Blowup or no blowup? The interplay between theory and numerics

Hou, Thomas Y.; Li, Ruo

doi:10.1016/j.physd.2008.01.018

Cited by 67 publications

(72 citation statements)

References 25 publications

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“…Of the several filtered/dealiased pseudospectral methods tested by Bustamante & Kerr (2008), the method chosen for the calculations here is a combination of the 2/3 dealiasing rule plus a 36th-order filtering that was first introduced without dealiasing (Hou 2008).…”

Section: Computational Method Domain and New Initial Conditionmentioning

confidence: 99%

“…At a meeting on the Euler equations in 2007 in Aussois, France, the conclusions on regularity of the two anti-parallel calculations given (Bustamante & Kerr 2008;Hou 2008) were different, even though both used initial conditions based on Kerr (1993). The flaws in the Kerr (1993) prescription have now been identified and corrected.…”

Section: Computational Method Domain and New Initial Conditionmentioning

confidence: 99%

“…Following previous work (Kerr 1993;Bustamante & Kerr 2008;Hou 2008), all of the calculations used anisotropic meshes, with most taking z = 2 x = 4 y. Fine resolution is most critical in z due to the flattening of the vortices, with the requirement that the z-position of sup |ω(t)|, z ∞ be more than 12 mesh points from the z = 0 dividing plane for sup |ω(t)| to converge.…”

Section: Computational Method Domain and New Initial Conditionmentioning

confidence: 99%

“…The existing analysis tools are unable to simultaneously cover the necessary range of scales in both space and time, while existing methods for mapping vortex tubes onto Eulerian meshes tend to generate ghost images unless ad hoc massaging is applied (Bustamante & Kerr 2008;Hou 2008). This has led to weak and conflicting conclusions that depend upon the numerical method used and the choice of analysis that is applied to the results.…”

Section: Introductionmentioning

confidence: 99%

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Bounds for Euler from vorticity moments and line divergence

Kerr

2013

J. Fluid Mech.

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The inviscid growth of a range of vorticity moments is compared using Euler calculations of anti-parallel vortices with a new initial condition. The primary goal is to understand the role of nonlinearity in the generation of a new hierarchy of rescaled vorticity moments in Navier-Stokes calculations where the rescaled moments obey D m D m+1 , the reverse of the usual Ω m+1 Ω m Hölder ordering of the original moments. Two temporal phases have been identified for the Euler calculations. In the first phase the 1 < m < ∞ vorticity moments are ordered in a manner consistent with the new Navier-Stokes hierarchy and grow in a manner that skirts the lower edge of possible singular growth with D 2 m → sup |ω| ∼ A m (T c − t) −1 where the A m are nearly independent of m. In the second phase, the new D m ordering breaks down as the Ω m converge towards the same super-exponential growth for all m. The transition is identified using new inequalities for the upper bounds for the −dD −2 m /dt that are based solely upon the ratios D m+1 /D m , and the convergent super-exponential growth is shown by plotting log(d log Ω m /dt). Three-dimensional graphics show significant divergence of the vortex lines during the second phase, which could be what inhibits the initial power-law growth.

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Section: Computational Method Domain and New Initial Conditionmentioning

confidence: 99%

Section: Computational Method Domain and New Initial Conditionmentioning

confidence: 99%

Section: Computational Method Domain and New Initial Conditionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Bounds for Euler from vorticity moments and line divergence

Kerr

2013

J. Fluid Mech.

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“…[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.

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144

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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An improved dealiasing scheme for the fourth‐order Runge‐Kutta method: Formulation, accuracy and efficiency analysis

Sinhababu

Ayyalasomayajula

2020

Numerical Methods in Fluids

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Summary In this paper, the Random Phase Shift Method (RPSM) dealiasing scheme has been developed with the classical fourth‐order explicit Runge‐Kutta (RK4) method. This scheme is implemented in different benchmark problems to verify its numerical accuracy and computational efficiency where strong gradients are present in the solution. The propagation of aliasing errors through the substeps of RK4 is derived to show the existence of the residual aliasing error terms which results in mild smoothing effect without dissipating the small‐scale flow structures. Smoothness and numerical stability in the solutions obtained from the RPSM scheme also remain well preserved even at under‐resolved conditions. Numerical results agree well with the analytical and the computed solutions from previous studies. RPSM scheme shows a slight delay in the formation of numerical singularity for the inviscid flows but the filtering‐based schemes suffer from early blow‐up problem. We observe that this scheme displays better resolving ability than higher‐order exponential smoothing spectral filter scheme in capturing the strong fronts accurately even at just resolved spatial grid resolutions. Three‐dimensional truncation‐based dealiasing scheme, spherical truncation (SPT) shows vortices generated due to the parasitic currents in the solution of the inviscid three‐dimensional Taylor Green (TG) vortex flows. RPSM displays only the accurate isocontours of vortical field at nearly same computational expenses as the SPT scheme.

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Blowup or no blowup? The interplay between theory and numerics

Cited by 67 publications

References 25 publications

Bounds for Euler from vorticity moments and line divergence

Bounds for Euler from vorticity moments and line divergence

Potentially singular solutions of the 3D axisymmetric Euler equations

An improved dealiasing scheme for the fourth‐order Runge‐Kutta method: Formulation, accuracy and efficiency analysis

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