the distribution sense.

Energy conservation for 2D Euler with vorticity in $L (\operatorname{log} L)^\alpha$

In these notes we discuss the conservation of the energy for weak solutions of the two-dimensional incompressible Euler equations. Weak solutions with vorticity in $L^\infty_t L^p_x$ with $p\geq 3/2$ are always conservative, while for less integrable vorticity the conservation of the energy may depend on the approximation method used to construct the solution. Here we prove that the canonical approximations introduced by DiPerna and Majda provide conservative solutions when the initial vorticity is in the class $L(\log L)^\alpha$ with $\alpha>1/2$.

Traveling vortex pairs for 2D incompressible Euler equations

In this paper, we study desingularization of vortices for the two-dimensional incompressible Euler equations in the full plane. We construct a family of traveling vortex pairs for the Euler equations with a general vorticity function, which constitutes a desingularization of a pair of point vortices with equal intensities but opposite signs. The results are obtained by using an improved vorticity method.

Nonlinear Stability of Planar Vortex Patches in an Ideal Fluid

In this paper, we prove nonlinear stability of planar vortex patches concentrated near a non-degenerate minimum point of the Robin function in a general bounded domain. These vortex patches are stationary solutions of the two-dimensional incompressible Euler equations. The result is obtained by showing that these concentrated vortex patches are in fact isolated maximizers of the kinetic energy among isovortical patches.

Growth of perimeter for vortex patches in a bulk

We consider the two-dimensional incompressible Euler equations. We construct vortex patches with smooth boundary on T2 and R2 whose perimeter grows with time. More precisely, for any constant M>0, we construct a vortex patch in T2 whose smooth boundary has length of order 1 at the initial time such that the perimeter grows up to the given constant M within finite time. The construction is done by cutting a thin slit out of an almost square patch. A similar result holds for an almost round patch with a thin handle in R2.

A Characteristic Mapping method for the two-dimensional incompressible Euler equations

We propose an efficient semi-Lagrangian method for solving the two-dimensional incompressible Euler equations with high precision on a coarse grid. This new approach evolves the flow map using a combination of the Characteristic Mapping (CM) method [1] the gradient-augmented level-set (GALS) method [2]. The flow map possesses a semigroup structure which allows for the decomposition of a long-time deformation into short-time submaps. This leads to a numerical scheme that achieves exponential resolution in linear time. Error estimates are provided and conservation properties are analysed. The computational efficiency and the high precision of the method are illustrated in the vortex merger, four-modes and random flow problems. Comparisons with the Cauchy-Lagrangian method [3] are also presented.

Spectral stability of smooth solitary waves for the Degasperis-Procesi equation

The Degasperis-Procesi equation is an approximating model of shallow-water wave propagating mainly in one direction to the Euler equations. Such a model equation is analogous to the Camassa-Holm approximation of the two-dimensional incompressible and irrotational Euler equations with the same asymptotic accuracy, and is completely integrable with the bi-Hamiltonian structure. In the present study, we establish existence and spectral stability results of localized smooth solitons to the Degasperis-Procesi equation on the real line. The stability proof relies essentially on refined spectral analysis of the linear operator corresponding to the second-order variational derivative of the local Hamiltonian of the Degasperis-Procesi equation.

A nonlocal shallow-water model with the weak Coriolis and equatorial undercurrent effects

We derive a quasilinear nonlocal shallow-water model equation for moderate-amplitude equatorial waves with effects of the weak Coriolis force and equatorial undercurrent. This mathematical modeling is analogous to the Camassa-Holm approximation of the two-dimensional incompressible and rotational Euler equations in the equatorial region. Moreover, we investigate the influences and interactions of the weak Coriolis force caused by the Earth rotation, vorticity and nonlocal higher nonlinearities on the wave-breaking phenomena. In certain cases, by applying the method of characteristics and extremal property of the solution to the Riccati-type differential inequality, we demonstrate that wave breaking phenomena occur only depending on a shape of wave initially in the local spacial point.

Two-front solutions of the SQG equation and its generalizations

The generalized surface quasi-geostrophic (GSQG) equations are transport equations for an active scalar that depend on a parameter $0<\alpha \le 2$. Special cases are the two-dimensional incompressible Euler equations ($\alpha = 2$) and the surface quasi-geostrophic (SQG) equations ($\alpha = 1$). We derive contour-dynamics equations for a class of two-front solutions of the GSQG equations when the fronts are a graph. Scalar reductions of these equations include ones that describe a single front in the presence of a rigid, flat boundary. We use the contour dynamics equations to determine the linearized stability of the GSQG shear flows that correspond to two flat fronts. We also prove local-in-time existence and uniqueness for large, smooth solutions of the two-front equations in the parameter regime $1<\alpha\le 2$, and small, smooth solutions in the parameter regime $0<\alpha\le 1$.

On the Convergence of the Spectral Viscosity Method for the Two-Dimensional Incompressible Euler Equations with Rough Initial Data

We propose a spectral viscosity method to approximate the two-dimensional Euler equations with rough initial data and prove that the method converges to a weak solution for a large class of initial data, including when the initial vorticity is in the so-called Delort class i.e. it is a sum of a signed measure and an integrable function. This provides the first convergence proof for a numerical method approximating the Euler equations with such rough initial data and closes the gap between the available existence theory and rigorous convergence results for numerical methods. We also present numerical experiments, including computations of vortex sheets and confined eddies, to illustrate the proposed method.

Inverse problem on conservation laws

The explicit formulation of the general inverse problem on conservation laws is presented for the first time. Within this problem, one aims to derive the general form of systems of differential equations that admit a prescribed set of conservation laws. The particular cases of the inverse problem on first integrals of ordinary differential equations and on conservation laws for evolution equations are studied. We also solve the inverse problem on conservation laws for differential equations admitting an infinite dimensional space of zeroth-order conservation-law characteristics. This particular case is further studied in the context of conservative first-order parameterization schemes for the two-dimensional incompressible Euler equations. We exhaustively classify conservative first-order parameterization schemes for the eddy-vorticity flux that lead to a class of closed, averaged Euler equations possessing conservation of generalized circulation, generalized momentum and energy.

Steady point vortex pair in a field of Stuart-type vorticity

A new family of exact solutions to the two-dimensional steady incompressible Euler equation is presented. The solutions provide a class of hybrid equilibria comprising two point vortices of unit circulation – a point vortex pair – embedded in a smooth sea of non-zero vorticity of ‘Stuart-type’ so that the vorticity $\unicode[STIX]{x1D714}$ and the stream function $\unicode[STIX]{x1D713}$ are related by $\unicode[STIX]{x1D714}=a\text{e}^{b\unicode[STIX]{x1D713}}-\unicode[STIX]{x1D6FF}(\boldsymbol{x}-\boldsymbol{x}_{0})-\unicode[STIX]{x1D6FF}(\boldsymbol{x}+\boldsymbol{x}_{0})$, where $a$ and $b$ are constants. We also examine limits of these new Stuart-embedded point vortex equilibria where the Stuart-type vorticity becomes localized into additional point vortices. One such limit results in a two-real-parameter family of smoothly deformable point vortex equilibria in an otherwise irrotational flow. The new class of hybrid equilibria can be viewed as continuously interpolating between the limiting pure point vortex equilibria. At the same time the new solutions continuously extrapolate a similar class of hybrid equilibria identified by Crowdy (Phys. Fluids, vol. 15, 2003, pp. 3710–3717).