Equilibrium position of a rigid sphere in a turbulent jet: A problem of elastic reconfiguration

The position of floating spheres trapped within an immersed turbulent water jet is investigated. Using the self-similarity properties of the jet velocity profile, the equilibrium problem is formulated in a rescaled space where the sphere is static and deformable. This approach is found to be related to a problem of elastic reconfiguration where elasticity arises here from the geometry of the flow instead of an actual deformation of a body.

Figure 1

Schematics of the trapped sphere experiment. Two cameras are positioned at ${90}^{\circ }$ to observe the motion of the sphere in two perpendicular planes. The free jet profile is visualized from camera 2 (not represented) by the injection of dye at the nozzle. A solid circle is inserted to represent the typical trapping of a floating centimeter scale sphere submitted to buoyant forces and fluid forces with magnitudes $F_A$ and $F_f$, respectively.

Figure 2

Left: Axial velocity $v_z$ of the jet as a function of the radial distance $r$. Eleven axial distances to the nozzle are considered for $z$ starting from 84 mm going up to 264 mm with a linear spacing of 18 mm. The flow rate at the nozzle is $Q=27$ mL ${\mathrm{s}}^{-1}$. Right: Product $z{v}_z$ (${\mathrm{m}}^2$ ${\mathrm{s}}^{-1}$) as a function of the normalized radial coordinate $r/z$ for the same data points. The solid line is a fitting function established after the profile $\mathcal{F}(r/z)\propto 1/{(1+{(r/tan\left(\alpha \right)z)}^2)}^2$ [24] with $\alpha =0.12$ rad.

Figure 3

Illustration of the self-similarity. Left: Three spheres with same drag for the same ratio $z_e/R$, where $z_e$ is the equilibrium distance to the nozzle and $R$ the sphere radius. Center: Three identical spheres at their equilibrium positions for three different flow rates. The cylindrical frame $\{r,z\}$ is represented at the origin of the jet. Right: Profiles in the rescaled frame $\{r^{\prime }=r/z_e,z^{\prime }=z/z_e\}$ of the spheres' surfaces impacted by the jet for the three corresponding positions of the central figure. The associated numbers indicate the radial portion of the sphere impacted by the jet in the scaled frame $\alpha z_e/R$, where $\alpha$ is the characteristic opening angle of the jet. This dimensionless number is interpreted as a Cauchy number [see Eq. (5)].

Figure 4

Reconfiguration number [Eq. (6)] as a function of the Cauchy number [Eq. (5)] for polypropylene spheres and a melting ice ball. The solid line corresponds to the equilibrium problem (3) with an empirical expression for the drag coefficient, $\tilde{c}(r/R)=1-{(r/R)}^{\eta }$, with $\eta =0.5$. The inset plot shows the equilibrium distance to the nozzle,$z_e$, as a function of the flow rate $Q$ for the same data points.