Towards a finite-time singularity of the Navier–Stokes equations Part 1. Derivation and analysis of dynamical system

Abstract

The evolution towards a finite-time singularity of the Navier–Stokes equations for flow of an incompressible fluid of kinematic viscosity $\unicode[STIX]{x1D708}$ is studied, starting from a finite-energy configuration of two vortex rings of circulation $\pm \unicode[STIX]{x1D6E4}$ and radius $R$, symmetrically placed on two planes at angles $\pm \unicode[STIX]{x1D6FC}$ to a plane of symmetry $x=0$. The minimum separation of the vortices, $2s$, and the scale of the core cross-section, $\unicode[STIX]{x1D6FF}$, are supposed to satisfy the initial inequalities $\unicode[STIX]{x1D6FF}\ll s\ll R$, and the vortex Reynolds number $R_{\unicode[STIX]{x1D6E4}}=\unicode[STIX]{x1D6E4}/\unicode[STIX]{x1D708}$ is supposed very large. It is argued that in the subsequent evolution, the behaviour near the points of closest approach of the vortices (the ‘tipping points’) is determined solely by the curvature $\unicode[STIX]{x1D705}(\unicode[STIX]{x1D70F})$ at the tipping points and by $s(\unicode[STIX]{x1D70F})$ and $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D70F})$, where $\unicode[STIX]{x1D70F}=(\unicode[STIX]{x1D6E4}/R^{2})t$ is a dimensionless time variable. The Biot–Savart law is used to obtain analytical expressions for the rate of change of these three variables, and a nonlinear dynamical system relating them is thereby obtained. The solution shows a finite-time singularity, but the Biot–Savart law breaks down just before this singularity is realised, when $\unicode[STIX]{x1D705}s$ and $\unicode[STIX]{x1D6FF}/\!s$ become of order unity. The dynamical system admits ‘partial Leray scaling’ of just $s$ and $\unicode[STIX]{x1D705}$, and ultimately full Leray scaling of $s,\unicode[STIX]{x1D705}$ and $\unicode[STIX]{x1D6FF}$, conditions for which are obtained. The tipping point trajectories are determined; these meet at the singularity point at a finite angle. An alternative model is briefly considered, in which the initial vortices are ovoidal in shape, approximately hyperbolic near the tipping points, for which there is no restriction on the initial value of the parameter $\unicode[STIX]{x1D705}$; however, it is still the circles of curvature at the tipping points that determine the local evolution, so the same dynamical system is obtained, with breakdown again of the Biot–Savart approach just before the incipient singularity is realised. The Euler flow situation ($\unicode[STIX]{x1D708}=0$) is considered, and it is conjectured on the basis of the above dynamical system that a finite-time singularity can indeed occur in this case.

Direct numerical and large-eddy simulation of trefoil knotted vortices

“…The result from this work shows that the high-Reynolds-number case results in smaller radii of curvature at both tipping points (namely, and mm) compared with the low-Reynolds-number case ( and mm), as illustrated in figure 9 and reported in table 4. This is as expected, as in the high-Reynolds-number case the tipping points experience more intense stretching, resulting in a higher local curvature, following the mechanism explained by Moffatt & Kimura (2019). A slight increase in curvature at tipping points is observed when the pair of tipping points approaches from far field; however, unlike what is seen in Moffatt & Kimura (2019), when tipping points are getting close enough and the bridging starts, the curvatures are flattened instead of increasing rapidly, and the vortex lines on both sides of the dividing plane tend to be more anti-parallel.…”

“…This is as expected, as in the high-Reynolds-number case the tipping points experience more intense stretching, resulting in a higher local curvature, following the mechanism explained by Moffatt & Kimura (2019). A slight increase in curvature at tipping points is observed when the pair of tipping points approaches from far field; however, unlike what is seen in Moffatt & Kimura (2019), when tipping points are getting close enough and the bridging starts, the curvatures are flattened instead of increasing rapidly, and the vortex lines on both sides of the dividing plane tend to be more anti-parallel. This process is depicted in figure 9( b ), where the radii of curvature reach a minimum value and then increase as the two vortex tubes approach each other.…”

“…The vortex centreline configuration at such a tipping point is nearly symmetric: as seen from figure 9 and table 4, the angles between the vectors normal to the vortex line and the line connecting two tipping points ( and , defined as inclination angles) being almost equal for both Reynolds number cases. Curvature at the tipping point, as studied by Moffatt & Kimura (2019), has been considered to influence the dynamics of reconnection. The result from this work shows that the high-Reynolds-number case results in smaller radii of curvature at both tipping points (namely, and mm) compared with the low-Reynolds-number case ( and mm), as illustrated in figure 9 and reported in table 4.…”

Self-attenuation of extreme events in Navier–Stokes turbulence

“…Given that intense vorticity is arranged in tubes with weak curvature, additional insight could be obtained by a simple kinematic analysis of stretching generated by such structures. To this end, we consider a simple axisymmetric vortex tube with a radius of curvature R c 27 – 29 . Utilizing a curvilinear polar coordinate system: ( , , ), which, respectively, correspond to unit vectors in the radial direction, the azimuthal direction and the direction tangent along the (curved) axis of the tube, we assume that the vorticity is of the form .…”

Separation scaling for viscous vortex reconnection

“…) – which definitely breaks the local assumption required for the scaling. Inspired by the recent works of Moffatt & Kimura (2019 a , b ) on the finite time singularity of Euler and N–S equations, we studied reconnection of two colliding slender vortex rings (the ratio between the initial vortex core size and the radius of the ring is approximately 0.01) and found that before reconnection follows a scaling when (Yao & Hussain 2020 a ). The main objective of the present work is to further elucidate the time scaling of minimum separation distance for (classical) viscous vortex reconnection.…”