arXiv subjects

Robert M. Kerr

Publications and source records attributed to Robert M. Kerr.

At least 19 recordsLinked to original sources

arXiv2024

Navier-Stokes bounds and scaling for compact trefoils in $(2\ell\pi)^3$ domains

For a perturbed trefoil vortex knot evolving under the Navier-Stokes equations, a sequence of $\nu$-independent times $t_m$ are identified corresponding to a set of scaled, volume-integrated vorticity moments $\nu^{1/4}{\it O}_{V1}$ with this hierarchy $t_\infty\le\dots\le t_m\dots t_1=t_x\approx40$ and ${\it O}_{Vm}=(\int_{V\ell}|\omega|^{2m}dV)^{1/2m}$. For $Z(t)={\it O}^2_{V1}(t)$ the volume-integrated enstrophy, convergence of $\sqrt{\nu}Z(t)$ at $t_x=t_1$ marks the end of the reconnection scaling phase. Physically, reconnection follows from the formation of a double vortex sheet, then a knot, which splits into spirals. $Z$ then accelerates, leading to approximate finite-time $\nu$-independent convergence of the energy dissipation rate $\epsilon(t)=\nu Z(t)$ at $t_\epsilon\sim 2t_x$ and sustained over a finite span $\Delta T_\epsilon\searrow 0.5 t_\epsilon$, giving Reynolds number independent finite-time, dissipation, $\Delta E_\epsilon=\int_{\Delta T_\epsilon}\epsilon dt$, and thus satisfying one definition for a {\it dissipation anomaly}. Evidence for transient Kolmogorov-like enstrophy spectra is found over ${\Delta T_\epsilon}$. A critical factor in achieving these temporal convergence laws is how the domain $V_\ell=(2\ell\pi)^3$ is increased as $\ell\sim\nu^{-1/4}$, for $\ell=2$ to 6, then to $\ell=12$, as $\nu$ decreases. $(2\ell\pi)^3$ domain compatibility with established $(2\pi)^3$ mathematics in appendix allows small $\nu$ Navier-Stokes solutions. Two spans of $\nu$ are considered. Over the first factor of 25 decrease in $\nu$, all of the $\nu^{1/4}{\it O}_{Vm}(t)$ converge to their respective $t_m$. For the next factor of 5 decrease in $\nu$, $\ell$ is increased to $\ell=12$, there is only convergence of $\nu^{1/4}\Omega_{V\infty}(t)$ to $t_\infty$ and later $\sqrt{\nu}Z(t)$ convergence at $t_1=t_x$ and $\epsilon(t)$ over $t\sim t_\epsilon$.

physics.flu-dyn↗

arXiv2023

Dependence of trefoil vortex knots upon the initial vorticity profile

Six sets of Navier-Stokes trefoil vortex knots in $(2π)^3$ 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 $ν$. Maps of the helicity density $h=u\cdotω$ 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 $ν$-independent convergence of $\sqrtνZ(t)$ at a common $t_x$. For those with the broadest wings, enstrophy growth accelerates during reconnection, leading to approximately $ν$-independent convergent finite-time dissipation rates $ε=νZ$. 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, $ε_h<0$, and small, but positive $h$. Later, the sum of these localized patches of $ε_h<0$ 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 $ε_h<0$ 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 $\sqrtνZ(t)$ phase begins and can explain accelerated growth of the enstrophy and evidence for finite-time energy dissipation $ΔE_ε$. Despite the initial symmetries.

physics.flu-dyn↗

arXiv2017

Scaling of Navier-Stokes trefoil reconnection

Perturbed, helical trefoil vortex knots and a set of anti-parallel vortices are examined numerically to identify the scaling of their helicity and vorticity norms during reconnection. For the volume-integrated enstrophy $Z=\intω^2 dV$, a new scaling regime is identified for both configurations where as the viscosity $ν$ changes, all $\sqrtνZ(t)$ cross at $ν$-independent times $t_x$, identified as when the first reconnection events end. Self-similar linear collapse of $B_ν(t)=(\sqrtνZ)^{-1/2}$ can be found for $t\lesssim t_x$ by linearly extrapolating $B_ν(t)$ to zero at critical times $T_c(ν)$, then plotting $(T_c(ν)-t_x)(B_ν(t)-B_x)$ where $B_x=B_ν(t_x)$. The size $\ell^3$ of the periodic domains must be increased as $ν$ is decreased to maintain this scaling as implied by known Sobolev space bounds. The anti-parallel calculations show that the linear collapse of $B_ν(t)$ begins with a quick, viscosity-independent exchange of the circulation $Γ$ between the original vortices and the new vortices. Up to and after the trefoil knots' first reconnection at time $t_x$, their helicity ${\cal H}$ is preserved, validating the experimental centreline helicity observation of Scheeler et al (2014a). Because the cubic Navier-Stokes velocity norm $L_3$ barely changes and the Navier-Stokes $\|ω\|_\infty$ are bounded by the Euler values, these flows are never singular. Despite this, the Navier-Stokes $Z$ can, for a brief period, grow faster than the Euler $Z$ and the following increase in the viscous energy dissipation rate $ε=νZ$ shows $ν$-independent convergence at $t\approx 2t_x$. Taken together, these results could be a new paradigm whereby smooth solutions without singularities or roughness could generate a $ν\to0$ {\it dissipation anomaly} (finite dissipation in a finite time) as $\ell\to\infty$, as seen in physical turbulent flows.

physics.flu-dyn↗

arXiv2017

Trefoil knot structure during reconnection

Three-dimensional images of evolving numerical trefoil vortex knots are used to study the growth and decay of the enstrophy and helicity. Negative helicity density ($h<0$) plays several roles. First, sheets of oppositely-signed helicity dissipation of equal magnitude on either side of the maximum of the enstrophy dissipation allow the global helicity ${\cal H}$ to be preserved through the first reconnection, as suggested theoretically (Laing et al 2015) and observed experimentally (Scheeler et al. 2014). Next, to maintain the growth of the enstrophy and positive helicity within the trefoil while ${\cal H}$ is preserved, $h<0$ forms in the outer parts of the trefoil so long as the periodic boundaries do not interfere. To prevent that, the domain size $\ell$ is increased as the viscosity $ν\to0$. Combined, this allows two sets of trefoils to form a new scaling regime with linearly decreasing $(\sqrtνZ(t))^{-1/2}$ up to common $ν$-independent times $t_x$ that the graphics show is when the first reconnection ends. During this phase there is good correspondence between the evolution of the simulated vortices and the reconnecting experimental trefoil of Kleckner and Irvine (2013) when time is scaled by their respective nonlinear timescales $t_f$. The timescales $t_f$ are based upon by the radii $r_f$ of the trefoils and their circulations $Γ$, so long as the strong camber of the experimental hydrofoil models is used to correct the published experimental circulations $Γ$ that use only the flat-plate approximation.

physics.flu-dyn↗

arXiv2016

Approach and separation of quantum vortices with balanced cores

Using two innovations, smooth, but distinctly different, scaling laws for the numerical reconnection of pairs of initially orthogonal and anti-parallel quantum vortices are obtained using the three-dimensional Gross-Pitaevskii equations, the simplest mean-field non-linear Schrödinger equation for a quantum fluid. The first innovation suppresses temporal fluctuations by using an initial density profile that is slightly below the usual two-dimensional steady-state Padé approximate profiles. The second innovation is to find the trajectories of the quantum vortices from a pseudo-vorticity constructed on the three-dimensional grid from the gradients of the wave function. These trajectories then allow one to calculate the Frenet-Serret frames and the curvature of the vortex lines. For the anti-parallel case, the scaling laws just before and after reconnection obey the dimensional $δ\sim|t_r-t|^{1/2}$ prediction with temporal symmetry about the reconnection time $t_r$ and physical space symmetry about the $x_r$, the mid-point between the vortices, with extensions of the vortex lines formng the edges of an equilateral pyramid. For all of the orthogonal cases, before reconnection $δ_{in}\sim(t-t_r)^{1/3}$ and after reconnection $δ_{out}\sim(t-t_r)^{2/3}$, which are respectively slower and faster than the dimensional prediction. In these cases, the reconnection takes place in a plane defined by the directions of the curvature and vorticity. To define the structure further, lines are drawn that connect the four arms that extend from the reconnection plane, four angles $θ_i$ between these arms are found, then summed, giving $\sumθ_i>360^\circ$. This implies that the overall structure is convex or hyperbolic, as opposed to the acute angles of the anti-parallel pyramid.

cond-mat.quant-gas↗

arXiv2015

Simulated Navier-Stokes trefoil reconnection

The evolution and self-reconnection of a perturbed trefoil vortex knot is simulated, then compared to recent experimental measurements (Scheeler et al. 2014a). Qualitative comparisons using three-dimensional vorticity isosurfaces and lines, then quantitative comparisons using the helicity. To have a single initial reconnection, as in the experiments, the trefoil is perturbed by 4 weak vortex rings. Initially there is a long period with deformations similar to the experiment during which the energy, continuum helicity and topological self-linking number are all preserved. In the next period, once reconnection has clearly begun, a Reynolds number independent fraction of the initial helicity is dissipated in a finite time. In contrast, the experimental analysis finds that the helicity inferred from the trajectories of hydrogen bubbles is preserved during reconnection. Since vortices reconnect gradually in a classical fluid, it is suggested that the essential difference is in the interpretation of the reconnection timescales associated with the observed events. Both the time when reconnection begins, and when it ends. Supporting evidence for the strong numerical helicity depletion is provided by spectra, a profile and visualisations of the helicity that show the formation of negative helicity on the periphery of the trefoil. A single case with the same trajectory and circulation, but a thinner core, replicates this helicity depletion despite larger Sobolev norms, showing that the reconnection timescale is determined by the initial trajectory and circulation of the trefoil, not the initial vorticity. This case also shows that the very small viscosity, $ν\rightarrow 0$ mathematical restrictions upon finite-time dissipative behavior do not apply to this range of modest viscosities.

physics.flu-dyn↗

arXiv2014

Regimes of nonlinear depletion and regularity in the 3D Navier-Stokes equations

The periodic $3D$ Navier-Stokes equations are analyzed in terms of dimensionless, scaled, $L^{2m}$-norms of vorticity $D_{m}$ ($1 \leq m < \infty$). The first in this hierarchy, $D_{1}$, is the global enstrophy. Three regimes naturally occur in the $D_{1}-D_{m}$ plane. Solutions in the first regime, which lie between two concave curves, are shown to be regular, owing to strong nonlinear depletion. Moreover, numerical experiments have suggested, so far, that all dynamics lie in this heavily depleted regime \cite{DGGKPV13}\,; new numerical evidence for this is presented. Estimates for the dimension of a global attractor and a corresponding inertial range are given for this regime. However, two more regimes can theoretically exist. In the second, which lies between the upper concave curve and a line, the depletion is insufficient to regularize solutions, so no more than Leray's weak solutions exist. In the third, which lies above this line, solutions are regular, but correspond to extreme initial conditions. The paper ends with a discussion on the possibility of transition between these regimes.

nlin.CD↗

arXiv2013

Incompressible hydrodynamic turbulence from a chain reaction of vortex reconnection events

From a new anti-parallel initial condition using long vortices, three-dimensional turbulence forms after two reconnection steps and the formation of at least one vortex ring. The long domain is needed in order to accommodate the multiple reconnections, which enhance vortex stretching rates and the generation of small-scale vortex structures within the vortex rings. In addition to making the initial vortices very long, new features introduced with this initial condition are an initial profile less likely to shed vortex sheets and an improved method for mapping the direction of the vorticity onto the three-dimensional mesh. To get to the turbulent state, the vortices progress through the following steps: First, until the first reconnection, the vortex dynamics is largely consistent with existing work on strong, possibly singular, growth of the vorticity in the Euler equations. Second, vortex reconnection at the junction of the primary symmetry planes meet. About half of the circulation from each vortex reconnects into two "bridges", leaving behind two "threads", with the bridges and threads arranged into 4 orthogonal pairs. In the third step, stretching is induced by new anti-parallel attractions due to the twisting that forms along the reconnected vortices. This extra stretching then pulls on the threads as they wind around the bridges, resulting in spirals over the entire vortices. In the fourth step, multiple vortex rings form through multiple reconnections, with each ring consisting of spiral vortices. A $k^{-5/3}$ energy spectra begins to form after the first ring separates and eventually covers one decade. It is argued that the spirals are the source of the $k^{-5/3}$ spectra that develops. Furthermore, a new hierarchy of rescaled vorticity moments is found where the lower-order moments bound the higher-order moments for all orders and all times.

physics.flu-dyn↗

arXiv2012

Bounds on a singular attractor in Euler using vorticity moments

A new rescaling of the vorticity moments and their growth terms is used to characterise the evolution of anti-parallel vortices governed by the 3D Euler equations. To suppress unphysical instabilities, the initial condition uses a balanced profile for the initial magnitude of vorticity along with a new algorithm for the initial vorticity direction. The new analysis uses a new adaptation to the Euler equations of a rescaling of the vorticity moments developed for Navier-Stokes analysis. All rescaled moments grow in time, with the lower-order moments bounding the higher-order moments from above, consistent with new results from several Navier-Stokes calculations.Furthermore, if, as an inviscid flow evolves, this ordering is assumed to hold, then a singular upper bound on the growth of these moments can be used to provide a prediction of power law growth to compare against. There is a significant period where the growth of the highest moments converges to these singular bounds, demonstrating a tie between the strongest nonlinear growth and how the rescaled vorticity moments are ordered. The logarithmic growth of all the moments are calculated directly and the estimated singular times for the different $D_m$ converge to a common value for the simulation in the best domain.

nlin.CD↗

arXiv2010

Numerical generation of a vortex ring cascade in quantum turbulence

A symmetric anti-parallel quantum pair of vortices is simulated using the three-dimensional Gross-Pitaevski equations. The initial development before cores interact directly demonstrates the traditional vortex dynamics of stretching, curvature and torsion in a manner consistent with a filament calculation and simulations of the classical, ideal Euler equations. Once the cores begin to interact, reconnection develops in the vacuum that forms between the pair. Out of the reconnection region, vortex waves are emitted with properties similar to waves in the local induction approximation. These waves propagate down the initial vortex and deepen. When they deepen far enough, secondary reconnections occur and vortex rings form. Near this time, spectra have a $k^{-3}$ regime. As the vortex rings fully separate, the high wavenumber spectra grow until, at the final time simulated, spectra in two directions develop nearly -5/3 subranges. This occurs without the dissipation of energy. Preliminary analysis of the flow of energies in spectral scale and physical space is discussed.

cond-mat.quant-gas↗

arXiv2008

3D Euler about a 2D Symmetry Plane

Initial results from new calculations of interacting anti-parallel Euler vortices are presented with the objective of understanding the origins of singular scaling presented by Kerr (1993) and the lack thereof by Hou and Li (2006). Core profiles designed to reproduce the two results are presented, new more robust analysis is proposed, and new criteria for when calculations should be terminated are introduced and compared with classical resolution studies and spectral convergence tests. Most of the analysis is on a $512 \times 128 \times 2048$ mesh, with new analysis on a just completed $1024 \times 256 \times 2048$ used to confirm trends. One might hypothesize that there is a finite-time singularity with enstrophy growth like $Ω\sim (T_c-t)^{-γ_Ω}$ and vorticity growth like $||ω||_\infty \sim (T_c-t)^{-γ}$. The new analysis would then support $γ_Ω\approx 1/2$ and $γ> 1$. These represent modifications of the conclusions of Kerr (1993). Issues that might arise at higher resolution are discussed.

physics.flu-dyn↗

arXiv2006

Computational Euler History

A new pseudospectral calculation of collapsing Euler vortices \cite{HouLi06} has called into question the long-term conclusions of singular behavior described earlier in \cite{Kerr93,Kerr05}. This review is designed to: improve the discussion of the detailed analysis of one test initial condition designed find sources of errors, to compare that with calculations showing no evidence of a singularity, and to document two sets of discussions. Those prior to 1993 between competing teams. And recent discussions of what is needed to reach more convincing conclusions.

physics.flu-dyn↗

arXiv2006

Quaternions and particle dynamics in the Euler fluid equations

Vorticity dynamics of the three-dimensional incompressible Euler equations is cast into a quaternionic representation governed by the Lagrangian evolution of the tetrad consisting of the growth rate and rotation rate of the vorticity. In turn, the Lagrangian evolution of this tetrad is governed by another that depends on the pressure Hessian. Together these form the basis for a direction of vorticity theorem. Moreover, in this representation, fluid particles carry ortho-normal frames whose Lagrangian evolution in time are shown to be directly related to the Frenet-Serret equations for a vortex line. The frame dynamics suggest an elegant Lagrangian relation regarding the pressure Hessian tetrad. The equations for ideal MHD are similarly considered.

nlin.CD↗

arXiv2002

Transient vortex events in the initial value problem for turbulence

A vorticity surge event that could be a paradigm for a wide class of bursting events in turbulence is studied to examine how the energy cascade is established and how this event could serve as a new test of LES turbulence models. This vorticity surge event is tied to the formation of the energy cascade in a direct numerical simulation by the traditional signatures of a turbulent energy cascade such as spectra approaching -5/3 and strongly Beltramized vortex tubes. A coherent mechanism is suggested by the nearly simultaneous development of a maximum of the peak vorticity $\|ω\|_\infty$, growth of the dissipation, the appearance of a helically aligned local vortex configuration and strong, transient oscillations in the helicity wavenumber spectrum. This coherence is also examined for two LES models, a traditional purely dissipative eddy viscosity model and a modern method (LANS$-α$) that respects the nonlinear transport properties of fluids. Both LES models properly represent the spectral energy and energy dissipation associated with this vorticity surge event. However, only the model that preserves nonlinear fluid transport properties reproduces the helical properties, including Beltrami-like vortex tubes.

nlin.CD↗

arXiv2001

The energy budget in Rayleigh-Benard convection

It is shown using three series of Rayleigh number simulations of varying aspect ratio AR and Prandtl number Pr that the normalized dissipation at the wall, while significantly greater than 1, approaches a constant dependent upon AR and Pr. It is also found that the peak velocity, not the mean square velocity, obeys the experimental scaling of Ra^{0.5}. The scaling of the mean square velocity is closer to Ra^{0.46}, which is shown to be consistent with experimental measurements and the numerical results for the scaling of Nu and the temperature if there are strong correlations between the velocity and temperature.

nlin.CD↗

arXiv2000

Helicity in Hydro and MHD Reconnection

Helicity, a measure of the linkage of flux lines, has subtle and largely unknown effects upon dynamics. Both magnetic and hydrodynamic helicity are conserved for ideal systems and could suppress nonlinear dynamics. What actually happens is not clear because in a fully three-dimensional system there are additional channels whereby intense, small-scale dynamics can occur. This contribution shows one magnetic and one hydrodynamic case where for each the presence of helicity does not suppress small-scale intense dynamics of the type that might lead to reconnection.

astro-ph↗

arXiv2000

An inertial range length scale in structure functions

It is shown using experimental and numerical data that within the traditional inertial subrange defined by where the third order structure function is linear that the higher order structure function scaling exponents for longitudinal and transverse structure functions converge only over larger scales, $r>r_S$, where $r_S$ has scaling intermediate between $η$ and $λ$ as a function of $R_λ$. Below these scales, scaling exponents cannot be determined for any of the structure functions without resorting to procedures such as extended self-similarity (ESS). With ESS, different longitudinal and transverse higher order exponents are obtained that are consistent with earlier results. The relationship of these statistics to derivative and pressure statistics, to turbulent structures and to length scales is discussed.

physics.flu-dyn↗

arXiv2000

New tests for a singularity of ideal MHD

Analysis using new calculations with 3 times the resolution of the earlier linked magnetic flux tubes confirms the transition from singular to saturated growth rate reported by Grauer and Marliani \cite{GrauerMar99} for the incompressible cases is confirmed. However, all of the secondary tests point to a transition back to stronger growth rate at a different location at late times. Similar problems in ideal hydrodynamics are discussed, pointing out that initial negative results eventually led to better initial conditions that did show evidence for a singularity of Euler. Whether singular or near-singular growth in ideal MHD is eventually shown, this study could have bearing on fast magnetic reconnection, high energy particle production and coronal heating.

physics.plasm-ph↗