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Diagnostics for the anti-parallel calculations during the first reconnection. a) The growth of Γz(t) (mirrored by the decay of Γy(0) = 4.6) and the collapse of ǫΓ (1.2), the circulation exchange between the symmetry planes, both of which first become noticeable at t ≈ 14. b) Contours of ωy on the y = 0 plane and ωz on the z = 0 plane at t = 16. c) Bν (t) and the B ∼ ν (t) dashed lines that are extrapolated from tangents at Bν | t=14 , with both sets of curves converging near tx = 22.85 with Bν (tx) ≈ 0.73. d) Aν (t) (2.2) collapse between when the u∞ position shifts to the y = z = 0 line at tr = 12.5 and tx, when the B ∼ ν (t) cross.
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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, where ν is viscosity, is identified before any ∈ = νZ dissipation scaling begins. Further rescaling shows linearly decreas...
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... do the anti-parallel rescaled enstrophies B ν (t) behave? These are given in figure 4c, with a change in behaviour at t ≈ 16 for all four ν as reconnection begins and a convergence of the solid lines at t ≈ 25 as the gap forms in the vorticity isosurface. ...
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... unlike the trefoils in figure 2a, these B ν (t) do not decrease linearly, so another method is needed to find t x , T c (ν) and A ν (t). The first step in figure 4a is to plot the t ∼ 16, ν 1e − 3 circulations Γ z on the x − y symmetry plane and the following integral of the dissipative terms along the y = z = 0 line ( Virk et al. 1995) This is the viscous circulation exchange ǫ Γ (t) (1.2) between Γ y on the x − z symmetry plane and Γ z on the x − y symmetry plane as illustrated in figure 4b with contours of ω y and ω z in the symmetry planes. ...
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... unlike the trefoils in figure 2a, these B ν (t) do not decrease linearly, so another method is needed to find t x , T c (ν) and A ν (t). The first step in figure 4a is to plot the t ∼ 16, ν 1e − 3 circulations Γ z on the x − y symmetry plane and the following integral of the dissipative terms along the y = z = 0 line ( Virk et al. 1995) This is the viscous circulation exchange ǫ Γ (t) (1.2) between Γ y on the x − z symmetry plane and Γ z on the x − y symmetry plane as illustrated in figure 4b with contours of ω y and ω z in the symmetry planes. ...
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... the ǫ Γ (t) have a ν-independent collapse that begins at t ≈ 14 > t r ≈ 12.5, t r defined in section 4, and lasts to t Γ ≈ 16.5, the time with maximum ǫ Γ , when the growing Γ z (t) for different ν cross and by which there is a small, but finite, exchange of circulation ∆Γ = Γ z (t Γ ) in figure 4a, an exchange becomes the visible gap in figure 3b at t = 24. Over this brief period, the exchange is consistent with singular Leray scaling (6.3) ( Leray 1934) and is suppressed if the domain size, particularly in z, is not increased as ν decreases, as for the two ν=3.125e-5 ...
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... the ǫ Γ (t) collapse phase was identified, it was realised that the B ν (t) do linearly decay, but only for the brief period (t r = 12.5) t (t Γ = 16.5). These can easily be extrapolated to t > t Γ = 16.5 to form extrapolated B ∼ ν (t), dashed lines in figure 4c, from which t x = 22.85 can be defined as when the B ∼ ν cross and the T c (ν) (2.1) as when B ∼ ν (T c ) = 0. Together in (2.2) with the B ν (t), these define the A ν (t) that collapse for t r t t x and a bit more in time in figure 4d, even if this collapse is not linear. ...
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... ω(t) ∞ can be bounded by the higher-order sup u(t) − v(t) H s ℓ using further ℓ- dependent Sobolev space embedding inequalities (Robinson et al. 2016), (5.1) and (5.2) can ensure that Z is bounded and the following will be suppressed as ν → 0: The decreasing B ν (t) (1.1) scaling and dissipation growth ǫ = νZ in figure 2 and the ǫ Γ (1.2) and A ν (t) (2.2) collapse in figure 4. ...
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... Leray scaling provide hints for the origins of the ǫ Γ (t Γ ) collapse as ν → 0 in figure 4a and the Z(t x ) = ν −1/2 /B 2 ν (t x ) ∼ ν −1/2 → ∞ scaling implied by figure 2a? To begin, the Leray estimates needed for scaling u L 3 are ...
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... the anti-parallel vortices, while the B ν (t) in figure 4c suggest that some type of collapse might exist, two steps are needed to find the collapsing A ν (t). 4a shows that the reconnection begins with a collapse of the circulation exchange rates ǫ Γ (1.2) for the shorter period of t r t t Γ . ...
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... this brief period, Z, ǫ Γ and u L 3 ℓ are all consistent with Leray scaling (6.2). Next, by extrapolating the t < t Γ B ν (t) to the t > t Γ B ∼ ν (t) in 4c, the t x and T c (ν) are found that (2.2) uses to create the collapsing A ν (t) in figure 4d. The possible lesson is that even if rescaling enstrophy as B ν (t) does not yield a perfect collapse, it can still be used as a diagnostic for finding self-similar behaviour. ...
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Five sets of Navier-Stokes trefoil vortex knots in (2π)3 domains show how the shape of their initial profiles, Gaussian/Lamb-Oseen or algebraic, and their widths influence their evolution, as defined by their enstrophy Z(t), helicity H(t), and changes in their dissipation-scale structures. Significant differences develop even when all have the same...
Citations
... A dimensionless parameter Se is introduced to characterize the secondary flow intensity (Song and Wang, 2016;Song et al., 2009). On the other hand, the enstrophy ε is used to characterize the vortex intensity (Kerr, 2018). For the former, the vorticity is integrated to represent the intensity of the vortex. ...
Improving the thermal efficiency of internal combustion engines requires a high compression ratio, a high thermal load, and high mechanical strength in the combustion chambers. However, they will promote excessive temperatures within the combustion chamber and increase the likelihood of knocking. As the piston reciprocates, the engine oil injected into the piston crown oil gallery can oscillate within the gallery to reduce the piston temperature. However, the interaction between the ambient air and the interface of the axial jet liquid column results in unstable fluctuations, significantly affecting the filling rate of the oil gallery. In this study, the fluctuation of the jet interface is analyzed by changing the Reynolds number (Re) and bending radius R of the nozzle. Two-dimensional/two-component and two-dimensional/three-component particle image velocimetry are used to measure the flow fields in the longitudinal section of two bent nozzles with small (R = 15 mm) and large (R = 60 mm) curvature radii and horizontal cross-sections at 4D, 2.5D, and 1D from the exit of each bent nozzle, respectively, to explore the causes of the fluctuating internal flow. By calculating the secondary flow intensity Se on the inner and outer wall sides of the nozzle, the inner and outer jet interface width changes, and the different fragmentation mechanisms of the jet interface, it is shown that the higher frequency of the high-speed core region at the outer wall side of the nozzle interior can drive the waveform on the outside of the jet interface to break and produce small-diameter droplets easily.
... Because the vorticity at a singularity must diverge [5], effort has been spent on understanding the behavior of interacting localized de-singularized vortex tubes, leading to the study of vortex reconnection and its role in turbulence. Various configurations have been proposed and investigated numerically for reconnection in classical turbulence (e.g., [6][7][8]), quantum turbulence (e.g., [9][10][11]), and for the existence of singularity (e.g., [12][13][14][15][16]). ...
... Phase I is defined by the inequalities (1.4) and (1.5), and the analysis of the present paper is restricted to this phase. We make no assertion concerning phase II, for which DNS is an indispensable mode of investigation (as, for example, in the paper of Kerr (2018), who has used DNS to study the phase II reconnection process for two prototypical vortex tube configurations (antiparallel and trefoil knot)). ...
An exact analytical solution is obtained for the dynamical system derived in Part 1 of this series (Moffatt & Kimura, J. Fluid Mech , vol. 861, 2019 a , pp. 930–967), which describes the approach of two initially circular vortices of finite but small cross-section symmetrically located on inclined planes. This exact solution, applicable in the inviscid limit, allows determination of the amplification $\mathcal {A}_{\omega }$ of the axial vorticity within the finite time $T$ during which the basic assumptions of the model continue to apply. It is first shown that, for arbitrarily prescribed $\mathcal {A}_{\omega }$ , it is possible to specify smooth initial conditions of finite energy such that, in the inviscid limit, this amplification is achieved within the time $T$ . When viscosity is included, an estimate is provided for the minimum vortex Reynolds number that is sufficient for the same result to hold. The predictions are broadly compatible with results from direct numerical simulations at moderate Reynolds numbers. Moreover, it is shown that one may come arbitrarily close to a finite-time singularity of the Navier–Stokes equation by appropriate choice of an initial, smooth, finite-energy velocity field; however, this approach to a singularity is ultimately thwarted through breach of the assumptions on which the dynamical system is based. Thus we make no claim here concerning realisation of a Navier–Stokes singularity. Moreover, we find that the conditions required to attain a large amplification $\mathcal {A}_{\omega }\gg 1$ during the time $T$ are far beyond those that can be realised in either experiment or direct numerical simulation.
... Trefoil vortex knots are an initial state that is inherently helical, self-reconnecting, and mathematically compact, meaning that they can be isolated far from boundaries. The goal of this paper is to revisit recent trefoil knots simulations [1][2][3][4] to ascertain why different initial vorticity profiles generate starkly contrasting answers to those questions. ...
... Before the results in Refs. [1][2][3][4], the most that numerics has been able to tell us about the role of helicity is that for single-signed helical Fourier modes, energy dissipation can be suppressed for a short time [5]. These flows then evolve into traditional decaying numerical turbulence: without any further insight into whether h has a role in either achieving, or suppressing, finite energy dissipation as the viscosity decreases. ...
... To illustrate the differences, Figs. 1 and 2 compare the Z (t ) and H(t ) evolution of two sets of calculations with the same circulation = 1 (7) and same three-fold symmetric trajectories, but representing these initial core profiles: respectively, evolution using a Gaussian/Lamb-Oseen (10) core profile [3] and that of a p r = 1 algebraic (9) core profile [2]. How do Z (t ) and H(t ) evolve for these cases? ...
Five sets of Navier-Stokes trefoil vortex knots in (2π)3 domains show how the shape of their initial profiles, Gaussian/Lamb-Oseen or algebraic, and their widths influence their evolution, as defined by their enstrophy Z(t), helicity H(t), and changes in their dissipation-scale structures. Significant differences develop even when all have the same three-fold symmetric trajectory, the same initial circulation and the same range of the viscosities ν. The focus is upon how the dynamics of helicity density h=u·ω affects reconnection and the evolution of enstrophy. h≲0 patches on the vorticity isosurfaces show where and how reconnection forms. For the Lamb-Oseen profile, the tightest and most linearly unstable, there is only a brief spurt of enstrophy growth as thin braids form at these positions; before being dissipated as the post-reconnection helicity H grows significantly. For the algebraic cases: as h<0 vortex sheets form prior to reconnection, there is ν-independent convergence of νZ(t) at a common tx. For those with the broadest wings, enstrophy growth accelerates after reconnection, leading to approximately convergent dissipation rates ε=νZ(t). Maps of terms from the budget equations onto centerlines illustrate the divergent behavior. Lamb-Oseen briefly forms six locations of centerline convergence with local negative dips in the helicity dissipation εh and vortical-helicity flux hf. These are the source of the following positive increase in the global H and suppression of enstrophy production. For the algebraic profiles there are only three locations of centerline convergence, each with spans of less localized εh<0 that could be the seeds for the h<0 vortex sheets and whose interactions can explain the later accelerated growth of the enstrophy, approximate ν-independent convergence of the energy dissipation rates ε, and evidence for finite-time energy dissipation ΔEε, despite the initial symmetries.
... Trefoil vortex knots are an initial state that is inherently helical, self-reconnecting, and mathematically compact, meaning that they can be isolated far from boundaries. The goal of this paper is to revisit recent trefoil knots simulations [1][2][3][4] to ascertain why different initial vorticity profiles generate starkly contrasting answers to those questions. ...
... Before the results in papers [1][2][3][4], the most that numerics has been able to tell us about the role of helicity is that for single-signed helical Fourier modes, energy dissipation can be suppressed for a short time [5]. These flows then evolve into traditional decaying numerical turbulence: without any further insight into whether h has a role in either achieving, or suppressing, finite energy dissipation as the viscosity decreases. ...
... To illustrate the differences, figures 1 and 2 compare Z(t) and H(t) for two sets of calculations with the same circulation Γ = 1 (7) and same three-fold symmetric trajectories, but representing different initial core profiles. Respectively, evolution using a Gaussian/Lamb-Oseen (10) core profile, as recently reported [3], and from a p r = 1 algebraic core (9) that has already provided evidence for a dissipation anomaly, finite ∆E (1) [2]. Another difference in their initialization is the vortex core width. ...
Six sets of Navier-Stokes trefoil vortex knots in $(2\pi)^3$ domains show how the shape of the initialprofile influences the evolution of the enstrophy $Z$, helicity ${\cal H}$ and dissipation-scale. Significant differences develop even when all have the same three-fold symmetric trajectory, the same initial circulation and the same range of the viscosities $\nu$. Maps of the helicity density $h=u\cdot\omega$ onto vorticity isosurfaces patches show where $h\lesssim0$ sheets form during reconnection. For the Gaussian/Lamb-Oseen profile helicity ${\cal H}$ grows significantly, with only a brief spurt of enstrophy growth as thin braids form then decay during reconnection. The remaining profiles are algebraic. For the untruncated algebraic cases,$h<0$ vortex sheets form in tandem with $\nu$-independent convergence of $\sqrt{\nu}Z(t)$ at a common $t_x$. For those with the broadest wings, enstrophy growth accelerates during reconnection, leading to approximately $\nu$-independent convergent finite-time dissipation rates $\epsilon=\nu Z$. By mapping terms from the budget equations onto centerlines, the origins of the divergent behavior are illustrated. Lamb-Oseen has six locations of centerline convergence form with local negative helicity dissipation, $\epsilon_h<0$, and small, but positive $h$. Later, the sum of these localized patches of $\epsilon_h<0$ leads to a positive increase in the global ${\cal H}$ and suppression of enstrophy production. For the algebraic profiles: There are only three locations of centerline convergence, each with spans of less localized $\epsilon_h<0$ and some $h<0$. Spans that could be the seeds for the $h<0$ vortex sheets that form in the lower half of the trefoil as the $\sqrt{\nu}Z(t)$ phase begins and can explain accelerated growth of the enstrophy and evidence for finite-time energy dissipation $\Delta E_\epsilon$. Despite the initial symmetries.
... Extreme flow behavior is often associated with vortex reconnection, and this problem was investigated in Kerr (2018) using initial conditions involving a trefoil vortex and two perturbed anti-parallel vortex tubes. In addition to identifying regions of selfsimilar behavior in these flows, that study also examined the time evolution of different solution norms during the reconnection event. ...
... Since these quantities play an important role in characterizing the extreme behavior in Navier-Stokes flows, in Figure 4a,b,c we show the time evolution of the norms u(t) L 3 , u(t) H 1/2 and ∇ × u(t) L ∞ in solutions corresponding to selected representative optimal initial conditions obtained by solving Problems 1 and 2. In most cases, these quantities reveal modest transient growth followed by decay. The transient growth of the norm u(t) L 3 in the flow with the initial condition obtained by solving Problem 1, cf. Figure 4a, is noteworthy as it was not observed in flows obtained by solving Problem 0 in Kang et al. (2020) or in Kerr (2018). ...
In this investigation, we perform a systematic computational search for potential singularities in 3D Navier–Stokes flows based on the Ladyzhenskaya–Prodi–Serrin conditions. They assert that if the quantity \(\int _0^T \Vert \textbf {u}(t) \Vert _{L^q(\Omega )}^p \, dt\), where \({2/p+3/q = 1}\), \(q > 3\), is bounded, then the solution \(\textbf {u}(t)\) of the Navier–Stokes system is smooth on the interval [0, T]. In other words, if a singularity should occur at some time \(t \in [0,T]\), then this quantity must be unbounded. We have probed this condition by studying a family of variational PDE optimization problems where initial conditions \(\textbf {u}_0\) are sought to maximize \(\int _0^T \Vert \textbf {u}(t) \Vert _{L^4(\Omega )}^8 \, dt\) for different T subject to suitable constraints. Families of local maximizers all having the form of vortex loops of different shape are found numerically using a large-scale adjoint-based gradient approach. Even in the flows corresponding to the optimal initial conditions determined in this way no evidence has been found for singularity formation, which would be manifested by unbounded growth of \(\Vert \textbf {u}(t) \Vert _{L^4(\Omega )}\). However, the maximum enstrophy attained in these extreme flows scales in proportion to \(\mathcal {E}_0^{3/2}\), the same as found by Kang et al. (2020) when maximizing the finite-time growth of enstrophy. In addition, we also consider sharpness of an a priori estimate on the time evolution of \(\Vert \textbf {u}(t) \Vert _{L^4(\Omega )}\) by solving another PDE optimization problem and demonstrate that the upper bound in this estimate could possibly be improved.
... The investigations [46][47][48][49] focused on the time evolution of vorticity moments and compared it against bounds on these quantities obtained using rigorous analysis. Recent computations [50] considered a 'trefoil' configuration meant to be defined on an unbounded domain (although the computational domain was always truncated to a finite periodic box). A simplified semi-analytic model of vortex reconnection was recently developed and analysed based on the Biot-Savart Law and asymptotic techniques [51,52]. ...
This review article offers a survey of the research program focused on a systematic computational search for extreme and potentially singular behaviour in hydrodynamic models motivated by open questions concerning the possibility of a finite-time blow-up in the solutions of the Navier–Stokes system. Inspired by the seminal work of Lu & Doering (2008 Ind. Univ. Math. 57 , 2693–2727), we sought such extreme behaviour by solving PDE optimization problems with objective functionals chosen based on certain conditional regularity results and a priori estimates available for different models. No evidence for singularity formation was found in extreme Navier–Stokes flows constructed in this manner in three dimensions. We also discuss the results obtained for one-dimensional Burgers and two-dimensional Navier–Stokes systems, and while singularities are ruled out in these flows, the results presented provide interesting insights about sharpness of different energy-type estimates known for these systems. Connections to other bounding techniques are also briefly discussed.
This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’.
... These processes may differ for flows more complex than fully developed turbulence (FDT) with, for FDT, stronger intermittency and possibly an inverse cascade of energy [55]. Direct numerical simulations of reconnection processes in fluids were performed for a trefoil knot configuration [56] (as well as for anti-parallel vortices), and helicity spectra and reconnection of vortex structures were recently computed using a vortex method [57]. ...
Helicity, a measure of the breakage of reflectional symmetry representing the topology of turbulent flows, contributes in a crucial way to their dynamics and to their fundamental statistical properties. We review several of their main features, both new and old, such as the discovery of bi-directional cascades or the role of helical vortices in the enhancement of large-scale magnetic fields in the dynamo problem. The dynamical contribution in magnetohydrodynamic of the cross-correlation between velocity and induction is discussed as well. We consider next how turbulent transport is affected by helical constraints, in particular in the context of magnetic reconnection and fusion plasmas under one- and two-fluid approximations. Central issues on how to construct turbulence models for non-reflectionally symmetric helical flows are reviewed, including in the presence of shear, and we finally briefly mention the possible role of helicity in the development of strongly localized quasi-singular structures at small scale.
This article is part of the theme issue ‘Scaling the turbulence edifice (part 2)’.
... Enstrophy (t * ) remains constant for very high Re at late times, suggesting a balance between enstrophy production and dissipation. The rapid increase of during reconnection is closely related to the idea of dissipation anomaly, an important topic in turbulent flows (Sreenivasan 1984, Frisch 1995, Kerr 2018a. ...
As a fundamental topology-transforming event, reconnection plays a significant role in the dynamics of plasmas, polymers, DNA, and fluids—both (classical) viscous and quantum. Since the 1994 review by Kida & Takaoka, substantial advances have been made on this topic. We review recent studies of vortex reconnection in (classical) viscous flows, including the physical mechanism, its relationship to turbulence cascade, the formation of a finite-time singularity, helicity dynamics, and aeroacoustic noise generation.
... The investigations [46][47][48][49] focused on the time evolution of vorticity moments and compared it against bounds on these quantities obtained using rigorous analysis. Recent computations [50] considered a "trefoil" configuration meant to be defined on an unbounded domain (although the computational domain was always truncated to a finite periodic box). A simplified semi-analytic model of vortex reconnection was recently developed and analyzed based on the Biot-Savart law and asymptotic techniques [51,52]. ...
This review article offers a survey of the research program focused on a systematic computational search for extreme and potentially singular behavior in hydrodynamic models motivated by open questions concerning the possibility of a finite-time blow-up in the solutions of the Navier-Stokes system. Inspired by the seminal work of Lu & Doering (2008), we sought such extreme behavior by solving PDE optimization problems with objective functionals chosen based on certain conditional regularity results and a priori estimates available for different models. No evidence for singularity formation was found in extreme Navier-Stokes flows constructed in this manner in 3D. We also discuss the results obtained for 1D Burgers and 2D Navier-Stokes systems, and while singularities are ruled out in these flows, the results presented provide interesting insights about sharpness of different energy-type estimates known for these systems. Connections to other bounding techniques are also briefly discussed.
