3.1. Flow velocity and current height

During a gravity current's propagation, the TC fluid velocity profiles along the slope are similar (Altinakar, Graf & Hopfinger Reference Altinakar, Graf and Hopfinger1996). The bed-parallel velocity profiles are characterized by two shear layers separated by a velocity maximum, which can be fitted by the empirical power relation and the semi-Gaussian relationship in the near-bed region and the jet region, respectively (Altinakar et al. Reference Altinakar, Graf and Hopfinger1996; Farizan et al. Reference Farizan, Yaghoubi, Firoozabadi and Afshin2019). The profile of the fluid bed-parallel velocity normalized by the maximum bed-parallel velocity ($u_{f}^{//}/u_{f}^{//,max}$) is plotted in figure 3, together with the experimental results (Altinakar et al. Reference Altinakar, Graf and Hopfinger1996; Nourmohammadi, Afshin & Firoozabadi Reference Nourmohammadi, Afshin and Firoozabadi2011; Farizan et al. Reference Farizan, Yaghoubi, Firoozabadi and Afshin2019; Hitomi et al. Reference Hitomi, Nomura, Murai, De Cesare, Tasaka, Takeda, Park and Sakaguchi2021) for comparison. In figure 2, the dimensionless distance of the current from the slope $Z_\uparrow$ ($y$-axis in the figure) is defined as

(3.1)\begin{equation} {Z_\uparrow} = \left\{\begin{array}{@{}ll@{}} {Z/{H_m},} & {Z \le {H_m}}\\ {(Z - {H_m})/(H -{H_m}),} & {Z > {H_m}} \end{array} \right., \end{equation}

where $Z$ denotes the distance from the slope and $H_m$ is the distance between the slope and the position where the maximum fluid bed-parallel velocity $u_f^{//,max}$ occurs. The layer-averaged height of current $H$ is expressed by (Ellison & Turner Reference Ellison and Turner1959)

(3.2)\begin{equation} {H } = {\left( {\int_0^{Z_{top} } {u_{f}^{/{/}} \,{\rm d}Z} } \right)^2}\left/\,\int_0^{Z_{top} } {{{( {u_{f}^{/{/}}} )}^2}\,{\rm d}Z} \right., \end{equation}

in which the upper limit of the integral $Z_{top}$ is located at the position where $u_{f}^{//}$ transitions from positive to negative. One can see that all the simulation results are consistent with the previous experimental results, which confirms the feasibility of our TC model.

Figure 3. Dimensionless average fluid bed-parallel velocity profiles at $t=\text {8--10}$ for six simulations.

For the binary mixture with the same total particle concentration, we define the coarse and fine particle fronts of the TC ($x_{front,C}$ and $x_{front,F}$) as the positions of the coarse and fine particles farthest moving, respectively. Figure 4 shows the temporal evolution of non-dimensional current front $x_{front}$ and the front velocity $u_{front}$ ($={\rm d}\kern 0.06em x_{front}/{\rm {d}}t$) for coarse and fine particles, and exhibits how the mean front position and mean front velocity over $t=4\unicode{x2013}6$ and $t=8\unicode{x2013}10$ respond to the variation in $\phi _F$. It can be observed that the entire TC front goes through a short acceleration process of $t\approx 2$, which is approximately consistent with the duration of $t\approx 1.5$ in Cantero et al. (Reference Cantero, Lee, Balachandar and Garcia2007). The maximum front velocity reaches approximately 0.5, consistent with the recognition in Gladstone et al. (Reference Gladstone, Phillips and Sparks1998) and Steenhauer et al. (Reference Steenhauer, Tokyay and Constantinescu2017). After that, the front slightly decelerates for a short time and then travels at a constant velocity, i.e. constant velocity regime in lock-exchange TC evolution (Huppert & Simpson Reference Huppert and Simpson1980). Finally, the TC progressively slows down. Such evolution agrees with previous research (Huppert & Simpson Reference Huppert and Simpson1980; Cantero et al. Reference Cantero, Lee, Balachandar and Garcia2007). The constant velocity regime in the monodisperse fine particle TC is noticeable at approximately $t=4\unicode{x2013}10$. In the binary cases, this constant regime lasts for a shorter times and becomes less obvious with the decreasing fine particle fraction. Monodisperse coarse particle TC basically does not show a constant velocity process. In our simulation, the movement of the front during the acceleration stage aligns with the previous results of the theoretical solution (Huppert & Simpson Reference Huppert and Simpson1980) and simulations of saline homogeneous gravity flows (Cantero et al. Reference Cantero, Lee, Balachandar and Garcia2007). During the slumping stage, the constant velocity of the saline gravity flow is slightly higher than that in the present study. Additionally, the presence of particles causes the TC to exit the slumping stage earlier than the saline gravity flow, which agrees with the findings in Gladstone & Woods (Reference Gladstone and Woods2000).

Figure 4. Time evolution of (a) the front position and (b) the front velocity for coarse and fine particles in all simulations. The symbols and lines represent the data for the coarse and fine components, respectively. The black dotted line in panel (a) indicates the theoretical solution for saline gravity flow from Huppert & Simpson (Reference Huppert and Simpson1980), and the black square in panels (a,b) indicate the simulated results of saline gravity flow in Cantero et al. (Reference Cantero, Lee, Balachandar and Garcia2007). (c) Mean front position and (d) mean front velocity that are averaged over different periods at different $\phi _F$.

Regardless of $\phi _F$, the front velocities of fine component and coarse component before $t= 6$ in figure 4(b,d) are quite similar, which makes the curves of the front positions of the two dispersed phases in figure 4(a) basically coincide. However, the front position of coarse particles and fine particles then separate gradually in figure 4(a). The main reason is that, as shown in figure 4(b), coarse particles decelerate faster than fine particles and the coarse particles even deposit finally on the slope. In addition, the front velocity for both coarse and fine particles is larger with greater $\phi _F$ (figure 4b,d). The front separation time of two dispersed phases is postponed and the front position can go farther as shown in figure 4(a). Above descriptions suggest that increasing the fraction of fine particles improves the TC's transport capacity significantly, which is consistent with the previous qualitative understanding (Gladstone et al. Reference Gladstone, Phillips and Sparks1998; Salaheldin et al. Reference Salaheldin, Imran, Chaudhry and Reed2000).

In order to capture the turbidity regions of two dispersed phases, we define the current height along the slope $h_{t,C}$ and $h_{t,F}$ for the coarse and fine particles, respectively. Figure 5 plots the instantaneous height at two specific moments ($t = 4$ and $t = 6$), the temporal variation of the current height at two chosen positions ($x=x_{front} - 0.8$ and $x=x_{front} - 1.6$), and the average height at two positions in two distinct time periods ($t=4\unicode{x2013}6$ and $t=13\unicode{x2013}15$). In figure 5(a,b), we can observe that the curves of the height along the slope basically coincide under all cases, whether it is fine component or coarse component. Additionally, there is an increasing noticeable difference in the thickness of the fine and coarse components, indicating a distinct vertical segregation of the two dispersed phases. As shown in figure 5(c,d), the thickness of the coarse and fine components near the front is basically the same within the acceleration stage. In dilute bidisperse suspensions, hydrodynamics assumes a dominant role, resulting in similar statistical behaviour for the same phase in the same total particle concentration, which is consistent with our previous study (Zhu et al. Reference Zhu, Qian, Lin and Yu2022). When entering the slumping stage, the decreasing speed of the thickness of the coarse component $h_{t,C}$ is greater than that of $h_{t,F}$, and $h_{t,C}$ decays to zero at $t= 10$. The value of $h_{t,F}$ gradually decreases and tends to unity, which is approximately half the initial height of the fluid–particle mixture ($h_{t,F} \approx 1$). This is consistent with the understanding of the flow regime of the lock-exchange TC (Bonnecaze et al. Reference Bonnecaze, Huppert and Lister1993; Necker et al. Reference Necker, Härtel, Kleiser and Meiburg2002; Kneller et al. Reference Kneller, Nasr-Azadani, Radhakrishnan and Meiburg2016; Kyrousi et al. Reference Kyrousi, Leonardi, Roman, Armenio, Zanello, Zordan, Juez and Falcomer2018). The unity thickness $h_{t,F}$ can be maintained for a period of time, especially in figure 5(c) where the sampling position is closer to the front. Furthermore, as shown in figure 5(c–f), the thicknesses of both the coarse and fine components are largely independent of $\phi _F$ in the early stage. In the later stage, the larger $\phi _F$ gives rise to the smaller thickness of the fine component.

Figure 5. Current height for coarse and fine component along the slope (a) at $t = 4$ and (b) at $t = 6$. Time evolution of the current height (c) at $x=x_{front} - 0.8$ and (d) at $x=x_{front} - 1.6$. Mean current height (e) at $x=x_{front} - 0.8$ and ( f) at $x=x_{front} - 1.6$ as a function of $\phi _F$ during $t=4\unicode{x2013}6$ and $t=13\unicode{x2013}15$.

3.2. Vortical coherent structures

Turbulent coherent structures can be shown with $Q$-criterion, which is given by the second invariant of fluid velocity gradient tensor $\boldsymbol {\nabla } \boldsymbol {u}_f$ (Hunt, Wray & Moin Reference Hunt, Wray and Moin1988),

(3.3)\begin{equation} Q = \tfrac{1}{2}( {\varOmega_{ij}}{\varOmega_{ij}} - {{S _{ij}}{S _{ij}}} ), \end{equation}

with ${S_{ij}} = ( {\partial u_{f,i} /\partial x_j + \partial u_{f,j} /\partial x_i } )/2$ and ${\varOmega _{ij}} = ( {\partial u_{f,i} /\partial x_j - \partial u_{f,j} /\partial x_i } )/2$ representing the symmetric and antisymmetric components of fluid velocity gradient tensor (strain rate tensor and rotation rate tensor), respectively. This criterion was often employed to investigate coherent vortical structures (Koohandaz et al. Reference Koohandaz, Khavasi, Eyvazian and Yousefi2020; Xie et al. Reference Xie, Hu, Pähtz, He and Cheng2022). Here we discuss how these flow patterns around the current head respond to the variation of the relative fine particle volume fraction $\phi _F$.

The vortical coherent structure by $Q=0.25$ together with particles in the $\phi _F=0.6$ case gives us a direct view for the time evolution of the bidisperse TC, as shown in figure 6. The coherent structures are rendered with the fluid bed-normal velocity. The coarse particles are drawn by dots in black and the fine ones are indicated by dots in grey, respectively. As the particle Reynolds number $Re_p$ is small, there only exists a large vortex due to the movement of the current front. The small vortex structures are mainly induced by the motion of the particle groups. In the initial stage ($t < 2$), the coarse and fine particles settle and move along the slope. The settling velocity of coarse particles is greater than that of fine particles, resulting in obvious coarse particle segregation downward as shown in figure 6(b). In addition, at $t = 6$, the front positions of coarse and fine particles are very close, which agrees well with the results in figure 4(a). However, the heights covered by coarse and fine particles are significantly different in the vertical direction, with the fine component reaching the higher position. The height of the fine particle layer can be seen to be larger than that of the coarse particles. The main reason is that the turbulent velocity is far from sufficient to counteract the particle settling velocity of the coarse component as compared with the fine one (Garcia & Parker Reference Garcia and Parker1993; McCaffrey et al. Reference McCaffrey, Choux, Baas and Haughton2003). As the TC continues to advance, most coarse particles settle and stop on the slope and only a small number follows in the current head at $t = 10$. At the current head, the high current intensity can still drive some coarse particles to travel along the slope. In the final stage ($t = 14$), all the coarse particles stop on the slope, but the fine particles still move along the slope. The lobe-and-cleft structures do not exist, which is mainly due to the weakened shear with the wall at low Reynolds number (Espath et al. Reference Espath, Pinto, Laizet and Silvestrini2014).

Figure 6. Vortical coherent structures captured by $Q=0.25$ in case of $\phi _F=0.6$ at (a) $t=2$, (b) $t=6$,(c) $t=10$, and (d) $t=14$. The coherent structures are rendered with the fluid bed-normal velocity. The coarse particles are also drawn by dots in black and the fine ones are indicated by dots in grey.

The mixing layer is expected to adjust in response to the initial relative particle volume fraction. Figure 7 shows the vortical coherent structures of the current head by $Q=0.25$ at $t=6$. The coarse or fine particles are coloured by black and grey, respectively. It is clearly observed that either the coarse or fine particle area is quite similar in all cases. Coarse particles with a larger settling velocity only cover half the height of fine particles, and even a large number of coarse particles have deposited on the slope. Moreover, in low particle concentration, the motion of coarse or fine particles is mainly influenced by the fluid hydrodynamics, which is also found in our previous studies (Zhu et al. Reference Zhu, Qian, Lin and Yu2022). The front position of the coarse or fine particles almost coincide in all cases, which is also observed in figure 4(a). For the mixing particle cases, the anticlockwise vortex near the upper interface is mainly caused by the fine particles and it enhances with the increasing fine particle concentration in terms of spatial scale. The relatively small vortex structures near the bottom boundary increase in number and size as the relative volume fraction of coarse particles increases. It implies that the increase of coarse particle proportion will enhance the local turbulent kinetic energy near the wall. Prior studies have shown that, for the small (point) particles, while coarse particles tend to enhance the turbulent kinetic energy, fine particles tend to attenuate it (Rashidi, Hetsroni & Banerjee Reference Rashidi, Hetsroni and Banerjee1990; Pan & Banerjee Reference Pan and Banerjee1996; Zhao, Andersson & Gillissen Reference Zhao, Andersson and Gillissen2013). The fluid in our simulation starts out static, and the enhancement of the local turbulent kinetic energy is probably attributed to more work done by coarse particles on the local fluid.

Figure 7. Vortical coherent structures captured by $Q=0.25$ of six cases at $t=6$. The coherent structures are rendered with the fluid bed-normal velocity. The coarse particles are also drawn by dots in black and the fine ones are indicated by dots in grey.

The three figures above evidently show that the two dispersed phases propagate in an approximately independent manner. Nonetheless, particle collisions and segregation-induced flows in the bidisperse transport process do establish some degree of connection between the two phases following their release, which will be discussed in detail in the following sections.

3.3. Bidisperse deposition process

The deposition of settling particles in the TC affects the shaping of the terrain during the propagation process. It is a common and meaningful method to analyse the deposition process through the deposition mass (Zgheib, Bonometti & Balachandar Reference Zgheib, Bonometti and Balachandar2015), deposition rate (Espath et al. Reference Espath, Pinto, Laizet and Silvestrini2014; Francisco, Espath & Silvestrini Reference Francisco, Espath and Silvestrini2017) and deposition profiles (de Leeuw et al. Reference de Leeuw, Eggenhuisen and Cartigny2018; Hu et al. Reference Hu, Tao, Li and He2020). Here, we attempt to figure out how the deposition of bidisperse TCs occurs from these behaviours, and we focus on the fine and coarse components independently.

The dimensionless deposition mass, $\tilde {m}_{depo}$, is defined as the ratio of the mass of the deposition particles to the initial mass of the corresponding particle components, the time evolution of which for six cases is shown in figure 8. Figure 9 plots the time evolution of non-dimensional particle deposition mass for the fine and coarse components (figure 9a,b), and their deposition rates ${\rm d} \tilde {m}_{depo}/{\rm d} t$ (figure 9c,d). For both fine and coarse particles, the total particle deposition mass increases continuously with the development of TC (figure 8) due to the overwhelming gravity of the particles compared with other forces hindering settlement. The deposition rate of fine particles increases approximately linearly before 14 dimensionless time, after which it remains constant or decreases slightly (figure 9c). The deposition rate of coarse component increases roughly linearly to a maximum value at $t= 7$, followed by a rapid decay of the deposition rate due to the reduction of transported coarse particles, and reaches zero around $t= 11$ (figure 9d). Before the coarse particles basically complete the settling process, the ratio of the deposition mass of the coarse particles to the fine particles is approximately equal to the ratio of their Stokes settling velocities ($\approx 4$). It is worth noting that since the deposition rate of the coarse component is always greater than that of the fine one until 10 dimensionless time. It makes the addition of total particle deposition mass dominated by the coarse component at around $t < 10$, with a very rapidly increasing trend. After $t = 10$, the deposition of the coarse component is basically completed, and the deposition of fine particles dominate the deposition behaviour of the current, which leads to a relatively slow increase in the mass of total particle deposition.

Figure 8. Time evolution of non-dimensional total particle deposition mass for six cases.

Figure 9. Time evolution of non-dimensional particle deposition mass for all cases for (a) fine component and (b) coarse component, and time evolution of non-dimensional particle deposition rate for all cases for(c) fine component and (d) coarse component. The black dashed line in panels (c) and (d) represent the linear prediction. The insets show the average particle deposition mass and particle deposition rate over $t=4\unicode{x2013}6$ at varying $\phi _F$.

As the relative volume fraction of fine particles $\phi _F$ increases, the deposition rate of the fine component increases gradually (figure 9c), thus a larger particle deposition mass for larger $\phi _F$ (figure 9a), while the deposition rate of the coarse one does not change roughly, so does $\tilde {m}_{depo}$ for the coarse component. In other words, this means that in the bidisperse TC, the increase of the coarse component can slow down the deposition process of the fine component. It may be related to the higher colliding particle proportion at larger coarse component fraction, which will be discussed in detail in the next subsection. The strong oscillation of ${\rm d}\tilde {m}_{depo}/{\rm d}{t}$ for coarse particles is due to the relatively small number of particles at this time, which causes a jump in the number of particles in statistics.

The average particle deposition height, $h_{depos}$, expresses the volume of deposition particles per unit of bed-parallel area. Figure 10 depicts the average deposition height of the fine component $h_{F,depos}$ and the average deposition height of the coarse component $h_{C,depos}$ at the end of the simulation duration. It can be seen that on the outside of the gate, the average deposition height tends to become thinner along the slope. The variation is qualitatively consistent with the experimental results by Gladstone et al. (Reference Gladstone, Phillips and Sparks1998) and the simulation results by Francisco et al. (Reference Francisco, Espath and Silvestrini2017). From the left wall to the gate, the deposition height of fine particles in figure 10(a) shows that the closer to the left wall or the closer to the gate, the larger the deposition height. The occurrence of the former increase is related to the aggregation of fine particles with smaller settling velocities near the left wall under the effect of invasion flow. For the coarse component (figure 10b), the average deposition height increases near the gate. Since both fine and coarse particles can be transported farther on average as $\phi _F$ increases (figure 4), the deposits can accordingly cover a wider area (figure 10a,b).

Figure 10. Average particle deposition height along the slope for six cases for (a) fine component and (b) coarse component.

3.4. Bidisperse particle statistics

Particle statistics during bidisperse TC segregation are discussed in this section, including particle velocities, particle concentration profiles and particle collisions.

3.4.1. Average bed-parallel velocities of transported particles

We perform a calculation of the average bed-parallel transported particle velocity for the fine and coarse components ($\overline {u^{//}_{pF}}$ and $\overline {u^{//}_{pC}}$) in figure 11. Figure 12 shows the average value of $\overline {u^{//}_{pF}}$ and $\overline {u^{//}_{pC}}$ over different periods at various $\phi _F$. One can easily see from the figures that the higher the fine particle proportion, the larger the average velocity of the fine or coarse particles. The average bed-parallel velocity of the transported fine or coarse particles increases, reaches the peak at around $t=4$ and then diminishes. In the bidisperse TC cases, the average bed-parallel velocity of the coarse particles, including the maximum velocity, is slightly higher than that of the fine particles during this acceleration phase ($t\le 4$). The possible reason is that the coarse particles settle faster compared with the fine particles, resulting in their bed-parallel velocity component being larger than fine particles. Afterward, the transported coarse particles slow down much more quickly than the transported fines before $t\approx 9$ due to the greater resistance against advancing. When the fine particle proportion is low ($\phi _F=0,0.2,0.4$), almost all coarse particles finally settle on the slope and the average bed-parallel transported particle velocity drops to zero. However, for the high fine particle proportion bidisperse cases ($\phi _F\ge 0.6$), the transported coarse particles speed up at approximately $t=9\unicode{x2013}12$ and even surpass the fine ones. Notably, the average velocity of transported coarse particles is currently close to the front velocity in figure 4(b). Thus, we can judge that the transported coarse particles are mostly located at the current front. We speculate that it is the strong flow near the front that keeps the coarse particles in motion.

Figure 11. Time evolution of the average bed-parallel velocity for transported coarse and fine particles in all simulations. The symbols and lines represent the data for the coarse and fine components, respectively.

Figure 12. Average bed-parallel velocity for transported coarse and fine particles which are averaged over different periods at varying $\phi _F$.

3.4.2. Particle vertical profiles

Figure 13 shows the non-dimensional average particle concentration profiles (fine, coarse and total components, $\alpha _{pF}/\phi _F$, $\alpha _{pC}/\phi _C$ and $\alpha _{p}$) and gradient Richardson number $Ri_g$ at $x= 4.80$ when $t=7.9\unicode{x2013}8.1$ in all simulations. A useful factor, $Ri_g$, to evaluate the stratification (Kneller et al. Reference Kneller, Nasr-Azadani, Radhakrishnan and Meiburg2016) is given by

(3.4)\begin{equation} Ri_g = \frac{\dfrac{\partial \alpha_p}{\partial z}}{\left(\dfrac{\partial u_f^{/{/}}}{\partial z}\right)^2}. \end{equation}

It is obvious that the fine and coarse components exhibit different characteristics while transporting. For the fine component (figure 13a), in the upper layers interacting with the ambient fluid, the non-dimensional particle concentration $\alpha _{pF}/\phi _F$ exhibits an upward exponential decrease. The increase in the fraction of fine particles helps the fine particles to be suspended at higher positions and increases the concentration gradient of fine component $\partial (\alpha _{pF}/\phi _F)/\partial z$ near the upper interface of the TC. In the lower layers, the non-dimensional concentration $\alpha _{pF}/\phi _F$ is uniformly distributed in the vertical direction, and is substantially equal to the initial particle concentration ($=$0.01). This kind of profile with different distribution of upper and lower layers can also be found in previous studies (Huang et al. Reference Huang, Imran and Pirmez2007; Kneller et al. Reference Kneller, Nasr-Azadani, Radhakrishnan and Meiburg2016). Due to the large settling velocity of the coarse component, the coarse particles are mainly transported in the lower layers of the current (figure 13b) as compared with the fine component and even form a deposited layer at the bottom more quickly. The concentration of coarse particles declines exponentially upwards, in line with the understanding of Hitomi et al. (Reference Hitomi, Nomura, Murai, De Cesare, Tasaka, Takeda, Park and Sakaguchi2021) and Sequeiros, Mosquera & Pedocchi (Reference Sequeiros, Mosquera and Pedocchi2018), which indicates an obviously stratified gradient of coarse component. With the increase of the $\phi _F$, the coarse particles can also be maintained at a slightly higher position, which means that the increase of fine particles can inhibit the settling of coarse particles to a certain extent. Such differential concentration profile of the coarse and fine components in the current head is attributed to their different settling velocities ($u_{T,C}>u_{T,F}$), which is consistent with the experimental understandings of Baas et al. (Reference Baas, McCaffrey, Haughton and Choux2005). In short, for the bidisperse TC, the upper layer is mainly dominated by the fine component, and the lower layer is coexistence of coarse and fine components. As can be observed in figure 13(c), for the monodisperse TCs, both coarse and fine particle concentrations firstly increase at the interfacial region and keep almost constant until near the slope. For the bidisperse TCs, we can observe the particle segregation. With increasing $\phi _F$, particles are less accumulated near the wall and more distributed in the TC head area, which can be also directly observed in figure 7.

Figure 13. Non-dimensional average particle concentration profiles and gradient Richardson number profiles at $x=4.80$ when $t=7.9\unicode{x2013}8.1$ for six cases. Concentration profiles of (a) fine component $\alpha _{pF}/\phi _F$, (b) coarse component $\alpha _{pC}/\phi _C$ and (c) total component $\alpha _p$ and (d) profiles of gradient Richardson number $Ri_g$.

Figure 13(d) shows that the gradient Richardson number increases at around $z-z_b=1$ and $z-z_b=0.5$, respectively, which is consistent with previous results in Nasr-Azadani, Meiburg & Kneller (Reference Nasr-Azadani, Meiburg and Kneller2018). The gradient Richardson number less than 0.25 at approximately $z-z_b=1$ is associated with the fine particle concentration profile. The growth of Kelvin–Helmholtz billows here is mainly inhibited by the considerable fluid viscosity at low Reynolds numbers (Koohandaz et al. Reference Koohandaz, Khavasi, Eyvazian and Yousefi2020). A very large $Ri_g$ at $z-z_b=0.5$ implies very strong stratification, which is dominated by the coarse particle concentration profile and fluid velocity profile.

3.4.3. Particle collision process

Particle segregation in the lock-exchange bidisperse TC mainly results from the different velocities of coarse and fine particles. The contact force, as the key to particle collisions, has showed its non-negligible effect on particle transport in our previous TC study (Xie et al. Reference Xie, Hu, Zhu, Yu and Pähtz2023). Here we employ different contact force thresholds $\eta _t$ to traverse and retrieve all the transported particles to obtain the contact force magnitude and the corresponding particle's proportion ($n_{co,C(F)}/N_{pC(F)}$), as shown in figure 14. In the monodisperse TC, the number of collision particles increases gradually due to similar falling velocity, reaches the maximum and then decreases because of the reduction of transported particles. However, in the bidisperse TC, the collision number increases rapidly during the segregation process, which is closely connected with the unequal settling velocities of coarse and fine particles in table 1. Both coarse particles and fine particles will be more accessible to collide with the other dispersed phase. When the threshold $\eta _t$ increases, the collision number decreases significantly except the $\phi _F=1$ case. We can confirm that the contact force in the pure coarse particle case is larger than half of the effective gravity force ($0.5|\boldsymbol {G}'|$), while that in the pure fine particle case is around $0.10|\boldsymbol {G}'|$ and $0.25|\boldsymbol {G}'|$. As for the bidisperse cases, we speculate that most collisions belong to the collisions between the coarse and fine particles and the magnitude of the force is mostly smaller than $0.25|\boldsymbol {G}_C'|$ ($2|\boldsymbol {G}_F'|$). In other words, the segregation enhances the collision on fine particles ($n_{co,F}/N_{pF}$ in figure 14e–h has been increased) and increases the proportion of colliding coarse particles with small contact force (figure 14a,b), whereas its ability to provide large contact force for coarse particles is weak (figure 14c,d). It is clear from the foregoing that the particle collisions are considerably affected by the segregation of two dispersed phases.

Figure 14. Collision transported particle number non-dimensionalized by the total particle number with different threshold $\eta _t$ for (a–d) coarse component and (e–h) fine component: (a,e) $\eta _t = 0.05 |\boldsymbol {G'}|$; (b, f) $\eta _t = 0.10 |\boldsymbol {G'}|$; (c,g) $\eta _t = 0.25 |\boldsymbol {G'}|$; (d,h) $\eta _t = 0.50 |\boldsymbol {G'}|$.

3.5. Particle transport properties in the TC process

The segregation process in bidisperse TC changes the transport properties of particles in the flow. Here we perform the statistics of the average particle Reynolds number (defined by (2.10)) of the transported fine and coarse particles ($\overline {Re_{p,F}}$ and $\overline {Re_{p,C}}$) as shown in figure 15(a). It characterizes the transport properties of the particles in the flow. The ratio of the two $\xi _{CF}=\overline {Re_{p,C}}/\overline {Re_{p,F}}$ is shown in figure 15(b). Note that we assume that when the mass of the transported particles of coarse component is less than 10 % of its initial mass, the component essentially finishes the settling process, and we define the exact time as $T_c$. The particle Reynolds number of the coarse component, as well as the forces exerted on them in the next subsection, is not plotted at $t > T_c$. The average particle Reynolds numbers of both fine and coarse particles decrease gradually (figure 15a), demonstrating a dropping slip velocity ($|\boldsymbol {u}_f - \boldsymbol {u}_p|$) and a rising drag coefficient $C_D$. In essence, it reflects that the two-phase movement tends to coordinate during the process of particles settling. The ratio $\xi _{CF}$ gradually falls and approaches 2 before the coarse component basically finishes settling, as shown in figure 15(b). It can give

(3.5)\begin{equation} \overline{{{| {{{\boldsymbol{u}}_f} - {{\boldsymbol{u}}_p}} |}_{@C}}} /\overline{{{| {{{\boldsymbol{u}}_f} - {{\boldsymbol{u}}_p}}|}_{@F}}} = \frac{{{d_{pF}}}}{{{d_{pC}}}}{\xi _{CF}} \to 1, \end{equation}

where subscripts $@F$ and $@C$ represent the slip velocity for transported fine and coarse particles, respectively. In that way, it follows that as TC evolves, remaining transported coarse particles will eventually maintain a motion state similar to that of the fine particles, even though the majority of the coarse particles have been deposited by gravity. This part of the transported coarse particles mainly participate in TC evolution near the very front after approximately $t = 8$. In addition, the relative fine particle volume fraction $\phi _F$ essentially has no impact on $\overline {Re_{p,F}}$ and $\overline {Re_{p,C}}$, and accordingly has no impact on the average particle Reynolds number ratio $\xi _{CF}$.

Figure 15. Time evolution of (a) the average particle Reynolds number of the transported fine and coarse particles ($\overline {Re_{p,F}}$ and $\overline {Re_{p,C}}$) and (b) the average particle Reynolds number ratio between coarse and fine transported particles $\xi _{CF}$.

We also investigate the average particle Reynolds number in the bed-parallel and bed-normal direction, $\overline {Re_{p}^{//}}$ and $\overline {Re_{p}^{\bot }}$, where the bed-parallel and bed-normal particle Reynolds number ($Re_{p}^{//}$ and $Re_{p}^{\bot }$) are separately given by velocity decomposition,

(3.6)$$\begin{gather} Re_p^{/{/}} = \frac{\rho_f d_p |\boldsymbol{u}_f^{/{/}} - \boldsymbol{u}_p^{/{/}}|}{\mu _f}, \end{gather}$$

(3.7)$$\begin{gather}Re_p^{\bot} = \frac{\rho_f d_p |\boldsymbol{u}_f^{\bot} - \boldsymbol{u}_p^{\bot}|}{\mu _f}. \end{gather}$$

As $\overline {Re_{p}^{//}}$ and $\overline {Re_{p}^{\bot }}$ for the coarse component and for the fine component are quite similar, we only plot the time variation of $\overline {Re_{p,F}^{//}}$ and $\overline {Re_{p,F}^{\bot }}$ for fine particles in figure 16. Here $\overline {Re_{p,F}^{\bot }}$ decreases during the whole simulation duration, suggesting that the bed-parallel velocity difference between the fluid and transported fine particles is shrinking. Note that the decay rate of $\overline {Re_{p,F}^{//}}$ in logarithmic coordinates gradually slows down at $t = 0\unicode{x2013}4$ and then approximately keeps constant (${\rm d}( {\lg (\overline {Re_{p,F}^{//}} )})/{\rm d}t \approx -3.76\times {10^{-2}}$). By comparison, $\overline {Re_{p,F}^{\bot }}$ exhibits an obvious increase before $t=6.5$, associated with the rapid settling due to the dominance of particle gravity. After that, $\overline {Re_{p,F}^{\bot }}$ remains unchanged (only decreases slightly).

Figure 16. Time evolution of average particle Reynolds number for fine component in the bed-parallel direction ($\overline {Re_{p,F}^{//}}$) and bed-normal direction ($\overline {Re_{p,F}^{\bot }}$).

Figure 17 plots the average value of $\overline {Re_{p,F}^{//}}$ and $\overline {Re_{p,F}^{\bot }}$ over two periods at varying relative fine particle volume fraction $\phi _F$, which, in addition to figure 16, depicts how $\overline {Re_{p,F}^{//}}$ and $\overline {Re_{p,F}^{\bot }}$ respond to $\phi _F$. As can be observed from figures 16 and 17, the influence of $\phi _F$ on $\overline {Re_{p,F}^{//}}$ and $\overline {Re_{p,F}^{\bot }}$ can be divided into three stages.

1. (i) Bidisperse transport stage (stage I). In this stage, the reduction of fine particle quantity (the increase of coarse particle quantity) promotes the increase of the bed-normal average particle Reynolds number $\overline {Re_{p,F}^{\bot }}$. This is due to the greater settling velocity of the coarse particles intensifying the bed-normal flow. This enhanced flow and the collision during the segregation process (as shown in figure 14) jointly boost the fine component's settlement, resulting in a smaller covered height depicted in figure 13(a).

2. (ii) Transition stage (stage II). In this stage, TC transitions from bidisperse transport to fine particle transport as the coarse particles basically settle. The impact of $\phi _F$ on $\overline {Re_{p,F}^{\bot }}$ progressively diminishes, while a reduction in $\phi _F$ leads to a corresponding decrease in $\overline {Re_{p,F}^{//}}$.

3. (iii) Monodisperse transport stage (stage III). This is a stage where the reduction of $\phi _F$ decreases $\overline {Re_{p,F}^{//}}$. The reason is the lower particle velocity in the case of smaller $\phi _F$ (figure 11) that corresponds to the smaller slip velocity. These facts demonstrate that the bed-parallel transport of fine particles in the smaller $\phi _F$ case has better synchronization with the flow in the later stages of the bidisperse TC.

Figure 17. Average particle Reynolds number for fine component at different $\phi _F$ that is averaged over $t = 4\unicode{x2013}6$ and $t = 10\unicode{x2013}12$: (a) bed-parallel direction and (b) bed-normal direction.

In brief, the alterations in particle transport properties resulting from variations in $\phi _F$ exhibit a strong correlation with the completion of the downward segregation process of the coarse component.

In order to intuitively exhibit the properties of particles following the flow, we depict the bed-parallel slice of fluid transverse velocity $v_f$ together with particles at different distances from the bed ($z-z_b = 0.04$ and 0.20) of three cases as shown in figure 18. As shown in the figure, the majority of particles’ transverse velocity conforms to that of the fluid. And the particle Stokes numbers $St$ of the coarse and fine components are both far less than unity in table 1. These facts reveal that both types of particles can effectively follow the flow. Thus, it is reasonable to conjecture that the transport motion of them would be similar, as mentioned above. Nonetheless, the coarse particles exhibit a higher $St$ value, indicating a relatively weaker capacity for transport.

Figure 18. Bed-parallel slice of the transverse fluid velocity $v_f$ at $t = 6$ at $z-z_b = 0.04$ ($4d_{pF}$; a–c) and at $z-z_b = 0.20$ ($20d_{pF}$; d–f) for three cases: (a,d) $\phi _F = 0$; (b,e) $\phi _F = 0.6$; and (c, f) $\phi _F = 1.0$. Here $z_b$ denotes the bed elevation. Large dots represent the coarse particles and small ones the fine particles. Red dots are positive transverse velocity particles ($v_p>0$) and black dots are negative transverse velocity particles ($v_p<0$).

3.6. Average force evolution

Figure 19(a,b) shows the time variation of dimensionless average forces on the particles in the bed-parallel and bed-normal direction of Case 4 ($\phi _F=0.6$), respectively. The dimensionless forces in the figure are for all transported particles, which is the sum of the force components of all transported particles divided by the total particle effective gravity. The figures include effective drag force $\boldsymbol {F}^{Ed}$ (the sum of effective gravity $\boldsymbol {G}'$ and drag force $\boldsymbol {F}^d$), lift force $\boldsymbol {F}^l$, added mass force $\boldsymbol {F}^{add}$, contact force $\boldsymbol {F}^c$ and total force ($\boldsymbol {F}^T=\boldsymbol {F}^{Ed}+\boldsymbol {F}^{l}+\boldsymbol {F}^{add}+\boldsymbol {F}^{c}$).

Figure 19. Time evolution of average (a) bed-parallel and (b) bed-normal forces on total transported particles for case $\phi _F=0.6$.

In the bed-parallel direction (figure 19a), the positive average total force first increases and then decreases to zero at $t=0\unicode{x2013}4$, which indicates that the particles are on average accelerated along the slope. At $t > 4$, the total force becomes negative, and the particles, on average, exhibit a gradual decay in velocity, consistent with our knowledge of TC on flat slopes (Xie et al. Reference Xie, Hu, Pähtz, He and Cheng2022). The effective drag force remains positive (except in the very beginning stages) and is the dominant factor promoting particles to transport along the slope, where the effective gravity plays a decisive role ($\overline {\boldsymbol {G}'_{//}}=\sin \theta \simeq 0.1$). The average lift force is positive at $t=0\unicode{x2013}2$, because most of the particles are subjected to the positive vorticity. After $t=2$, the negative vorticity induced by the bottom wall gradually becomes prominent, and more particles enter the negative vorticity region, so that the lift force becomes considerably negative. The added mass and contact forces are approximately negligible in the bed-parallel direction as compared with other forces. In the bed-normal direction indicated in figure 19(b), the average total force is negative at first ($t=0\unicode{x2013}2.3$), then becomes positive at $t=2.3\unicode{x2013}4.9$ and eventually turns back to negative after $t=4.9$ due to the overwhelming particle gravity. The positive average contact force provided by the bottom wall comparatively resists the negative effective drag force, whereas the lift force and added mass force are secondary.

It is worthwhile to explore the dynamic process of the coarse and fine components separately, which is crucial for the evolution of the bidisperse TC. Figure 20 shows the time evolution of the average forces for the transported fine and coarse particles in the bed-parallel and bed-normal direction.

Figure 20. In case $\phi _F=0.6$, time evolution of average forces on fine transported particles (a) in the bed-parallel direction and (b) in the bed-normal direction, and time evolution of average forces on coarse transported particles (c) in the bed-parallel direction and (d) in the bed-normal direction.

In the bed-parallel direction, before the coarse component roughly finishes the settling process (approximately $t<8.4$), the evolution trend of the total force of the coarse and fine particles is approximately exactly the same. It explains why the forward positions of the two components in figure 4(a) highly coincide. The main forces of the two components are still effective drag force and lift force, which is consistent with the overall understanding of TC in figure 19(a). For the fine components (figure 20a), between 0 and 4 dimensionless time, the effective drag force is roughly 0, which means that the effective gravity and drag force along the slope are in balance, and the lift force is positive. Subsequently, the effective drag force gradually increases and becomes the factor that drives the particles forward, while the lift force becomes negative and becomes the inhibitory factor. For the coarse component (figure 20c), the effective drag force acts as the dominant factor promoting the advancement from the very beginning. Since the coarse particles have a larger settling velocity, they will enter the negative vorticity region of the lower layer more quickly, and thus the lift force remains negative except for the very initial moment. Interestingly, the contact force of the coarse particles shows a negative value before the settling process is completed (figure 20c), while the fine particle shows a positive value (figure 20a). Combining the average contact force of the TC as a whole (figure 19a), it is not difficult to conclude that in the evolution of the bidisperse TC, the fine particles are pushed forward by the coarse particles until the coarse particles basically complete the settlement. After that, the contact force on the fine particles also becomes negative due to the collision of sediment and the bottom slope.

In the vertical direction (figure 20b,d), the total force evolution of the coarse grains is also highly similar with that of the fine grains. The effective drag force of the coarse particles is negative, and due to the prominent effective gravity, the particles sink, while the contact force remains negative. At $t=0\unicode{x2013}7$ in figure 20(b), the fine particles receive a downward contact force from the collision with the coarse particles. The purpose of this contact force is to make the fine particles sink as quickly as possible in synchronization with the coarse particles. This causes the slip velocity of the fine particles to be very large, and the upward drag force exceeds the effective gravity of the particle, making the effective drag force positive. After $t > 7$, the fine particle group as a whole generally exhibits the settling with a constant velocity ($\overline {\boldsymbol {F}^{T}_{\bot }}\approx 0$), in which the contact force becomes positive and the effective drag force turns to negative.

Given that, in the bed-parallel direction, the fine and coarse particles are predominantly governed by the effective drag force ($\overline {\boldsymbol {F}^{Ed}_{//,F}}$ and $\overline {\boldsymbol {F}^{Ed}_{//,C}}$), lift force ($\overline {\boldsymbol {F}^{l}_{//,F}}$ and $\overline {\boldsymbol {F}^{l}_{//,C}}$) and contact force ($\overline {\boldsymbol {F}^{c}_{//,F}}$ and $\overline {\boldsymbol {F}^{c}_{//,C}}$), we here explore how these components respond to $\phi _F$ as shown in figure 21. It is clear that the influence of $\phi _F$ on these force components can be divided into two parts according to the critical point $t=T_c$. During segregation in bidisperse transport at time $t< T_c$, the contact force exerted on fine particles is positively correlated with the coarse particle proportion, as noted by the exacerbated coarse–fine collisions in figure 14. The negative contact force experienced by coarse particles rises in tandem with their quantity at $4< t< T_c$. It results in a more rapid deceleration and creates an increased velocity difference between the two dispersed phases. As a consequence, this collision process tends to provoke increase/decrease in the velocity differences between the fine particles/coarse particles and fluid phase, resulting in corresponding alterations in the bed-parallel negative drag force experienced by the fine and coarse particles. Thus, fine and coarse particles’ effective drag forces are observed to decrease and increase, respectively. The lift force at this time is not sensitive to the change in $\phi _F$ (figure 21b,e). To sum up, variations in contact force are counterbalanced by variations in effective drag force, which is expressed as

(3.8)\begin{equation} \Delta \overline{{\boldsymbol{F}}_{/{/},F}^{Ed}} (\Delta {\phi _F}) \sim \Delta \overline{{\boldsymbol{F}}_{/{/},F}^{c}} (\Delta {\phi _F}) \quad\!\! {\rm{and}} \quad \!\! \Delta \overline{{\boldsymbol{F}}_{/{/},C}^{Ed}} (\Delta {\phi _F}) \sim \Delta \overline{{\boldsymbol{F}}_{/{/},C}^{c}} (\Delta {\phi _F}), \quad\!\! t< T_c . \end{equation}

After $t=T_c$, the TC is dominated by the transport of fine particles. Even if the coarse component has been deposited, its impact on the movement of the fine particle will persist at $t>T_c$. The negative contact force $\overline {{\boldsymbol {F}}_{//,F}^{c}}$ gradually rises in figure 21(c) as $\phi _F$ drops because fine particles are moved closer to the bottom wall. At the same time, the weakening of the flow at a smaller $\phi _F$ suggests the weakening of the vorticity field, which makes the negative lift gradually decrease in figure 21(b). The variation in lift force equalizes the variation in contact force brought on by altering $\phi _F$, i.e.,

(3.9)\begin{equation} \Delta \overline{{\boldsymbol{F}}_{/{/},F}^{l}} (\Delta {\phi _F}) \sim \Delta \overline{{\boldsymbol{F}}_{/{/},F}^{c}} (\Delta {\phi _F}), \quad t>T_c . \end{equation}

Figure 21. Time evolution of (a,d) average effective drag force, (b,e) average lift force and (c, f) average contact force in the bed-parallel direction: (a–c) fine component and (d–f) coarse component.

3.7. Energy conversion process

We analyse the energy transformation process of TC. It is generally recognized that the TC is initially driven by the particle gravitational potential energy (Meiburg et al. Reference Meiburg, Radhakrishnan and Nasr-Azadani2015). During the evolution of the TC, the particle gravitational potential energy is gradually transformed into the fluid potential energy, fluid kinetic energy, particle kinetic energy and dissipated energy due to the fluid viscosity and particle collision process. Particle gravitational potential energy $E_p^p$, fluid potential energy $E_p^f$, particle kinetic energy $E_k^p$ and fluid kinetic energy $E_k^f$ can be explicitly calculated as follows:

(3.10)$$\begin{gather} E_p^p(t) =\sum_{i=1}^{N_p} m_i |\boldsymbol{g}| z_{p,i}, \end{gather}$$

(3.11)$$\begin{gather}E_k^p(t) = \sum_{i=1}^{N_p} \left( \frac{1}{2} m_i |\boldsymbol{u}_{p,i}|^2 + \frac{1}{2} I |\boldsymbol{\omega}_{p,i}|^2 \right), \end{gather}$$

(3.12)$$\begin{gather}E_p^f(t) =\int _\varOmega \alpha_f \rho_f |\boldsymbol{g}| z \,{\rm d} V, \end{gather}$$

(3.13)$$\begin{gather}E_k^f(t) = \int _\varOmega \frac{1}{2} \alpha_f \rho_f |\boldsymbol{u}_f|^2 \,{\rm d} V, \end{gather}$$

where $N_p$ is the number of particles, $z_{p,i}$ is the elevation of the particle $i$ and $\varOmega$ represents the whole simulation domain. The relationship among the energy components in the TC system can be given as follows (Xie et al. Reference Xie, Hu, Pähtz, He and Cheng2022):

(3.14)\begin{equation} \Delta {E_{Diss}} ={-} [ {\Delta E_p^p + \Delta E_p^f + \Delta E_k^p + \Delta E_k^f} ], \end{equation}

where $\Delta {E_{Diss}}$ is the dissipated energy and $\Delta$ denotes the energy discrepancy between the current time and the initial time.

Figure 22 depicts the time evolution of change of each energy component, $\Delta E_p^{p,\ast }$, $\Delta E_p^{f,\ast }$, $\Delta E_k^{p,\ast }$, $\Delta E_k^{f,\ast }$ and $\Delta E_{Diss}^{\ast }$, where the energy component with symbol $\ast$ represents that the energy is non-dimensionalized by the relative initial particle gravitational potential energy with reference to the plane $z=1$. We plot the available potential energy $E_{p,avail}^\ast =-\Delta E_p^{p,\ast } - \Delta E_p^{f,\ast }$ in figure 22(c). Figure 22(a,b,c, f) shows that the trend changes of the curves of the non-dimensionalized potential energy terms and the dissipated energy term are quite similar. The particle gravitational potential energy decreases gradually in all cases with particles settling and transporting downstream, except in the monodisperse coarse particle TC. In the case of $\phi _F=0$, the particle gravitational potential energy declines first, and it remains essentially unchanged at $t>12$ because most of coarse particles have settled on the slope. As shown in figure 22(d,e), the fluid kinetic energy and the particle kinetic energy gradually increase from dimensionless time 0 to 4, and then gradually decrease at $t>4$. The fact that the particle kinetic energy reaches its maximum $t=4$ is related to the change of the average longitudinal total force of the transported particles from positive to negative (figure 19). After $t=4$, the particle phase begins to exert a force on the fluid phase of TC, so as to promote its advancement. During this transition, TC enters the slumping stage with a constant velocity. In general, in the TC evolution, the majority of the particle gravitational potential energy is converted into fluid potential energy and dissipated energy, only a small amount of energy is converted into the kinetic energy of the fluid and particle phases, which is consistent with previous understandings (Xie et al. Reference Xie, Hu, Pähtz, He and Cheng2022, Reference Xie, Hu, Zhu, Yu and Pähtz2023).

Figure 22. Time evolution of energy components for all cases: (a) particle gravitational potential energy $\Delta E^{p,\ast }_p$; (b) fluid potential energy $\Delta E^{f,\ast }_p$; (c) available potential energy $E_{p,avail}^{\ast }$; (d) fluid kinetic energy $\Delta E^{f,\ast }_k$; (e) particle kinetic energy $\Delta E^{p,\ast }_k$; and ( f) dissipated energy $\Delta E^{\ast }_{Diss}$.

Figure 23 plots the above energy components as a function of $\phi _F$ at five selected moments. At $t<12$, an increase in the proportion of coarse particles boosts the consumption rate of the particle gravitational potential energy in figure 23(a) due to the higher settling velocity of the coarse particles, resulting in more available potential energy $E_{p,avail}^\ast$, as shown in figure 23(c). The conversion amount of the particle gravitational potential energy in each case is approximately equal around $t=12$ when most of the coarse particles have completed the settling process (figure 9b). From figure 23(d,e), it is generally observed that the kinetic energy of fluid and particle phases tends to be greater at lower $\phi _F$ before $t=4$. It is due to the faster settling of coarse particles. When $t\ge 8$, most of the coarse particles are settled, and an increase in $\phi _F$ results in higher kinetic energy of the fluid and particle phases, as shown in figure 23(d,e). Figure 23( f) shows that the amount of energy dissipated grows with the proportion of coarse particles at the early stage, which may be associated with the increase in the number of small vortex structures (figure 7).

Figure 23. Energy components as a function of $\phi _F$ at five instants ($t = 2,4,8,12,16$): (a) particle gravitational potential energy $\Delta E^{p,\ast }_p$; (b) fluid potential energy $\Delta E^{f,\ast }_p$; (c) available potential energy $E_{p,avail}^{\ast }$; (d) fluid kinetic energy $\Delta E^{f,\ast }_k$; (e) particle kinetic energy $\Delta E^{p,\ast }_k$; and ( f) dissipated energy $\Delta E^{\ast }_{Diss}$.

To depict the conversion for available potential energy, figure 24 plots the time evolution of fluid kinetic energy, particle kinetic energy and dissipated energy that are divided by the available potential energy ($\Delta E^{f,\ast }_k/E_{p,avail}^{\ast }$, $\Delta E^{p,\ast }_k/E_{p,avail}^{\ast }$ and $\Delta E^{\ast }_{Diss}/E_{p,avail}^{\ast }$). Figure 24 also shows how these quantities vary with $\phi _F$ at different times. The available potential energy is primarily converted into fluid kinetic energy and dissipated energy in the early stages and primarily into dissipated energy in the latter stages, whereas the amount converted into particle kinetic energy in the whole process is very low. At $t=0\unicode{x2013}2$, the conversion proportion from available potential energy to fluid kinetic energy increases (figure 24a), while the conversion proportion to dissipated energy decreases (figure 24c). Subsequently, the change trends of the two reverse. Throughout the simulation period, the conversion proportion to particle kinetic energy continuously declines (figure 24b). As $\phi _F$ increases, the conversion proportion to fluid kinetic energy and particle kinetic energy increases (figure 24d,e), while that to dissipated energy decreases (figure 24f). This implies that the increase in fine particle proportion helps to maintain the TC flowing and slow down the dissipation of energy, which agrees with figures 22 and 23.

Figure 24. Time evolution of energy conversion proportion per available potential energy: (a) fluid kinetic energy $\Delta E^{f,\ast }_k/E_{p,avail}^{\ast }$; (b) particle kinetic energy $\Delta E^{p,\ast }_k/E_{p,avail}^{\ast }$; and (c) dissipated energy $\Delta E^{\ast }_{Diss}/E_{p,avail}^{\ast }$. (d) Fluid kinetic energy, (e) particle kinetic energy and ( f) dissipated energy per available potential energy as a function of $\phi _F$ at five selected instants ($t = 2,4,8,12,16$).

To address the energy evolution process of the coarse and fine components in the bidisperse TC, we employ the ratio of the energy component change per unit particle mass of fine to coarse components ($\psi ^\varSigma _{p,FC}$ for gravitational potential energy and $\psi ^\varSigma _{k,FC}$ for particle kinetic energy) and the ratio of the energy component change rate per unit particle mass of fine to coarse components ($\psi ^{\partial t}_{p,FC}$ for gravitational potential energy and $\psi ^{\partial t}_{k,FC}$ for gravitational potential energy). These variables are given by

(3.15)$$\begin{gather} \psi _{p,FC}^\varSigma = \frac{{\Delta E_p^{pF}}}{{\Delta E_p^{pC}}}\left/\,\frac{{{\phi _F}}}{{{\phi _C}}}\right., \quad \psi _{k,FC}^\varSigma = \frac{{\Delta E_k^{pF}}}{{\Delta E_k^{pC}}}\left/\,\frac{{{\phi _F}}}{{{\phi _C}}}\right., \end{gather}$$

(3.16)$$\begin{gather}\psi _{p,FC}^{\partial {t}} = \frac{{\partial E_p^{pF}/\partial {t}}}{{\partial E_p^{pC}/\partial {t}}}\left/\,\frac{{{\phi _F}}}{{{\phi _C}}}\right., \quad \psi _{k,FC}^{\partial {t}} = \frac{{\partial E_k^{pF}/\partial {t}}}{{\partial E_k^{pC}/\partial {t}}}\left/\,\frac{{{\phi _F}}}{{{\phi _C}}}\right., \end{gather}$$

with the superscript $pF$ and $pC$ for energy denoting the fine and coarse components, respectively.

Figure 25 depicts the temporal evolution of $\psi ^\varSigma _{p,FC}$, $\psi ^\varSigma _{k,FC}$, $\psi ^{\partial t}_{p,FC}$ and $\psi ^{\partial t}_{k,FC}$. Notably, the curves in the figure only illustrate the ratios in the bidisperse transport stage. The figure makes it clear that these ratios are independent of $\phi _F$. It suggests that these parameters can be employed to uniformly characterize the energy conversion of the bidisperse TC. We can observe from figure 25(a) that $\psi ^\varSigma _{p,FC}$ is always less than unity. This means that the gravitational potential energy converted from the fine particles is always smaller than that from the coarse particles; in other words, the coarse component is more critical. With the TC evolution, $\psi ^\varSigma _{p,FC}$ gradually increases and approaches unity. In figure 25(c), the change rate ratio $\psi ^{\partial t}_{p,FC}<1$ at $t=0\unicode{x2013}4$, implying that the conversion rate of potential energy of coarse particles is dominant. At $t=4\unicode{x2013}6$, $\psi ^{\partial t}_{p,FC}\approx 1$ means that the conversion rate of potential energy of fine particles is comparable with that of coarse particles, and subsequently the conversion rate of fine particles is larger.

Figure 25. Ratio of the change in (a) particle gravitational potential energy and (b) particle kinetic energy per unit mass of fine to coarse particles versus the dimensionless time, $\psi ^\varSigma _{p,FC}$ and $\psi ^\varSigma _{k,FC}$. Time evolution of the ratio of the change rate in (c) particle gravitational potential energy and (d) particle kinetic energy per unit mass of fine to coarse particles, $\psi ^{\partial t}_{p,FC}$ and $\psi ^{\partial t}_{k,FC}$.

Here $\psi ^\varSigma _{k,FC}$ keeps increasing from approximately 0.1 to 10 shown in figure 25(b), exhibiting a transition from the initial predominance of coarse particle motion to that of fine particle transport. The particle kinetic energy generally increases first and then decreases (figure 22d); however, compared with fine particles, coarse particles experience kinetic energy decay earlier in general. As a result, there is an ‘inverse energy change stage ($\psi _{p,FC}^{\partial {t}}<0$)’ at $t = 3.2\unicode{x2013}3.6$ in figure 25(d) where the kinetic energy of the fine particles $E^{pF}_k$ increases while $E^{pC}_k$ decreases. It is interesting that when the kinetic energies of the coarse and fine particles are equal (approximately $t=3.6$ in figure 25b), the kinetic energy of the coarse particles $E^{pC}_k$ starts to decline. Following this ($t>3.6$), the kinetic energy of the fine particles $E^{pF}_k$ similarly enters the decreasing stage due to the decrease in the kinetic energy of the coarse particles $E^{pC}_k$ and the slowing down of the potential energy input (decrease in $\partial \Delta E^p_p/\partial t$ in figure 22a).