Impact of a small ellipticity on the sustainability condition of developed turbulence in a precessing spheroid

We experimentally investigate the flow transition to developed turbulence in a precessing spheroid with a small ellipticity. Fully developed turbulence appears through a subcritical transition when we fix the Reynolds number (the spin rate) and gradually increase the Poincaré number (the precession rate). In the transitional range of the Poincaré number, two qualitatively different turbulent states (i.e., fully developed turbulence and quiescent turbulence with a spin-driven global circulation) are stable and they are connected by a hysteresis loop. This discontinuous transition is in contrast to the continuous transition in a precessing sphere, for which neither bistable turbulent states nor hysteresis loops are observed. The small ellipticity of the container makes the global circulation of the confined fluid more stable, and it requires much stronger precession of the spheroid, than a sphere, for fully developed turbulence to be sustained. Nevertheless, once fully developed turbulence is sustained, its flow structures are almost identical in the spheroid and sphere. The argument [Lorenzani and Tilgner, J. Fluid Mech. 492, 363 (2003); Noir et al., Geophys. J. Int. 154, 407 (2003)] on the basis of the analytical solution [Busse, J. Fluid Mech. 33, 739 (1968)] of the steady global circulation in a weak precession range well describes the onset of the fully developed turbulence in the spheroid.

Figure 1

A precessing oblate spheroid with ellipticity 0.1. We drive the precession by spinning, with an angular velocity ${\mathbf{\Omega }}_s$, the container on a turntable rotating with ${\mathbf{\Omega }}_p$. The minor axis of the spheroid is parallel to the spin axis and the two axes are perpendicular $\left({\mathbf{\Omega }}_s\perp {\mathbf{\Omega }}_p\right)$.

Figure 2

Experimental apparatus. An acrylic cylindrical container with a spheroidal or spherical cavity spins (with the angular velocity ${\mathbf{\Omega }}_s)$ on a turntable which rotates with ${\mathbf{\Omega }}_p$. A laser sheet, which rotates together with the turntable, for flow visualization and PIV is always on the equatorial plane of the cavity. Visualized flow is recorded by a camera fixed on the turntable.

Figure 3

Flow visualizations by using reflective flakes of two turbulent states in the oblate spheroid at a common set of parameters $(\text{Re}=4.0\times {10}^4$ and $\text{Po}=0.055)$. (a) Quiescent turbulence, which is dominated by a spin-driven global circulation. (b) Developed turbulence. Movies are available in the Supplemental Material [38].

Figure 4

[(a), (b)] Mean velocity and [(c), (d)] turbulence intensity on the equatorial plane evaluated by the PIV over 200 spin periods. The parameters are same as in Fig. 3: $\text{Re}=4.0\times {10}^4$ and $\text{Po}=0.055$. We observe [(a), (c)] QT and [(b), (d)] DT. The vertical arrow in panels (a) and (b) indicates the wall spin velocity $a{\mathrm{\Omega }}_s$. The turbulence intensity in panels (c) and (d) is defined by the rms value of a velocity component normalized by the magnitude of the local mean velocity. The region where PIV data are absent is shaded in light blue in panels (c) and (d). The dot in panels (a) and (b) indicates the location for $U_x$ to be calculated in Fig. 7.

Figure 5

The downward triangles $(▿)$ denote the maximum $\text{Po}$ for QT to be sustained when we increase $\text{Po}$ for a given $\text{Re}$, while the upward triangles $\left(▵\right)$ denote the minimum $\text{Po}$ for DT to be sustained when we decrease $\text{Po}$. These are plotted on the basis of visualizations of flow in the oblate spheroid. The gray curve denotes ${\text{Po}}^{\left(t\right)}$ at which the inclination angle $\phi$ (Fig. 6) of the steady spin-driven global circulation becomes a decreasing function of $\text{Po}$ for a fixed $\text{Re}$. For comparison, the maximum $\text{Po}$ for steady flows and the minimum $\text{Po}$ for periodic flows are plotted as a function of $\text{Re}$ by open and closed squares [29], respectively. The dashed line indicates the scaling $\text{Po}\propto {\text{Re}}^{-3/10}$.

Figure 6

The cosine of the angle $\phi$ between the axis of the spin-driven steady circulation and the spin axis as a function of $\text{Po}$ for five different values of the Reynolds number: $\text{Re}=5\times {10}^3,{10}^4,2\times {10}^4,4\times {10}^4$, and $8\times {10}^4$ (from right to left). We evaluate $\phi$ on the basis of Eq. (6), which is valid for steady flow in a low-$\text{Po}$ range. Black curves, for the oblate spheroid $\left(\eta =0.1\right)$; gray curves, for a sphere. Circles on the black curves denote the points at which $d\phi /d\text{Po}=0$.

Figure 7

The temporal average $U_x$ of a horizontal velocity component at a point (at the distance $0.75a$ on the precession axis) shown in Figs. 4 and 4 on the equatorial plane as a function of $\text{Po}$. This quantity distinguishes the flow states. Namely, smaller (i.e., larger in magnitude) values of $U_x$ correspond to QT or laminar flows, whereas the larger values correspond to DT. Red curves (with closed circles) are the results with increasing $\text{Po}$; blue curves (with open circles) are those with decreasing $\text{Po}$. (a) Results for the oblate spheroid and (b) sphere. A hysteresis loop is observed for the spheroid but not for the sphere. We fix the Reynolds number at $\text{Re}=4.0\times {10}^4$. In panel (a), ${\text{Po}}^{\left(t\right)}\left(\approx 0.0528\right)$ is indicated by the vertical dashed line.

Figure 8

Flake visualization of flow in a precessing sphere. The Reynolds number is fixed at $\text{Re}=4.0\times {10}^4$ and the Poincaré number is gradually increased as (a) $\text{Po}=0.02$, (b) 0.024, (c) 0.028, and (d) 0.032. Movies are available in the Supplemental Material [38].

Figure 9

Mean velocity field in (a) the oblate spheroid $\left(\eta =0.1,\text{Po}=0.07\right)$ and (b) a sphere $\left(\text{Po}=0.055\right)$. The Reynolds number is common $\left(\text{Re}=4.0\times {10}^4\right)$. The mean flow structures look very similar to each other despite the difference of cavity shapes. Vertical thick arrows indicate the wall speed $\left(a{\mathrm{\Omega }}_s\right)$ on the equatorial plane.