Bounds for Euler from vorticity moments and line divergence

Kerr, Robert M.

doi:10.1017/jfm.2013.325

Cited by 36 publications

(46 citation statements)

References 14 publications

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“…. ; are the vorticity moment integrals (21). Clearly, the divergence of the time integral of any finite-order Ω 2m implies the blowup of R ts 0 kωk ∞ dt, and in our case Ω 4 = O(t s − t) −1 , which fulfills the criterion.…”

Section: Numerical Resultsmentioning

confidence: 58%

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Potentially singular solutions of the 3D axisymmetric Euler equations

Luo

Hou

2014

Proc. Natl. Acad. Sci. U.S.A.

137

166

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The question of finite-time blowup of the 3D incompressible Euler equations is numerically investigated in a periodic cylinder with solid boundaries. Using rotational symmetry, the equations are discretized in the (2D) meridian plane on an adaptive (moving) mesh and is integrated in time with adaptively chosen time steps. The vorticity is observed to develop a ring-singularity on the solid boundary with a growth proportional to ∼(t s − t) −2.46 , where t s ∼ 0.0035056 is the estimated singularity time. A local analysis also suggests the existence of a self-similar blowup. The simulations stop at τ 2 = 0.003505 at which time the vorticity amplifies by more than (3 × 10 8 )-fold and the maximum mesh resolution exceeds (3 × 10 12 ) 2 . The vorticity vector is observed to maintain four significant digits throughout the computations.W hether initially smooth solutions to the 3D incompressible Euler equationscan develop a singularity in finite time is one of the most fundamental problems in mathematical fluid dynamics. Standing open for more than 250 y and closely related to the Clay Millennium Prize Problem on the Navier-Stokes equations, the problem has received great attention from not only the mathematics but also the physics and engineering communities, where the formation of singularities in inviscid (Euler) flows is believed to be relevant to the creation of small scales in viscous turbulent flows (1-3). The finite-time blowup problem has been studied extensively from both mathematical and numerical points of view. On the mathematical side, a number of useful blowup/nonblowup criteria have been obtained over the years, which have greatly facilitated the numerical search of a finite-time singularity (4-10). On the numerical side, interesting numerical simulations suggesting the existence of a finite-time singularity have been reported from time to time (see, for example, refs. 11-17), but in essentially every case, evidence against a singularity was found in subsequent studies (see, for example, refs. 18-21). This casts doubt on the validity of the claimed singularity and leaves the question of finite-time blowup an unresolved puzzle. By focusing on flows with rotational symmetry and other special properties, we have identified, through careful numerical studies, a class of potentially singular solutions to the 3D axisymmetric Euler equations in a radially bounded, axially periodic cylinder (see Eqs. 2 and 3 below). The simulations take advantage of the reduced computational complexity in cylindrical geometries, and use a specially designed adaptive mesh to achieve a maximum mesh resolution of over (3 × 10 12 ) 2 near the point of the singularity. This results in a computed vorticity vector with four digits of accuracy (up to the stopping time) and with a (3 × 10 8 )-fold increase in magnitude. The numerical data are checked against all major blowup/nonblowup criteria to confirm the validity of the singularity. A careful local analysis also suggests the existence of a self-similar blowup in the meridian plane.We...

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Section: Numerical Resultsmentioning

confidence: 58%

“…[11][12][13][14][15][16][17], but in essentially every case, evidence against a singularity was found in subsequent studies (see, for example, refs. [18][19][20][21]. This casts doubt on the validity of the claimed singularity and leaves the question of finite-time blowup an unresolved puzzle.…”

mentioning

confidence: 99%

Potentially singular solutions of the 3D axisymmetric Euler equations

Luo

Hou

2014

Proc. Natl. Acad. Sci. U.S.A.

137

166

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“…This is the period when this calculation has nearly Euler dynamics and the effects of viscosity compared to nonlinear growth are minimal. The growth of the D m (t) in true Euler dynamics is the topic of another paper (Kerr 2013b). …”

Section: The First Set Of Simulationsmentioning

confidence: 99%

Vorticity moments in four numerical simulations of the 3D Navier–Stokes equations

et al. 2013

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The issue of intermittency in numerical solutions of the 3D Navier-Stokes equations on a periodic box [0, L] 3 is addressed through four sets of numerical simulations that calculate a new set of variables defined by D m (t) = ( , giving rise to a suggestion that a depletion of nonlinearity is occurring which could be the cause of Navier-Stokes regularity. The first simulation is of very anisotropic decaying turbulence; the second and third are of decaying isotropic turbulence from random initial conditions and forced isotropic turbulence at fixed Grashof number respectively; the fourth is of very-high-Reynolds-number forced, stationary, isotropic turbulence at up to resolutions of 4096 3 .

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“…Despite some ambiguity in interpreting the initial data used by [38], they managed to advance the solution up to t = 19, which is beyond the singularity time T = 18.7 alleged by [38]. By using newly developed analytic tools based on rescaled vorticity moments, Kerr also confirmed in a very recent study [39] that solutions computed from initial data analogous to that used in [38] eventually converge to superexponential growth and hence are unlikely to lead to a singularity. In a later work, Hou and Li [33] also repeated the computation of [5] and found that the singularity reported by [5] is likely an artifact due to insufficient resolution.…”

Section: Introduction the Celebrated Three-dimensional (3d) Incomprementioning

confidence: 98%

“…Representative work in this direction include [27,45], which studied Euler flows with swirls in axisymmetric geometries, the famous computation of Kerr and his collaborators [38,8,39], which studied Euler flows generated by a pair of perturbed antiparallel vortex tubes, and the viscous simulations of [5], which studied the 3D Navier-Stokes equations using Kida's high-symmetry initial data. Other interesting pieces of work are [10,47], which studied axisymmetric Euler flows with complex initial data and reported singularities in the complex plane.…”

Section: Introduction the Celebrated Three-dimensional (3d) Incomprementioning

confidence: 99%

Toward the Finite-Time Blowup of the 3D Axisymmetric Euler Equations: A Numerical Investigation

Luo¹,

Hou²

2014

Multiscale Model. Simul.

102

150

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Abstract. Whether the three-dimensional incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to this long-standing open question from a numerical point of view by presenting a class of potentially singular solutions to the Euler equations computed in axisymmetric geometries. The solutions satisfy a periodic boundary condition along the axial direction and a no-flow boundary condition on the solid wall. The equations are discretized in space using a hybrid 6th-order Galerkin and 6th-order finite difference method on specially designed adaptive (moving) meshes that are dynamically adjusted to the evolving solutions. With a maximum effective resolution of over (3 × 10 12 ) 2 near the point of the singularity, we are able to advance the solution up to τ 2 = 0.003505 and predict a singularity time of ts ≈ 0.0035056, while achieving a pointwise relative error of O(10 −4 ) in the vorticity vector ω and observing a (3 × 10 8 )-fold increase in the maximum vorticity ω ∞ . The numerical data are checked against all major blowup/non-blowup criteria, including Beale-Kato-Majda, Constantin-Fefferman-Majda, and Deng-Hou-Yu, to confirm the validity of the singularity. A local analysis near the point of the singularity also suggests the existence of a self-similar blowup in the meridian plane.

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Bounds for Euler from vorticity moments and line divergence

Cited by 36 publications

References 14 publications

Potentially singular solutions of the 3D axisymmetric Euler equations

Potentially singular solutions of the 3D axisymmetric Euler equations

Vorticity moments in four numerical simulations of the 3D Navier–Stokes equations

Toward the Finite-Time Blowup of the 3D Axisymmetric Euler Equations: A Numerical Investigation

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