Dynamics of a reactive spherical particle falling in a linearly stratified fluid

Motivated by numerous geophysical applications, we have carried out laboratory experiments of a reactive (i.e., melting) solid sphere freely falling by gravity in a stratified environment, in the regime of large Reynolds (Re) and Froude numbers. We compare our results to nonreactive spheres in the same regime. First, we confirm, for larger values of Re, the stratification drag enhancement previously observed for low and moderate Re. We also show an even more significant drag enhancement due to melting, much larger than the stratification-induced one. We argue that the mechanism for both enhancements is similar, due to the specific structure of the vorticity field sets by buoyancy effects and associated baroclinic torques, as deciphered for stratification by Zhang et al. [J. Fluid Mech. 875, 622 (2019)]. Using particle image velocimetry, we then characterize the long-term evolution (at time $t\gg 1/N$, with $N$ the Brünt-Väisälä frequency) of the internal wave field generated by the wake of the spheres. The measured wave field is similar for both the reactive and inert spheres: indeed, each sphere fall might be considered as a quasi-impulsive source of energy in time and the horizontal direction, as the falling time (the sphere radius) is much smaller than $N$ (than the tank width). Internal gravity waves are generated by wake turbulence over a broad spectrum, with the least damped component being at the Brünt-Väisälä frequency and the largest admissible horizontal wavelength. About 1% of the initial potential energy of each sphere is converted into kinetic energy in the internal waves, with no significant dependence on the Froude number over the explored range.

Figure 1

Diagram of the seven regimes of wakes observed in the experiments or numerical studies [1]. The solid and empty large markers denote our experiments with the reactive and plastic spheres, respectively. This figure is a modification of Fig. 4 of Hanazaki et al. [21].

Figure 2

Schema of the experimental setup for the small tank. The 50 FPS camera (1) allows us to perform PIV measurements over the whole tank. We use a 50 mm lens on the 500 FPS camera to track the rapid dynamics of the sphere wake in a reduced domain. A green laser of 1 W is used here.

Figure 3

(a) Density profile of the four different linear stratifications in the small tank. One stratification ($N=0.142$ Hz) has been used with backlighting to track the sphere position (red solid line and squares). Two different stratifications ($N=0.094\text{and}0.136$ Hz) have been used for PIV measurements (dashed lines and circles). The last stratification ($N=0.196$ Hz) has been used with the two measurement techniques (black line and diamonds). (b) Density profile in the large tank, only used with the backlighting technique: the linear stratification, of the same depth as in the small tank, is enclosed between the top and bottom homogeneous layers, with a noticeable density jump around the bottom interface.

Figure 4

Phase diagram of binary mixtures of water (${\mathrm{H}}_2\mathrm{O}$) and sodium chloride (NaCl). (s) and (l) denote the solid and the liquid phases, respectively. The subscripts $e$, $pe$, and $tp$ denote the eutectic, peritectic, and triple points of the phase diagram, respectively. The NaCl $2{\mathrm{H}}_2\mathrm{O}$ phase is a hydrated salt and is called hydrohalite. The fluid with a salt concentration of 25 wt% (red star) is cooled below the eutectic temperature (vertical red arrow) where a solid mixture is formed by pure ice and hydrohalite (horizontal dotted red arrows). The blue lines indicate the liquid density.

Figure 5

Experiment in a small beaker of a sphere melting in a turbulent fluid using a magnetic stirrer.

Figure 6

Evolution of the (a) radius, (b) fluid temperature, and (c) fluid density for two similar experiments (red and black) of a reactive sphere melting in well-mixed pure water. Thick solid lines [in (a)] and dotted lines with circles [in (b) and (c)] denote measurements. Thin solid lines correspond to the model presented in Sec. 3a with a kinematic viscosity $9\times {10}^{-7}{\mathrm{m}}^2$ ${\mathrm{s}}^{-1}$ and a velocity 35 cm ${\mathrm{s}}^{-1}$.

Figure 7

Evolution of the radius normalized by its initial value [5 (red), 7.9 (blue), 14 mm (black), respectively] during the fall with a kinematic viscosity of $9\times {10}^{-7}{\mathrm{m}}^2$ ${\mathrm{s}}^{-1}$ (dotted lines). The falling velocity and time are based on an experiment in the large tank with a stratification $N=0.164$ Hz (see Fig. 9).

Figure 8

Evolution of the velocity during the fall of the plastic spheres for two different radii, (a), (b) 14.3 and (c), (d) 3.2 mm, in (a), (c) a homogeneous fluid and in (b), (d) a linear stratified layer with $N=0.164$ Hz. The left column shows the horizontal and vertical position of each sphere, together with the norm of the velocity, $U\left(z\right)=\sqrt{u_z^2+2u_x^2}$ ($u_z$ and $u_x$ are the vertical velocity and the horizontal velocity, respectively), as a color scale. Note that in (b) and (d), the stratification starts just above the top of the figure and ends at the bottom of the figure, i.e., at $z=25$ cm. The middle column represents the evolution of the norm of the velocity $U\left(z\right)$ for each sphere. The lines are colorized when the spheres are in the stratified layer, starting from $t=0$ at the top interface. The right column shows the distribution of the norm of the velocity, $U\left(z\right)$, of all spheres through the fall. The Reynolds number Re and the Froude number Fr are calculated with the median of the velocity distribution.

Figure 9

Same as Fig. 8 for the reactive spheres [(a), (b) 14, (c) 7.9, and (d) 5 mm, respectively] in the large tank with (a), (c) a homogeneous fluid and with (b), (d) a linear stratification of $N=0.164$ Hz. Note that (c) is for a different radius than (d).

Figure 10

Drag coefficient $C_d^m$ as a function of the Reynolds number Re for three 5-mm-radius reactive spheres falling in a stratified layer ($N=0.164$ Hz), considering various hypotheses for the sphere radius and the fluid viscosity. The solid line corresponds to Eq. (12) and the gray area denotes the $+6$% and $-4$% estimated uncertainty on this equation [52]. Small dot markers denote instantaneous values during the fall of each sphere, while the large squares show the corresponding median values and standard deviations.

Figure 11

Drag coefficient $C_d^m$ as a function of the Reynolds number Re. The solid line corresponds to Eq. (12) and the gray area denotes the $+6$% and $-4$% estimated uncertainty on this equation (Clift et al. (1978) [52]). The diamonds and squares denote the experiments in the small and large tanks, respectively, and the empty and filled symbols stand for the plastic and reactive spheres, respectively. (a) is for a homogeneous fluid and (b) is for a stably stratified one.

Figure 12

Measured drag coefficient as a function of the Froude number Fr at all $z$ positions for (a) plastic spheres and (b) reactive spheres. The color scale denotes the Reynolds number Re. (For interpretation of the references to color in this figure legend, the reader is referred to the online version of this paper.)

Figure 13

Added drag coefficient $C_d^m/C_d^H-1$ as a function of the Richardson number Ri. Our experimental results are the squares and diamonds (filled or empty for reactive or plastic spheres, respectively). The filled and empty triangles denote the previous numerical results of [16, 21, 22]. The pentagon denotes an Argo float [2]. The crosses and stars correspond to the numerical results of Yick et al. [14]. The dashed lines represent the high-Reynolds-number regime suggested by Zhang et al. [15] for different values of Re. Here, all data are for a Schmidt number of 700. The value of the Froude number for each data point is scaled with the color bar. (For interpretation of the references to color in this figure legend, the reader is referred to the online version of this paper.)

Figure 14

Evolution of the kinetic energy in the stably stratified layer with $N=$ 0.196 Hz when (a) a plastic sphere and (b) a reactive sphere of 14 mm radius fall. The blue line is smoothed with a moving average to remove all high-frequency noise ($>2$ Hz). The black line is filtered with a low-pass filter with a cutoff frequency $f_c=0.25$ Hz to keep only the signature of propagating internal waves. An exponential fit is computed for the signal between 20 and 130 s. (For interpretation of the references to color in this figure legend, the reader is referred to the online version of this paper.)

Figure 15

Spatial mean of the frequency spectrum of the vertical velocity field for three different stratifications with Brünt-Väisälä frequency (vertical bold black line) $N=$ 0.094, 0.136, and 0.196 Hz. Dashed and solid lines correspond to the reactive and plastic spheres respectively. Blue, black, and red denote three different radius 14, 7.9, and 5 mm for the reactive spheres and 14.2, 7.9, 4.8 mm for the plastic spheres.

Figure 16

Spatially averaged spectrogram of the vertical velocity in the stably stratified layer with $N=$ 0.196 Hz (black line) during and after the fall of a 14-mm-radius reactive sphere. Spectrum energy is an arbitrary logarithmic scale. The vertical red line denotes the launch time of the sphere.

Figure 17

(a) The attenuation ${\omega }_I^{E_k}$, (b) the initial wave amplitude $A_0$, and (c) the ratio between the initial wave kinetic energy $E_k(t=0)$ and the potential energy $E_p$ as a function of the radius of the spheres for three different stratifications with a Brünt-Väisälä frequency $N=0.094$ Hz (red), 0.136 Hz (green), and 0.196 Hz (black). The empty and full diamonds denote the plastic and reactive spheres, respectively. The error bars denote the standard deviation based on several experiments performed with the same spheres (between two and four launches for each size and each stratification). In (b), the solid black line shows an $a^3$ slope. (For interpretation of the references to color in this figure legend, the reader is referred to the online version of this paper.)

Figure 18

(a), (b) Spatiotemporal diagram of the vertical velocity induced by a plastic sphere (14.3 mm radius) falling in a stably stratified layer with $N=0.196$ Hz, (a) as a function of $x$ at middle tank depth and (b) as a function of $z$ in the middle of the tank. The vertical velocity is filtered with a pass-band filter around $N$. (c) Power density spectrum for $k_x$ for the two radially symmetric windows highlighted in (a).