Chaotic Blowup in the 3D Incompressible Euler Equations on a Logarithmic Lattice

Campolina, Ciro S.; Mailybaev, Alexei A.

doi:10.1103/physrevlett.121.064501

Cited by 23 publications

(42 citation statements)

References 56 publications

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“…The existence of SSs for Euler or Navier-Stokes is still controversial. In the Euler case, Chae proved that there is no backward SS (Chae 2007), but self-similar blowup behaviours with h = −1 were found on logarithmically spaced lattice (Campolina & Mailybaev 2018), or during reconnection of tent vortices (Kimura & Moffatt 2018). In the Navier-Stokes case (see Bradshaw & Tsai (2018) for a recent review), the SS necessarily has an h = −1 exponent (in agreement with the h = −1 rescaling symmetry, see Section 4.1) and blows up in finite time.…”

Section: Beyond Kolmogorov Using Multi-fractalsmentioning

confidence: 98%

Beyond Kolmogorov cascades

2019

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The large-scale structure of many turbulent flows encountered in practical situations such as aeronautics, industry, meteorology is nowadays successfully computed using the Kolmogorov–Kármán–Howarth energy cascade picture. This theory appears increasingly inaccurate when going down the energy cascade that terminates through intermittent spots of energy dissipation, at variance with the assumed homogeneity. This is problematic for the modelling of all processes that depend on small scales of turbulence, such as combustion instabilities or droplet atomization in industrial burners or cloud formation. This paper explores a paradigm shift where the homogeneity hypothesis is replaced by the assumption that turbulence contains singularities, as suggested by Onsager. This paradigm leads to a weak formulation of the Kolmogorov–Kármán–Howarth–Monin equation (WKHE) that allows taking into account explicitly the presence of singularities and their impact on the energy transfer and dissipation. It provides a local in scale, space and time description of energy transfers and dissipation, valid for any inhomogeneous, anisotropic flow, under any type of boundary conditions. The goal of this article is to discuss WKHE as a tool to get a new description of energy cascades and dissipation that goes beyond Kolmogorov and allows the description of small-scale intermittency. It puts the problem of intermittency and dissipation in turbulence into a modern framework, compatible with recent mathematical advances on the proof of Onsager’s conjecture.

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Section: Beyond Kolmogorov Using Multi-fractalsmentioning

confidence: 98%

Beyond Kolmogorov cascades

2019

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“…In it is estimated that conclusions regarding finite-time singularities using the analyticity strip method would require spatial resolutions of (16 k) 3 to (32 k) 3 for the Kida-Pelz initial data. A recent study by Campolina & Mailybaev (2018) suggests that the resolution available via classical direct numerical simulation is not sufficient to investigate blowup.…”

Section: State Of the Art And Limitations In Tracing Finite-time Singularitiesmentioning

confidence: 99%

Numerical evidence of anomalous energy dissipation in incompressible Euler flows: towards grid-converged results for the inviscid Taylor–Green problem

et al. 2021

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The well-known energy dissipation anomaly in the inviscid limit, related to velocity singularities according to Onsager, still needs to be demonstrated by numerical experiments. The present work contributes to this topic through high-resolution numerical simulations of the inviscid three-dimensional Taylor–Green vortex problem using a novel high-order discontinuous Galerkin discretisation approach for the incompressible Euler equations. The main methodological ingredient is the use of a discretisation scheme with inbuilt dissipation mechanisms, as opposed to discretely energy-conserving schemes, which – by construction – rule out the occurrence of anomalous dissipation. We investigate effective spatial resolution up to $8192^3$ (defined based on the $2{\rm \pi}$ -periodic box) and make the interesting phenomenological observation that the kinetic energy evolution does not tend towards exact energy conservation for increasing spatial resolution of the numerical scheme, but that the sequence of discrete solutions seemingly converges to a solution with non-zero kinetic energy dissipation rate. Taking the fine-resolution simulation as a reference, we measure grid-convergence with a relative $L^2$ -error of $0.27\,\%$ for the temporal evolution of the kinetic energy and $3.52\,\%$ for the kinetic energy dissipation rate against the dissipative fine-resolution simulation. The present work raises the question of whether such results can be seen as a numerical confirmation of the famous energy dissipation anomaly. Due to the relation between anomalous energy dissipation and the occurrence of singularities for the incompressible Euler equations according to Onsager's conjecture, we elaborate on an indirect approach for the identification of finite-time singularities that relies on energy arguments.

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“…Given that such flows evolving near the edge of regularity involve the formation of very small flow structures, these computations typically require the use of state-of-the-art computational resources available at a given time. The computational studies focused on the possibility of finite-time blow-up in the 3-D Navier-Stokes and/or Euler system include Brachet et al (1983), Pumir & Siggia (1990), Brachet (1991), Kerr (1993), Pelz (2001), , Ohkitani & Constantin (2008), Ohkitani (2008), Grafke et al (2008), Gibbon et al (2008), Hou (2009), Orlandi, Pirozzoli & Carnevale (2012), Bustamante & Brachet (2012), Orlandi et al (2014) and Campolina & Mailybaev (2018), all of which considered problems defined on domains periodic in all three dimensions. The investigations by Donzis et al (2013), Kerr (2013b) and Gibbon et al (2014) and Kerr (2013a) focused on the time evolution of vorticity moments and compared it against bounds on these quantities obtained using rigorous analysis.…”

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confidence: 99%

Maximum amplification of enstrophy in three-dimensional Navier–Stokes flows

2020

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This investigation concerns a systematic search for potentially singular behavior in 3D Navier-Stokes flows. Enstrophy serves as a convenient indicator of the regularity of solutions to the Navier Stokes system -as long as this quantity remains finite, the solutions are guaranteed to be smooth and satisfy the equations in the classical (pointwise) sense. However, there are no estimates available with finite a priori bounds on the growth of enstrophy and hence the regularity problem for the 3D Navier-Stokes system remains open. In order to quantify the maximum possible growth of enstrophy, we consider a family of PDE optimization problems in which initial conditions with prescribed enstrophy E 0 are sought such that the enstrophy in the resulting Navier-Stokes flow is maximized at some time T . Such problems are solved computationally using a large-scale adjoint-based gradient approach derived in the continuous setting. By solving these problems for a broad range of values of E 0 and T , we demonstrate that the maximum growth of enstrophy is in fact finite and scales in proportion to E 3/2 0 as E 0 becomes large. Thus, in such worst-case scenario the enstrophy still remains bounded for all times and there is no evidence for formation of singularity in finite time. We also analyze properties of the Navier-Stokes flows leading to the extreme enstrophy values and show that this behavior is realized by a series of vortex reconnection events.

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Chaotic Blowup in the 3D Incompressible Euler Equations on a Logarithmic Lattice

Cited by 23 publications

References 56 publications

Beyond Kolmogorov cascades

Beyond Kolmogorov cascades

Numerical evidence of anomalous energy dissipation in incompressible Euler flows: towards grid-converged results for the inviscid Taylor–Green problem

Maximum amplification of enstrophy in three-dimensional Navier–Stokes flows

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