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

Donzis, Diego A.; Gibbon, John; Gupta, A.; Kerr, Robert M.; Pandit, Rahul; Vincenzi, Dario

doi:10.1017/jfm.2013.409

Cited by 33 publications

(68 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…First, nearly continuous higher-order vorticity moments and their production terms were saved at run-time for most of the major calculations. Using this data, the higher-order vorticity moment analysis of Donzis et al (2013) and Kerr (2013b) can be repeated to see if this alternative picture of the t r t t x colllapse yields new hints for the origins and generality of the B ν (t) (1.1) scaling. Then, to find what connects the distant periodic boundaries to the evolving compact structures, the diagnostics introduced by Kerr (2017) should be applied to all of the ν 3.125-e5 trefoil calculations.…”

Section: Discussionmentioning

confidence: 99%

“…This was done by determining higher-order vorticity moments ω L p ℓ , which when rescaled as suggested by (Gibbon 2010(Gibbon , 2012 are invariably inversely ordered from the Hölder (5.2) expectation (Donzis et al 2013). The case where this ordering is strongest is an earlier version of the anti-parallel calculation here.…”

Section: Self-similar Anti-parallel Collapsementioning

confidence: 99%

“…Furthermore, because this configuration has been studied extensively, existing high-resolution anti-parallel results can fill some of the gaps in the interpretation of the trefoil results, including what type of Euler (ν = 0) evolution is expected (Kerr 2013b) and how the enstrophy scaling could be connected to the hierarchy of vorticity moments found in several other calculations (Donzis et al 2013). The disadvantages of anti-parallel are that the global helicity is identically zero and the initial integral norms increase as the domain is increased, making comparisons to the mathematics difficult.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Enstrophy and circulation scaling for Navier–Stokes reconnection

Kerr

2018

J. Fluid Mech.

Self Cite

View full text Add to dashboard Cite

As reconnection begins and the enstrophy Z grows for two configurations, helical trefoil knots and anti-parallel vortices, two regimes of self-similar collapse are observed. First, during trefoil reconnection a new √ νZ scaling, ν viscosity, is identified before any ǫ = νZ dissipation scaling begins. Further rescaling shows linearly decreasing B ν (t) = (at configuration dependent crossing times t x . Gaps in the vortex structures identify the t x as when reconnection ends and collapse onto ν-independent curves can be obtained using). The critical times T c (ν) are identified empirically by extrapolating the linear B ν (t) regimes to B ∼ ν (T c ) = 0, yielding an A ν (t) collapse collapse that forms early as ν varies by 256. These solutions are regular or non-singular, as shown by decreasing cubic velocity norms u L 3 ℓ . For the anti-parallel vortices, first there is an exchange of circulation, from Γ y (y = 0) to Γ z (z = 0), mediated by the viscous circulation exchange integral ǫ Γ (t), which is followed by a modified B ν (t) collapse until the reconnection ends at t x . Singular Leray scaling and mathematical bounds for higherorder Sobolev norms are used to help explain the origins of the new scaling and why the domain size ℓ has to increase to maintain the collapse of A ν (t) and ǫ Γ as ν decreases. BackgroundWhen numerical calculations are used to explore fundamental questions about the nonlinear growth of turbulence, the numerics should be appropriate for how those questions were formulated mathematically. For questions that are posed in Sobolev spaces, corresponding to truncated Fourier series, periodic calculations would appear ideal. However, when the question is posed in the whole space, that is R 3 , localised aperiodic initial states would be more appropriate.This report will apply a Fourier-based code to two configurations with global helicities H at opposite extremes. Aperiodic, maximal H, trefoil vortex knots and H ≡ 0 antiparallel vortices, both with significant perturbations. Trefoil vortex knots have been the subject of recent experiments (Scheeler et al. 2014) and simulations (Kerr 2017) and the type of perturbed anti-parallel calculations used here have a long history (Melander & Hussain 1989; Kerr 2013a,b). The trefoil calculations were originally designed to address the intriguing experimental claim that the global helicity (1.6) was preserved during reconnections, which Kerr (2017) has confirmed through the first reconnection.Equally significant, as ν is decreased and the domains are increased, the simulated volume-integrated enstrophies Z, defined by (1.5), linearly converge at a ν-independent time t = t x when scaled using B ν (t) = √ νZ(t) −1/2 with a common B x = B ν (t x ) .(1.1)The three-dimensional graphics showed the first reconnection ending at t x .

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Self-similar Anti-parallel Collapsementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Enstrophy and circulation scaling for Navier–Stokes reconnection

Kerr

2018

J. Fluid Mech.

Self Cite

View full text Add to dashboard Cite

show abstract

“…This becomes stronger as m increases. Two new related papers are Kerr (2013), which shows the details of the physical space structures responsible for the D m (t) fluctuations in figure 2, and Donzis et al (2013), which shows that this hierarchy is found for all the Navier-Stokes simulations it has been tested on, including forced and decaying isotropic calculations.…”

Section: Applying the D M To Numerical Datamentioning

confidence: 95%

Bounds for Euler from vorticity moments and line divergence

Kerr

2013

J. Fluid Mech.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…Such depletion is a well-documented phenomenon for both Euler and Navier-Stokes cases [1,6,17,21,24,30,31,34]. The question is how to capture and exploit this favourable phenomenon in a rigorous manner.…”

Section: Introductionmentioning

confidence: 99%

Depletion of nonlinearity in the pressure force driving Navier–Stokes flows

Tran

2015

Nonlinearity

View full text Add to dashboard Cite

Abstract. The dynamics of the velocity norms ||u|| L q , for q ≥ 3, in Navier-Stokes flows is studied. The pressure term that drives this dynamics has a high degree of nonlinear depletion, which owes its origin to a genuine negative correlation between |u| and |∇|u||, among other things. Under viscous effects, such depletion may give rise to mild growth of ||u|| L q . We explore the possibility of non-singular growth of ||u|| L q .

show abstract

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

Cited by 33 publications

References 39 publications

Enstrophy and circulation scaling for Navier–Stokes reconnection

Enstrophy and circulation scaling for Navier–Stokes reconnection

Bounds for Euler from vorticity moments and line divergence

Depletion of nonlinearity in the pressure force driving Navier–Stokes flows

Contact Info

Product

Resources

About