Rotating Euler Equations Research Articles

On the stabilizing effect of rotation in the 3d Euler equations

AbstractWhile it is well known that constant rotation induces linear dispersive effects in various fluid models, we study here its effect on long time nonlinear dynamics in the inviscid setting. More precisely, we investigate stability in the 3d rotating Euler equations in with a fixed speed of rotation. We show that for any , axisymmetric initial data of sufficiently small size ε lead to solutions that exist for a long time at least and disperse. This is a manifestation of the stabilizing effect of rotation, regardless of its speed. To achieve this we develop an anisotropic framework that naturally builds on the available symmetries. This allows for a precise quantification and control of the geometry of nonlinear interactions, while at the same time giving enough information to obtain dispersive decay via adapted linear dispersive estimates.

Singularity for the one-dimensional rotating Euler equations of Chaplygin gases

This paper is focused on the singularity formation of smooth solutions for the one-dimensional rotating Euler equations of Chaplygin gases, which is a nonhomogeneous quasilinear hyperbolic system with linearly degenerate characteristic fields. We overcome the influence of the rotation terms and show that the density itself of the smooth solution tends to infinity in finite time for a kind of initial data.

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.

Stochastic 2D rotating Euler flows with bounded vorticity or white noise initial conditions

In this paper, we consider the stochastic two-dimensional (2D) rotating Euler equations with [Formula: see text] initial conditions and white noise initial conditions, respectively. The existence and uniqueness of the equations with [Formula: see text] vorticity are proved. The stability of [Formula: see text] tending to 0 for [Formula: see text] initial conditions will be proved. At last, the case of white noise initial conditions is considered.

Long time existence of classical solutions for the rotating Euler equations and related models in the optimal Sobolev space

Koh Y et al (2014 J. Differ. Equ. 256 704–44) proved the long time existence of classical solutions to the 3D rotating Euler equations for initial data in the Sobolev space (s > 7/2). Here we improve their assumed regularity and weaken the lower bound of the rotating speed by establishing a new low-frequency Strichartz estimate and taking two different measures to dominate the Lip norm of the velocity. As an application, we derive similar results for some related models.

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.

Global solvability of the rotating Navier-Stokes equations with fractional Laplacian in a periodic domain

We consider existence of global solutions to equations for three-dimensional rotating fluids in a periodic frame provided by a sufficiently large Coriolis force. The Coriolis force appears in almost all of the models of meteorology and geophysics dealing with large-scale phenomena. In the spatially decaying case, Koh et al. (J Differ Equ 256:707–744, 2014) showed existence for the large times of solutions of the rotating Euler equations provided by the large Coriolis force. In this case the resonant equation does not appear anymore. In the periodic case, however, the resonant equation appears, and thus the main subject in this case is to show existence of global solutions to the resonant equation. Research in this direction was initiated by Babin et al. (Indiana Univ Math J 48:1133–1176, 1999) who treated the rotating Navier-Stokes equations on general periodic domains. On the other hand, Golse et al. (Bursting Dynamics of the 3D Euler Equations in Cylindrical Domains. Instability in Models Connected with Fluid Flows. I. Springer, New York, pp 301–338. arXiv:0704.0337v1 , 2008) considered bursting dynamics of the resonant equation in the case of a cylinder with no viscosity. Thus we may not expect to show global existence of solutions to the resonant equation without viscosity in the periodic case. In this paper we show existence of global solutions for fractional Laplacian case (with its power strictly less than the usual Laplacian) in the periodic domain with the same period in each direction. The main ingredient is an improved estimate on resonant three-wave interactions, which is based on a combinatorial argument.

Long time existence of classical solutions for the 3D incompressible rotating Euler equations

We consider the initial value problem of the 3D incompressible rotating Euler equations. We prove the long time existence of classical solutions for initial data in $H^s(\mathbb{R}^3)$ with $s$ > 5/2. Also, we give an upper bound of the minimum speed of rotation for the long time existence when initial data belong to $H^{7/2}(\mathbb{R}^3)$.

Nonlinear Exact Solutions of the 2-Dimensional Rotational Euler Equations for the Incompressible Fluid* *Supported by the National Natural Science Foundation of China under Grant No. 11301269, Jiangsu Provincial Natural Science Foundation of China under Grant No. BK20130665, the Fundamental Research Funds KJ2013036 for the Central Universities, Student Research Training under Grant No. 1423A02 of Nanjing Agricultural University, and the Research

In this paper, the Clarkson-Kruskal direct approach is employed to investigate the exact solutions of the 2-dimensional rotational Euler equations for the incompressible fluid. The application of the method leads to a system of completely solvable ordinary differential equations. Several special cases are discussed and novel nonlinear exact solutions with respect to variables x and y are obtained. It is of interest to notice that the pressure p is obtained by the second kind of curvilinear integral and the coefficients of the nonlinear solutions are solitary wave type functions like tanh(kt/2) and sech (kt/2) due to the rotational parameter k ≠ 0. Such phenomenon never appear in the classical Euler equations wherein the Coriolis force arising from the gravity and Earth's rotation is ignored. Finally, illustrative numerical figures are attached to show the behaviors that the exact solutions may exhibit.

The gradient of potential vorticity, quaternions and an orthonormal frame for fluid particles

The gradient of potential vorticity (PV) is an important quantity because of the way PV (denoted as q) tends to accumulate locally in the oceans and atmospheres. Recent analysis has shown that the vector quantity B = ∇q × ∇θ for the three-dimensional incompressible rotating Euler equations evolves according to the same stretching equation as for the vorticity and the magnetic field in magnetohydrodynamics. For example, it allows various analogies, such as stretching dynamics, helicity, superhelicity and cross helicity. In addition, using quaternionic analysis, the dynamics of the B-vector naturally allow the construction of an orthonormal frame attached to fluid particles; this is designated as a quaternion frame. The alignment dynamics of this frame are particularly relevant to the three-axis rotations that particles undergo as they traverse regions of a flow when the PV gradient ∇ q is large.

On Pressureless Gases Driven by a Strong Inhomogeneous Magnetic Field

We are interested in the life span and the asymptotic behavior of the solutions to a system governing the motion of a pressureless gas that is submitted to a strong, inhomogeneous magnetic field ${\varepsilon}^{-1} B(x)$ of variable amplitude but fixed direction; this is a first step in the direction of the study of rotating Euler equations. This leads to the study of a multidimensional Burgers-type system on the velocity field $u_{\varepsilon}$, penalized by a rotating term ${\varepsilon}^{-1} u_{\varepsilon}\wedge B(x)$. We prove that the unique, smooth solution of this Burgers system exists on a uniform time interval $[0,T]$. We also prove that the phase of oscillation of $u_{\varepsilon}$ is an order one perturbation of the phase obtained in the case of a pure rotation (with no nonlinear transport term), ${\varepsilon}^{-1}B(x)t$. Finally, going back to the pressureless gas system, we obtain the asymptotics of the density as ${\varepsilon}$ goes to zero.

Rotation prevents finite-time breakdown

We consider a two-dimensional (2D) convection model augmented with the rotational Coriolis forcing, U t + U·∇ x U=2 kU ⊥, with a fixed 2 k being the inverse Rossby number. We ask whether the action of dispersive rotational forcing alone, U ⊥, prevents the generic finite-time breakdown of the free nonlinear convection. The answer provided in this work is a conditional yes. Namely, we show that the rotating Euler equations admit global smooth solutions for a subset of generic initial configurations. With other configurations, however, finite-time breakdown of solutions may and actually does occur. Thus, global regularity depends on whether the initial configuration crosses an intrinsic, O(1) critical threshold (CT), which is quantified in terms of the initial vorticity, ω 0=∇× U 0, and the initial spectral gap associated with the 2×2 initial velocity gradient, η 0≔ λ 2(0)− λ 1(0), λ j (0)= λ j (∇ U 0). Specifically, global regularity of the rotational Euler equation is ensured if and only if 4kω 0(α)+η 2 0(α)<4k 2,∀α∈ R 2 . We also prove that the velocity field remains smooth if and only if it is periodic. An equivalent Lagrangian formulation reconfirms the CT and shows a global periodicity of velocity field as well as the associated particle orbits. Moreover, we observe yet another remarkable periodic behavior exhibited by the gradient of the velocity field. The spectral dynamics of the Eulerian formulation [SIAM J. Math. Anal. 33 (2001) 930] reveals that the vorticity and the divergence of the flow evolve with their own path-dependent period. We conclude with a kinetic formulation of the rotating Euler equation.

Asymptotic of the Solutions of Hyperbolic Equations with a Skew-Symmetric Perturbation

Using methods introduced by S. Schochet inJ. Differential Equations114(1994), 476–512, we compute the first term of an asymptotic expansion of the solutions of hyperbolic equations perturbated by a skew-symmetric linear operator. That result is first applied to two systems describing the motion of geophysic fluids: the rotating Euler equations and the primitive system of the quasigeostrophic equations. Finally in the last part, we study the slightly compressible Euler equations by application of that same result.

Investigation of Crossflow Shocks on Delta Wings in Supersonic Flow

In the study described, a finite difference approach to solving the rotational Euler equations, explicitly fitting shocks as a boundary, is applied to a variety of geometrical shapes in the lower supersonic Mach number regime. It is shown how special techniques based on the physics of the flow can be used to circumvent a variety of numerical difficulties encountered with the conical flow problem that are primarily associated with the initial value characteristics of the hyperbolic scheme, causing embedded shock-induced entropy and crossflow layers to develop on the body surface. UMEROUS investigations over the past several decades have dealt with the problem of supersonic flow over thin wings. Many of the earlier methods used linearized potential flow theory and slender body assumptions to obtain solutions. The failure of linearized potential flow solutions for flows where significant supersonic crossflow regions exist is fairly well known (e.g., Refs. 1-3). More recently, nonlinear methods, such as the method of characteristi cs4 and the method of lines, 5 were developed which were adequate at small angles of attack and simple, but not very thin, cross- sectional shapes. Nevertheless, these nonlinear methods did not broach the topic of crossflow embedded shocks which occur on thin bodies, even at relatively small angles of attack. Under certain geometric and flow conditions, a large ex- pansion occurs around the leading edge, causing an embedded region of supersonic crossflow to occur on the leeward side of the body. Since the supersonic region has no advance warning of the symmetry plane crossflow stagnation condition, an embedded shock forms, causing the crossflow to become subsonic and capable of satisfying this boundary condition. A shock-capturing finite-difference approach was developed and applied to complex shapes with embedded shocks in the high supersonic to hypersonic Mach number regime.6'7 This shock-capturing approach uses the con- servation form of the Euler governing equations in order to resolve the embedded shocks with a minimal amount of numerical instability. In contrast to the aforementioned nonlinear methods in- tegrating the rotational form of Euler equations, a technique was developed to integrate the irrotational equations for supersonic conical flow. A relaxation method was applied to the conical full potential flow equation in Ref. 3 that relied heavily on numerical transonic techniques. Numerical solutions were computed successfully for circular/elliptic cones and thin winglike cross sections from low to high in- cidence at freestream Mach numbers between 1.2 and 3. This method captures the shocks over a few mesh points, ap- parently without numerical instabilities. Using the nonlinear

Some numerical solutions of the rotational Euler equations.

Numerical solution by the method of successive over-relaxation is presented for a class of linear partial differential equations of the elliptic type. The equations solved govern the twodimensional and axisymmetric flow of an incompressible, inviscid, rotational fluid. The boundary conditions chosen, which may be of the mixed type, are shown to give rise to solutions having application to flows in some pure fluidic devices. Numerical experiments are used to determine optimum relaxation parameters, and many numerical solutions are presented as examples. Nomenclature = mesh size p = pressure q = velocity vector r = radial coordinate (axisymmetric flow) Uj v, w = velocity components x = axial coordinate (two-dimensional flow) y = normal coordinate (two-dimensional flow) z = normal coordinate (axisymmetric flow) v = kinematic viscosity p = density = angular coordinate (axisymmetric flow) ^ = stream function O = vorticity component co = over-relaxation parameter e = iteration tolerance able, an IBM 1130, dictated a choice of boundary conditions such that the governing equations are linear in form. This problem was chosen for its practical applications, particularly in flow situations where the diffusive processes may be neglected but the rotational property of the flowfield is important. To be sure, the rotational flow is generated by the action of viscosity; however, in problems concerned with the interaction of two flows, or of flows and solid boundaries, dissipative phenomena may be negligible in some neighborhood of the interaction. As an example, a freejet impinging on a flat surface is treated analytically along with other inviscid rotational flows in Ref. 1. The boundary condition chosen in this investigation will be shown to give rise to a flowfield whose solution may be applicable to problems in the pure fluidic devices field.