2.1. Experimental set-up

Experiments are performed by continuously sieving sugar above a cylindrical water tank of height $38.5 \,\,{\textrm{cm}}$ and diameter $28.5\,\,{\textrm{cm}}$ containing $24.5 \,{\textrm{l}}$ of initially fresh water ( $\rho _0=1000 \,{\textrm{kg m}}^{-3}$ , $\nu = 10^{-6} \,{\textrm{m}}^{2}\,{\textrm{s}}^{-1}$ ). The tank is filled long before the experiments to ensure that water is at room temperature, i.e. 22 $^\circ$ C on average. A sieve of radius $R_{\textit{sieve}} = 8.25 \,\,{\textrm{cm}}$ is placed $8.15\,\,{\textrm{cm}}$ above the water’s free surface. A motor controls the horizontal sinusoidal or triangular oscillations of the sieve above the tank with a controlled amplitude and frequency. No accumulation of sugar at the free surface nor any noticeable delay was observed before the penetration of sugar grains underwater, likely promoted by dissolution.

To visualise sugary water, we cook our own fluorescent sugar using rhodamine. Appendix A provides details to minimise clogging of the sieves when grinding and sorting sugar grains by size. The size of grains is constrained by the mesh size of the sieves that are used to separate them, with a mesh size in the list $\{56,125,140,180,224,450,1000\}{\,\unicode{x03BC} {\textrm{m}}}$ . In the following, we assume that grains are spherical, with a characteristic initial radius $r_p$ calculated as the arithmetic average of the minimum and maximum radii within a range listed in table 1. The assumption that grains are spherical is motivated by the fact that observations with a microscope reveal that sugar grains have an aspect ratio near unity. This assumption only modifies our calculations of the grains’ settling velocity and dissolution rate by prefactors of order unity. A key point is that a new calibration of the mass rate is required for each experiment.

Table 1. Average radius of sugar grains and particle Reynolds number $\textit{Re}_{\kern-1pt p} = 2r_p w_s/\nu$ based on their settling velocity $w_s$ , depending on the range of diameters they belong to (see (2.4)).

Figure 1. (a) Photograph of the experimental set-up sketched in (b); the tank is seeded with orange fluorescent PIV particles. (b) The camera with a green filter records the green light reflected by ordinary sugar grains until they fully dissolve – see the green dots in the triangular region called the dissolution layer. The camera with an orange filter records the motion of orange passive tracers (not sketched for visibility). The central shaded region corresponds to a plume of sugary water (solute is invisible for the cameras). Solid arrows represent downwelling in the plume, while dotted arrows represent the lateral recirculation. (c) Dyed sugar is sieved in the same laser sheet and recorded by the camera with an orange filter, allowing the tracking of sugar in both its solid and dissolved phases. The central shaded region represents a solutal plume, now visible for the camera.

The visualisations (see figure 1 a) are performed in a vertical laser sheet with a half-angle of divergence of $30^\circ$ , using a Powell lens and a laser of wavelength $532 \,{\textrm{nm}}$ with a power of $450 \,{\textrm{mW}}$ (Laser Quantum $532\,{\textrm{nm}}$ CW laser $2 \,{\textrm{W}}$ ). To dewarp the deformation induced by the cylindrical tank thanks to an accurate coordinate system, a chequerboard of $1\times 1 \,{\textrm{cm}}^2$ black and white squares was used underwater in the plane of the laser sheet. When ordinary sugar grains are sieved, they reflect the laser beam so they appear in green. Consequently their motion is recorded by a PointGrey camera equipped with a green filter (band-pass filter from Edmund Optics, CWL $532\,{\textrm{nm}}$ , FWHM $10\,{\textrm{nm}}$ ) with a frame rate of $50 \,{\textrm{fps}}$ . If the tank is seeded with orange fluorescent tracers (Cospheric, UVPMS-BO-1.03), their motion is recorded by a second synchronous camera equipped with an orange filter (high-pass filter above $570\,{\textrm{nm}}$ ). This configuration is sketched in figure 1(b). The velocity fields are quantified using DPIVSoft (Meunier & Leweke Reference Meunier and Leweke2003) with $32\times 32\,{\textrm{px}}^2$ windows overlapping by 50 %, giving a typical resolution of $2.7\,{\textrm{mm}}$ horizontally and $3.8\,{\textrm{mm}}$ vertically.

When dyed sugar grains are used, they appear orange due to their fluorescence. The presence of the coloured molecules inside the sugar ensures that throughout the whole process of dissolution, these molecules are released in water, that is why sugary water also appears in orange in the laser sheet. The motions of both dyed sugar grains and sugary water are recorded at $50\,{\textrm{fps}}$ by another identical camera that is equipped with an orange filter (high-pass filter above $570\,{\textrm{nm}}$ ); see figure 1(c).

2.2. Main three dynamical ingredients

2.2.1. Collective drag

When a single grain falls in quiescent water, it only drags fluid in its vicinity. Conversely, a group of multiple grains may act like a local negative buoyancy at the scale of the group of grains. Averaging the drag force of all grains in a unit volume, and assuming a balance between buoyancy and drag (Kriaa et al. Reference Kriaa, Favier and Le Bars2023), the volumetric drag force reads

(2.1) \begin{equation} \boldsymbol{f}_{\textit{drag}}(\boldsymbol{x},t) = [\rho _p-\rho (\boldsymbol{x},t)] \phi (\boldsymbol{x},t) \boldsymbol{g} , \end{equation}

where $\phi (\boldsymbol{x},t)$ is the local volume fraction of grains, $\boldsymbol{g}$ is gravity, $\rho _p$ is the density of a sugar grain and $\rho (\boldsymbol{x},t)\geqslant \rho _0$ is the local density of sugary water, which we denote $\rho _0$ in the absence of sugar. Equation (2.1) contains no information about the size of particles nor their spacing, since it is derived in the diluted regime ( $\phi \ll 1$ ) by modelling particles as a continuum (note in our experiments the grain volume fraction is in the range $4\times 10^{-5}-8\times 10^{-4}$ , as estimated from (4.3) in § 4.4). The averaging procedure that leads to (2.1) requires that the number density $\sim \phi r_p^{-3}$ of particles be sufficiently large for the averaged drag to be meaningful (Mellado et al. Reference Mellado, Stevens, Schmidt and Peters2010; Chou & Shao Reference Chou and Shao2016). A mathematically equivalent condition is that the interparticle distance

(2.2) \begin{equation} l_{\textit{inter}} \propto r_p \phi ^{-1/3} \end{equation}

of a uniform suspension be sufficiently small for the hydrodynamic perturbations of neighbouring grains to overlap, enabling them to collectively drag the interstitial fluid, as concluded in past experimental (Harada, Mitsui & Sato Reference Harada, Mitsui and Sato2012) and numerical (Yamamoto Reference Yamamoto2015) studies. These conclusions are further supported by several studies that show that despite their fundamentally long-range hydrodynamical interactions, the perturbations induced by particles with negligible (Subramanian & Koch Reference Subramanian and Koch2008; Guazzelli & Morris Reference Guazzelli and Morris2011; Pignatel, Nicolas & Guazzelli Reference Pignatel, Nicolas and Guazzelli2011) or finite inertia (Koch Reference Koch1993; Daniel et al. Reference Daniel, Ecke, Subramanian and Koch2009) only extend up to some finite, typical length scale.

It means that for a given $\phi$ , larger grains being farther away, they are expected to drag fluid less efficiently. Future sections will confirm that our largest grains only drag fluid in localised wakes, as observed when using particle image velocimetry (PIV) particles, whereas the smallest grains set in motion all the fluid contaminated with sugar. In reference to hydrodynamical interactions that lead grains to behave ‘fluid-like’ (Harada et al. Reference Harada, Mitsui and Sato2012), we refer to this behaviour as ‘collectivity’.

2.2.2. Gravitational drift

Consider the fall of a single supposedly spherical sugar grain of radius $r_p$ on a time scale much lower than its dissolution time scale. Neglecting its inertia (see details in Kriaa et al. Reference Kriaa, Favier and Le Bars2023), the momentum equation reduces to a balance between its buoyancy and drag:

(2.3) \begin{equation} \boldsymbol{v}_p(\boldsymbol{x}_p,t) = \boldsymbol{v}(\boldsymbol{x}_p,t) + w_s(r_p)\boldsymbol{e}_z, \quad {\textrm{with}} \quad \boldsymbol{e}_z = \boldsymbol{g}/g. \end{equation}

This equation means that the grain velocity $\boldsymbol{v}_p(\boldsymbol{x}_p,t)$ is equal to the local fluid velocity $\boldsymbol{v}(\boldsymbol{x}_p,t)$ with the addition of a gravitational drift along the direction of gravity at the terminal velocity $w_s(r_p)$ . The latter is parameterised as (Clift, Grace & Weber Reference Clift, Grace and Weber2005; Crowe et al. Reference Crowe, Schwarzkopf, Sommerfeld and Tsuji2011)

(2.4) \begin{equation} w_s \equiv \frac {w_s^{\,\textit{Stokes}}}{1+0.15\textit{Re}_{\kern-1pt p}^{0.687}}, \end{equation}

where

(2.5) \begin{equation} \textit{Re}_{\kern-1pt p} \equiv \frac {2r_pw_s}{\nu } \end{equation}

is the particle Reynolds number and

(2.6) \begin{equation} w_s^{\,\textit{Stokes}} \equiv \frac {2gr_p^2(\rho _p-\rho _0)}{9\nu \rho _0} \end{equation}

is the Stokes velocity. In (2.6), $\nu$ is the kinematic viscosity of clear water. We neglect the change in density and viscosity of the background fluid associated with dissolved sugar. Indeed, the accumulation of solute in the tank during long experiments in the quasi-steady regime leads to an estimated final uniform solute concentration of $1.36\ \,{\textrm{kg m}}^{-3}$ , corresponding to an increase of the viscosity of only 0.3 % (Swindells et al. Reference Swindells, Snyder, Hardy and Golden1958). The values used for our calculations are listed in table 2. Note that (2.4) and (2.6) are valid for spheres. Again, we expect these expressions to be good approximations of the actual settling velocity of our sugar grains, up to a prefactor of order unity.

Table 2. Main properties of sugar and sugary water.

When modelling multiple grains as a continuum, (2.1) showed that the drag force contains no information about the particle size. Yet, due to the settling term in (2.3), the advection of the volume fraction $\phi (\boldsymbol{x},t)$ reads

(2.7) \begin{equation} \partial _t \phi + \boldsymbol{v} \boldsymbol{\cdot }\boldsymbol{\nabla }\phi = \kappa _p {\nabla}^2 \phi - w_s \frac {\partial \phi }{\partial z}, \end{equation}

where $\kappa _p$ is an effective particle diffusivity. The last term of (2.7) quantifies the gravitational drift of particles that causes a partial decoupling between the volume fraction $\phi (\boldsymbol{x},t)$ and the fluid parcels at the same location $\boldsymbol{x}(t)$ . Due to this partial decoupling, an identical drag force induces lower velocities on a fluid initially at rest when the settling velocity is larger, i.e. when the grains are larger (Kriaa et al. Reference Kriaa, Favier and Le Bars2023). In parallel, using larger grains implies that their number density is lower when $\dot {m}$ is fixed, but their wakes are larger and faster. Whether this changing structure of the flow in the vicinity of grains enhances or reduces the velocity induced on the fluid is not accounted for here.

2.2.3. Dissolution

Assuming that the dissolution rate of a spherical sugar grain is uniform at all times, the grain remains spherical and its mass varies in time following mass conservation, i.e.

(2.8) \begin{equation} \frac {\textrm{d}}{{\textrm{d}}t} \left [ \frac {4\pi r_p^3 \rho _p}{3}\right ] = -4\pi r_p^2 \kappa \left ( \frac {\partial \mathcal{C}}{\partial n}\right )_i, \end{equation}

with $\kappa$ the diffusivity of sucrose in water and $n$ the normal to the grain–fluid interface, positive in the direction of the fluid. The concentration gradient, evaluated at the grain–water interface, scales like the ratio of the concentration difference between the bulk fluid and the concentration at the interface, over the thickness of the solutal boundary layer. We assume a linear relation between sugar concentration and density (Philippi et al. Reference Philippi, Berhanu, Derr and Courrech du Pont2019; Cohen et al. Reference Cohen, Berhanu, Derr and Courrech du Pont2020; Pegler & Wykes Reference Pegler and Wykes2020), i.e.

(2.9) \begin{equation} \mathcal{C}(\boldsymbol{x},t) = \frac {\rho (\boldsymbol{x},t)-\rho _0}{\rho _{\textit{sat}}-\rho _0}\mathcal{C}_{\textit{sat}}, \end{equation}

with $\rho _{\textit{sat}}$ and $\mathcal{C}_{\textit{sat}}$ respectively the density and concentration of water at saturation. Combining (2.8) and (2.9), we get

(2.10) \begin{equation} \underbrace {-\dot {r}_p}_{{\textrm {recession speed}}} = \overline {k}\frac {\mathcal{C}_{\textit{sat}}}{\rho _p} \frac {\rho _i(t) - \rho (t)}{\rho _{\textit{sat}} - \rho _0}, \end{equation}

where the mass transfer coefficient $\overline {k}$ is used as a substitute of, and is defined as, the ratio between the diffusivity of sugar over the size of the concentration boundary layer. The last fraction in (2.10) shows that dissolution is delayed when the ambient density $\rho (t)$ gets close to the density $\rho _i$ at the grain–water interface (Liu, Ning & Ecke Reference Liu, Ning and Ecke1996). The solute must diffusive away from the sugar grain for dissolution to continue; if the ambient is saturated, we have $\rho (t)=\rho _i=\rho _{\textit{sat}}$ and dissolution ceases ( $\dot {r}_p=0$ ).

The spatial average of the mass transfer coefficient $\overline {k}$ parameterises the influence of settling on the size of the solutal boundary layer, which controls the mass transfer. Solutions to the boundary layer equations around a sphere suggest that, for a particle Reynolds number $\textit{Re}_{\kern-1pt p} \gg 1$ , we expect the dimensionless mass transfer coefficient $2r_p\overline {k}/\kappa$ to scale like $Sc^{1/3}\textit{Re}_{\kern-1pt p}^{1/2}$ (e.g. Ranz & Marshall Reference Ranz and Marshall1952; Lochiel & Calderbank Reference Lochiel and Calderbank1964; Rice & Jones Reference Rice and Jones1979), whereas we expect it to scale like $Sc^{1/3}\textit{Re}_{\kern-1pt p}^{1/3}$ when $\textit{Re}_{\kern-1pt p} \ll 1$ (e.g. Levich Reference Levich1962; Lochiel & Calderbank Reference Lochiel and Calderbank1964; Ulvrová et al. Reference Ulvrová, Coltice, Ricard, Labrosse, Dubuffet, Velímský and Šrámek2011), with $Sc=\nu /\kappa$ the Schmidt number. Theoretical calculations as well as comparisons with experiments enable us to estimate the prefactors, which are known within a few tens of percent. For a detailed discussion about the mass transfer correlations and the sources of uncertainties, we refer the reader to Lochiel & Calderbank (Reference Lochiel and Calderbank1964) and He et al. (Reference He, Cui, Chin, Darnige, Claudin and Semin2024). In practice, we choose standard values and impose a transition of $\overline {k}$ from the former law to the latter as

(2.11) \begin{equation} \overline {k} = \frac {\kappa }{2r_p} Sh,\quad \text{with} \quad Sh = \begin{cases} 2 + 0.6 Sc^{1/3} \textit{Re}_{\kern-1pt p}^{1/2} & \text{if}\ \textit{Re}_{\kern-1pt p} \geqslant 1,\\ 2 + {1.02} Sc^{1/3} \textit{Re}_{\kern-1pt p}^{1/3} & \text{if}\ \textit{Re}_{\kern-1pt p}\lt 1, \end{cases} \end{equation}

where $Sh$ is the Sherwood number, which verifies $Sh=2$ for pure diffusion (i.e. $\textit{Re}_{\kern-1pt p}=0$ ) due to its normalisation by the diffusive time scale $2r_p/\kappa$ based on the particle diameter. At this stage, determination of the recession speed of the grain–water interface only requires knowledge of the fluid density $\rho _i(t)$ at this interface. As widely assumed in the literature (e.g. Pegler & Wykes Reference Pegler and Wykes2020), we consider that $\rho _i(t)=\rho _{\textit{sat}}$ at all times.

For a sugar grain falling in fresh water ( $\rho =\rho _0$ ), the recession speed $\dot {r}_p$ only depends on $\overline {k}$ whose dependence on $r_p$ can be computed using (2.11). Although the mass transfer coefficient is larger for smaller grains ( $\overline {k} \rightarrow \infty$ when $r_p \rightarrow 0$ ), its variations are limited to about $45\,\%$ in the range $30\hbox{-}500{\,\unicode{x03BC} {\textrm{m}}}$ . So varying $r_p$ does not modify the dissolution rate $\dot {r}_p$ by orders of magnitude. However, for a given volume fraction $\phi$ , smaller grains are more numerous and have in total a larger surface area in contact with water. In a unit volume of fluid with volume fraction $\phi$ , the mass of sugar that dissolves per unit time is the integral of (2.10) over the surface area of one grain, multiplied by the number of grains in the unit volume. This total mass rate reads

(2.12) \begin{equation} \frac {{\textrm{d}}m_{\textit{tot}}}{{\textrm{d}}t} \propto \kern-40pt \underbrace {\frac {\phi }{\frac {4}{3} \pi r_p^3}}_{{\textrm{number of grains per unit volume}}} \kern-40pt 4\pi r_p^2 \rho _p | \dot {r}_p | \propto \frac {\overline {k}}{r_p}, \end{equation}

which is valid under the assumption that saturation effects can be neglected (such effects likely depend on the radius $r_p$ since smaller particles are closer to each other).

According to (2.12), the total mass flux ${\textrm{d}}m_{\textit{tot}}/{\textrm{d}}t$ increases as $r_p$ is lowered (figure 2) due to the surface-to-volume ratio being proportional to $r_p^{-1}$ . Smaller sugar grains are therefore expected to impart their buoyancy to water faster.

Figure 2. Evolution of the ratio $\overline {k}/r_p$ that controls the total mass transfer due to a collection of grains of radius $r_p$ and uniform volume fraction. The line is continuous when the parameterisation of Levich (Reference Levich1962) applies ( $\textit{Re}_{\kern-1pt p}\lt 1$ ); it is dashed when the parameterisation of Ranz & Marshall (Reference Ranz and Marshall1952) applies ( $\textit{Re}_{\kern-1pt p}\geqslant 1$ ).

3.1. Qualitative overview of representative cases

3.1.1. Rectilinear fall of large sugar grains

We start with the end-member of large sugar grains (see supplementary movie 1 available at https://doi.org/10.1017/jfm.2025.10766). Figure 3(a) shows snapshots of $363 \,{\unicode{x03BC}\textrm {m}}$ -sized dyed sugar grains settling in quiescent water when sieving with a mass rate $\dot {m}=0.052\, \,\textrm g\, \,\textrm s^{-1}$ , i.e. typically 163 sugar grains per second. Streaks of rhodamine highlight the rectilinear fall of this category of grains. The onset of fluid motions at the scale of the sieve is not immediate; the eventual convergence of rhodamine streaks towards the centre reveals that a converging flow gradually emerges, yet, the trajectories of large grains are barely affected and the latter keep settling vertically. The flow convergence and the continuous stochastic deposition of new sugar wakes lead to the overlap of dyed sugar wakes. This enhances the local negative buoyancy and, therefore, accelerates downward fluid motions. As for all grain sizes, the transient leads to a quasi-steady flow; the transient and quasi-steady states will be analysed in §§ 4.2 and 4.4, respectively. Importantly, the last snapshot in figure 3(a) still preserves a layered structure of vertical wakes, evidencing the laminar and essentially irrotational nature of the columnar flow that develops.

Figure 3. Snapshots of dyed sugar grains falling during the onset of fluid motions. The mean initial radius $r_p$ , the average mass rate $\dot {m}$ , the time interval between snapshots $\Delta t$ and the total height of the images are (a) $r_{p} = 363\,\unicode{x03BC} \textrm{m}, \dot{m} = 0.052\,\textrm{g}\,\textrm{s}^{-1}, \Delta t = 6\,\textrm{s}, \textrm{height}: 38\,\textrm{cm} $ , (b) $r_{p} = 45\,\unicode{x03BC} \textrm{m}, \dot{m} = 0.120\,\textrm{g}\,\textrm{s}^{-1}, \Delta t = 10\,\textrm{s}, \textrm{height}: 38\,\textrm{cm} $ and (c) $r_{p} = 101\,\unicode{x03BC} \textrm{m}, \dot{m} = 0.115\,\textrm{g}\,\textrm{s}^{-1}, \Delta t = 5.2\,\textrm{s}, \textrm{height}: 23\,\textrm{cm} $ .. The blue and red curves in (c) respectively show the edge of the solute advected radially inward in the plume, and the edge of the sugar grains raining out from this radially converging flow.

3.2. Onset through a rayleigh–Taylor-like instability

The key contribution to the transition G1 $\rightarrow$ G2 $\rightarrow$ G3) is the densification of the sugary layer at the top of the tank. The Rayleigh–Taylor-like instability is driven by the density difference between the ambient and the top sugar layer, which is at first order given by the Atwood number

(3.1) \begin{equation} \mathcal{A} \equiv \frac {\rho _{\textit{eff}}-\rho _0}{\rho _{\textit{eff}}+\rho _0}, \end{equation}

where $\rho _{\textit{eff}}\gt \rho _0$ is the effective density that accounts for the total mass of sugar (solid and dissolved) in the top layer. The effective density of the sugary layer is given by assuming a uniform dilution of the total mass of sugar sieved since the start of the experiment in the sugary layer, whose surface area is $\pi R_{\textit{sieve}}^2$ and whose depth corresponds at first order to the distance travelled by the first grains sieved with constant velocity $w_s$ , so that

(3.2) \begin{equation} \rho _{\textit{eff}} = \rho _0 + \frac {\dot {m}}{\pi R_{\textit{sieve}}^2w_s}, \end{equation}

which leads to the expression of the Atwood number

(3.3) \begin{equation} \mathcal{A} = \frac {\rho _{\textit{eff}}-\rho _0}{\rho _{\textit{eff}}+\rho _0} = \left [1 + \frac {2\rho _0 \pi R_{\textit{sieve}}^2w_s}{\dot {m}} \right ]^{-1}. \end{equation}

Note that (3.3) considers that buoyancy arises from the mass loading of solid particles. In practice, buoyancy is shared between solid grains and the solute, with a decrease of mass loading and an increase of solute concentration with depth. This neglects an 8.8 % change of density between solid sugar and saturated sugary water (see table 2). We show that such subtle aspects are not crucial for the large-scale dynamics; accounting for phase change is only necessary when quantifying the deceleration of shrinking sugar grains to model the final depth of the dissolution layer in § 5.2.

To determine the growth rate of the instability, consider a canonical linear Rayleigh–Taylor instability growing at an interface between two uniform semi-infinite layers, with gravity pointing in the direction of the lightest layer. Viscosity, whose dependence on sugar concentration is neglected in this section for the sake of simplicity, damps the instability and selects a mode of maximum growth rate (Alqatari et al. Reference Alqatari, Videbæk, Nagel, Hosoi and Bischofberger2020; Magnani et al. Reference Magnani, Musacchio and Boffetta2021)

(3.4) \begin{equation} \sigma _{\textit{max}} = \sqrt {\mathcal{A} gk_{\textit{max}} + (\nu k_{\textit{max}}^2)^2} - \nu k_{\textit{max}}^2 , \end{equation}

with wavenumber (Plesset & Whipple Reference Plesset and Whipple1974; Mikaelian Reference Mikaelian1993; Magnani et al. Reference Magnani, Musacchio and Boffetta2021)

(3.5) \begin{equation} k_{\textit{max}} = \left ( \frac {\mathcal{A}g}{8\nu ^2} \right )^{1/3} . \end{equation}

The more concentrated the initial sugary layer, the larger the Atwood number, the faster the growth of the instability (see (3.4)). Figures 3(b) and 3(c) enable us to measure the distance between mushrooms on their first (respectively second) snapshot. For $r_p=45\,\mathrm{\unicode{x03BC} m}$ and $\dot {m}=0.120\, \,\textrm g\, \,\textrm s^{-1}$ (respectively $r_p=101\ \,\mathrm{\unicode{x03BC} m}$ and $\dot {m}=0.115\ \,\textrm g\,\textrm s^{-1}$ ), we measure a distance between mushrooms of $0.53{-}0.64\ \,\mathrm{cm}$ (respectively $0.66{-}1.2\ \,\mathrm{cm}$ ) while the estimate $2\pi /k_{\textit{max}}$ from (3.5) yields $0.56\ \,\mathrm{cm}$ (respectively $0.93\ \,\mathrm{cm}$ ). Thus, our observations are compatible with (3.5), although the mushrooms in figures 3(b) and 3(c) are probably already in the nonlinear regime of the instability.

The present context slightly differs from the canonical configuration. The sugary layer develops just below the water free surface, hence, it may feel the influence of confinement. Alqatari et al. (Reference Alqatari, Videbæk, Nagel, Hosoi and Bischofberger2020) have shown that the most unstable mode is $k_{\textit{max}}$ only when the dense layer is thicker than $2\pi /k_{\textit{max}}$ . Otherwise, the present confinement stabilises the system and modifies the wavelength of the instability that now scales like the thickness of the sugary layer. Another difference is that the present instability is forced by a dispersed phase rather than a dense fluid. Yet the forcing conditions and the phenomenology are similar, as previously observed in the literature (Kimura Reference Kimura1988; Climent & Magnaudet Reference Climent and Magnaudet1999; Caballina, Climent & Dušek Reference Caballina, Climent and Dušek2003; Mudde Reference Mudde2005; Mezui, Obligado & Cartellier Reference Mezui, Obligado and Cartellier2022). A particular example can be found in Kimura (Reference Kimura1988) where bubbles were injected at the bottom of a two-dimensional cell ( $40\times 8\times 2 \,\mathrm{cm^3}$ ). They observed that the positively buoyant bubble-laden layer near the bottom developed a Rayleigh–Taylor-like instability whose mushrooms grew all the faster as bubbles accumulated more in this unstable layer, which is consistent with an increase of the bubble volume fraction in this layer.

When does the instability kick in? Large grains from group G1 show no instability because sugary mushrooms do not have enough time to grow before sugar grains decouple from them. The instability only develops if the sugary mushrooms grow faster than particles rain out of them (Carey Reference Carey1997; Jacobs et al. Reference Jacobs, Goldin, Collins, Piggott, Kramer, Melosh, Wilson and Allison2015; Lemus et al. Reference Lemus, Fries, Jarvis, Bonadonna, Chopard and Lätt2021). A first-order estimate can be based on the maximum growth rate of the actual Rayleigh–Taylor instability in (3.4) with the wavenumber given by (3.5). The instability emerges provided that the inverse growth rate $\sigma _{\textit{max}}^{-1}$ is comparable to or lower than the settling time over the scale of mushrooms $2\pi /k_{\textit{max}}w_s$ , i.e.

(3.6) \begin{equation} w_s \leqslant \frac {2\pi \sigma _{\textit{max}}}{k_{\textit{max}}}. \end{equation}

In the following, we simply call the case of equality the ‘marginal stability’ at the onset of the flow, which we aim to characterise in the parameter space $(\dot {m},r_p)$ . Equation (3.6) shows that, for a given Atwood number, a lower settling velocity favours the growth of the instability because raining out of sugary mushrooms is slower.

The solid dark line in figure 4 shows the curve of marginal stability in the parameter space $(\dot {m}, r_p)$ . Experimental measurements are shown as coloured symbols: dark squares are experiments that never lead to the Rayleigh–Taylor-like instability (G1), green circles are experiments with a delayed onset of the instability after a phase of settling (G2) and red diamonds are experiments with the immediate formation of mushrooms (G3). Despite the non-ideal experimental conditions of grains penetrating through the air–water interface with small yet visible perturbations of this interface during sieving, experiments of category G3 lie below the solid line. Some experiments of category G2 lie in this same region near the marginal curve, suggesting some reduction of the growth rate due to settling or confinement.

Figure 4. Regime diagram in the parameter space $(\dot {m}, r_p)$ grouping experiments according to the phenomenology observed at the onset of convection: G1, G2, G3 (see text in § 3.1.4 for further details). The solid dark line is the curve of marginal Rayleigh–Taylor stability from (3.6). The background colour shows values of the initial Rouse number as defined in (3.8). The dashed dark line indicates the transition between the grains that settle faster than the starting plume characteristic velocity ( $\mathcal{R}_0\gt 1$ ) and those that settle slower ( $\mathcal{R}_0\lt 1$ ). The dotted blue line delineates the region of collectivity ( $\mathcal{L}\lt 10$ below) and the region where grains behave individually ( $\mathcal{L}\gt 10$ above).

Note that the transition between G1 and G3 is actually gradual, which explains the existence of regime G2. Several studies in the literature have focused on the forcing of a Rayleigh–Taylor instability by settling particles in horizontally periodic numerical domains with a suspension of particles overlying a layer of clear fluid (Chou & Shao Reference Chou and Shao2016; Magnani et al. Reference Magnani, Musacchio and Boffetta2021). They showed that the initial growth rate of the instability is reduced by the settling of particles, all the more so as particles settle faster, in agreement with the linear stability analysis of Burns & Meiburg (Reference Burns and Meiburg2012) in two dimensions. Burns & Meiburg (Reference Burns and Meiburg2012) showed that during the growth of the particle-laden mushrooms, particles advect vorticity away from the interface that results in a smearing of the vorticity profile at the interface, thus damping the instability all the more efficiently as the settling velocity increases. These observations contribute to the gradual transition from G3 to G1 when $r_p$ increases. Considering our experiments of group G2 above the solid dark line, initially grains just settle since $w_s(r_{p0}) \gt 2\pi \sigma _{\textit{max}}/k_{\textit{max}}$ . But grains decelerate as they shrink due to dissolution. On the other hand, the top sugar layer gets denser with time due to the continuous addition of new grains. We therefore expect the effective density $\rho _{\textit{eff}}$ to gradually increase until the instability emerges after a finite delay.

3.3. Critical size decoupling from the starting plume

As previously observed in other contexts of two-phase (Caballina et al. Reference Caballina, Climent and Dušek2003) and one-phase starting plumes (see figure 4 in Friedl et al. Reference Friedl, Haertel and Fannelop1999), the acceleration of our sugary plumes eventually leads them to a phase of propagation with constant velocity. Balancing the leading contributions of inertia and buoyancy and neglecting any mutiplicative constant of order unity, the fluid velocity is expected to scale like the square root of buoyancy multiplied by the typical plume width, at first order given by the radius at injection $R_{\textit{sieve}}$ , as verified in the literature for bubble plumes (Caballina et al. Reference Caballina, Climent and Dušek2003) and bubble columns (Mezui et al. Reference Mezui, Obligado and Cartellier2022). From (3.2), this constant free-fall velocity reads

(3.7) \begin{equation} U_{\textit{onset}} = \left [\frac {\rho _{\textit{eff}}-\rho _0}{\rho _0}g R_{\textit{sieve}} \right ]^{1/2} = \left [\frac {\dot {m}g}{\pi R_{\textit{sieve}} w_s\rho _0} \right ]^{1/2}, \end{equation}

where we recall that $\rho _{\textit{eff}}$ is the total density incorporating both solid and dissolved sugar. The large values of the plume Reynolds number $2R_{\textit{sieve}} U_{\textit{onset}} /\nu$ in the range $10^3$ – $10^5$ suggest that buoyancy is balanced by inertia rather than viscous forces. We are interested in comparing this reference velocity with the settling velocity of our sugar grains under the form of an ‘initial Rouse number’, relevant when no flow has yet developed in the tank:

(3.8) \begin{equation} \mathcal{R}_0 \equiv \frac {w_s}{U_{\textit{onset}}} = \left [\frac {w_s^3 \pi R_{\textit{sieve}} \rho _0}{\dot {m}g} \right ]^{1/2}\ . \end{equation}

The Rouse number $\mathcal{R}_0$ characterises the coupling of sugar grains with the starting plume: grains accompany the starting plume while it sinks when $\mathcal{R}_0\lt 1$ , whereas grains verifying $\mathcal{R}_0\gt 1$ fall ahead of the starting plume because they settle faster than its characteristic velocity. Note the so-called Rouse number is usually defined as a ratio of settling velocity to friction velocity in the context of sediment resuspension. Here, we adopt the term ‘Rouse’ – as used in studies of inertial particles in turbulent flows (e.g. Deguen, Olson & Cardin Reference Deguen, Olson and Cardin2011; Monchaux & Dejoan Reference Monchaux and Dejoan2017) – to distinguish $\mathcal{R}_0$ from the Stokes number, which quantifies the inertia of particles independently of gravity. The isocontour $\mathcal{R}_0=1$ is shown as a dashed line in figure 4 in the parameter space $(\dot {m},r_p)$ : grains in the range $r_p\gt 200 {\,\unicode{x03BC} {\textrm{m}}}$ are expected to fall ahead of the sugar plume, as consistently observed with our experiments in group G1.

Some nuances are neglected in this first-order reasoning. As long as the plume has not finished accelerating, its downward velocity is still lower than the estimate $U_{\textit{onset}}$ , so the actual Rouse number is larger than the estimate $\mathcal{R}_0$ . Therefore, grains even slightly smaller than predicted in figure 4 might decouple from the starting plume. Note however that grains that lie just above the isocontour $\mathcal{R}_0 = 1$ tend to fall ahead of the plume only temporarily: these grains decelerate due to dissolution while the plume is accelerating, hence, the latter catches up with these grains, as already visible in the first three snapshots of figure 3(c).

3.4. Collective versus individual forcing

So far (3.2)–(3.8) have been derived from a field theory that may poorly describe the behaviour of particles that behave individually, i.e. when their interparticle distance is larger than a critical threshold; see § 2.2.1. Then, is the region of transition from red to dark symbols in figure 4 only due to an insufficient Atwood number of the sugary layer, or is it due to a transition from a collective to an individual behaviour of sugar grains that are too decoupled from the flow to produce a fluid-like instability? Answering this question requires us to quantify the transition from collective to individual behaviour by defining the critical interparticle distance over which particles are capable of interacting. For a single particle settling in a quiescent fluid, this critical length $l_{\textit{inter,c}}$ is defined as the distance from the particle where advection and momentum diffusion balance out. For very low particulate Reynolds numbers, it is the inertial screening length $\nu /w_s$ (Subramanian & Koch Reference Subramanian and Koch2008; Guazzelli & Morris Reference Guazzelli and Morris2011; Pignatel et al. Reference Pignatel, Nicolas and Guazzelli2011) whereas for small yet finite particulate Reynolds numbers, it corresponds to the particle radius $r_p$ (Koch Reference Koch1993; Daniel et al. Reference Daniel, Ecke, Subramanian and Koch2009). For our largest particle Reynolds numbers, particles may have laminar wakes whose typical size is also given by the particle size $r_p$ . Therefore, the critical interparticle distance is estimated as

(3.9) \begin{equation} \ l_{\textit{inter,c}} = \max \left \{ \frac {\nu }{w_s}, r_p \right \} \quad\text{for all } r_p, \end{equation}

and particles are expected to behave individually if their interparticle distance is larger than $l_{\textit{inter,c}}$ , typically by a factor $O(10)$ to be conservative. The estimate of this threshold has been validated in previous experiments (Tsuji, Morikawa & Terashima Reference Tsuji, Morikawa and Terashima1982) where spheres of large particle Reynolds numbers ( $\textit{Re}_{\kern-1pt p}\sim 10^2$ – $10^3$ ) could interact hydrodynamically up until $l_{\textit{inter}}\sim 10r_p$ , in agreement with the estimate $10l_{\textit{inter,c}}$ for this regime of large particle Reynolds numbers.

From the expression of the effective volume fraction

(3.10) \begin{equation} \phi _{\textit{eff}} = \frac {\dot {m}}{\pi R_{\textit{sieve}}^2 w_s \rho _p} , \end{equation}

(2.2) and (3.10) can be combined to estimate the interparticle distance. Then, the dimensionless interparticle distance

(3.11) \begin{equation} \mathcal{L} = \frac {l_{\textit{inter}}}{l_{\textit{inter,c}}} \end{equation}

is calculated in the whole parameter space to quantify collective ( $\mathcal{L} \ll 10$ ) and individual ( $\mathcal{L}\gg 10$ ) behaviours (see the blue dotted line in figure 4 whose slope changes when $r_p w_s/\nu =1$ ). Figure 4 shows that experiments of group G2 lie in the region of transition, typically in the range $\mathcal{L} \simeq 5-15$ , which includes the curve of marginal stability for the Rayleigh–Taylor instability. Therefore, the transition from G1 to G3 and the transition from individuality to collectivity overlap in the parameter space, which prevents us from disentangling them as the radius is decreased for a fixed mass flux.

Of course the previous computation of $\mathcal{L}$ is a crude estimate, especially as it neglects the fact that sugar grains are dissolving in the sugary layer, hence, their interparticle distance may vary little but their critical length scale of interaction $l_{\textit{inter,c}}$ is expected to vary as their size decreases. Accounting for this evolution in time and space then requires us to model grains that are sieved at different times, cross different solute concentrations along their fall, and reach a certain depth with different sizes and relative motion. This is beyond the scope of this paper. Additional discussions can be found in Kriaa (Reference Kriaa2023); the present calculations essentially guide the interpretations of measurements in the next sections.

3.5. Different pathways to large scales

Consider a fixed mass rate $\dot {m}$ . Where grains have not fully dissolved, the action of drag is longer lasting for smaller grains due to their slower drift. A smaller initial radius implies a smaller inter-grain distance, thus enhancing the ability of grains to collectively drag interstitial fluid. This favours the forcing of a large-scale flow when grains are sufficiently small, as shown in figure 5(a). This space–time diagram is a stack of the horizontal profiles of light intensity recorded at the front of the sugary layer when being tracked in time and depth (see explanations in Appendix B.1). The top of the graph shows tiny grain-scale filaments that coalesce and eventually organise as one large macroscale. This coarsening is initially due to the fact that many length scales are unstable when the sugary layer grows, including large ones of a small growth rate whose emergence is therefore delayed; eventually, the coarsening is due to the formation of the plume in which small grains are advected.

Figure 5. When tracking the front position of a plume of dyed sugar during the transient, at any time, the profile of light intensity can be extracted at the plume front. Plots (a) and (b) are space–time diagrams that stack such horizontal profiles in time from the start of an experiment (at the top of the diagrams) to the end of the transient (at the bottom of the diagrams). Blue regions are devoid of sugar, while red regions are concentrated in solid or dissolved sugar. See Appendix B.1 for details about the processing. Radii and mass rates are (a) $r_{p} = 101\,\unicode{x03BC} \textrm{m}, \dot{m} = 0.12\,\textrm{g}\,\textrm{s}^{-1}$ and (b) $r_{p} = 363\,\unicode{x03BC} \textrm{m}, \dot{m} = 0.52\,\textrm{g}\,\textrm{s}^{-1}$ .

Consider now some large grains ( $\mathcal{R}_0 \gt 1$ ) sieved with a low mass rate. The first grains settle vertically in almost perfectly still water (see how localised the sugary wakes are in figure 5 b). The stochastic forcing by drag and by the deposition of negatively buoyant solute in the wake of every grain contributes to the eventual emergence of fluid motions even when grains are large and the mass rate is low. In experiments with large sugar grains, this macroscopic motion happens after a long time lag, after several grains have hit the bottom of the tank (see figure 3 a). Hence, we observe a transition in time from small-scale wakes that move individually to a large-scale plume that emerges gradually at all heights due to the combination of wakes. Note that dissolution is not necessary to this transition, which we recover when sieving glass spheres instead of sugar grains, and which has previously been observed in bubble-laden flows (Mazzitelli & Lohse Reference Mazzitelli and Lohse2009).

4.1. Overview

To provide a synthetic view of the characteristic features of the flow when varying parameters $(\dot {m}, r_p)$ , figure 6 presents space–time diagrams of the light intensity recorded during the fall of dyed sugar. These profiles are obtained by a horizontal average of the light intensity captured by the camera at different times, before stacking them as Hovmoller diagrams. For sufficiently small sizes (figures 6 a and 6 b), all diagrams evidence a concave downward sinking of the buoyant material due to the growth of the sugary layer, whose front position is anticipated to increase first exponentially in time for the linear growth then quadratically for the nonlinear growth of a Rayleigh–Taylor instability (Boffetta & Mazzino Reference Boffetta and Mazzino2017; Magnani et al. Reference Magnani, Musacchio and Boffetta2021). Then, the concavity evens out around a depth $z=0.1 \,\textrm {m}$ and leads to a downward fall with apparent constant velocity. The transient plunging of sugar in experiments having $r_p=363 {\,\unicode{x03BC} {\textrm{m}}}$ shows no sharp front (figure 6 c). This observation is due to the fact that sugar does not accumulate in a top sugary layer when sugar grains are too large. Consequently, large grains fall in isolation, no bright concentrated region falls as a whole, so no sharp front of well-defined trajectory is visible in the space–time diagrams.

Figure 6. Space–time diagrams of the horizontally averaged light intensity recorded during the fall of dyed sugar grains of initial radius (a) $80 \,{\unicode{x03BC} {\textrm{m}}}$ ( $\dot{m} = 0.10 \textrm{g}\,\textrm{ s}^{-1}$ ), (b) $101\, {\unicode{x03BC} {\textrm{m}}}$ ( $\dot{m} = 0.12 \textrm{g}\,\textrm{ s}^{-1}$ ) and (c) $363\, {\unicode{x03BC} {\textrm{m}}}$ ( $\dot{m} = 0.05 \textrm{g}\,\textrm{ s}^{-1}$ ). Note the difference of ordinates for each diagram. The colourbar is common for all graphs, each being normalised by its maximum light intensity.

The texture of the space–time diagrams provides additional information. In figure 6(a) for $r_p=80 {\,\unicode{x03BC} {\textrm{m}}}$ , one can follow the detachment and downward sinking of some successive turbulent puffs of buoyant material (grains or solute). Conversely, for $r_p=101 {\,\unicode{x03BC} {\textrm{m}}}$ (figure 6 b), the diagram is smooth with no puffs since the plume is more laminar. Because large grains decouple from fluid motions, different structures must be distinguished in figure 6 c. Straight stripes reveal the rapid fall of large grains and suggest that dissolution little affects their settling velocity over the distance $\sim 0.3 \,\textrm {m}$ . The constant slope of the stripes from the start to the end of the experiment shows that fluid motions also have little influence on the grains, which is consistent with § 3.3. Figure 6(c) also contains blurred shapes of vertical extension $\sim 0.1 \,\textrm {m}$ that fall in the tank at much lower velocity than the large grains. These large-scale structures correspond to collections of wakes of sugar that gradually fall coherently due to the continuous drag and deposition of new wakes by the falling grains. These fluid motions are slow and delayed compared with the fast fall of sugar grains.

To quantify the flow, we show results obtained with ordinary white sugar and PIV particles (see the set-up in figures 1 a and 1 b). Except when $r_p=363 {\,\unicode{x03BC} {\textrm{m}}}$ , fluid motions emerge below the free surface and propagate down the tank as a plume. We adopt the following convention: $v$ is reserved for an Eulerian fluid velocity measured from PIV, whether it be horizontal $v_x$ or vertical $v_z$ ; a time derivative $\dot {z}$ of the $z$ position is reserved for Lagrangian velocities when tracking the propagation of a structure, whether it be solid grains, dissolved sugar or a region of fluid motions.

Figure 7 shows the time evolution of vertical profiles of the vertical velocity magnitude averaged over the horizontal extent of the field of view, $\langle |v_z| \rangle _x$ , for three different particle radii. When sieving starts at $t=0$ , dark blue levels are due to the absence of any initial fluid motion; after a certain delay, fluid motions appear at the top of the tank, they gradually intensify, and sink downward until reaching the bottom of the field of view. Due to the non-penetration of the fluid at the top free surface and the bottom wall, vertical velocities vanish near these boundaries.

Figure 7. Space–time diagrams of vertical profiles of the horizontally averaged magnitude of the vertical fluid velocity $\langle |v_z| \rangle _x$ . From left to right, the three panels show the influence of increasing the particle size in three experiments of comparable mass rates, namely (a) $ r_{p} = 80\,\unicode{x03BC} \textrm{m} $ and $\dot {m}=0.59 \,\textrm g\,\textrm s^{-1}$ , (b) $ r_{p} = 169\,\unicode{x03BC} \textrm{m} $ and $\dot {m}=0.57 \,\textrm g\,\textrm s^{-1}$ , (c) $ r_{p} = 363\,\unicode{x03BC} \textrm{m} $ and $\dot {m}=0.64 \,\textrm g\,\textrm s^{-1}$ . The colourbar is common for the three graphs. The black and white dotted curves show the maximum depth where solid grains are still visible. The red and white dotted lines show the convergence of isocontours of the space–time diagrams (see Appendix B.2).

Figures 7(a)–7(c) immediately reveal the decrease of the typical fluid velocities during the transient and in the quasi-steady regime when $r_p$ increases. This observation is consistent with the dilution of the buoyant material when $w_s$ increases ((3.2) and (3.7)), with the lower efficiency of collective drag (§ 2.2.1) and delayed transfer of buoyancy through dissolution (§ 2.2.3) when $r_p$ increases, and with the favoured decoupling of larger grains from the starting plume (§§ 2.2.2 and 3.3). Time fluctuations in figure 7(a) are due to the turbulent nature of the plume when $r_p=45 {\,\unicode{x03BC} {\textrm{m}}}$ , by contrast with the more laminar plumes when $r_p \geqslant 169 {\,\unicode{x03BC} {\textrm{m}}}$ in figures 7(b) and 7(c).

4.2. Propagation of the plume front

Hovmoller diagrams similar to figure 7 reveal that a vigorous transient almost systematically develops and then relaxes to a somewhat quieter quasi-steady regime. The existence of a vigorous transient before a quasi-steady regime has already been observed in experiments of inert particles continuously sieved in water (Zürner et al. Reference Zürner, Toupoint, De Souza, Mezouane and Monchaux2023), and it is reminiscent of downbursts in atmospheric flows (e.g. Srivastava Reference Srivastava1985, Reference Srivastava1987). These studies observed that the fluid motions initially developed through the propagation of a front. This observation is recovered in the present experiments for sufficiently small grains (figure 7 a). However, fluid motions do not propagate as a front when grains decouple substantially from the starting plume due to their large Rouse number $\mathcal{R}_0 \gg 1$ (figure 7 c).

To be quantitative, we define the front of the downward-propagating fluid motions as the curve along which different isocontours of velocity converge in the space–time diagrams of $\langle |v_z| \rangle _x$ (see Appendix B.2 for more details), as shown by the red and white dash-dotted lines in figure 7. We want to compare this front position to the maximum depth where sugar grains complete their dissolution. To this end, the same procedure is implemented on Hovmoller diagrams of the light intensity visualised when filming the sugar grains (photographs taken by the camera using a green filter). The resulting black and white dotted curves in figure 7 correspond to the front of the dissolution layer, which is the zone where sugar grains are still visible before complete dissolution.

In figure 7(a) the little time lag between the front of the dissolution layer (black and white dashed line) and the front of fluid motions (red and white dash-dotted line) is due to small grains falling together with the plume front. Conversely, a time lag appears when sieving larger grains: grains fall before the emergence of fluid motions (see figure 7 b). This time lag is a measure of the delayed onset of fluid motions that require an accumulation of stochastic forcings of several grains and the deposition of several sugary wakes before a plume develops. For the largest grains, no front can be defined due to the absence of a proper convergence of the isocontours (figure 7 c).

Figure 8. Average sinking velocity of the front of the starting plume for all experiments where a front of fluid motions can be defined ( $r_p \leqslant 169 {\,\unicode{x03BC} {\textrm{m}}}$ ).

The red and white curves have been computed for all experiments. We observe that the sinking front always propagates with an approximately constant velocity after a few centimetres below the region of flow convergence near the free surface (see figures 22 a–22 c in Appendix B.2). An affine law is fitted on each of these trajectories; its slope provides the constant propagation velocity $\dot {z}_{\textit{sinking}}$ of the front in figure 8. The latter confirms that the sinking is faster when the mass rate increases. The initial grain size may exert a non-monotonic influence on sinking velocity; however, the current data do not allow for a definitive conclusion.

4.3. Time before the quasi-steady regime

Figure 8 suggests that the duration of the transient is essentially controlled by the mass rate. To confirm this, we search for a definition of the time $t_{\textit{QS}}$ elapsed before the quasi-steady regime is reached. To this end, figure 9 shows the probability density function (PDF) of the vertical fluid velocities. The PDF is essentially large for positive values of $v_z$ , which correspond to the downward flow in the plume. Yet, since the PDF is obtained in the whole field of view, the upward recirculation leads to large values of the PDF for moderately negative (i.e. small upward) velocities, corresponding to the recirculation on either side of the plume.

Figure 9. Probability density function (PDF) of vertical fluid velocities $v_z$ in the whole laser sheet at each time from the start of an experiment (bottom of the diagram) to the end (top of the diagram). For this example, $r_p=45 {\,\unicode{x03BC} {\textrm{m}}}$ and $\dot {m}=0.09\, \,\textrm g\, \,\textrm s^{-1}$ . The horizontal dashed line corresponds to the time when the PDF reaches quasi-steadiness; the quasi-steady regime stops when sieving ends at $t=70 \,\textrm {s}$ . For details about the dark line and about processing, see Appendix B.3.

After some finite duration, the PDF becomes quasi-steady: this gives one criterion to define the end of the transient (see the horizontal dashed line). Yet, at low mass rates the plume front can reach the bottom of the tank a few seconds later. Then, the time when the plume reaches the bottom wall is a good estimate of the end of the transient, as assessed from the evolution of the average kinetic energy within the plume. This time is defined with no ambiguity by visualising large sugar grains hitting the bottom wall, or by visualising changes in the refractive index due to the presence of density gradients once the front of the plume reaches the reflection of the laser sheet on the bottom wall. Then, we adopt a conservative definition of $t_{\textit{QS}}$ : it is the maximum between the time of quasi-steadiness of the PDF of vertical fluid velocities and the time for the plume to hit the bottom wall.

Figure 10(a) confirms that the duration of the transient is essentially controlled by the mass rate. However, larger grains lead to a larger time lag between their fall and the downward propagation of the starting plume. This is shown in figure 10(b) where $t_{\textit{QS}}$ is compared with the characteristic time of sedimentation over the height of the tank. When grains are large ( $r_p \geqslant 169 {\,\unicode{x03BC} {\textrm{m}}}$ ), steadiness is reached after the first grains have hit the bottom of the tank. Conversely, the plume enables small grains to reach the bottom of the tank before the sedimentation time scale since $t_{\textit{QS}}\lt H_{\textit{tank}}/w_s$ . The maximum vertical kinetic energy

(4.1) \begin{equation} E_{\,kz,\textit{max}}=\underset {t}{\max } \left ( \frac {\int _x \int _z \frac {1}{2} v_z(x,z)^2 \,\mathrm{d}z \,\mathrm{d}x}{\int _x \int _z \,\mathrm{d}z \,\mathrm{d}x} \right ) \end{equation}

is shown in figure 10(c) where we observe that this energy expectedly increases with the mass rate. More importantly, we note that the intensity of the ‘downburst-like’ transient is considerably reduced for the largest grains, confirming their slow and inefficient forcing of fluid motions.

Figure 10. Characterisation of the end of the transient. (a) Time elapsed before the quasi-steady regime is reached, and (b) the same time $t_{\textit{QS}}$ normalised using the sedimentation time scale $H_{\textit{tank}}/w_s$ over the height of the tank $H_{\textit{tank}}$ . (c) Maximum vertical kinetic energy. The colourbar is common for all figures and shows the injected particle radius.

4.4. Quasi-steady flow

In this section, quantities are averaged in time during the quasi-steady regime. We focus on the flow inside the central plume, whose typical structure is shown with an example in figure 11(a). The plume is the region that concentrates downward fluid velocities near the centre of the tank. To minimise the influence of the upward recirculation, the plume centreline $x_{\textit{c,plume}}(z)$ is defined at all heights as the centroid of the downward fluid velocities $\tilde {v}_z=\max (v_z,0)$ along the horizontal direction (see the thick white line in figure 11 a). Then, the plume radius is defined as a weighted standard deviation with respect to the plume centreline, i.e.

(4.2) \begin{equation} \sigma _{\textit{x,plume}}(z) = \gamma \sqrt {\frac {\int \tilde {v}_z (x-x_{\textit{c,plume}})^2 dx}{\int \tilde {v}_z dx}}, \end{equation}

with $\gamma$ a constant of order unity. In our experiments we empirically set $\gamma =3/2$ as a compromise for the plume radius to include most of the downward fluid motions without extending as far as the regions of recirculations (see the black and white dashed lines in figure 11 a). The plume is defined at all heights as the region where $|x-x_{\textit{c,plume}}|\leqslant \sigma _{\textit{x,plume}}(z)$ .

Figure 11. Plume structure for an experiment of properties $r_p=45 {\,\unicode{x03BC} {\textrm{m}}}$ , $\dot {m}=0.09\, \,\textrm g\, \,\textrm s^{-1}$ . (a) Time-averaged plume structure during the quasi-steady regime. The time-averaged vertical fluid velocities $\langle v_z \rangle _t$ are normalised by the maximum fluid velocity at every height $z$ . The top convergence and bottom divergence have been cropped to focus on the region of the plume that is least influenced by the boundaries. (b) Superimposition of the normalised profiles of the time-averaged positive vertical velocity $\tilde {v}_z=\max (v_z,0)$ at each depth, with abscissas normalised by the plume width at the same depth.

In our experiments the plume width does not vary substantially with depth, for several reasons. When using large grains, the flow largely departs from a plume and is laminar (see figure 3 a); when using intermediate grains with a moderate mass flux $\dot {m}$ , most of the plume is laminar as well (shear instabilities only emerge about a decimetre deep in figure 3 c). In both cases, little growth is expected. When using sufficiently small grains, and for large values of $\dot {m}$ , the flow is turbulent. The plume source essentially injects buoyancy; the volume and momentum fluxes feeding the plume originate from the convergence of the fluid below the source and its subsequent downwelling. Hence, compared with a pure plume that grows linearly with depth, our plumes have a momentum deficit near the source, which explains the formation of a constriction in this region, as visible in figure 3(b) (see a discussion about this specific regime of plumes in, e.g. Hunt & Kaye Reference Hunt and Kaye2005). Below the constriction, the plume grows with depth (see figure 3 b), but it is constrained by the lateral recirculation and it cannot grow over a significant distance due to the dimensions of the sieve and tank. These observations on the plume growth are consistent with expectations from the literature on plumes having a momentum deficit (Hunt & Kaye Reference Hunt and Kaye2005; Schellenberg, Newton & Hunt Reference Schellenberg, Newton and Hunt2023). The collapse of velocity profiles (figure 11 b) is also consistent with the literature – which, for a negligible recirculation, should be Gaussian and therefore vanish to zero outside the plume. Yet, quantifying the growth rate of the turbulent plumes, and removing the influence of the recirculation on velocity profiles, requires dedicated additional experiments in a larger tank.

The vertical velocity is averaged over the whole plume and results are shown for all experiments in figure 12(a). Scatter is due to turbulence in the flow and to the sensitivity of the plume to the recirculations that tend to constrain the plume orientation and deflect it from the vertical. Nevertheless, measurements show clear trends: fluid motions are more vigorous when the mass rate is larger, and conversely tend to weaken as the grain size increases. This is consistent with previous sections (e.g. figure 7) and, in particular, with the analysis of the onset of the flow. However, the velocity $U_{\textit{onset}}$  (3.7) is not an appropriate estimate for the plume velocity because it depends excessively on the grain size: $U_{\textit{onset}}\propto w_s^{-1/2}$ , which gets infinitely large for vanishingly small grains. This inconsistency originates from the assumption that the cylindrical sugary layer dilutes in a volume growing with a velocity $w_s$ , which only holds before fluid motions appear. Afterwards, fluid motions dilute grains and dissolved sugar by advecting them away from the free surface; this reduces both the plume buoyancy and the fluid inertia, which we assume to be in balance in the quasi-steady regime.

Figure 12. (a) Average vertical velocity in the plume. (b) Evolution of the characteristic plume velocity $U_{\textit{plume}}$ in the parameter space. White lines are logarithmically equispaced isocontours. The red line corresponds to the isovalue $w_s/U_{\textit{plume}}=1$ and the dark line corresponds to the isovalue $w_s/U_{\textit{onset}}=1$ . (c) Measurements of the average plume velocity normalised by $U_{\textit{plume}}$ .

To refine our estimate of the plume velocity, let us denote $U_{\textit{plume}}$ the constant velocity that is expected in the plume by balancing the fluid inertia and the plume effective buoyancy. Let us assume that the radius of the plume is constant and equal to $R_{\textit{sieve}}$ as a first-order estimate. During any time change $\Delta t$ , the mass $\dot {m}\Delta t$ that is sieved dilutes in a volume of plume $\pi R_{\textit{sieve}}^2(U_{\textit{plume}}+w_s)\Delta t$ due to the combined transport of sugar by the settling of grains and the advection by the flow. Note that we neglect the side fluxes associated with the global recirculation and with the entrainment and mixing of the surrounding pure water. Therefore, the effective density is now expected to scale as

(4.3) \begin{equation} \rho _{\textit{eff,QS}} = \rho _0 + \frac {\dot {m}}{\pi R_{\textit{sieve}}^2 (U_{\textit{plume}}+w_s)} . \end{equation}

Following the reasoning of § 3.3 and again neglecting any prefactor of order unity, (3.7) now reads

(4.4) \begin{equation} U_{\textit{plume}} = \left [\frac {\rho _{\textit{eff,QS}}-\rho _0}{\rho _0}g R_{\textit{sieve}} \right ]^{1/2} = \left [\frac {\dot {m}g}{\pi R_{\textit{sieve}}(U_{\textit{plume}}+w_s)\rho _0} \right ]^{1/2} . \end{equation}

The large values of the plume Reynolds number $2R_{\textit{sieve}} U_{\textit{plume}} /\nu$ in the range 1100–5500 confirm that buoyancy is balanced by inertia rather than viscous forces. Terms can be recast in the previous equation and we have to solve for $U_{\textit{plume}}(\dot {m},w_s)$ in the following cubic equation:

(4.5) \begin{equation} U_{\textit{plume}}^3 + U_{\textit{plume}}^2 w_s = \frac {\dot {m}g}{\pi R_{\textit{sieve}}\rho _0} . \end{equation}

The two asymptotes are informative. When the settling velocity is very large compared with the plume velocity, only the term of leading order $U_{\textit{plume}}^2w_s$ is retained on the left-hand side of (4.5) so we recover the expression of $U_{\textit{onset}}$ in (3.7). In the limit of vanishing settling velocities, grains are so small that they behave as tracers, so the fluid velocity $U_{\textit{tracers}}$ should be independent of $w_s$ ; it is indeed the case when simplifying (4.5) for $w_s \ll U_{\textit{plume}}$ , which yields

(4.6) \begin{equation} U_{\textit{tracers}} = \left (\frac {\dot {m}g}{\pi R_{\textit{sieve}}\rho _0} \right )^{1/3} . \end{equation}

Equation (4.6) is the typical velocity that can be expected from dimensional analysis (Friedl et al. Reference Friedl, Haertel and Fannelop1999; Caballina et al. Reference Caballina, Climent and Dušek2003). Since $U_{\textit{tracers}}$ is independent of the particle size, it solves the problem of the divergence of $U_{\textit{onset}}$ in the limit of vanishing settling velocities.

For all experiments, the exact real root of (4.5) is found using the library Numpy in Python, and results are shown in figure 12(b). The values of $U_{\textit{plume}}$ are consistent with the fluid velocities measured in experiments and evidence the expected trends: they increase as sugar grains are smaller and as the mass flux increases. The velocity $U_{\textit{plume}}$ depends less and less on $r_p$ as the grain size vanishes, as expected from the expression of $U_{\textit{tracers}}$ in (4.6).

The accuracy of the prediction $U_{\textit{plume}}$ is confirmed by rescaling the measurements of the plume velocities in figure 12(c). All data points lie near unity, with only a few measurements that depart from unity by a factor $\sim 2$ . This good agreement raises a question: In the quasi-steady regime, once the flow has developed in the plume, could it be that the specificities of small versus large grains average out, since we observe that they have little influence on the average plume velocity? Indeed, fluid velocities are essentially controlled by the balance between the inertia and the buoyancy that dilutes in the downward stream. The question remains open, but note that this last observation may only hold for the average downward plume velocity, since previous sections showed that discrepancies between small and large grains do remain in the quasi-steady regime, in particular, the turbulent nature of the flow when $r_p$ is small, compared with the laminar nature of the flow when $r_p$ is large.

5.1. Transient sinking of sugar grains

The evolution of the dissolution layer is consistent with the observations made with figures 3 and 6. The first sugar grains that fall in the tank propagate downward in quiescent water; successive grains fall in the wake of previous grains and dissolve, until the sugary layer pushes the underlying fluid downward and radially outward; this forces a recirculation that leads to a constriction of the flow favouring the downward acceleration of fluid motions. The latter carry sugar grains to larger depths until a quasi-steady state is reached: the dissolution layer has reached its equilibrium depth. The same behaviour is recovered in all cases except when $r_p = 363 {\,\unicode{x03BC} {\textrm{m}}}$ and the mass rate is too low. When laden with small grains, the dissolution layer initially accelerates before sinking with a constant velocity; the subsequent quasi-steady regime shows oscillations due to successive turbulent puffs of grains that never reach the bottom of the tank. When grains are sufficiently large to decouple from the starting plume, they fall ahead of the plume cap, hence, no clear phase of acceleration appears. Such large grains fall with an approximately constant velocity, down to the bottom of the tank when the mass rate is large; see Kriaa (Reference Kriaa2023) for more details.

How fast does the dissolution layer form? To quantify the sinking velocity of the dissolution layer, the light intensity of each photograph is averaged over the horizontal extent of the field of view and the resulting vertical profiles are stacked in time to build space–time diagrams. Figures 13(a)–13(c), illustrate the influence of the mass rate on the growth of the dissolution layer for $r_p=101 {\,\unicode{x03BC} {\textrm{m}}}$ . The larger the mass rate, the larger the volume fraction in the dissolution layer, and hence, the larger the horizontally averaged light intensity. Also, a larger mass rate leads to a larger effective buoyancy in the plume, and hence, a shorter transient and a flow with larger fluctuations.

Figure 13. Space–time diagrams of horizontally averaged light intensity for $r_p=101 {\,\unicode{x03BC} {\textrm{m}}}$ . Mass rates are (a) $\dot{m} = 0.18\,\textrm{g}\,\textrm{s}^{-1}$ , (b) $\dot{m} = 0.36\,\textrm{g}\,\textrm{s}^{-1}$ and (c) $\dot{m} = 0.71\,\textrm{g}\,\textrm{s}^{-1}$ . The black and white dotted curves show the bottom of the dissolution layer.

The convergence of isocontours of light intensity in the space–time diagrams enables us to define the front of the dissolution layer, as shown by a black and white dashed line in figure 13. In the present section we focus on the sinking of the dissolution layer in the transient, which ends when the front position of the dissolution layer no longer increases in time but only fluctuates around a constant value. Figures 13(a)–13(c) show that the dissolution layer sinks with a low initial velocity in the first ten centimetres before accelerating. In the second phase the slope of the black and white dashed line is comparable to that of the puffs of grains in the quasi-steady regime, suggesting that this slope is characteristic of the fluid motions in the plume once quasi-steadiness is reached. Since our interest is presently on the transient, we analyse the initial low sinking velocity before the change of slope. This sinking velocity of the dissolution layer is denoted $\dot {z}_{\textit{DL}}$ . It is measured by fitting a linear law on the dashed curve for each experiment, and results are shown in figure 14(a). The sinking velocity of sufficiently small grains ( $r_p\lt 169 {\,\unicode{x03BC} {\textrm{m}}}$ ) increases essentially with the mass rate with little influence of the grain size. Conversely, for a fixed mass rate, the radius $r_p$ noticeably modifies the sinking velocity when $r_p \geqslant 169 {\,\unicode{x03BC} {\textrm{m}}}$ , with sudden jumps when $r_p$ varies from $101$ to $169 {\,\unicode{x03BC} {\textrm{m}}}$ , and then from $169$ to $363 {\,\unicode{x03BC} {\textrm{m}}}$ . These jumps are due to the gravitational decoupling between the starting plume and these large sugar grains whose settling velocity is much larger than the fluid velocities. This is shown in figure 14(b): unlike smaller grains, those in the range $r_p\geqslant 169 {\,\unicode{x03BC} {\textrm{m}}}$ verify $\dot {z}_{\textit{DL}}/ w_s \simeq 1$ .

Figure 14. (a) Sinking velocity $\dot {z}_{\textit{DL}}$ of the dissolution layer as detected from Hovmoller diagrams. (b) The same data are rescaled by the settling velocity $w_s$ and shown as a function of the initial radius. For $r_p\geqslant 169 {\,\unicode{x03BC} {\textrm{m}}}$ , data collapse on the dashed black curve of $\dot {z}_{\textit{DL}}/ w_s=1$ . (c) Comparison of the sinking velocity $\dot {z}_{\textit{DL}}$ with the sum of the settling velocity $w_s$ and the measured fluid velocity within the plume during the quasi-steady regime $v_{\textit{z,plume}}$ .

This last conclusion means that the influence of the plume velocities is almost negligible for large grains ( $r_p \geqslant 169 {\,\unicode{x03BC} {\textrm{m}}}$ ). Yet, the fluid velocities are not perfectly negligible. Figure 14(a) shows that the larger the mass rate, the larger the plume velocity $v_{\textit{z,plume}}$ , the faster the grains fall since their velocity reads $v_p = v_{\textit{z,plume}}+w_s$ (see (2.3)). This is confirmed in figure 14(c) which shows the normalised velocity $\dot {z}_{\textit{sinking}}/(v_{\textit{z,plume}}+w_s)$ for all experiments that included PIV measurements. As expected, the rescaled velocities are of order unity and major jumps due to the particle size have now disappeared. We still observe a departure from unity by a factor $\sim 1/2$ . A key aspect that likely contributes to this departure is the fact that during the transient, the plume velocity is still developing, so the use of the steady-state velocity $v_{\textit{z,plume}}$ in the normalisation is an anticipation that overestimates the role of advection in carrying grains faster downstream.

A slight increase of the normalised velocity $\dot {z}_{\textit{DL}}/(v_{\textit{z,plume}}+w_s)$ is still observable as the mass flux increases, especially for small grains. This remaining trend can originate from different physical aspects that have been neglected so far, which are mentioned in § 6.

5.2. Equilibrium depth

The transient ceases when the depth of the dissolution layer no longer increases and only fluctuates around a constant value. This depth is therefore averaged over the whole quasi-steady regime, and the resulting measurements of the equilibrium depth $z^\infty _{\textit{DL}}$ are shown in figure 15. Measurements for the size $r_p=363 {\,\unicode{x03BC} {\textrm{m}}}$ are irrelevant and not shown because such grains always hit the bottom of the tank before full dissolution. For $r_p\lt 363 {\,\unicode{x03BC} {\textrm{m}}}$ , figure 15 shows that for a fixed mass flux, the larger the grains the deeper the equilibrium depth. This can be captured from simple arguments. In the following we distinguish between the initial radius of grains ( $r_{p0}$ ) and their radius $r_p$ at any later time.

Figure 15. Equilibrium depth of the dissolution layer for all experiments performed with ordinary white sugar.

Consider the fall of a supposedly spherical and isolated grain of initial radius $r_{p0}$ in clear motionless water. Let us assume that at any time, the buoyancy and drag of the grain are in balance so that the grain falls at its terminal velocity $w_s(t) = w_s(r_p(t))$ . The time evolution of the grain can be solved numerically by integrating the two following equations:

\begin{align*} \kern120pt \begin{cases} \dot {r}_p & = -\overline {k}(t) \frac {\mathcal{C}_{\textit{sat}}}{\rho _p}, \qquad \quad \kern120pt \text{(}\text{5.1}\textit{a}\text{)}\\ \dot {z}_p & = w_s(r_p(t)).\qquad \kern132pt \text{(}\text{5.1}\textit{b}\text{)}\end{cases} \end{align*}

Here $z_p$ is the time-dependent vertical position of the grain centre of mass and (5.1a ) follows from the mass balance in (2.10) after assuming that the concentration at the grain interface is the saturation concentration so that $\rho _i=\rho _{\textit{sat}}$ (see § 2.2.3). Solutions of (5.1a ) and (5.1b ) are obtained by numerical integration with the initial conditions $r_p (t=0)=r_{p0}$ and $z_p(t=0)=0$ . Measurements of the equilibrium depth of the dissolution layer $z^\infty _{\textit{DL}}$ are compared with the numerical solutions $z_{\textit{max}}$ of (5.1a ) and (5.1b ) in figure 16(a). Almost all data points lie above unity, meaning sugar grains fall deeper than the depth $z_{\textit{max}}$ . Additionally, the smaller the grains, the farther away they fall compared with $z_{\textit{max}}$ , even more so as the mass rate increases. These observations are due to the downward fluid motions that carry sugar grains deep in the tank and enhance their maximum depth of dissolution. This effect is even more pronounced for smaller grains due to the large contrast between their small settling velocity and the large fluid velocity.

Figure 16. (a) Comparison between the measured equilibrium depth of the dissolution layer $z^\infty _{\textit{DL}}$ and the numerical value $z_{\textit{max}}$ of the maximum distance travelled by an isolated grain settling in clear still water. (b) Comparison between $z^\infty _{\textit{DL}}$ and the prediction $z_{\textit{max}} + U_{\textit{plume}} t_{\textit{max}}$ , with the solid dark line corresponding to the first bisector. The colourbar is common for all graphs; the size $r_p$ is implicitly the initial grain radius $r_{p0}$ that is used to compute $z_{\textit{max}}$ and $t_{\textit{max}}$ .

To account for this effect, we need to take into account the advection by the plume velocity. According to (2.3), this advection can be taken into account by adding a constant plume velocity on the right-hand side of (5.1b ). Therefore, any grain is expected to travel a distance that is the sum of the advection by the plume, and the distance $z_{\textit{max}}$ of complete dissolution that the grain travels by gravitational drift with respect to the fluid. Figure 16(b) compares the measurements $z_{\textit{DL}}^\infty$ with the prediction $z_{\textit{max}} + U_{\textit{plume}} t_{\textit{max}}$ using (4.5), and shows that this prediction captures reasonably well the contributions of both grain settling and advection by the plume to set the maximum dissolution depth.

The prediction $z_{\textit{max}} + U_{\textit{plume}} t_{\textit{max}}$ underestimates the dissolution depth. A missing contribution is the constriction of the plumes, which results in an acceleration of fluid with the spontaneous evolution of the plume to a state of lower momentum deficit (Hunt & Kaye Reference Hunt and Kaye2005). Although a depth-dependent description of the plumes is beyond the scope of the present paper, recent studies (Meunier & Nadal Reference Meunier and Nadal2018; Webb, Wise & Hunt Reference Webb, Wise and Hunt2023) provide models that can be adapted to account for this constriction and improve the prediction. Other missing contributions may originate from microphysical aspects that we have neglected. In the solutal boundary layer, sucrose is highly concentrated up to saturation at the grain surface. As a result, its diffusivity can reduce by an order of magnitude (English & Dole Reference English and Dole1950), delaying dissolution. Another consequence is that the solutal boundary layer is negatively buoyant and much more viscous than clear water, up to 770 times more at the grain surface (Davies Wykes et al. Reference Wykes, Megan, Huang, Hajjar and Ristroph2018). Consistently, we observed (not shown here) that grains dissolve in a surrounding envelope of solute that accompanies them during their fall (see Kerr Reference Kerr1995). The large sugar concentration in the envelope may delay dissolution, but this envelope may also enhance the grains’ speed, especially when their radius vanishes. Although a simple heuristic model incorporating these effects can match our measurements, additional observations are necessary to quantify the competition between these microscale effects, which is beyond the scope of the present study.

These same elements may explain the remaining increase of the sinking velocity $\dot {z}_{\textit{sinking}}/(v_{\textit{z,plume}}+w_s)$ with $\dot {m}$ (figure 14 c).