3.1. A Lagrangian-based mathematical description of fragmentation

Previous work on the movement of volume in fragmentation cascades applies Eulerian descriptions, focusing on volume flux. To find the equilibrium between entrainment and fragmentation, Gaylo et al. (Reference Gaylo, Hendrickson and Yue2021) balance the flux of volume in and out of the set of bubbles of a given range of sizes. To evaluate locality, Chan et al. (Reference Chan, Johnson and Moin2021b) study the flux of volume from bubbles larger than a given size to those smaller. Eulerian descriptions are useful to model the volume flow in and out of specified bubble sizes, but to derive $\tau _c$ we need to understand how any specific air volume flows through the entire cascade. For this, a Lagrangian description is more direct.

Consider how a single Lagrangian particle of air $p$ moves through a fragmentation cascade, illustrated in figure 1. Let $a_{p}(t)$ be the effective radius of the bubble that contains $p$ at time $t$. Treating the interface between fluids as zero-thickness, one bubble breaks up into two instantaneously, thus $a_{p}(t)$ is a step function. From this $a_{p}(t)$, we have a simple expression for $\tau _c$: defining time for a particle such that $a_p(0)=a_{max}$, our interest is the expected time for $p$ to reach the Hinze scale,

(3.2)\begin{equation} \tau_c \equiv \mathbb{E}\{\min\{t \;:\; a_p(t)\le a_{H}\}\} {.} \end{equation}

We now develop a relationship between this Lagrangian description and the previous fragmentation statistics. Incorporating the measurement interval $T$, we define the volume ratio between the bubble containing $p$ at time $t$ and the bubble containing $p$ at time $t+T$ as

(3.3)\begin{equation} v_R(t;T) \equiv [a_p(t+T)/a_p(t)]^3 {.} \end{equation}

If the bubble containing $p$ at time $t$ does not fragment over the measurement interval $T$, then $v_R=1$. If the bubble does fragment, then $v_R$ depends on the size of the daughter bubble that $p$ ends up in. Noting that the probability $p$ ends up in a given daughter is equivalent to $v^*$, the ratio of the volume of the daughter to that of the parent, the probability distribution function for $v_R$ given that fragmentation occurs, $f_{V_R \mid \text {frag}}$, is related to the previous fragmentation statistics through

(3.4)\begin{equation} f_{V_R \mid \text{frag}}(v_R; t, T) = \bar{m}(a_p(t); T) \; v_R \; f^*_V(v_R; a_p(t), T) {.} \end{equation}

We assume these statistics are scale invariant and introduce the non-dimensional parameter $T^*=T \varepsilon ^{1/3} a_p(t)^{-2/3}$. This gives

(3.5)\begin{equation} f_{V_R \mid \text{frag}}(v_R; T^*) = \bar{m}(T^*) \; v_R \; f^*_V(v_R; T^*) {,} \end{equation}

and any moment $n$ of the distribution is given by

(3.6) \begin{equation} \mathbb{E}\{[v_R(T^*)]^n \mid \text{frag}\} = \bar{m}(T^*) \int_0^1 {v^*}^{n+1} f^*_V(v^*;T^*) \,\mathrm{d} v^* {.} \end{equation}

Figure 1.

(a) Schematic of the Lagrangian description showing the path of a Lagrangian air particle $p$ through a sequence of fragmentations from large to small radii, $a_0, a_1, \ldots, a_n$, of the bubble containing $p$; and (b) the function $a_p(t)$ describing the evolution of this bubble radius. Describing the radius of the bubble containing $p$ as a function of time allows a propagation speed of $p$ through the cascade to be defined.

3.2. Defining the volume-propagation speed in a fragmentation cascade

To obtain $\tau _c$, we derive a metric that measures the speed at which Lagrangian air particles move through fragmentation cascades. To derive a speed, we must first define the ‘location’, $x(t)$, of a Lagrangian air particle $p$ within the cascade. In this case location refers to some scalar bubble-size metric within the cascade rather than a physical spatial coordinate. We define $x(t)$ to describe the location of $p$ within the fragmentation cascade such that the associated speed $s(t)\equiv \dot {x}(t)$ is constant for $a_p(t) > a_H$. A constant speed is necessary for many of the properties that will follow and, as a result of the scaling in (2.4), is achieved only by $x(t)\propto a_p(t)^{2/3}$. We choose

(3.7)\begin{equation} x(t) \equiv{-} \varepsilon^{{-}1/3} {a_p(t)}^{2/3}{,} \end{equation}

which has dimensions of time, so that, in addition to being constant, the time derivative of $x(t)$,

(3.8)\begin{equation} s(t) ={-}\frac{2}{3} \varepsilon^{{-}1/3} {a_p(t)}^{{-}1/3} \frac{\,\mathrm{d} }{\,\mathrm{d} t} a_p(t) {,} \end{equation}

is also positive and non-dimensional.

Because $a_p(t)$ is a step function, the derivative in (3.8) is ill-behaved. Thus, to evaluate $s(t)$ we consider its time-averaged value over a measurement interval $T$,

(3.9)\begin{equation} \langle s(t) \rangle_{T} \equiv \frac{1}{T} \int_{t}^{t+T} s(t') \,\mathrm{d} t' {.} \end{equation}

This gives

(3.10)\begin{equation} \langle s(t) \rangle_{T} = \frac{x(t+T) - x(t)}{T} = \frac{\varepsilon^{{-}1/3} a_p(t)^{2/3}}{T}(1-[v_R (t;T)]^{2/9}) {,} \end{equation}

where (3.3) defines the volume ratio $v_R(t;T)$. Furthermore, we perform an ensemble average to get $\mathbb {E}\{\langle s(t) \rangle _{T}\}$, the expected time-averaged speed for an ensemble of (independent) Lagrangian air particles. Noting that $\langle s(t) \rangle _{T}=0$ if no fragmentation occurs over the interval $T$,

(3.11)\begin{equation} \mathbb{E}\left\{\langle s(t) \rangle_{T}\right\} = \frac{p_{{\rm frag}}(a_p(t); T)}{\varepsilon^{1/3} a_p(t)^{{-}2/3} T} (1-\mathbb{E}\{[v_R(t;T)]^{2/9} \mid \text{frag}\}) {.} \end{equation}

The no-hysteresis assumption, along with (2.4), gives

(3.12)\begin{equation} \mathbb{E}\left\{\langle s(t) \rangle_{T}\right\} = C_{\varOmega}(We) \frac{1-\exp[-\varOmega(a_p(t)) T]}{\varOmega(a_p(t)) T} (1-\mathbb{E}\{[v_R(t;T)]^{2/9} \mid \text{frag}\}) {.} \end{equation}

Recalling that, by assumption, these statistics are scale invariant, we introduce $T^*$ and apply (3.6) to obtain

(3.13)\begin{equation} \mathbb{E}\left\{\langle s(t) \rangle_{T^*}\right\} = C_{\varOmega,\infty} \frac{1-\exp[- C_{\varOmega,\infty}\, T^* ]}{C_{\varOmega,\infty}\,T^* } \left[1-\bar{m}(T^*) \int_0^{1} {v^*}^{11/9} f_V^*(v^*; T^*) \,\mathrm{d} v^* \right]{.} \end{equation}

The limit $T^*\to 0$ gives the expected instantaneous speed,

(3.14)\begin{equation} \bar{s} \equiv \lim_{T^*\to 0} \mathbb{E}\left\{\langle s(t) \rangle_{T^*}\right\} = C_{\varOmega,\infty} \left[1-\bar{m} \int_0^{1} {v^*}^{11/9} f_{V}^*(v^*) \,\mathrm{d} v^* \right] {,} \end{equation}

where $\bar {m}$ and $f_{V}^*(v^*)$ describe the fragmentation statistics for $T^*\to 0$ and are equivalent to those in (2.1).

Hereafter, we refer to $\bar {s}$ as the volume-propagation speed of a fragmentation cascade. Although the size locations of individual Lagrangian air particles in the cascade follow step functions, by commuting time averaging and ensemble averaging, we are able to obtain an average instantaneous speed for particles in the cascade. This speed $\bar {s}$ can be related to fragmentation statistics measured over finite intervals $T$ (3.13), or the instantaneous statistics used by PBE (3.14). The relationship between the two is explored in § 3.4. In § 3.3 we use $\bar {s}$ to provide $\tau _c$.

3.3. Describing convergence time, $\tau _c$

As intended, our choice of the definition of location within the cascade, $x(t)$, makes $\bar {s}$ constant for $a_p(t)>a_H$. This constant speed means that, despite $x(t)$ being a step function, after a sufficient number of steps, we can treat fragmentation as a continuous process and apply the approximation $x(t) \approx t \bar {s}$ with reasonable (statistical) accuracy. Thus, we can approximate $\tau _c$ as the distance in $x$ between $a_{max}$ and $a_{H}$ divided by this speed,

(3.15)\begin{equation} \tau_c= \frac{\left(\varepsilon^{{-}1/3} {a_{max}}^{2/3} \right)-\left(\varepsilon^{{-}1/3} {a_H}^{2/3}\right)}{\bar{s}} {.} \end{equation}

Non-dimensionalising $\tau ^*_c=\tau _c \,\varepsilon ^{1/3} {a_{max}}^{-2/3}$ and defining $We_{max}$ to be the $We$ associated with $a_{max}$,

(3.16a,b)\begin{equation} \tau^*_c=C_{\tau} [1-(We_{max}/We_H)^{{-}2/5}] {;}\quad C_{\tau} \equiv 1/\bar{s} {.} \end{equation}

Despite the approximation used to derive (3.15) from $\bar {s}$ in (3.14), (3.16) is expected to be valid for $We^*=We_{max}/We_H$ not small (where multiple fragmentation events are generally necessary to reach $a_H$). This is confirmed by Monte Carlo simulations of prescribed fragmentation statistics (figure 2).

Figure 2.

The effect of $We^*$ on $\tau ^*_c$ as modelled by (3.16) (solid line) compared with Monte Carlo simulations of daughter distributions: $\bullet$, A; $+$, blue, B; $\times$, yellow, C; $\square$, red, D; $\circ$, green, E; $\circ$, violet, F (see table 1), where (3.1) is used to model the Hinze scale. The $95\,\%$ confidence interval on all $\tau ^*_c$ is $<1\,\%$.

Table 1.

Daughter distributions used in Monte Carlo simulations and corresponding daughter-distribution constants $C_{f}$ defined by (3.17) versus ${C_{f}}^\star$ defined by Gaylo et al. (Reference Gaylo, Hendrickson and Yue2021, (4.3)). Note, a constant to ensure $\int f_V^*(v^*)\,\mathrm {d}v^*=1$ is omitted for brevity.

For $We\sim \infty$ we recover the same $\tau _c\propto \varepsilon ^{-1/3} {a_{max}}^{2/3}$ scaling as previous work which assumes identical fragmentation (Deike et al. Reference Deike, Melville and Popinet2016). This scaling of $\tau _c$ is like $\tau _\ell$, demonstrating that the fragmentation rate is a dominant factor in determining $\tau _c$. Our propagation speed-based analysis provides the scaling constant $C_\tau$ which quantifies the contribution of fragmentation rate, as well as fragmentation statistics $\bar {m}$ and $f^*_V(v^*)$. For large-but-finite $We$, (3.16) captures the effect of the $We$-driven separation between $a_{max}$ and $a_H$ on the value of $\tau _c$; however, we note that the scaling or $\tau _c$ with $We$ will be more complex for small $We$ ($We \sim We _H$) as we have not incorporated the effect of finite-$We$ on fragmentation rate, such as modelled by (2.5), into our propagation speed-based analysis. In § 4.5, DNS shows for what sufficiently large $We$ this effect is negligible.

Although primarily driven by fragmentation rate, $\tau _c$ is also related to the fragmentation statistics $\bar {m}$ and $f^*_V(v^*)$ (Qi et al. Reference Qi, Mohammad Masuk and Ni2020), which is now quantified by the scaling constant $C_\tau$. To describe these relationships, we follow Gaylo et al. (Reference Gaylo, Hendrickson and Yue2021) and isolate the effect of $f_{V}^*$ from $\bar {m}$ through a daughter-distribution constant $C_{f}$, defined as the ratio between $C_{\tau }$ and a $C_{\tau }$ found using the same $\bar {m}$ but identical fragmentation, $f_{V}^*(v^*)=\delta (v^*-1/\bar {m})$, where $\delta$ is the Dirac delta function. This gives

(3.17a,b)\begin{equation} C_\tau = \frac{C_{f}/C_{\varOmega,\infty}}{1-\bar{m}^{{-}2/9}} {;}\quad C_{f} = \frac{1-\bar{m}^{{-}2/9}}{1-\bar{m} \int_0^{1} {v^*}^{11/9} f_{V}^*(v^*) \,\mathrm{d} v^*}{.} \end{equation}

In table 1 we compare this $C_{f}$ for general fragmentation cascades to the similar constant (hereafter denoted as ${C_{f}}^\star$) derived by Gaylo et al. (Reference Gaylo, Hendrickson and Yue2021) for the special case of power-law entrainment. The values are nearly equivalent, and, noting that $(9/2)(\ln \bar {m})^{-1} \approx (1-\bar {m}^{-2/9})^{-1}$, (3.17) predicts similar $\tau _c$ as Gaylo et al. (Reference Gaylo, Hendrickson and Yue2021) for their special case.

3.4. Measurement-interval independence of volume-propagation speed

A consequence of $\bar {s}$ being constant for $a_p(t)>a_H$ is that the time-averaged value and the instantaneous speed are equal, $\mathbb {E}\{\langle s(t) \rangle _{T}\}=\bar {s}$, so long as $a_p(t+T)>a_H$. Thus, to obtain $\bar {s}$ we must choose a $T$ such that $\Pr \{a(t+T)>a_H\}\approx 1$. For measurements of an initial parent-bubble radius $a=a_p(t)$, we define an upper bound $T_{U}$ as the interval we expect $a_p(t+T_{U})\sim a_H$ and require $T \ll T_{U}$. Through the same arguments used to derive $\tau _c$, this upper bound is

(3.18)\begin{equation} T \ll \varepsilon^{{-}1/3} {a}^{2/3} C_\tau [1-(We/We_H)^{{-}2/5}] {,} \end{equation}

or simply $T \ll \tau _c$ for $a=a_{max}$. From Monte Carlo simulations of prescribed fragmentation statistics measuring initial bubbles $a=a_{max}$, figure 3 confirms that $\mathbb {E}\{\langle s \rangle _{T}\}$ gives an exact, $T$-independent measurement of $\bar {s}$ for $T \ll \tau _c$.  $T_{U}$ provides an upper bound on $T$ for experiments or simulations, although we point out that it is an a posteriori measure because $C_\tau$ is derived from $\bar {s}$.

Figure 3.

Measurements of $\mathbb {E}\{\langle s\rangle _{T}\}$ from Monte Carlo simulations of daughter distributions A–F (see table 1) at a range of $T/\tau _c$, normalised by $\bar {s}$ calculated using (3.14). Colours based on $We^*$: green, $2$; red, $50$; blue, $100$; magenta, $200$, where (3.1) is used to model the Hinze scale. The $95\,\%$ confidence interval on $\mathbb {E}\{\langle s\rangle _{T}\}$ for $T/\tau _c<1$ is $<3\,\%$.

Finally, $T$-independence means $\,\mathrm {d} \mathbb {E}\{\langle s(t) \rangle _{T}\}/\,\mathrm {d} T=0$. As can been seen by taking the derivative of (3.13) with $T^*$, this bounds how scale-invariant fragmentation statistics $\bar {m}(T^*)$ and $f^*_V(v^*; T^*)$ can depend on $T^*$ and provides insight into the relationship between $\bar {m}(T^*)$ and $f^*_V(v^*; T^*)$ measured at large $T^*$ versus the theoretical $T^*\to 0$ limiting case used in PBE. This is useful because a finite relaxation time $\tau _r$ implies a lower bound ($T> \tau _r$) for measuring fragmentation statistics that are compatible with the PBE no-hysteresis assumption.

4.1. Methodology

For DNS, we solve the three-dimensional, incompressible, immiscible, two-phase, Navier–Stokes equations using a second-order finite-volume scheme on a uniform Cartesian grid. Phases are captured by the conservative volume-of-fluid method (Weymouth & Yue Reference Weymouth and Yue2010), and surface tension is calculated using a height-function-based continuous-surface-force method (Popinet Reference Popinet2009). More detail on the DNS solver is provided by Campbell (Reference Campbell2014) and Yu et al. (Reference Yu, Hendrickson, Campbell and Yue2019). During the simulation, normals-based informed component labelling (Hendrickson, Weymouth & Yue Reference Hendrickson, Weymouth and Yue2020) identifies bubbles, the air volumes of which are then tracked using Eulerian label advection  (ELA) (Gaylo et al. Reference Gaylo, Hendrickson and Yue2022).

To develop the initial turbulent velocity field for the simulation, we use a linear forcing method (Lundgren Reference Lundgren2003; Rosales & Meneveau Reference Rosales and Meneveau2005) on a triply periodic cubic domain, length $L=5.28$, to develop single-phase HIT with a (non-dimensionalised) characteristic turbulent dissipation rate $\varepsilon =1$, velocity fluctuation $u_{{rms}}=1$ and Reynolds number $Re_T = {u_{{rms}}^4}/{\varepsilon \nu _{w}}=200$. Using the single-phase HIT as the initial velocity field, we perform simulations with an ensemble of different initial air–water bubble populations (density ratio $\rho _w/\rho _a=1000$, viscosity ratio $\mu _w/\mu _a=100$, void fraction $1\,\%$) at a range of turbulent Weber numbers, $We_T = {\rho _w u_{{rms}}^5}/{\varepsilon \sigma }$. Although the abrupt introduction of bubbles to single-phase HIT is non-physical, numerical simulations rapidly adjust (Yu et al. Reference Yu, Hendrickson, Campbell and Yue2019; Rivière et al. Reference Rivière, Mostert, Perrard and Deike2021). Populations are created by randomly distributing (without overlap) spherical bubbles with radii between $3L/256$ and $15L/256$ following $N(a)\propto a^{-10/3}$. By repeating the random generation and distribution of bubble populations in the initial HIT velocity field, unique but statistically similar initial bubble populations are generated to provide statistical variation between our ensemble simulations.

During the evolution, linear forcing is applied to regions of water to maintain $\varepsilon \approx 1$ (Rivière et al. Reference Rivière, Mostert, Perrard and Deike2021). Figure 4 shows the evolution of a sample simulation and figure 5 shows the evolution of the ensemble bubble-size distribution both for $We_T=100$. We note that, with our focus on bubbles $a>a_H$, the transition to a distinct a power-law regime for $N(a< a_H)$ is not captured (Deane & Stokes Reference Deane and Stokes2002). Over a measurement interval $t^n$ to $t^{n+1}=t^n+T$, ELA provides the unique, volume-conservative volume-tracking matrix, where each element $a_{ij}$ describes the volume that moved from a parent bubble $j$ that is identified at $t^n$ to a bubble $i$ identified at $t^{n+1}$ (Gaylo et al. Reference Gaylo, Hendrickson and Yue2022). From volume-tracking matrices, fragmentation statistics $\mathbb {E}\{\langle s \rangle _{T} \}$ and $p_{{\rm frag}}(a; T)$ can easily be computed. We study fragmentation statistics for parent bubbles of radii $a_0< a<1.2 a_0$, where $a_0=7 L/256$ provides a balance between the number of observed fragmentation events per simulation and resolution of the daughter bubbles. While this simulation is inherently transient, figure 5 illustrates that for this range of bubbles a quasisteady period exists. By initialising the bubbles to follow an equilibrium fragmentation cascade $N(a)\propto a^{-10/3}$ (Garrett et al. Reference Garrett, Li and Farmer2000), the fragmentation of bubbles $a>a_0$ maintains the population of bubbles $a\sim a_0$ for $t/t_\ell <3$, where $t_\ell ={(0.42)}^{-1} \varepsilon ^{-1/3} {a_0}^{2/3}$ is an a priori estimate of $\tau _\ell$ (Martínez-Bazán et al. Reference Martínez-Bazán, Montañés and Lasheras1999a). To exclude the fragmentation of the initial set of spherical bubbles (see figure 4), we study fragmentation over $1< t/t_\ell$. Thus, by measuring fragmentation statistics over $1< t/t_\ell <3$, we measure a quasisteady population of parent bubbles that are realistically formed by a fragmentation cascade.

Figure 4.

Volume-of-fluid $f=0.5$ isosurface for one of the $We_T=100$ simulations at (a) $t/t_\ell =0$; (b) $t/t_\ell =1$; (c) $t/t_\ell =3$.

Figure 5.

Average bubble-size distribution $N(a)$ for $We_T=100$ simulations at times: red, $t/t_\ell =0$; blue, $t/t_\ell =1$; green, $t/t_\ell =3$. Here $N(a)\propto a^{-10/3}$ is provided for reference over the range of initialised spherical bubbles (dashed line) and the range of measured parent bubbles, $a_0< a<1.2 a_0$ (solid line).

4.2. Grid independence

The choice of cell size, $\varDelta$, is driven by resolving the relevant scales of turbulence and surface tension. For turbulence, we compare the grid with the Kolmogorov microscale, $\eta \sim \varepsilon ^{-1/4} {\nu _w}^{3/4}$, where $\varDelta /\eta \lesssim 1$ ensures turbulence is resolved. For surface tension, we consider the cell Weber number $We_\varDelta = \rho _w u_{{rms}}^2 \varDelta /4 {\rm \pi}\sigma$, which estimates the ratio between the grid and the minimum characteristic radius of curvature of an interface deformed by inertial turbulence. Here $We_\varDelta <1$ ensures surface tension forces are resolved by the grid (Popinet Reference Popinet2018). We also consider $\varDelta /a_H$, comparing the grid to the Hinze scale: with $\varepsilon$ and $u_{{rms}}$ fixed ${We_\varDelta }^{3/5} \propto \varDelta /a_H$. Based on these metrics we find $L/\varDelta =256$ resolves turbulence and surface tension for our entire range of $We_T$ (see table 2).

With no clear lower limit to the ratio between the daughter-bubble and parent-bubble volume ($v^*$), grid resolution limitations require us to filter out daughter bubbles of radius $a<2 \varDelta$. Figure 6 shows that the bubble-size distribution of filtered bubbles, $N(a>2\varDelta )$, is grid-independent. For $L/\varDelta =256$ and parent bubbles $a_0=7 L/256$, $a<2 \varDelta$ corresponds to $v^*<0.02$. While this filter prevents us from measuring the full range of possible daughter bubbles, especially sub-Hinze daughters, we expect this to have little effect on the statistics of interest for two reasons. First, sub-Hinze bubble production by fragmentation happens concurrently with the production of large daughter bubbles (Rivière et al. Reference Rivière, Ruth, Mostert, Deike and Perrard2022), so excluding small daughters should not affect the measured rate of fragmentation used to obtain $\tau _r$ and $\tau _\ell$. Second, for $\tau _c$, the integral of the daughter-size distribution in (3.17) weights local daughter production ($v^*\sim 1/\bar {m}$) over non-local daughter production ($v^*\ll 1$), making the contribution of the excluded small daughters small. This is related conceptually to locality, which suggests $v^*\ll 1$ can be neglected when modelling the cascade (Chan et al. Reference Chan, Johnson and Moin2021b,Reference Chan, Johnson, Moin and Urzayc).

Figure 6.

Average bubble-size distribution $N(a>2\varDelta )$ for $We_T=200$ at time $t/t_\ell =3$ from simulations with grids: magenta, $L/\varDelta =128$; green, $L/\varDelta =192$; black, $L/\varDelta =256$; blue, $L/\varDelta =384$. Horizontal axis is normalised by $\varDelta =L/256$ and $N(a)\propto a^{-10/3}$ is provided for reference over the range of initialised spherical bubbles (dashed line) and the range of measured parent bubbles, $a_0< a<1.2 a_0$ (solid line).

To confirm that we resolve turbulence and surface tension, that the filter has a negligible effect, and (more broadly) that the statistics we measure are independent of the grid, we perform a convergence study for $We_T=200$ using three additional grids, $L/\varDelta =128$, $192$ and $384$. The results of this convergence study (see figure 7) show that our measurements of fragmentation statistics $\mathbb {E}\{\langle s \rangle _{T} \}$ and $p_{{\rm frag}}(a; T)$ (from which the time scales will be calculated) are grid independent for $L/\varDelta \ge 256$.

Figure 7.

Grid-convergence study for (a) fragmentation rate constant $C_\varOmega$ and (b) convergence constant $C_\tau$ based on simulations of $We_T=200$ (parent bubbles $We=50$–$71$) with different grids, measured using $T/t_\ell =0.4$. Error bars indicate $95\,\%$ confidence interval.

4.3. Estimating relaxation time, $\tau _r$

For each simulation, we use six instances of ELA with different measurement intervals $T$. Using (2.4) and (2.7), we calculate $C_\varOmega (We; T)$ from each $p_{{frag}}(a;T)$. Figure 8(a) shows how $T$ affects the measured value of $C_\varOmega$, where we use $T/t_\ell =0.4$ as a reference value for each $We$. If the no-hysteresis assumption were valid for all $T$, $C_\varOmega$ would be a constant for each $We$. Figure 8(a), however, shows a strong dependence on small $T$. We observe that this dependence is approximately exponential, which provides an empirical definition of the relaxation time $\tau _r$ as well as the hysteresis strength $A$:

(4.1)\begin{equation} C_\varOmega(We; T)/C_\varOmega(We; T\sim\infty) = 1 + A \exp[{-}T/\tau_r] {.} \end{equation}

We observe that $\tau _r$ scales like $\tau _\ell$ rather than, say, bubble natural period, $We^{-1/2} \varepsilon ^{-1/3} a^{2/3}$. Thus, we define the scaling constant $C_r$ and write $\tau _r=C_r \varepsilon ^{-1/3} {a}^{2/3}$. This scaling suggests that, for $We>We_H$, the physical mechanisms for the decay of hysteresis are not related to surface tension. Future, more detailed, studies of the dynamics of individual bubbles are necessary to understand hysteresis and identify the mechanisms for its decay. For our statistical study, our concern is to determine when hysteresis can be neglected. Least-squares regression of the combined data for all $We$ gives $C_r\approx 0.11$. Hereafter, we measure all results with $T/t_\ell =0.4$ (corresponding to $T/\tau _r \approx 8$), which guarantees that effect of hysteresis on our estimation of $\tau _\ell$ and $\tau _c$ is negligible.

Figure 8.

Measured (a) fragmentation-rate constant $C_\varOmega$ normalised by $(C_\varOmega )_{ref}$, the value measured using $T/t_\ell =0.4$ and (b) the convergence constant $C_\tau$ for $We$ of ($\circ$) $101$–$142$; ($\times$) $50$–$71$; ($\square$) $25$–$36$; ($\vartriangle$) $13$–$18$; ($\triangledown$) $6.3$–$8.9$. In (a), variance-weighted least-squares fit of all data to (4.1) (dashed line) gives $C_r=0.11$ and $A=2.2$ ($R^2=0.954$). In (b), error bars indicate $95\,\%$ confidence interval and the estimated large-$We$ value of $C_\tau =9$ (dashed line) is included for reference.

4.4. Estimating bubble lifetime, $\tau _\ell$

We now seek the expected bubble lifetime, $\tau _\ell$. Figure 9(a) shows our measurements of $C_\varOmega (We)$ and their fit to (2.5). We find the Hinze-scale $We_H=6.9$, similar to $We_H=4.7$ measured by Martínez-Bazán et al. (Reference Martínez-Bazán, Montañés and Lasheras1999a) and $We_H=2.7\unicode{x2013}7.8$ by Risso & Fabre (Reference Risso and Fabre1998). However, we obtain $C_{\varOmega,\infty }=1.4$, greater than $C_{\varOmega,\infty }=0.42$ measured by Martínez-Bazán et al. (Reference Martínez-Bazán, Montañés and Lasheras1999a) and $C_{\varOmega,\infty }=0.95$ from HIT simulations by Rivière et al. (Reference Rivière, Mostert, Perrard and Deike2021). An important distinction between our fragmentation rate measurements and previous experimental and numerical measurements is that we measure bubbles that have been formed as the daughters of previous fragmentation, so the bubbles are already distorted by fragmentation. The effect of this distinction can be demonstrated by measuring the fragmentation statistics over an earlier time in our simulation, $0< t/t_\ell <1$, when (as opposed to the later time $1< t/t_\ell <3$) many parent bubbles which started spherical have not yet fragmented. When we measure this earlier time range (denoted by $({\cdot })_{t< t_\ell }$), we obtain a similar $(We_H)_{t< t_\ell }=7.0$ but an appreciably smaller $(C_{\varOmega,\infty })_{t< t_\ell }=0.88$ ($R^2=0.974$). As our interest is bubbles within fragmentation cascades, our value of $C_{\varOmega,\infty }\approx 1.4$ is more relevant for bubbles formed by fragmentation. Note that $1/C_{\varOmega,\infty }$ is an order of magnitude larger than $C_r$ (i.e. $\tau _\ell \gg \tau _r$), which confirms that the PBE no-hysteresis assumption is reasonable when modelling fragmentation cascades.

Figure 9.

(a) Fragmentation rate constant $C_\varOmega$ and (b) convergence constant $C_\tau$ as functions of $We$, measured using $T/t_\ell =0.4$. Error bars indicate $95\,\%$ confidence interval. In (a), variance-weighted least-squares fit to (2.5) (dashed line) gives $We_H=6.9$ and $C_{\varOmega,\infty }=1.4$ ($R^2=0.890$). In (b), the estimated large-$We$ value of $C_\tau =9$ (dashed line) is included for reference.

4.5. Estimating convergence time, $\tau _c$

We now seek the convergence time, $\tau _c$. As shown in § 3, the time-averaged speed $\mathbb {E}\{\langle s \rangle _{T} \}$, available from ELA, gives a $T$-independent measurement of $C_\tau$ so long as (3.18) is satisfied. Figure 9(b) shows the value of $C_\tau$ we obtain over a range of $We$. We find that the model developed in § 3, which as a result of large-$We$ assumptions predicts a constant $C_\tau$, is accurate for $We\gg We _H$, or more specifically $We>30$, where we measure $C_\tau \approx 9$. To validate that our measurement is $T$-independent, we also measure $C_\tau$ using a range of $T$ for $We=$ $50$–$71$ (figure 8b). As expected, for $T\lesssim \tau _r$ we see a dependence on $T$ due to hysteresis, but for $T\gg \tau _r$ $C_\tau$ is independent of $T$. Using $C_\tau = 9$, (3.18) gives $T/T_U<0.2$ for $We>30$, so we do not expect any effect of the Hinze scale driven upper bound on $T$-independence described in § 3.4.