3.1. Effect of adhesion on breakage

In figure 3(a–c), we show the temporal evolution of the number of overall collisions $N_C$, the number of sticking collisions $N_S$, rebound events $N_R$ and breakage events $N_B$ for $St = 5.8$ and three different values of adhesion parameter $Ad_n$, which is defined as

(3.1)\begin{equation} Ad_n = \frac{\gamma}{\rho_p \bar{v}_{n}^2 r_p}, \end{equation}

where $\bar {v}_{n} = \sqrt {\langle v_n^{2}\rangle }$ is the square root of the average value of $v_n^{2}$ over all collision events. The particles are considered to have collided at the minimum separation distance $h = h_{min} = 2 \times 10^{-4} r_p$ and the impact velocity $v_n$ is calculated for each collision event at this moment. The values of $v_n$ are different for different collision events and $\bar {v}_{n}$ here can be regarded as an effective value to measure the kinetic energy of colliding particles. When the adhesion is extremely weak ($Ad_n = 0.73$), $N_C$ increases linearly with time. It indicates that the collision kernel $\varGamma$ almost keeps as a constant, which is consistent with previous DNS results for non-adhesive particles (Wang et al. Reference Wang, Wexler and Zhou2000). Here, $N_R$ is close to $N_C$ and both $N_S$ and $N_B$ are nearly zero. Agglomerates therefore can barely be formed given such a weak adhesion. For the case with a relatively stronger adhesion ($Ad_n = 7.3$), agglomeration between colliding particles can be clearly observed. However, the agglomeration at this $Ad_n$ value is still quite limited, since the sticking probability is small (${\sim }0.4$). When $Ad_n$ further increases to $70$, adhesion plays a dominant role. As illustrated in figure 3(c), $N_S \approx N_C$, implying that almost all collisions lead to the agglomeration of colliding particles. Moreover, $N_C$ no longer increases linearly with time in this case, which confirms previous results that intense agglomeration will push the system away from statistical equilibrium.

Figure 3.

Temporal evolution of the number of collisions $N_C$, the number of sticking collisions $N_S$ and rebound collisions $N_R$, and the number of breakage events $N_B$ for $St = 5.8$ and $(a)$$Ad_n = 0.73$, $(b)$$Ad_n = 7.3$ and $(c)$$Ad_n = 70$.

Another interesting result observed in figure 3 is that the breakage of agglomerates is not obvious when the adhesion is either too weak or too strong. When $Ad_n = 0.73$, the breakage is limited by the small number of bonds that can be formed upon collisions. In contrast, the contacting bond formed at $Ad_n = 70$ is too strong to be broken by the fluid stress or the impact of a third particle. A considerable number of breakage events can only be observed at a moderate value of $Ad_n$.

We normalize the number of sticking collisions $N_S$, rebound collisions $N_R$ and breakage events $N_B$ with the total number of collisions $N_C$ and plot them against $Ad_n$ in figure 4. Three different regimes can be identified: a rebound regime with $\hat {N}_R > 95\,\%$; a sticking regime with $\hat {N}_S > 95\,\%$; and a transient regime between the above two regimes. The critical $Ad_n$ values dividing the three regimes are approximately $1.5$ and $35$. Simulation results for different $St$ collapse, implying that the possibility of occurrence of sticking, rebound and breakage events can be well quantified by the dimensionless adhesion number $Ad_n$.

Figure 4.

Normalized number of sticking collisions $\hat {N}_S$ (blue), rebound collisions $\hat {N}_R$ (red) and breakage events $\hat {N}_B$ (purple) over the entire simulation as functions of $Ad_n$. Results for three different Stokes numbers are shown: circles, $St = 2.9$; triangles, $St = 5.8$; and diamonds, $St = 12$.

3.2. Formulation of breakage rate

In the current subsection, we focus on the formulation of the rate of collision-induced breakage of agglomerates. In turbulent flow laden with particles, the growth or collision-induced breakage of agglomerates results from two successive processes. First, the turbulent flow brings two initially separate agglomerates (or particles) close enough to initiate collisions. Second, the two colliding agglomerates will either merge into a large one, rebound from each other or break up into fragments.

For the first step (i.e. collision), we introduce the classic statistical model of the collision rate in particle-laden turbulence. The collision rate for agglomerates of size $i$, $\dot {n}_{C}(i)$, can be expressed as

(3.2)\begin{equation} \dot{n}_{C}(i)= \sum_{j=1}^{\infty} \varGamma(i, j) n(j)n(i), \end{equation}

where $\varGamma (i,j)$ is the collision kernel between agglomerates of size $i$ and agglomerates of size $j$ and $n(i)$ is the average number concentration of size group $i$. For homogenous isotropic turbulence, the collision kernel $\varGamma (i, j)$ has been modelled by (Zhou, Wexler & Wang Reference Zhou, Wexler and Wang2001)

(3.3)\begin{equation} \varGamma(i, j)=2 {\rm \pi}R_{i j}^{2}\left\langle\left|w_{r}\right|\right\rangle g\left(R_{i j}\right), \end{equation}

where $R_{ij}$ is the radius of the effective collision spheres (known as ECSs) for agglomerates of size $i$ and $j$, $\langle |w_{r}|\rangle = \bar {v}_n$ is the average radial relative velocity and $g(R_{ij})$ is the radial distribution function at the distance of contact. The collision kernel $\varGamma (i,j)$ has been evaluated for non-interacting particles with different values of Stokes number in several previous studies. For monodisperse spherical particles (i.e. $i=j=1$), the collision kernel, normalized by the collision kernel for zero-inertia particles $\varGamma _0 (1,1) =(8 {\rm \pi}\epsilon / 15 v)^{1 / 2}(2 r_{p})^{3}$, increase from $1$ to ${\sim }10$ as $St$ increase from $0$ to ${\sim }1$ and does not obviously change when $St$ further increases (Saffman & Turner Reference Saffman and Turner1956; Sundaram & Collins Reference Sundaram and Collins1997; Wang et al. Reference Wang, Wexler and Zhou2000; Zhou et al. Reference Zhou, Wexler and Wang2001). In our simulation, the values of $\varGamma (1,1) / \varGamma _{0}(1,1)$ are $7.0$, $10.2$, $11.1$, $11.0$ for $St = 1.4, 2.9, 5.8$, $12$, respectively. These values are quite close to the previous DNS results for non-interacting particles (Wang et al. Reference Wang, Wexler and Zhou2000). The effective collision radius for an agglomerate with $i$ primary particles and that with $j$ primary particles can be calculated as $R_{ij}=R_g(i)+R_g(j)$, where $R_g(i)$ is the gyration radius for the agglomerates with $i$ primary particles (Jiang & Logan Reference Jiang and Logan1991; Flesch, Spicer & Pratsinis Reference Flesch, Spicer and Pratsinis1999; Chen et al. Reference Chen, Li and Marshall2019a).

The breakage rate due to the collisions with other particles or agglomerates can be expressed as the product of the collision rate $\dot {n}_{C}(i)$ and the fraction of collision events resulting in breakage $\varPsi$ (Kellogg et al. Reference Kellogg, Liu, LaMarche and Hrenya2017) as follows:

(3.4)\begin{equation} f_{br}(i) = \frac{\varPsi \dot{n}_{C}(i)} {n(i)} = \varPsi \sum_{j=1}^{\infty} \varGamma(i, j) n(j). \end{equation}

Here, the fraction of breakage events $\varPsi$ is defined as the ratio of the breakage number to the overall collision number and should include the influence of both turbulent transport and particle scale interactions. In prior work, a critical breakage velocity $v_{b,crit}$ was introduced, assuming that agglomerate breaks when the magnitude of the normal relative velocity $v_n$ satisfies $v_n > v_{b,crit}$. The fraction of breakage events $\varPsi$, therefore, can be calculated as $\varPsi =\int _{v_{b, {crit}}}^{\infty } P_C(v_{n})\,\textrm {d} v_{n}$, with $P_C(v_n)$ being the probability density function (p.d.f.) of normal impact velocity (Kellogg et al. Reference Kellogg, Liu, LaMarche and Hrenya2017; Liu & Hrenya Reference Liu and Hrenya2018). Here, we introduce a new statistical framework to calculate $\varPsi$ in terms of well known impact velocity distributions $P_C(v_n)$. This formulation is expected to be more general than the previous model based on the critical breakage velocity. For collision events with impact velocity $v_n$, the fine-grained probability of breakage is recorded as $\psi (v_n)$. Thus, the distribution of velocity for a breakage event is given by

(3.5)\begin{equation} P_B(v_n)=\frac{P_C(v_n) \psi\left(v_{n}\right)}{\int_0^{\infty} P_C(v) \psi\left(v\right)\, \mathrm{d} v}, \end{equation}

where the denominator is the normalization coefficient. Here, $\psi (v)$ can be regarded as a transfer function, which relates the probability distribution of breakage to the impact velocity distribution.

For particles with a given adhesion value, $\psi (v_n)$ is expected to be zero as $v_n$ tends to zero (sticking regime), and rises to unity as $v_n$ increases, given that all colliding agglomerates will break when the impact velocity is sufficiently large. Knowing the value of $\psi (v_n)$, one can directly obtain the fraction of breakage $\varPsi$ through

(3.6)\begin{equation} \varPsi = \int_0^{\infty} P_C(v_n) \psi\left(v_n\right)\, \mathrm{d} v_n. \end{equation}

Substituting (3.6) into (3.4) further gives the breakage rate.

To validate the statistical framework above and to give a specification of the transfer function $\psi (v_n)$, we obtain the statistics of doublet breakage from DNS–DEM simulation and compare them with the theoretical descriptions in (3.4). The breakage of doublets has been widely adopted as the prototype of agglomerates that break into two fragments. For doublets, the breakage rate in (3.4) reduces to

(3.7)\begin{equation} f_{br}(2) = \frac{\varPsi \dot{n}_{C}(2)}{n(2)} = \varPsi \sum_{j=1}^{\infty} \varGamma(2, j) n(j). \end{equation}

At the early stage of agglomeration most particles remain as singlets (Liu & Hrenya Reference Liu and Hrenya2018; Chen et al. Reference Chen, Li and Marshall2019a) and the equation above can be further simplified as

(3.8)\begin{equation} f_{br}(2) \approx \varPsi \varGamma(1, 2) n(1) = n(1)S_{12} \varPsi \varGamma(1,1). \end{equation}

On the right-hand side of the equation, we relate the singlet–doublet collision kernel $\varGamma (1,2)$ to the singlet–singlet kernel through $\varGamma (1,2) = S_{12}\varGamma (1,1)$, where the constant $S_{12}$ is the correction for collisional cross-sectional areas for singlet–doublet collisions. Here, $\varGamma (1,1)$ for particles with different $St$ values has been well modelled from the ghost particle approach. Although the expression in (3.8) only gives low-order statistics for the breakage of doublets, it provides valuable insights: the breakage rate scale linearly to the number concentration and the effect of turbulent transport are included in both $\varGamma (1,1)$ and the breakage fraction $\varPsi$; contacting interactions affects the breakage rate by changing $\varPsi$ through the transfer function $\psi (v_n)$ in (3.6).

In order to obtain the transfer function $\psi (v_n)$, we track all the collision events in the simulation and record whether the collision leads to the breakage of the agglomerate according to the criterion in § 2.4. The p.d.f. of the impact velocity $P_C(v_n)$ for singlet–doublet collision events are then measured at different $St$ and $Ad$ values (as shown in figure 5a–c). For the cases with weak adhesion ($Ad = 0.64$), most particles remain as singlets and the number of singlet–doublet collision events that can be observed within a large-eddy turnover time is quite limited. We thus run three simulations with different initial random positions of particles to obtain more collision events. It ensures a good statistic on the collision velocity for singlet–doublet collision events and breakage events. For a given value of $St$, varying $Ad$ does not obviously affect $P_C(v_n)$. In contrast, a strong dependence on $St$ can be observed. For collisions that result in the breakage of a doublet, we also plot the corresponding p.d.f. of the impact velocity, $P_B(v_n)$, in figure 5(d–f). One can easily find a strong correlation between $P_B(v_n)$ and $Ad$. Particles with stronger adhesion tend to stick together upon collisions. The breakage events, therefore, are more likely to happen with a higher impact velocity.

Figure 5.

Probability density functions of the collision velocity (normal component) $v_n$ for singlet–doublet collision events (a–c) and collision-induced breakage events (d–f). Statistics are made over approximately a large-eddy turnover time $t\in [15, 25]$. Different columns are results for different Stokes numbers: $St = 2.9$ (a,d); $St = 5.8$ (b,e); and $St = 12$ (c,f). For each Stokes number, we show results from different $Ad$ values: $Ad = 0.64$ (squares); $Ad = 1.3$ (circles); $Ad = 6.4$ (upward triangles); and $Ad = 12$ (downward triangles).

We then calculate the transfer function $\psi (v_n)$ inversely from $P_C(v_n)$ and $P_B(v_n)$ according to (3.5). As shown in figure 6(a), despite the inconsistency in $P_C(v_n)$, $\psi (v_n)$ for different $St$ collapses nicely. In contrast, the adhesion strongly affects $\psi (v_n)$. Although, there is considerable scatter in the data at large $v_n$ due to the limited sample size of the energetic collision events, the transfer function $\psi (v_n)$ at a given $Ad$ value is roughly linear to the collision velocity $v_n$. The results in figure 6(a) suggest that the transfer function may only depend on the short-range contacting interactions, whereas the effects of turbulent transport and hydrodynamic interactions are included in the p.d.f. of the impact velocity $P_C(v_n)$. To validate the argument above, we run simulations with different particle radius (ranging from 0.0075 to 0.0125) and with/without the hydrodynamic damping force (2.5) at a fixed $St$ value. As seen in figure 6(a), the measured transfer function $\psi (v_n)$ does not show obvious dependence on the particle size and the hydrodynamic interaction, confirming that the transformation function $\psi (v_n)$ is determined by the short-range contacting interactions.

Figure 6.

(a) Transfer function $\psi (v_{{n}})$ versus collision velocity $v_n$ for different Stokes numbers: $St = 2.9$ (squares); $5.8$ (circles); and $12$ (triangles). Also for different $Ad$ values: $Ad = 1.3$ (light blue); $6.4$ (yellow); and $12$ (dark blue). Results with different particle radius (ranging from 0.0075 to 0.0125) and with/without the hydrodynamic damping force (2.5) at $St = 2.9$ are also included. Scatters are results calculated from p.d.f. in figure 5, and dashed lines are linear fittings from (3.9). (b and c) Fitting parameters $(v_{C2} - v_{C1})^{-1}$ and $v_{C1}$ as functions of $Ad$. Legends are the same as in panel (a).

According to the results in figure 6(a), we propose a linear relationship between $\psi$ and $v_n$ as follows:

(3.9)\begin{equation} \psi(v_n)=\begin{cases} {0,} & {\text{for } v_n < v_{C1},} \\ {\dfrac{1}{v_{C2} - v_{C1} } (v_n - v_{C1}),} & {\text{for } v_{C1} \leq v_n \leq v_{C2},}\\ {1,} & {\text{for } v_n > v_{C2}.} \end{cases}\end{equation}

Two typical values of collision velocity $v_{C1}$ and $v_{C2}$ are indicated by (3.9). Breakage does not happen when the collision velocity between two agglomerates, $v_n$, is smaller than $v_{C1}$. On the other hand, if $v_n > v_{C2}$, the colliding doublets always break. We then fit the measured values of the transfer function $\psi (v_n)$ (linear part) using (3.9) for all the cases presented in figure 6(a) and plot the fitting parameters $v_{C1}$ and the slope $(v_{C2} - v_{C1})^{-1}$ as a function of $Ad$. It is seen that the fitted values of the slope for different cases centre around a logarithmic curve (figure 6b), which reads

(3.10)\begin{equation} (v_{C2} - v_{C1})^{-1} = -2.1 \ln \left( \frac{Ad}{13}\right). \end{equation}

Several interesting features are indicated by (3.10). First, the slope diverges in the small adhesion limit ($Ad \to 0$), indicating that there is a critical collision velocity separating the breakage and non-breakage collisions. This is in accordance with the theoretical model proposed by Liu & Hrenya (Reference Liu and Hrenya2018), in which a Heaviside function $H(v - v_{b,crit})$ is proposed to transform the p.d.f. of normal impact velocity $P_C(v_n)$ into the p.d.f. of impact velocity for breakage events $P_B(v_n)$. We show here that such a transfer function is reasonable only when the adhesive interaction is extremely weak. As $Ad$ increases, the slope of $\psi (v_n)$ considerably decreases and there is no sharp transition between breakage and non-breakage collision velocities. Although the data points for the minimum breakage velocity $v_{C1}$ are relatively dispersed when plotted as a function of $Ad$, a quadratic curve, $v_{C1} = a\,Ad^2$ with $a = 7.4\times 10^{-4}$, can roughly describe the variation of $v_{C1}$ (see figure 6c). At large adhesion limit $v_{C1}$ diverges, implying that all collisions give rise to the growth of agglomerates when the adhesion is sufficiently strong.

To further validate the model of the transfer function, we present an example of the model prediction for cases with $St= 2.9$ in figure 7(a). First, the p.d.f. of the normal collision velocity $P_C(v_n)$ is measured from the simulation with small $Ad$ value ($1.3$). The breakage fraction $\varPsi$ is then calculated by substituting (3.9) and the measured $P_C(v_n)$ into (3.6). One can also adopt models of $P_C(v_n)$ obtained from simulations with non-interacting particles to estimate the breakage fraction $\varPsi$ (Salazar & Collins Reference Salazar and Collins2012; Saw et al. Reference Saw, Bewley, Bodenschatz, Ray and Bec2014; Bhatnagar, Gustavsson & Mitra Reference Bhatnagar, Gustavsson and Mitra2018). Such approximation does not bring large errors since $P_C(v_n)$ is almost independent of adhesive interactions (see figure 5). The result generated from the model together with predictions for $St = 1.4$, $5.8$ and $11.5$ is plotted as a dashed line in figure 7(b). We see that the model predictions are in accordance with DNS–DEM simulations.

Figure 7.

(a) Probability density functions $P_C(v_n)$ of the normal collision velocity for $St = 2.9$ and $Ad = 1.3$ (left-hand axis) and the transfer function $\psi (v_n)$ modelled by (3.9) (right-hand axis). Colour code spans from blue to yellow with increasing $Ad$ (from $0.1$ to $10$). (b) Fraction of collision-induced breakage of doublets $\varPsi$ at different $Ad$ values. Points are DNS–DEM results and dashed lines are results calculated from $P_C(v_n)$ and the modelled $\psi (v_n)$ (3.9).

The deviation between the model and the simulations in figure 7(b) may result from the linear assumption of the transfer function $\psi (v_n)$ (3.9), in which a sharp transition is assumed between the linear part $((v_n-v_{C1})/(v_{C2}-v_{C1}))$ and unity. The simulation data in figure 6(a), in contrast, shows a much slower approach to unity, indicating that the model in (3.9) overestimates $\psi (v_n)$ when $v_n \to v_{C2}$. Despite this deviation, our simplified model well captures the variation of breakage fraction $\varPsi$ with adhesion $Ad$. Moreover, the Stokes number dependence of $\varPsi$ can be observed in figure 7(b). Since the breakage fraction $\varPsi$ here is calculated from a universal transfer function, the $St$ number dependence of $\varPsi$ originates from the difference in $P_C(v_n)$: the hydrodynamic damping force significantly reduces the relative approaching velocity of colliding particles with small $St$.

The collision-induced breakage rate of the doublets $f_{b r}(2)$ is calculated from (3.8) and compared with DNS–DEM results in figure 8(a). Quantitative agreement is observed, indicating that the analytical model well captures the effects of the particle inertia and the adhesive interaction on the breakage. Since the adhesion parameter $Ad$ does not include the effect of particle inertia, there is considerable distinction in results for different $St$ at the same $Ad$. We stress again that particle inertia affects the breakage rate through its influence on the statistics of the collision velocity. One simple way to include both effects of particle inertia and the adhesion is to use the modified adhesion parameter $Ad_{n}$ (see (3.1)), which scales the adhesion using $St$-dependent average velocity $\bar {v}_{n} = \sqrt {\langle v_n^{2}\rangle }$. The normalized breakage rate, when plotted as a function of $Ad_{n}$, collapses nicely onto the exponential curve (see figure 8b)

(3.11)\begin{equation} \frac{f_{br}\tau_k}{r_p^3 n(1)} = 86\exp(-0.12Ad_n). \end{equation}

The result indicates that $\bar {v}_{n} = \sqrt {\langle v_n^{2}\rangle }$ is an appropriate choice to scale the effect of adhesion and the collision-induced breakage rate can be well estimated once $Ad_n$ is known.

Figure 8.

(a) Normalized breakage rate $f_{br}\tau _k/(r_p^3 n(1))$ for doublets as a function of $Ad$. The scatters are DNS–DEM results and the dashed lines are predictions from (3.8), in which the breakage fraction $\varPsi$ is calculated from the p.d.f. of normal collision velocity through $\varPsi =\int _{0}^{\infty } P_{C}(v_n) \psi (v_n) \,\mathrm {d} v$ (see (3.6)) and the transfer function $\psi (v_n)$ is modelled by (3.9). (b) Normalized breakage rate as a function of $Ad_n$. The dashed line is exponential fittings using (3.11).

It should be noted that the model in (3.8) is valid only for early stage agglomeration, since the transfer function is derived for singlet–doublet collisions. Both agglomerate size and structure may affect the formulation of the transfer function. Predicting the breakage rate for agglomerates with arbitrary size and structures through first principles is practically impossible. It is thus normally accepted to describe the breakage rate using an exponential or a power-law function, in which the parameters are related to agglomerate size and particle–particle interactions. Agglomerate size dependence of the breakage rate will be discussed in § 3.3.

It is of great importance to know how the breakage rate scales with particle size $r_p$ and particle number density $n$. We measure the doublet breakage at different particle sizes ($r_p = 0.005\text {--}0.015$) and particle numbers ($N = 19\,600\text {--}40\,000$) for typical $St$ and $Ad$ values (shown in figure 9). The results are plotted in a scaled form: $\hat {f}_{br} = f_{br}(r_p)/f_{br}(r_{p,0})$, $\hat {r}_p = r_p/r_{p,0}$ in figure 9(a) and $\hat {f}_{br} = f_{br}(n(1))/f_{br}(n_{m}(1))$, $\hat {n}(1) = n(1)/n_{m}(1)$ in figure 9(b). Here, $f_{br}(r_{p,0})$ is the doublet breakage rate for the case with $r_{p,0} = 0.01$ and $f_{br}(n_{m}(1))$ is the breakage rate for the case with the maximum value of singlet number density $n_{m}(1)$. As displayed in figure 9, DNS–DEM results follow the power laws $\hat {f}_{br} \propto \hat {r}^2_p$ and $\hat {f}_{br} \propto \hat {n}^1(1)$ when particle size and singlet number density are varied.

Figure 9.

Scaled breakage rate $\hat {f}_{br}$ as a function of (a) scaled particle size $\hat {r}_p$ and (b) scaled number density of singlet $\hat {n}(1)$ at $St = 2.9$, $5.8$$12$ and $Ad = 1.3$, $3.8$. Dashed lines in (a and b) indicate power law functions with exponents $2$ and $1$, respectively.

The $n(1)$ dependence is easy to understand from (3.8). The $\hat {r}^2_p$ scaling originates from the $r_p$ dependence of the collision kernel $\varGamma (1,1)$ in (3.8). For inertial particles ($St \gg 1$), the approaching velocity of colliding particles is decorrelated from the local fluid gradient, thus is not affected by the particle size. The $\hat {r}^2_p$ scaling enters $\varGamma (1,1)$ through the effective collision area. We note that the size scaling here is valid for particles that are smaller than or comparable to the Kolmogorov scale. It may not hold for particles that are considerably larger than the Kolmogorov scale. In the latter case, particle-resolved simulations would be needed to precisely calculate the flow around and forces on particles (Ernst, Dietzel & Sommerfeld Reference Ernst, Dietzel and Sommerfeld2013; Liu & Wu Reference Liu and Wu2019; Peng, Ayala & Wang Reference Peng, Ayala and Wang2019; Wang et al. Reference Wang, Wan, Peng, Liu and Wang2019).

3.3. Agglomerate size dependence of the breakage rate

The breakage rate of agglomerates with size $A$ are calculated from DNS–DEM simulations according to

(3.12)\begin{equation} f_{br}(A) = \frac{N_{br}(A)}{N(A)\Delta t}, \end{equation}

where $N(A)$ is the number of agglomerates of size A averaged over the time range $t\in [30,40]$, $N_{br}(A)$ is the breakage number of agglomerates with size $A$ and $\Delta t = 10$. As shown in figure 10(a), a stronger adhesion promotes the formation of larger agglomerates. In contrast, the number of breakages decreases with $Ad$ (see figure 10b). To provide meaningful statistics, we only calculate $f_{br}(A)$ when $N_{br}(A)$ is larger than $20$. The results are normalized by the mean shear rate $G$ and plotted as a function of size $A$ in figure 10(c). It is seen that the breakage rate depends linearly on the agglomerate size with the slope being a function of $Ad_n$. Fitting the data at different $Ad_n$ and $St$ according to

(3.13)\begin{equation} \frac{f_{b r}(A)}{G}=\zeta(St, Ad_n)\, A +\chi \end{equation}

gives us the values of the slope $\zeta (St,Ad_n)$. As shown in figure 10(d), when plotted as a function of $Ad_n$, $\zeta$ for different $St$ centres around a universal curve, which is analogous to the $f_{br}$ dependence in figure 8(b). The universal curve has an power-law form: $\zeta = 0.012Ad_n^{-0.81}$. These results once again confirm that the modified adhesion parameter $Ad_n$ is an appropriate choice to reflect both effects of the particle inertia and adhesive interactions on the breakage.

Figure 10.

(a) Number of agglomerates of size $A$ averaged over the time range $t\in [30,40]$. (b) Number of breakages of agglomerates of size $A$ measured within $t\in [30,40]$ for $St = 5.8$. (c) Breakage rate, normalized by the shear rate $G$, as a function of agglomerate size $A$. The dashed lines are linear fittings from (3.13). (d) The fitting values of the slope $\zeta$ for the linear relationship between $f_{br}/G$ and $A$ at different $Ad_n$ and $St$ values. The dashed line indicates the power function $\zeta = 0.012Ad_n^{-0.81}$.

3.4. Role of flow structure

In this subsection, we quantify the correlation between structures of turbulence and the breakage of agglomerates with different $St$ and $Ad$ values. We identify the flow structures based on the second invariant of the velocity gradient tensor $\mathcal {Q}=(\mathcal {R}^{2}-\mathcal {S}^{2}) / 2$, where the strain rate tenor $\mathcal {S}=(\mathcal {A}+\mathcal {A}^{\mathrm {T}}) / 2$ and the rotation rate tensor $\mathcal {R}=(\mathcal {A}-\mathcal {A}^{\mathrm {T}}) / 2$ are symmetric and antisymmetric parts of the velocity gradient tensor $\mathcal {A}=\tau _{k} \boldsymbol {\nabla } \boldsymbol {u}$ (normalized by the Kolmogorov time $\tau _{k}$), respectively. Figure 11(a) presents the countour plots of $Q$, showing the vortex tubes with $Q > 3.3\sqrt {\langle Q^2 \rangle }$ and straining sheets with $Q < -2.5\sqrt {\langle Q^2 \rangle }$, and the corresponding two-dimensional slice at $y=0$. One can clearly see the red vortex tubes surrounded by blue straining sheets (vortex-strain worm-rolls), which implies that intense structures typically occur near each other (Picardo et al. Reference Picardo, Agasthya, Govindarajan and Ray2019).

Figure 11.

(a) Countour plot of $Q$ and the two-dimensional slice at $y=0$. Vortex tubes with $Q > 3.3\sqrt {\langle Q^2 \rangle }$ are coloured in red and straining sheets with $Q <-2.5\sqrt {\langle Q^2 \rangle }$ are coloured in blue. (b) Average $Q$, sampled by singlet–doublet collision events and (c) average $Q$ sampled by doublet breakage events as functions of $St$. (d) Average $Q$ for breakage events as a function of $Ad$ at different $St$ values: $St= 1.4$ (squares); $St = 2.9$ (circles); $St = 5.8$ (upward triangles); and $St = 12$ (downward triangles). The straight dashed lines are linear fittings.

We calculate the average $\mathcal {Q}$, sampled by singlet–doublet collisions, at different $St$ and $Ad$ values in figure 11(b). The results for non-interacting particles based on ghost collision approximations are also included (Picardo et al. Reference Picardo, Agasthya, Govindarajan and Ray2019) (only data at $St > 0.5$ are shown here). One can notice that as $St$ increases from $0.5$ to $20$, $\mathcal {Q}$ increases from a negative value to zero, implying that finite-inertia particles ($St\sim 1$) tend to collide in the straining zone whereas particles with large inertia collide uniformly. According to Picardo et al. (Reference Picardo, Agasthya, Govindarajan and Ray2019), decreasing $St$ also leads to the approach to zero of $\mathcal {Q}$ and the largest absolute value of $\mathcal {Q}$ occurs at $St \approx 0.3$. Such flow structure dependence is due to two aspects. First, particles with finite inertia ($St \approx 1$) tend to accumulate in straining regions outside vortices due to the centrifugal effect (known as preferential concentration). Moreover, particle inertia also increases the relative approaching velocity between particles. Such an effect also prevails in straining zones (Picardo et al. Reference Picardo, Agasthya, Govindarajan and Ray2019). Here, we show that varying particle–particle contacting interactions ($Ad$) does not obviously affect the structure dependence of collisions.

The average $\mathcal {Q}$, sampled by singlet–doublet collision-induced breakage events, shows a strong dependence on $Ad$ (figure 11c). Doublets with larger $Ad$ value are more difficult to break thus needing higher impact velocities. For particles with moderate inertia ($St \approx 1$), violent collisions are more likely caused by particles ejected rapidly from strong vortices and happen in straining sheets (with smaller negative $\mathcal {Q}$) that envelope the vortices. As $St$ increases, the relative velocity between colliding particles becomes less sensitive to the underlying flow, both collision events and breakage events distribute more uniformly in the flow. As shown in figure 11(d), the relationship between $\mathcal {Q}$ and $Ad$ at given $St$ can be well described by linear functions.