Turbulence and jet-driven zonal flows: Secondary circulation in rotating fluids due to asymmetric forcing

We report on experiments and modeling on a rotating confined liquid that is forced by circumferential jets coaxial with the rotation axis, wherein system-scale secondary flows are observed to emerge. The jets are evenly divided in number between inlets and outlets and have zero net mass transport. For low forcing strengths the sign of this flow depends on the sign of a sloped end cap, which simulates a planetary β plane. For increased forcing strengths the secondary flow direction is insensitive to the slope sign, and instead appears to be dominated by an asymmetry in the forcing mechanism, namely, the difference in radial divergence between the inlet and outlet jet profiles. This asymmetry yields a net radial velocity that is affected by the Coriolis force, inducing secondary zonal flow.

Figure 1

Apparatus picture (a) and schematic views of the side (b) and top (c). At the inner and outer cylinder radii ($r_i$ and $r_o$) the fluid height varies ($\mathrm{\Delta }h$) due to a sloped end cap, the slope of which may be reversed to change the sign of β. The side schematic (b) includes a typical LDV diagnostic point, while the top view (c) pictures the forcing mode, $m=3$ with jets (centerline radius $r_j=12.45\mathrm{cm}$) paired, as in the experiment. Inlets for fluid extraction are shown in yellow (light gray) while outlets for fluid injection are shown in green (dark gray).

Figure 2

(a) Raw velocity data for different radii, from smallest to largest ($r=7.89$ to 15.19 cm). The forcing pumps are active from 60 to 180 s. The intermediate radial locations may be seen on the abscissa of panel (b). (b) Solid body rotation is evident when the pumps are off (dashed line), but during forcing a departure from this profile due to secondary flow may be observed. Here $\beta =-0.18$ and $z=19.4\mathrm{cm}$; these data are typical for $z$ above the midplane. The vertical lines in panel (b) between 12 and 13 cm represent the inner and outer boundaries of the jets. Error bars represent one standard deviation.

Figure 3

Azimuthal velocity departures from solid body flow during forcing. Opposing zonal flows are observed with low forcing near the β plane with +β in red (dark gray) and -β in blue (dashed, black). The $\beta =0$ case is shown in light gray. Stronger and more height-dependent induced flows, which are independent of β, appear with higher forcing. All data are with $m=3$ and $\mathrm{\Omega }=10.5\mathrm{rad}/\mathrm{s}$. Data in line with the forcing jets have been removed for clarity.

Figure 4

Simulation results on zonal flows due to jet asymmetry with $\beta =0$. (a) Azimuthal velocity difference, averaged over the azimuthal direction, for the scaled-down $\mathrm{\Omega }=50-\mathrm{rpm}$ simulation, demonstrating a prograde flow for $r>0$ near the orifices, driven by the stronger radial motions associated with fluid extraction. A retrograde flow is found well below the midplane (here at $z=4.75\mathrm{cm}$); note that the axial profile is sensitive to both $\mathrm{\Omega }$ and $\nu$. (b) Jet outflow/inflow radial profiles of axial velocity $u_z\left(\theta \right)$ and their narrowing with $\mathrm{\Omega }$, especially for extraction. The area under each curve remains constant, reflecting zero net flux. (c) Joint effect of rotation upon the secondary flow maximum magnitude, $\max \left\{u_{\theta }\right\}$, yielding an approximate 2/3 power law scaling as described in the text.