Sustaining mechanism of small-scale turbulent eddies in a precessing sphere

It has been known for a long time that fully developed turbulence is sustained in a precessing container. The aim of the present study is to reveal the sustaining mechanism of turbulence in a precessing sphere by means of laboratory experiments. We conduct experiments using a Newtonian fluid (water) and viscoelastic fluids (dilute solutions of surfactant, cetyltrimethyl ammonium chloride, and polymers, polyethylene oxide) to understand the sustaining mechanism of turbulence of Newtonian fluids by examining turbulence modifications due to the surfactant and polymer additives. When the Reynolds number based on the spin angular velocity and radius of the sphere is fixed, the most developed turbulence is sustained with the Poincaré number (the precession rate) being about 0.1. The key ingredient of the developed turbulence is a pair of large-scale vortex tubes which robustly exists in the flow. Assuming that these vortex tubes sustain small-scale turbulent eddies through an energy cascading process, we can explain all our experimental observations. Concerning the turbulence modification by the additives, the time-scale criteria by Lumley [J. Polymer Sci.: Macromol. Rev. 7, 263 (1973)] and the refined theory by Tabor and de Gennes [Europhys. Lett. 2, 519 (1986)] explain the experimental result that the pair of large-scale vortex tubes survives even when small-scale turbulent eddies are drastically suppressed by the surfactant additive.

Figure 1

Precessing sphere. We examine the case that the axes of the spin and precession are perpendicular to each other. Here ${\mathbf{\Omega }}_s$ and ${\mathbf{\Omega }}_p$ denote the spin and precession angular velocities, respectively.

Figure 2

Schematics of a pair of large-scale vortex tubes in turbulence of a Newtonian fluid for $\text{Po}=0.1$. (Here $\text{Po}$ indicates the precession rate; see Sec. 2a for the definition.) These are depicted on the basis of DNSs [50, 51]. Black arrows are streamlines of the averaged velocity field of turbulence. Red and blue curves show the cross-sectional flows on the $x\text{−}y$ and $z\text{−}x$ planes, respectively. The vortices are anticyclones with angular velocities approximately antiparallel to ${\mathbf{\Omega }}_p$.

Figure 3

Experimental apparatus. Two stepper motors drive the spin and precession of the acrylic container. A laser sheet visualizes the flow on the equatorial plane of the spherical container. A camera fixed on the turntable records visualized flows through the observation window at the bottom of the container. It is mounted on a linear stage to record flows at different locations on the equatorial plane.

Figure 4

Visualization with using aluminum flakes of turbulence for $\text{Re}=8.02\times {10}^4$ and $\text{Po}=0.1$. (a) Water, (b) CTAC solution (50 ppm), and (c) PEO solution (50 ppm). Movies are available in the Supplemental Materials [68].

Figure 5

Snapshots of the $z$-component vorticity, ${\omega }_z$, normalized by ${\mathrm{\Omega }}_s$ for $\text{Re}=8.02\times {10}^4$ and $\text{Po}=0.1$. (a) Water, (b) CTAC solution (50 ppm), and (c) PEO solution (50 ppm). The left, center, and right columns show the regions that $-1\le x\lesssim -0.2,-0.4\lesssim x\lesssim 0.4$, and $-0.2\lesssim x\le 1$, respectively.

Figure 6

Temporal average of ${({\omega }_z/{\mathrm{\Omega }}_s)}^2$ for $\text{Re}=8.02\times {10}^4$ and $\text{Po}=0.1$. (a) Water, (b) CTAC solution (50 ppm), and (c) PEO solution (50 ppm).

Figure 7

Temporally averaged velocity field on the equatorial plane for $\text{Re}=8.02\times {10}^4$ and $\text{Po}=0.1$. (a) Water, (b) CTAC solution (50 ppm), and (c) PEO solution (50 ppm). The vertical thick arrow indicates the wall speed, $a{\mathrm{\Omega }}_s$, on the equatorial plane. The direction of the spin of the container is counterclockwise.

Figure 8

Reynolds-number dependence of the temporally averaged value of ${({\omega }_z/{\mathrm{\Omega }}_s)}^2$ for $\text{Po}=0.1$. (a, d, g) $\text{Re}=1.01\times {10}^4$, (b, e, h) $2.03\times {10}^4$, and (c, f, i) $4.01\times {10}^4$. (a–c) Water, (d–f) CTAC solution (50 ppm), and (g–i) PEO solution (50 ppm).

Figure 9

Reynolds-number dependence of the temporally averaged velocity fields for $\text{Po}=0.1$. (a, d, g) $\text{Re}=1.01\times {10}^4$, (b, e, h) $2.03\times {10}^4$, and (c, f, i) $4.01\times {10}^4$. (a–c) Water, (d–f) CTAC solution (50 ppm), and (g–i) PEO solution (50 ppm). The vertical thick arrow indicates the wall speed, $a{\mathrm{\Omega }}_s$, on the equatorial plane. The direction of the spin of the container is counterclockwise.

Figure 10

Two possibilities of the Reynolds-number dependence of the turbulence modification. Right panels show the shear rate, $\dot{\gamma }$, as a function of (a) the distance from the wall and (b) the size of eddies for two different Reynolds numbers, ${\text{Re}}_{\mathrm{low}}<{\text{Re}}_{\mathrm{high}}$. (a) First possibility: small-scale turbulent eddies in a central region of the sphere are sustained by being advected from the wall. If this is the case, the region without the suppression of these eddies expands toward the center of the sphere as $\text{Re}$ increases because the shear rate, $\dot{\gamma }$, of the flow gets smaller with the distance from the wall and the turbulence suppression occurs at the location where the time scale, ${\dot{\gamma }}^{-1}$, of the flow coincides with the characteristic time, $\tau$, of the fluid. The left panel depicting the viscosity, $\eta$, of the CTAC solution as a function of shear rate [65, 66] shows that $\eta$ gets larger and takes a maximum value around ${\dot{\gamma }}^{-1}\approx \tau$. Note that $\dot{\gamma }$ at a given location is larger for higher $\text{Re}$ in our experimental setup. (b) Second possibility in terms of (b-1) the elastic theory and (b-2) non-Newtonian viscosity: the turbulence is sustained by an energy cascading process in the bulk. If this is the case, only small eddies are reduced for lower $\text{Re}$, but larger ones are also reduced for higher $\text{Re}$ because the suppression occurs at the length scale, ${\ell }_c$, which is larger for higher $\text{Re}$ in our experiments. The precise meaning of the truncation scale, ${\ell }_c$, is different in (b-1) the elastic theory and in (b-2) the viscous theory, but this Reynolds-number dependence is common.

Figure 11

Instantaneous profile of shear rate, $\dot{\gamma }$, of turbulence of a Newtonian fluid for $\text{Po}=0.1$. DNS results for our experimental setup $(a=90$ mm and $\nu ={10}^{-6}{\mathrm{m}}^2$/s). (a) $\text{Re}=1\times {10}^4$ and (b) $2\times {10}^4$. Contour levels are in a logarithmic scale. Recall that the reciprocal of the characteristic time of the CTAC solution is about $10{\mathrm{s}}^{-1}$ (see Sec. 2c).

Figure 12

Shear rate, $\dot{\mathrm{\Gamma }}$, of temporally averaged velocity fields of turbulence of a Newtonian fluid for $\text{Po}=0.1$. DNS results for our experimental setup $(a=90$ mm and $\nu ={10}^{-6}{\mathrm{m}}^2$/s). (a) $\text{Re}=5\times {10}^3$, (b) $1\times {10}^4$, and (c) $2\times {10}^4$.

Figure 13

Visualization with using mica flakes coated with titanium dioxide of turbulence for $\text{Po}=0.02$ and $\text{Re}=8.02\times {10}^4$. (a) Water and (b) CTAC solution (50 ppm). Movies are available in the Supplemental Materials [68].

Figure 14

Visualization with using aluminum flakes of turbulence for $\text{Po}=0.2$. (a, e) $\text{Re}=1.01\times {10}^4$, (b, f) $\text{Re}=2.03\times {10}^4$, (c, g) $4.01\times {10}^4$, and (d, h) $8.02\times {10}^4$. (a–d) Water and (e–h) CTAC solution (50 ppm).