Steady circulation induced by inertial modes in a librating cylinder

The paper is devoted to an experimental study of steady flows excited by an oscillating motion of fluid in a cylinder, the rotation rate of which periodically changes (librations). The rotational oscillations lead to the appearance of a system of averaged toroidal vortices in a viscous Stokes boundary layer close to the cylinder side wall. The number of vortices depends on the frequency of librations, σ. If σ exceeds twice the rotation rate, the steady flow structure has the form of two vortices located close to the cavity ends, i.e., corner flow. If $\sigma <2$ at some libration frequencies, inertial modes—natural modes of rotating fluid oscillations in the closed cavity—are excited. In this case additional averaged vortices arise. The number of vortices is determined by the axial wave number of the inertial mode. With a decrease in the dimensionless frequency of fluid oscillation, the vortices increase in size, which leads to a transformation of the vortex structure. In this case, the flow near the ends is a superposition of the flows excited by inertial mode and corner flow. At the same time, the vortices which are the most distant from the periphery do not feel the corner flow. It is shown that the viscous dissipation of the oscillating motion in the fluid bulk leads to a linear decrease in the velocity of steady circulation with the dimensionless frequency.

Figure 1

Scheme of the experimental setup.

Figure 2

The velocity field averaged over the period of librations in the axial section of the cylinder (a) and track visualization of the flow structure (b) at $\sigma =2.2$, $\varepsilon =0.10$, and $E=1.8\times {10}^{-3}$. The color shows the φ component of the averaged vorticity field, $\overline{\mathrm{ro}{\mathrm{t}}_{\phi }\mathbf{u}}$.

Figure 3

Instantaneous and averaged velocity field at $\sigma =1.28$ {4,1} and $\omega =3900$ (a), (b), and at $\sigma =1.56$ {6,1} and $\omega =4700$ (d), (e). The amplitude of librations and the Ekman number are $\varepsilon =0.12$ and $E=3.3\times {10}^{-4}$ respectively. Panels (c) and (f) show the schemes of steady circulation in the Stokes layer.

Figure 4

(a) The distribution of the averaged velocity modulus $|u_z|$ along the cylinder side wall at $\sigma =1.28$, $\varepsilon =0.12$, and $E=1.6\times {10}^{-4}$; (b) dependence of the maximum fluid velocity in the vortices on the amplitude of librations $\varepsilon$ at a fixed value of the axial wave number at $n=4$ and different values of radial number $m$.

Figure 5

Distribution of the dimensionless averaged velocity $V=|{\overline{u}}_z^m|/\left({\mathrm{\Omega }}_{\mathrm{rot}}R{\varepsilon }^2\right)$ over the libration frequency $\sigma$ at $E=1.6\times {10}^{-4}$; labels 2,1, 6,2, etc. mean the numbers of inertial modes $\left\{n,m\right\}$.

Figure 6

Averaged velocity field and its track visualization at $\sigma =0.77$ (mode {2,1}), $\varepsilon =0.15$, and $\omega =330$ (a), (e); $\sigma =1.28$ {4,1}, $\varepsilon =0.15$, and $\omega =540$ (b), (f); $\sigma =1.56$ {6,1}, $\varepsilon =0.15$, and $\omega =660$ (c), (g); $\sigma =1.72$ {8,1}, $\varepsilon =0.20$, and $\omega =730$ (d), (h). The Ekman number for all cases is $E=2.4\times {10}^{-3}$. In panels (a)–(d) the color shows the φ component of the vorticity field averaged over the libration period, $\overline{\mathrm{ro}{\mathrm{t}}_{\phi }\mathbf{u}}$; solid curves with arrows in panel (e) schematically illustrate the structure of the steady circulation.

Figure 7

Averaged velocity field and its track visualization at $\sigma =1.28$ {4,1}, $\varepsilon =0.15$, $\omega =110$, and $E=1.2\times {10}^{-2}$.

Figure 8

Dependence of the velocity of steady circulation on the Ekman number $E$.

Figure 9

Dependence of the dimensionless velocity of steady circulation excited by the fluid oscillations in the fluid bulk, $V$ (points: 1), and the velocity of the corner flow, $V_c$ (rhombuses: 2), on the frequency of librations $\sigma$ at $E=2.4\times {10}^{-3}$. Shading indicates the viscosity dissipated modes.

Figure 10

Dependence of averaged velocity on the dimensionless frequency of fluid oscillations, $\omega$.

Figure 11

The velocity of corner flow versus the dimensionless frequency $\omega$.