A Model of Interacting Navier-Stokes Singularities

Abstract

We introduce a model of interacting singularities of Navier-Stokes equations, named pinçons. They follow non-equilibrium dynamics, obtained by the condition that the velocity field around these singularities obeys locally Navier-Stokes equations. This model can be seen as a generalization of the vorton model of Novikov that was derived for the Euler equations. When immersed in a regular field, the pinçons are further transported and sheared by the regular field, while applying a stress onto the regular field that becomes dominant at a scale that is smaller than the Kolmogorov length. We apply this model to compute the motion of a pair of pinçons. A pinçon dipole is intrinsically repelling and the pinçons generically run away from each other in the early stage of their interaction. At a late time, the dissipation takes over, and the dipole dies over a viscous time scale. In the presence of a stochastic forcing, the dipole tends to orientate itself so that its components are perpendicular to their separation, and it can then follow during a transient time a near out-of-equilibrium state, with forcing balancing dissipation. In the general case where the pinçons have arbitrary intensity and orientation, we observe three generic dynamics in the early stage: one collapse with infinite dissipation, and two expansion modes, the dipolar anti-aligned runaway and an anisotropic aligned runaway. The collapse of a pair of pinçons follows several characteristics of the reconnection between two vortex rings, including the scaling of the distance between the two components, following Leray scaling tc-t.

Keywords: non-equilibrium dynamics; singularity; turbulence.

Figure 1

Streamlines (white curves) of velocity field around a pinçon of intensity $\gamma =0.6$ in a plane that contains the axis of the pinçon $\mathbf{\gamma }=\gamma {\mathit{e}}_z$, which is represented by the black arrow. The field is axisymmetric around that axis. The color represents the norm of rescaled velocity (a) and vorticity (b) fields, where $\nu$ is the kinematic viscosity and $r_0$ some insignificant length scale. The coordinates $x,y,z$ are also nondimensionalized by $r_0$. There is no azimuthal component of velocity, while the vorticity is purely azimuthal with respect to the axis $\mathbf{\gamma }$.

Figure 8

Time evolution of the distance r, the pinçons angles ${\theta }_1$, and ${\theta }_2$ (one line for each), total intensity ${\gamma }_1^2+{\gamma }_2^2$ and anisotropy $({\gamma }_1^2-{\gamma }_2^2)/({\gamma }_1^2+{\gamma }_2^2)$ in the two expansion cases. (a) case of repelling dipolar expansion; (b) case of aligned expansion. The points are colored by the value of the pair mutual angle, $\phi$.

Figure 9

(a) Time evolution of the distance r, the pinçons angles ${\theta }_1$, and ${\theta }_2$ (one line for each), total intensity ${\gamma }_1^2+{\gamma }_2^2$ and anisotropy $({\gamma }_1^2-{\gamma }_2^2)/({\gamma }_1^2+{\gamma }_2^2)$ in the case of explosive collapse. (b) Phase space for a pair of pinçons in the initial stage of the dynamics, just after pinçons creation. The phase space is r, $\gamma ={\gamma }_1^2+{\gamma }_2^2$ and $\xi =\xi \left({\theta }_1\right)\xi \left({\theta }_2\right)$. The points are colored by the value of the pair mutual angle $\phi$ in all the figures.

Figure 2

Parameters of a pinçon as a function of its intensity $\gamma$. (a) intensity of the force produced by the pinçon at its location. The black dashed line has equation $y=16\pi \gamma$; (b) generalized momentum of a pinçon. The black dashed line has equation $y=8\gamma /3$.

Figure 3

(a) Schematic geometry of pinçons creation at reconnection; (b) geometry of the dipole: two pinçons located at ${\mathbf{x}}_{\alpha }$ and ${\mathbf{x}}_{\beta }$, and such that initially ${\mathbf{\gamma }}_{\alpha }+{\mathbf{\gamma }}_{\beta }=0$. By convention, the angle $\theta$ is the angle between ${\mathbf{\gamma }}_{\alpha }$ and $\mathbf{r}={\mathbf{x}}_{\alpha }-{\mathbf{x}}_{\beta }$. The (a) is adapted from Figure 3 of [34].

Figure 4

Dynamics of a dipole of pinçon for various initial conditions and (a) without friction, corresponding to the initial stage of the dynamics, just after pinçons creation; (b) with friction coefficient 0.7, corresponding to the late stage dynamics. The radius is initially fixed to $r=1$, the dipole intensity is initially set to $\gamma =0.1$ and the initial dipole orientation is fixed at different values between 0 and $\pi$ (identified by different colors). The panel represents the time evolution of the different quantities: E: Interaction energy; $r^2$: Square of Dipole separation; $\gamma$: Dipole intensity; $\theta$: Dipole orientation. Black dashed lines on $\theta \left(t\right)$ figure correspond to $arccos\left(\frac{1}{\sqrt{3}}\right),\frac{\pi }{2}$, and $\pi -arccos\left(\frac{1}{\sqrt{3}}\right)$.

Figure 5

Phase space for a dipole of pinçons color-coded by the time, shown on the color bar. (a) Without friction, corresponding to the initial stage of the dynamics, just after pinçons creation; (b) With friction—coefficient 0.7—corresponding to the late stage dynamics. The radius is initially fixed to $r=1$, the dipole intensity is initially set to $\gamma =0.1$ and the initial dipole orientation is fixed at different values between 0 and $\pi$.

Figure 6

Comparison with the weak limit approximation: solid lines: complete model; dashed line: weak limit; (a) without friction; (b) with friction, coefficient 0.7. The panel represents the time evolution of the different quantities: E: Interaction energy; $r^2$: Square of Dipole separation; $\gamma$: Dipole intensity; $\theta$: Dipole orientation. Black dashed lines on $\theta \left(t\right)$ figure correspond to $arccos\left(\frac{1}{\sqrt{3}}\right),\frac{\pi }{2}$, and $\pi -arccos\left(\frac{1}{\sqrt{3}}\right)$.

Figure 7

Effect of noise in the weak limit dissipative case for dissipation coefficient 0.7 (a) on the dynamics. The panel represents the time evolution of the different quantities: E:Interaction energy; $r^2$ Square of Dipole separation; $\gamma$: Dipole intensity; $\theta$: Dipole orientation. Black dashed lines on a $\theta \left(t\right)$ figure correspond to $arccos\left(\frac{1}{\sqrt{3}}\right),\frac{\pi }{2}$, and $\pi -arccos\left(\frac{1}{\sqrt{3}}\right)$; (b) on the phase-space, color-coded by the time, shown on the color bar.The radius is initially fixed to $r=1$, the dipole intensity is initially set to $\gamma =0.1$ and the initial dipole orientation is fixed at different values between 0 and $\pi$. The intensity of the noise is $\mu =0.009$.

Figure 10

(a) Histogram of the values of the power law exponent $\delta$; (b) squared distance rescaled by for the explosive collapse cases. The black dashed line corresponds to the Leray scaling with $\delta =1/2$. We see that, for $t/t_c$ close to 1, most of the curves follow a power law with a power exponent $\delta$ close to $1/2$.

Figure 11

Time evolution of the different variables characterizing the pair of pinçons for seven cases with the same initial configuration but with different forcing characteristic time coefficient $\lambda ={\tau }_{\nu }/{\tau }_{\mathrm{forcing}}=r_0^2/\left(\nu {\tau }_{\mathrm{forcing}}\right)$ and dissipation coefficient $\rho =(\psi \left(0\right)/C_{\psi }){(r_0/\eta )}^2$. We see that, on the one hand, when dissipation is high and the forcing time is short, the pinçons die very fast with no close interaction. On the other hand, if the dissipation is low, the dynamics are very similar to the case without dissipation, and the pinçons still collapse in an explosive manner with their intensities tending to 1. Two intermediate cases are found where we have both the collapse dynamics and a separation dynamics without explosion.

Figure 12

Dynamics of a pair of pinçons as a function of time in the plane defined by ${\mathbf{\gamma }}_1$ and $\mathbf{r}=r{\mathbf{e}}_z$ for two different forcing characteristic time coefficient ($\lambda$) and dissipation coefficient $\rho$: (a) $(\rho ,\lambda )=(0.18,1.00)$ and (b) $(\rho ,\lambda )=(1.00,0.10)$. The vectors gives the projection of ${\mathbf{\gamma }}_1$ and ${\mathbf{\gamma }}_2$ in the plane and the color codes the time, from $t=0$ (dark blue) to $t=t_{\mathrm{final}}$ (dark red), as well as the coordinate of the points on the vertical axis x. In both cases, the two pinçons (blue and red points) move initially towards each other (the distance is read on the horizonal axis z) and then their orientations change and they repel each other. In the case of (a), the transition between the two phases has a configuration similar to the dipole with one of the pinçons axis abruptly turning from an angle close to 0 to an angle close to $\pi$, then the pinçons die very fast. In the case of (b), the dynamics are smoother with hardly any change in the dipole relative orientation, only the axis angles change slowly, and the pinçons survive a long time with a stable configuration.

Figure 13

Time evolution of the maximum of rescaled velocity field (a) and vorticity field (b) around a pair of pinçons with same initial configurations and different forcing characteristic time coefficient $\lambda ={\tau }_{\nu }/{\tau }_{\mathrm{forcing}}$ and dissipation coefficient $\rho$. The rescaling is the same as in Figure 1, with $r_0$ the initial separation between the pinçons. The values correspond to a moving average over 15 time steps.