Three-dimensional flow structures of turbulence in precessing spheroids

We conduct direct numerical simulations of turbulence sustained in slowly precessing spheroids with the absolute value of the ellipticity $\eta$ being between zero (i.e., a sphere) and 0.2. Using a flexible grid generation algorithm, we can effectively simulate flows in an arbitrarily shaped container. This enables us to investigate the ellipticity dependence of the precession-driven flow in spheroids with the spin and precession axes being at a right angle. The numerical results are in excellent agreement with experimental data under the same flow conditions. In particular, we numerically realize hysteresis loops, which have been well known since the seminal experiments by Malkus [Science 160, 259 (1968)], connecting two qualitatively different states in a precessing spheroid with non-negligible ellipticity $\left|\eta \right|$ larger than about $10/\sqrt{\text{Re}}$ (where $\text{Re}$ denotes the Reynolds number defined by using the spin angular velocity). Our numerical simulations reveal the three-dimensional turbulent flow structures in these states. One is a high-energy state where the mean flow is approximated by a uniform-vorticity flow. The other is a low-energy state with twisted mean-flow streamlines, which lead to fully developed turbulence when the Reynolds number is high enough. The mean-flow structures in the low-energy state are common irrespective of the ellipticity; namely, the main component of the mean flow is a circulation about the axis perpendicular both to the spin and precession axes, but the torsion of the mean-flow streamlines is larger for smaller $\eta$. For sufficiently high Reynolds numbers, the low-energy state and therefore developed turbulence are sustained for the Poincaré number (the precession rate) larger than about $\left|\eta \right|/2$. On the other hand, stronger precession leads to the significant reduction of turbulence in a central region of the container. Hence, a container with smaller $\left|\eta \right|$ is adequate to sustain developed turbulence with weak precession.

Figure 1

Precessing spheroid with the spin and precession axes being at a right angle. Numerical grid is shown on (a) the cavity wall and (b) the equatorial plane. For visibility, the number of the grid points is reduced. The coordinates $(x,y,z)$ fixed on the rotating frame (the precession frame) are defined so that the spin and precession axes are parallel to the $z$ and $x$ axes, respectively.

Figure 2

Comparison of the temporally averaged (over 50 spin periods) velocity field on the equatorial plane of (a), (b) the precessing sphere and (c), (d) spheroid with $\eta =0.1$. Results of (a), (c) the DNSs with the number of grid points being ${98}^3$ and (b), (d) laboratory experiments. Parameters are common: $\text{Re}={10}^4$ and $\text{Po}=0.1$. The length of the vectors is normalized so that $a{\mathrm{\Omega }}_s$ becomes unity. The DNS results are plotted, by interpolation, only on the points where the experimental data exist.

Figure 3

The temporal average $E$ of the kinetic energy in the precession frame as a function of $\text{Po}$. The energy $E$ attains the maximum for the solid-body rotational flow ($\text{Po}=0$). We examine four different values ($\eta =0$, black curve; 0.1, dark gray; 0.15, gray, 0.2, light gray) of the ellipticity of the spheroidal container. The Reynolds number is fixed at $\text{Re}={10}^4$. Upper triangles are the results of DNSs with increasing $\text{Po}$, and lower triangles are those with decreasing $\text{Po}$. We observe hysteresis loops for $\eta =0.15$ and 0.2.

Figure 4

Temporally averaged velocity fields in the bistable states for $\text{Re}={10}^4$ and $\text{Po}=0.085$ in the precessing spheroid with $\eta =0.15$. We show the mean velocity fields (in the precession frame) on three cross sections of the flow in the (a)–(c) high-$E$ state and (d)–(f) low-$E$ state. It is seen in panel (c) that the fluid velocity in the high-$E$ state is as fast as the velocity $a{\mathrm{\Omega }}_s$ of the cavity wall on the equatorial plane.

Figure 5

(a), (b) The contour on the equatorial plane and (c), (d) the isosurface of the average $(u_x^{\prime }+u_y^{\prime }+u_z^{\prime })/3$ of root mean squares of three velocity components for $\text{Re}={10}^4$ and $\text{Po}=0.085$ in the precessing spheroid with $\eta =0.15$. (a), (c) High-$E$ and (b), (d) low-$E$ states. Note the difference in the contour levels in panels (a) and (b). The threshold of the isosurface is (c) 0.014 and (d) 0.07.

Figure 6

Three-dimensional flow structures in a $\text{Po}$ range around the hysteresis loop for $\text{Re}={10}^4$ and $\eta =0.15$. The Poincaré number is (a) $\text{Po}=0.08$, (b), (c) 0.085, and (d) 0.09. For $\text{Po}=0.08$ and 0.09, we observe only flows in the (a) high- and (d) low-$E$ states, respectively. The precession with $\text{Po}=0.085$ sustains bistable states with (b) high and (c) low energy. Panel (e) shows the theoretical prediction [2] of the angle $\phi$ between the spin ($z$) axis and the vorticity vector of the steady uniform-vorticity flow. The dotted vertical lines in panel (e) indicate the three values ($0.08,0.085$, and 0.09) of $\text{Po}$. The red and blue straight lines in panels (a)–(d) indicate the direction of the uniform vorticity vector of the steady flow in the high- and low-$E$ states, respectively.

Figure 7

Isosurfaces (blue objects) of the instantaneous enstrophy field in the (a) high-$E$ state and (b) low-$E$ state. The threshold is set to be (a) $2.5{\mathrm{\Omega }}_s^2$ and (b) $12{\mathrm{\Omega }}_s^2$. Structures in the boundary layer near the wall ($\sqrt{x^2+y^2+{[z/(1-\eta )]}^2}>0.9a$) are not shown. Red curves are the streamlines shown in Fig. 6.

Figure 8

Laminarization in a central region along the precession axis for higher $\text{Po}$. We draw streamlines of the temporally averaged field in a hemisphere ($x>0$). The Reynolds number is fixed at $\text{Re}={10}^4$, and the Poincaré number is changed as (a) $\text{Po}=0.2$, (b) 0.3, (c) 0.4, (d) 0.5, (e) 1, and (f) 2. The ellipticity of the cavity is $\eta =0.15$.

Figure 9

Streamlines of mean flow in the spheroids with different ellipticity: (a) $\eta =0$, (b) 0.1, and (c) 0.2. Flow parameters are common: $\text{Po}=0.1$ and $\text{Re}={10}^4$. Streamlines are drawn from the spin axis, and colored in red for those towards $y>0$ and in blue for those towards $y<0$. Red (or blue) streamlines are right-hand (or left-hand) spirals.

Figure 10

Streamlines of mean flow in (a) the precessing sphere and prolate spheroids with the ellipticity (b) $\eta =-0.1$ and (c) $-0.2$. Flow parameters are common: $\text{Po}=0.1$ and $\text{Re}={10}^4$. Panel (a) shows the same streamlines as in Fig. 9 but from a different viewpoint. Note that we rotate the objects so that we can see the twisted vortices; the line of sight in panel (c) is different with the angle $\pi /4$ (around the $z$ axis) from panels (a) and (b).