The recently proposed low degree-of-freedom model of Moffat and Kimura [1,2] for describing the approach to finite-time singularity of the incompressible Euler fluid equations is investigated. The model assumes an initial finite-energy configuration of two vortex rings placed symmetrically on two tilted planes. The Hamiltonian structure of the inviscid limit of the model is obtained. The associated noncanonical Poisson bracket [3] and two invariants, one that serves as the Hamiltonian and the other a Casimir invariant, are discovered. It is shown that the system is integrable with a solution that lies on the intersection for the two invariants, just as for the free rigid body of mechanics whose solution lies on the intersection of the kinetic energy and angular momentum surfaces. Also, a direct quadrature is given and used to demonstrate the Leray form for finite-time singularity in the model. To the extent the Moffat and Kimura model accurately represents Euler's ideal fluid equations of motion, we have shown the existence of finite-time singularity.
We present a formal, approximate model for singularity formation in classical fluid dynamics in three dimensions. The construction utilizes an approximation of local two-dimensionality to study an anti-parallel hairpin vortex structure with a cross-section equivalent to the 2D Chaplygin-Lamb dipole vortex. The model exhibits a finite time Euler singularity at an isolated point, with only finite local stretching of vortex lines. The model also suggests an associated Navier-Stokes problem, which exhibits a finite-time point singularity, provided that a Reynolds number is sufficiently large. The …
Some classical and recent results on the Euler equations governing perfect (incompressible and inviscid) fluid motion are collected and reviewed, with some small novelties scattered throughout. The perspective and emphasis will be given through the lens of infinite-dimensional dynamical systems, and various open problems are listed and discussed.
We consider the 3D incompressible Euler equations in vorticity form in the following fundamental domain for the octahedral symmetry group: $\{ (x_1,x_2,x_3): 0<x_3<x_2<x_1 \}.$ In this domain, we prove local well-posedness for $C^α$ vorticities not necessarily vanishing on the boundary with any $0<α<1$, and establish finite-time singularity formation within the same class for smooth and compactly supported initial data. The solutions can be extended to all of $\mathbb{R}^3$ via a sequence of reflections, and therefore we obtain finite-time singularity formation for the 3D Euler equations in $\…
The search of finite-time singularity solutions of Euler equations is considered for the case of an incompressible and inviscid fluid. Under the assumption that a finite-time blow-up solution may be spatially anisotropic as time goes by such that the flow contracts more rapidly into one direction than into the other, it can be shown that the dynamics of an axially symmetric flow with swirl may be approximated to a simpler hyperbolic system. By using the method of characteristics, it can be shown that generically the velocity flow exhibits multi-valued solutions appearing on a rim at a finite d…
We prove that bounded weak solutions of the compressible Euler equations will conserve thermodynamic entropy unless the solution fields have sufficiently low space-time Besov regularity. A quantity measuring kinetic energy cascade will also vanish for such Euler solutions, unless the same singularity conditions are satisfied. It is shown furthermore that strong limits of solutions of compressible Navier-Stokes equations that are bounded and exhibit anomalous dissipation are weak Euler solutions. These inviscid limit solutions have non-negative anomalous entropy production and kinetic energy di…
Euler-Lagrange variational principle is used to obtain analytical and numerical flow relations in cylindrical tubes. The method is based on minimizing the total stress in the flow duct using the fluid constitutive relation between stress and rate of strain. Newtonian and non-Newtonian fluid models; which include power law, Bingham, Herschel-Bulkley, Carreau and Cross; are used for demonstration.
In this paper, we prove the existence of smooth initial data for the 2D free boundary incompressible Euler equations (also known for some particular scenarios as the water wave problem), for which the smoothness of the interface breaks down in finite time into a splash singularity or a splat singularity.
It has been known since work of Lichtenstein [42] and Gunther [29] in the 1920's that the $3D$ incompressible Euler equation is locally well-posed in the class of velocity fields with Hölder continuous gradient and suitable decay at infinity. It is shown here that these local solutions can develop singularities in finite time, even for some of the simplest three-dimensional flows.
We prove local well-posedness in regular spaces and a Beale-Kato-Majda blow-up criterion for a recently derived stochastic model of the 3D Euler fluid equation for incompressible flow. This model describes incompressible fluid motions whose Lagrangian particle paths follow a stochastic process with cylindrical noise and also satisfy Newton's 2nd Law in every Lagrangian domain.
For all $ε>0$, we prove the existence of finite-energy strong solutions to the axi-symmetric $3D$ Euler equations on the domains $ \{(x,y,z)\in\mathbb{R}^3: (1+ε|z|)^2\leq x^2+y^2\}$ which become singular in finite time. We further show that solutions with 0 swirl are necessarily globally regular. The proof of singularity formation relies on the use of approximate solutions at exactly the critical regularity level which satisfy a $1D$ system which has solutions which blow-up in finite time. The construction bears similarity to our previous result on the Boussinesq system \cite{EJB} though a nu…
In this paper, we present strong numerical evidences that the incompressible axisymmetric Euler equations with degenerate viscosity coefficients and smooth initial data of finite energy develop a potential finite-time locally self-similar singularity at the origin. An important feature of this potential singularity is that the solution develops a two-scale traveling wave that travels towards the origin. The two-scale feature is characterized by the scaling property that the center of the traveling wave is located at a ring of radius $O((T-t)^{1/2})$ surrounding the symmetry axis while the thic…
The equations for a self-similar solution of an inviscid incompressible fluid are mapped into an integral equation which hopefully can be solved by iteration. It is argued that the exponent of the similarity are ruled by Kelvin's theorem of conservation of circulation. The end result is an iteration with a nonlinear term entering in a kernel given by a 3D integral (in general 3D flow) or 2D (for swirling flows), which seems to be within reach of present day computational power. Because of the slow decay of the similarity solution at large distances the total energy is diverging and recent math…
On solutions of the Euler equation for incoherent fluid on a rotating sphere
INFN, Sezione di Lecce（意大利国家核物理研究所莱切分部）；Dipartimento di Matematica “G. Peano”, Università di Torino（都灵大学皮亚诺数学系）；INFN, Sezione di Torino（意大利国家核物理研究所都灵分部）
In a particle physics dynamics, we assume a uniform distribution as the physical measure and a measure-theoretic definition of entropy on the velocity configuration space. This distribution is labeled as the physical solution in the remainder of the article. The dynamics is governed by an assumption of a Lagrangian formulation, with the velocity time derivatives as the momenta conjugate to the velocity configurations. From these definitions and assumptions, we show mathematically that a maximum entropy production principle selects the physical measure from among alternate solutions of the Navi…
Air Drag Controls the Finite-Time Singularity of Euler's Disk
School of Engineering and Applied Sciences, Harvard University（哈佛大学工程与应用科学学院）；The Racah Institute of Physics, The Hebrew University of Jerusalem（耶路撒冷希伯来大学拉卡物理研究所）；Department of Physics of Complex Systems, Weizmann Institute of Science（魏茨曼科学委员会复杂物理系）
In this work we found the new class of exact stationary solutions for 2D-Euler equations. Unlike of already known solutions, the new one contain complex singularities. We consider as complex, point singularities which have the vector field index greater than one. For example, the dipole singularity is complex because its index is equal to two. We present in explicit form a large class of exact localized stationary solutions for 2D-Euler equations with the singularity which index is equal to three. The obtained solutions are expressed in terms of elementary functions. These solutions represent …
This paper addresses the construction and the stability of self-similar solutions to the isentropic compressible Euler equations. These solutions model a gas that implodes isotropically, ending in a singularity formation in finite time. The existence of smooth solutions that vanish at infinity and do not have vacuum regions was recently proved and, in this paper, we provide the first construction of such smooth profiles, the first characterization of their spectrum of radial perturbations as well as some endpoints of unstable directions. Numerical simulations of the Euler equations provide evi…
The document describes a numerical algorithm to simulate plasmas and fluids in the 3 dimensional space by the Euler method, in which the spatial meshes are fixed to the space. The plasmas and fluids move through the spacial Euler mesh boundary. The Euler method can represent a large deformation of the plasmas and fluids. On the other hand, when the plasmas or fluids are compressed to a high density, the spatial resolution should be ensured to describe the density change precisely. The present 3D Euler code is developed to simulate a nuclear fusion fuel ignition and burning. Therefore, the 3D E…
We show that the ideal (nondissipative) form of the dynamical equations for the Lipps-Hemler formulation of the anelastic fluid model follow as Euler-Poincaré equations, obtained from a constrained Hamilton's principle expressed in the Eulerian fluid description. This establishes the mathematical framework for the following properties of these anelastic equations: the Kelvin-Noether circulation theorem, conservation of potential vorticity on fluid parcels, and the Lie-Poisson Hamiltonian formulation possessing conserved Casimirs, conserved domain integrated energy and an associated variational…
Numerical simulations of the incompressible Euler equations are performed using the Taylor-Green vortex initial conditions and resolutions up to $4096^3$. The results are analyzed in terms of the classical analyticity strip method and Beale, Kato and Majda (BKM) theorem. A well-resolved acceleration of the time-decay of the width of the analyticity strip $δ(t)$ is observed at the highest resolution for $3.7<t<3.85$ while preliminary 3D visualizations show the collision of vortex sheets. The BKM criterium on the power-law growth of supremum of the vorticity, applied on the same time-interval, i…
We develop a Fourier-Chebyshev pseudospectral direct numerical simulation (DNS) to examine a potentially singular solution of the radially bounded, three-dimensional (3D), axisymmetric Euler equations [G. Luo and T.Y. Hou, Proc. Natl. Acad. Sci. USA, 111.36 (2014)]. We demonstrate that: (a) the time of singularity is preceded, in any spectrally truncated DNS, by the formation of oscillatory structures called tygers, first investigated in the one-dimensional (1D) Burgers and two-dimensional (2D) Euler equations; (b) the analyticity-strip method can be generalized to obtain an estimate for the (…
The Euler equations describing two-dimensional steady flows of an inviscid fluid are studied. These equations are reduced to one equation for the stream function and then, using the Hirota function, solutions of three nonlinear elliptic equations are found. %: Sine-Gordon, Sinh-Gordon and Tzitzéica. The solutions found are interpreted as sources in a rotating fluid, jets, chains of sources and sinks, vortex structures. We propose a new simple method for constructing solutions in the form of rational expressions of elliptic functions. It is shown that the flux of fluid across a closed curve is …
A new simulation method for solving fluid-structure coupling problems has been developed. All the basic equations are numerically solved on a fixed Cartesian grid using a finite difference scheme. A volume-of-fluid formulation (Hirt and Nichols (1981, J. Comput. Phys., 39, 201)), which has been widely used for multiphase flow simulations, is applied to describing the multi-component geometry. The temporal change in the solid deformation is described in the Eulerian frame by updating a left Cauchy-Green deformation tensor, which is used to express constitutive equations for nonlinear Mooney-Riv…
In this paper, a validation and uncertainty quantification (VUQ) framework for the Eulerian-Eulerian two-fluid-model based multiphase-computational fluid dynamics solver (MCFD) is formulated. The framework aims to answer the question: how to evaluate if a MCFD solver adequately represents the underlying physics of a multiphase system of interest? The proposed framework is based on total data-model integration (TDMI) approach that uses Bayesian method to inversely quantify the uncertainty of the solver predictions with the support of multiple experimental datasets. The framework consists of six…
The Navier-Stokes equations are the governing equations of fluid flows. They are deemed to embody all physics in a flow of Newtonian fluids like water, especially when we assume the fluid is incompressible. Fluid flows are usually described in the Eulerian perspective wherein we focus on the fluid's quantities at a fixed point in space as time processes, rather than the motion of each individual particle. The latter perspective is indeed the Lagrangian perspective which is widely used in solid mechanics like elasticity. Thus, the Navier-Stokes equations should be and indeed they are always use…
The mechanism for singularity formation in an inviscid wall-bounded fluid flow is investigated. The incompressible Euler equations are numerically simulated in a cylindrical container. The flow is axisymmetric with swirl. The simulations reproduce and corroborate aspects of prior studies reporting strong evidence for a finite-time singularity. The analysis here focuses on the interplay between inertia and pressure, rather than on vorticity. Linearity of the pressure Poisson equation is exploited to decompose the pressure field into independent contributions arising from the meridional flow and…
Here we present a transformation that maps the Schrodinger equation of quantum mechanics to the incompressible Euler equations of fluid mechanics. The transformation provides a wave solution and a potential function based on fluid properties that satisfy the Schrodinger equation given that the fluid velocity potential and pressure satisfy the Euler equations. Interestingly, in our transformation, the equivalent of quantum potential becomes the physical surface tension. This is contrary to the Madelung transformation that maps the Schrodinger equation to the compressible Euler equations where t…
The dynamics of singularity formation on the interface between two ideal fluids is studied for the Kelvin-Helmholtz instability development within the Hamiltonian formalism. It is shown that the equations of motion derived in the small interface angle approximation (gravity and capillary forces are neglected) admit exact solutions in the implicit form. The analysis of these solutions shows that, in the general case, weak root singularities are formed on the interface in a finite time for which the curvature becomes infinite, while the slope angles remain small. For Atwood numbers close to unit…
A 1934 paper by Leray posed the question of the regularity of solutions of the dynamical equations for incompressible inviscid fluids with smooth initial data. Since there has been many attempts to answer this question. Leray examined the possibility of self-similar solutions becoming singular in finite time at a definite space-time location. We reexamine this question in the light of a thorough analysis of the equations for the self-similar solution in axisymmetric geometries with a dependence on a logarithm of time besides the one due to the transformation to self-similar variables.
We revisit, both numerically and analytically, the finite-time blowup of the infinite-energy solution of 3D Euler equations of stagnation-point-type introduced by Gibbon et al. (1999). By employing the method of mapping to regular systems, presented in Bustamante (2011) and extended to the symmetry-plane case by Mulungye et al. (2015), we establish a curious property of this solution that was not observed in early studies: before but near singularity time, the blowup goes from a fast transient to a slower regime that is well resolved spectrally, even at mid-resolutions of $512^2.$ This late-ti…
