Helical vortices: Quasiequilibrium states and their time evolution

The time evolution of a viscous helical vortex is investigated by direct numerical simulations of the Navier-Stokes equations where helical symmetry is enforced. Using conservation laws in the framework of helical symmetry, we elaborate an initial condition consisting in a finite core vortex, the time evolution of which leads to a generic quasiequilibrium state independent of the initial core size. Numerical results at different helical pitch values provide an accurate characterization in time for such helical states, for which specific techniques have been introduced: helix radius, angular velocity, stream function–velocity–vorticity relationships, and core properties (size, self-similarity, and ellipticity). Viscosity is shown to be at the origin of a small helical velocity component, which we relate to the helical vorticity component. Finally, changes in time of the flow topology are studied using the helical stream function and three-dimensional Lagrangian orbits.

Figure 1

Local polar (blue) and helical (red) basis.

Figure 2

Definition of planes ${\mathrm{\Pi }}_0$ and ${\mathrm{\Pi }}_{\perp }$, as well as of local bases and coordinates used for vortex characterization. The dark spots represent the vortex core cut by either plane. Note that the sketch is done for negative values of $\eta$ and $\psi$.

Figure 3

Helical vortex of reduced pitches (a) $L=1$ and (b) $L=0.25$: isocontours of ${\omega }_{B_A}$ in the plane ${\mathrm{\Pi }}_{\perp }$. Both vortices have comparable core sizes.

Figure 4

Helical vortex of pitch $L=0.25$ at $\text{Re}=5000$ for $a_0=0.06$. Vorticity contours in the ${\mathrm{\Pi }}_{\perp }$ plane during the relaxation process, for times $\overline{t}\equiv t\mathrm{\Gamma }/2\pi a_0^2=0,10,...,80$. Contour levels are ${\omega }_{B_A}/{\omega }_{B_A}^{\left(0\right)}\left(0\right)=0.5,{10}^{-1},{10}^{-2},...,{10}^{-6}$.

Figure 5

Helical vortex of pitches $L=1$ and $L=0.25$ at $\text{Re}=5000$. (a) Time evolution of the radial position $r_A$ of the maximum of vorticity ${\omega }_B$ as a function of $\tau$, for three different initial conditions (IC), with core sizes $a_0=0.05,0.1,0.15$ corresponding to initial times $\tau =3.125$, 12.5, and 28.125. (b) Time evolution of the squared core size $a^2\left(\tau \right)$ for the two core sizes $a_0=0.05$ and 0.1 (the black dashed line is the two-dimensional diffusion law $4\tau /\text{Re}$). (c) Same as (b), but for the largest initial core size $a_0=0.15$.

Figure 6

Helical vortex for different values of $L$ between 0.25 and 3. Time evolution of the squared core size $a^2$ at $\text{Re}=5000$. The black dashed line shows the two-dimensional diffusion law $4\tau /\text{Re}$.

Figure 7

Helical vortex of pitch $L=0.5$ at $\text{Re}=5000$. (a) Axisymmetric part of the helical vorticity ${\omega }_{B_A}^{\left(0\right)}\left(\rho \right)$ at times $\tau =22.5,32.5,\cdots ,172.5$ (amplitude decreases with time). (b) Same profiles normalized in amplitude by the maximum value at each time, as a function of the similarity variable $\overline{\rho }=\rho /a$. The black dashed line shows ${\tilde{\omega }}_{B_A}^{\left(0\right)}\left(\overline{\rho }\right)=e^{-{\overline{\rho }}^2}$.

Figure 8

Helical vortex of reduced pitch $L=0.5$ at $\text{Re}=5000$. (a) Axisymmetric part of the quantity $u_H\left(\rho \right)$ at times $\tau =22.5,32.5,\cdots ,172.5$ (the profile spreads as time increases). (b) Profiles rescaled by the maximum absolute value at each time, as a function of the similarity variable $\overline{\rho }=\rho /a$. The black dashed line shows ${\tilde{u}}_H^{\left(0\right)}=-e^{-{\overline{\rho }}^2}$.

Figure 9

Helical vortex for different values of $L$ between 0.25 and 3, at $\text{Re}=5000$. (a) Helix radius $r_A$ as a function of $\tau$. (b) Same as (a) over a longer time period. (c) Angular velocity $\mathrm{\Omega }$ as a function of $\tau$; the dashed lines are the values ${\mathrm{\Omega }}_c$ predicted by the cutoff theory.

Figure 10

Helical vortex at $\text{Re}=5000$ of reduced pitch (a) and (b) $L=0.25$ at $\tau =92.5$ and and (c) and (d) $L=1$ at $\tau =1750$. (a) and (c) Isocontours in the ${\mathrm{\Pi }}_0$ plane (black lines) of the corotating stream function ${\mathrm{\Psi }}_R\equiv \mathrm{\Psi }+\frac{1}{2}{r}^2\mathrm{\Omega }$ ($\mathrm{\Omega }$ is the angular velocity of the vortex) superimposed on top of the quantity $\alpha {\omega }_B$. (b) and (d) Representation in the meridional plane ${\mathrm{\Pi }}_z$.

Figure 11

Helical vortex of pitch $L=0.5$ at $\text{Re}=5000$ for times (a) and (b) $\tau =62.5$ and (c) and (d): $\tau =162.5$. (a) and (c) Quantity $\alpha {\omega }_B$ (in color) with isocontours (black lines) of ${\mathrm{\Psi }}_R$, the corotating stream function. (b) and (d) Helical velocity $u_H$ (in color) with isocontours (white lines) of ${\mathrm{\Psi }}_R$. Representation in the ${\mathrm{\Pi }}_0$ plane.

Figure 12

Helical vortex of reduced pitch $L=0.5$ at $\text{Re}=5000$. Scatter plots of (a) $(\alpha {\omega }_B,{\mathrm{\Psi }}_R)$ and (b) $(u_H,\alpha {\omega }_B)$ at $\tau =62.5$ (blue), 112.5 (black), and 162.5 (red).

Figure 13

Helical vortex at $\text{Re}=5000$ for different values of $L$ between 0.25 and 3. (a) Time evolution of ellipticity ${\mu }_0$. Solid lines show values determined from the DNS and circles the ellipticity predicted by Eq. (51). (b) Time evolution of $\varepsilon$ measured through the DNS.

Figure 14

(a)–(e) Streamline topology in the rotating frame for a helical vortex with different core sizes $a$ at fixed $L=0.5$. For each value of $a$, streamlines are represented in planes ${\mathrm{\Pi }}_0$ (left) and ${\mathrm{\Pi }}_z$ (right). Critical points are pinpointed as solid circles. (f) Sketch of the three flow regions defined by the homoclinic orbits.

Figure 15

Rotating frame: streamline topology for one helical vortex of core size $a=0.3$ and for different $L$. Representation in planes ${\mathrm{\Pi }}_0$ (left) and ${\mathrm{\Pi }}_z$ (right). Critical points are pinpointed as closed circles.

Figure 16

Lagrangian orbits (solid black lines) in the corotating frame, around a helical vortex (green tube) for cases (a) $L=0.5$ and $a=0.06$, (b) $L=0.5$ and $a=0.25$, and (c) $L=0.25$ and $a=0.25$. The orbit is initiated in the vicinity of the vortex core (left), in the vicinity of the $z$ axis (center), and in the potential outer region (right).

Figure 17

Helical vortex of reduced pitches (a) $L=1$ and (b) $L=0.25$: monopolar vorticity component ${\omega }_{B_A}^{\left(0\right)}\left(\rho \right)$ (solid line) and its Gaussian fit (dashed line).

Figure 18

Helical vortex of reduced pitches $L=1$ (solid line) and $L=0.25$ (dashed line). Fitted core size $a$ (top curves in red) and dispersion radius $a_d$ (blue) as functions of ${\rho }_{\mathrm{cut}}/\pi Ld$. The fitted core size $a$ is found to be much less sensitive to ${\rho }_{\mathrm{cut}}$ than the dispersion radius.

Figure 19

Helical vortex for $L=0.25$ and $L=1$ at $\text{Re}=5000$. Two different core radii are considered: $a=0.22$, corresponding to $\tau =62.5$, and $a=0.30$, corresponding to $\tau =112.5$. (a) Shift $\varepsilon {\rho }^{\left(1\right)}$ of elliptical streamlines as a function of ${\rho }^{\left(0\right)}/a$. (b) Ellipticity $\mu$ as a function of ${\rho }^{\left(0\right)}/a$.