Conical shear-driven parametric instability of steady flow in precessing spheroids

We investigate experimentally the instability of steady flow of fluid confined in weakly precessing spheroids. It is known that the conical-shear instability (CSI) proposed by Lin, Marti, and Noir [Phys. Fluids 27, 046601 (2015)] grows in a precessing sphere, and the elliptical and shearing instabilities [Kerswell, Geophys. Astrophys. Fluid Dyn. 72, 107 (1993)] can grow in precessing spheroids. Previous theories predict that when $\text{Re}\lesssim {\eta }^{-5}$, where $\text{Re}$ is the Reynolds number defined by the spin angular velocity and the equatorial radius of the spheroid and $\eta$ is its ellipticity, CSI dominates the other two instabilities even in a spheroid, and that, in particular, when $\eta <0.17$ the critical Poincaré number (i.e., the critical precession rate) ${\text{Po}}^{\left(c\right)}$ is proportional to ${\text{Re}}^{-3/10}$ for $200{\eta }^{-2}\ll \text{Re}\lesssim {\eta }^{-5}$. To experimentally verify these predictions, we measure a long time-series of fluid velocity to accurately estimate the power spectrum. Then, we determine ${\text{Po}}^{\left(c\right)}$ as a function of $\text{Re}$ and show the qualitative change of its scaling depending on $\eta$. Our experimental results perfectly support the theoretical predictions, implying that CSI can grow in precessing spheroids.

Figure 1

Three precessing containers used in the experiments: (a) sphere ($\eta =0$), (b) prolate spheroid ($\eta =-1/9$), and (c) oblate spheroid ($\eta =1/10$). Precession frame $(x,y,z)$ defined as in (a) rotates at ${\mathit{\Omega }}_p$ with respect to the laboratory frame. We consider the case in which ${\mathit{\Omega }}_p\perp {\mathit{\Omega }}_s$.

Figure 2

Power spectrum of (a) solid-body rotational flow ($\text{Po}=0$), (b) steady flow ($\text{Po}=0.0046$), and (c) unsteady flow ($\text{Po}=0.0048$) in the precessing sphere ($\text{Re}=4.0\times {10}^4$). The frequency is normalized by spin period $T_s$. Gray and blue peaks represent the steady circulations of the container and fluid, respectively. The red peak observed in (c) corresponds to the unsteady mode.

Figure 3

Power spectrum for an unsteady flow in the oblate spheroid ($\text{Re}=4.0\times {10}^4$ and $\text{Po}=0.0425$). Gray and blue peaks represent the steady circulations of the container and fluid, respectively. The red peak corresponds to the unsteady mode.

Figure 4

Flow states in the precessing sphere (circles), prolate spheroid ($\eta =-1/9$, squares), and oblate ($\eta =1/10$, triangles). Closed and open symbols denote steady and unsteady flows, respectively. Solid, dotted, and dashed lines indicate the scalings ${\text{Re}}^{-3/10}, {\text{Re}}^{-4/5}$, and ${\text{Re}}^{-1/2}$, respectively.

Figure 5

The magnitude ${\omega }^{\left(c\right)}$ of the angular velocity of the approximately uniform vorticity flow at the critical point $\text{Po}={\text{Po}}^{\left(c\right)}$. Lines (black, the sphere; red, the prolate spheroid; blue, the oblate spheroid) are the estimation on the basis of the spectral peaks (such as the blue peaks in Figs. 2 and 3). Symbols ($•$, the sphere; $\blacksquare$, the prolate spheroid; $▴$, the oblate spheroid) are the theoretical predictions by Eq. (5).

Figure 6

The magnitude ${\epsilon }^{\left(c\right)}$ of the differential rotation at the critical point $\text{Po}={\text{Po}}^{\left(c\right)}$. Symbols are the same as in Fig. 4. Solid lines indicate the scaling ${\epsilon }^{\left(c\right)}\sim {\text{Re}}^{-3/10}$.

Figure 7

Flow states in the precessing sphere. Closed and open circles denote steady and unsteady flows, respectively. The results were obtained by PIV (black), LDV (red), and visualizations (blue). The dotted line is the same as that in Fig. 4, which is proportional to ${\text{Re}}^{-4/5}$.