Modified Taylor vortices

The classic problem reported by Taylor [Philos. Mag. 46, 671 (1923)] on the viscous decay of a planar vortex grid is here extended in three ways. First we present solutions for the vortex grid modulated in the direction normal to the grid. Second, we find a solution for viscous decay of vortices on a disk and its modulation along the axial direction. Finally, we provide solutions for the viscous decay of vortices on a cylindrical surface and its modulation along the radial direction. The exponentially decaying vortices in these modulated vortex systems represent exact unsteady three-dimensional solutions of the Navier-Stokes equations.

Figure 1

Taylor vortices plotted for $a=2\pi$ and $b=\pi$.

Figure 2

Variation of the stream-function contours with $x$ and $z$ at $y=0$ for the modulated vortices on a plane with $d=2$.

Figure 3

(a) Vortices on a disk plotted for $n=4$ and $r=2$. (b) Variation of the stream-function contours with $r$ and $z$ on a disk for $\theta =\pi /2,k=1$, and $n=4$.

Figure 4

Contours for the stream function $\psi =sin\left(3\theta \right)cos\left(\pi z\right)$ on a cylinder showing the presence of vortices.

Figure 5

Solutions $H\left(\rho \right)$ of Eq. (41) for $n=4,\kappa =1$: (a) $\beta =2$, (b) $\beta =1$.

Figure 6

Contours of the stream function for $\beta =2$ and $n=4$ in the plane (a) $z=0$, (b) $\theta =\pi /3$.

Figure 7

Contours of the stream function in the plane $z=0$ for $n=4$ and (a) $\beta =1.4$, (b) $\beta =0.95$.

Figure 8

Dashed red lines correspond to the asymptotic solutions and the continuous blue ones are the full solutions.

Figure 9

The dashed red lines are $\tilde{H}$, the dotted blue lines are the full solutions $H$, and the continuous gray curves correspond to the approximate solutions given by Eq. (55).