Low-to-moderate Reynolds number swirling flow in an annular channel with a rotating end wall

This paper presents a new method for solving analytically the axisymmetric swirling flow generated in a finite annular channel from a rotating end wall, with no-slip boundary conditions along stationary side walls and a slip condition along the free surface opposite the rotating floor. In this case, the end-driven swirling flow can be described from the coupling between an azimuthal shear flow and a two-dimensional meridional flow driven by the centrifugal force along the rotating floor. A regular asymptotic expansion based on a small but finite Reynolds number is used to calculate centrifugation-induced first-order correction to the azimuthal Stokes flow obtained as the solution at leading order. For solving the first-order problem, the use of an integral boundary condition for the vorticity is found to be a convenient way to attribute boundary conditions in excess for the stream function to the vorticity. The annular geometry is characterized by both vertical and horizontal aspect ratios, whose respective influences on flow patterns are investigated. The vertical aspect ratio is found to involve nontrivial changes in flow patterns essentially due to the role of corner eddies located on the left and right sides of the rotating floor. The present analytical method can be ultimately extended to cylindrical geometries, irrespective of the surface opposite the rotating floor: a wall or a free surface. It can also serve as an analytical tool for monitoring confined rotating flows in applications related to surface viscosimetry or crystal growth from the melt.

Figure 1

Partial view of the channel geometry under consideration. The floor is rotating slowly while the inner and outer side walls (at $r_i$ and $r_o$, respectively) are maintained stationary.

Figure 2

Truncation threshold $N$ required to achieve an absolute precision better than $\epsilon ={10}^{-3}$ on the calculated azimuthal velocity at leading order, $v_{\theta ,0}(r,h/r_o)$, as a function of the normalized distance from the rotating floor $z=z^{\prime }h/r_o$, with ($▴$, $s=0.004$) and without ($\blacksquare$, $s=0$) relaxed boundary conditions at the corners.

Figure 3

Azimuthal velocity profiles at leading order $v_{\theta ,0}(r,z)$ as a function of the normalized distances $r$ and $z$ where $s$ = 0 and geometrical aspect ratios $r_i/r_o=0.8$ and $h/r_o=0.2$ (line curves). Comparison with profiles computed with boundary conditions relaxed by gap conditions ($•$ $s=0.001$, $\blacksquare$ $s=0.002$, and $▴$ $s=0.004$).

Figure 4

Sensitivity of azimuthal velocity profiles at leading order $v_{\theta ,0}(r,z)$ to the $s$ gap relaxation parameter for two different depths ($z=0.01h/ro$ and $z=0.1h/ro$): ($•$) $s=0$, ($▴$) $s=0.002$, ($\blacksquare$) $s=0.004$, ($⧫$) $s=0.006$, and ($▾$) $s=0.01$. (A) Near the inner side wall ($r\sim \frac{r_i}{r_o}$). (B) Near the outer side wall ($r\sim 1$). Geometry: $r_i/r_o$ = 0.8 and $h/r_o=0.2$.

Figure 5

Comparison along a free liquid surface among ($-$) the azimuthal velocity profile, as calculated at $\mathcal{O}\left({\mathrm{Re}}^2\right)$ in this paper, $v_{{\theta }_0}(r,h/r_o)+{\mathrm{Re}}^2{v}_{{\theta }_2}(r,h/r_o)$; ($\diamond$) the azimuthal velocity, $v_{\theta }(r,h/r_o)$, as calculated from DNS by LH98 [16]; and ($\circ$) the velocity as measured by Manheimer and Schechter [25], for the following parameters: $\mathrm{Re}=146$, $r_i/r_o=0.8$ and $h/r_o=0.13$, $s=0.004$).

Figure 6

Ratio of maximum radial velocity over maximum azimuthal velocity, $v_{r_{\max }}/v_{{\theta }_{\max }}$, along the liquid surface, as a function of vertical aspect ratio for $\mathrm{Re}=1$, $r_i/r_o=0.8$, and $s=0.004$. Horizontal and vertical dotted lines are drawn to identify the flow pattern calculated in LH98 [16].

Figure 7

Maximum radial and azimuthal components of the velocity along the liquid surface, $v_{r_{\max }}$ (right axis) and $v_{{\theta }_{\max }}$ (left axis), respectively, as a function of the Reynolds number. Comparison between velocities at successive $\mathcal{O}\left({\mathrm{Re}}^n\right)$ orders ($n=0$, 1, 2) and DNS calculations of LH98 [16] for $\frac{r_i}{r_o}=0.8$, $\frac{h}{r_o}=0.2$, and $s=0.004$.

Figure 8

Streamlines for different values of the vertical aspect ratio with the following horizontal aspect ratio and gap width: $r_i/r_o=0.8$ and $s=0.004$. (a) $h/r_o=0.06$. (b) $h/r_o=0.14$. (c) $h/r_o=0.19$. (d) $h/r_o=0.3$. All values along the streamlines when multiplied by ${10}^6$ give the stream function ${\psi }_1$.

Figure 9

Streamlines for different values of the horizontal aspect ratio and a fixed vertical aspect ratio, $h/r_o=0.2$: (a) $r_i/r_o=0.8$ and $s=0.004$, (b) $r_i/r_o=0.6$ and $s=0.008$, (d) $r_i/r_o=0.4$ and $s=0.012$, (d) $r_i/r_o=0.1$ and $s=0.018$. All values along the streamlines when multiplied by ${10}^6$ give the stream function ${\psi }_1$.

Figure 10

Comparison along the liquid surface between ($\square$) the experimental data of Hirsa et al. [14] and (–) the azimuthal velocity, $v_{\theta ,0}(r,h/r_o)$, as calculated at zeroth order $[Z=Z_k\left(v_{\theta },\omega ,\psi \right)=0$ in (A2) and (A3)] with $\frac{r_i}{r_o}=0.77$, $\frac{h}{r_o}=0.11$, $\mathrm{Bo}=0.027$, and $\mathrm{Re}=250$.