3.1. Current front and morphological patterns

Figure 5 presents the time evolution of the current front position $\tilde x_{\textit{front}} - \tilde L_g$ and front velocity $\tilde u_{\textit{front}}$ , calculated as $\tilde u_{\textit{front}} = \mathrm{d} \tilde x_{\textit{front}} / \mathrm{d} \tilde t$ , of the particle and salinity flow. In order to access the duration of the slumping (constant-velocity) phase, $\tilde t_{sp}$ , the maximum front velocity is considered the beginning of the slumping stage, and a 10 % decay from the maximum velocity marks its end. The inset of figure 5( $b$ ) shows the relationship between the duration of the slumping stage and the time scale ratio $\tilde S$ . Initially, the front velocity of the current undergoes a brief acceleration, followed by a slight decrease or near-constant phase ( $\tilde u_{\textit{front}} \approx 0.6$ ), and finally decelerates. It is consistent with the typical comprehension of the lock-exchange gravity current (Huppert & Simpson Reference Huppert and Simpson1980; Cantero et al. Reference Cantero, Lee, Balachandar and Garcia2007; Huang et al. Reference Huang, Imran and Pirmez2008). Before $\tilde t=12$ , the simulated results of gravity current are quantitatively consistent with the experimental results and numerical results reported in previous studies (Necker et al. Reference Necker, Härtel, Kleiser and Meiburg2002; Dai Reference Dai2014; Gadal et al. Reference Gadal, Mercier, Rastello and Lacaze2023). For the saline flow front, the simulated results at $\tilde t\lt 15$ are quite consistent with the previous results (Necker et al. Reference Necker, Härtel, Kleiser and Meiburg2002; Dai Reference Dai2014). At $\tilde t\gt 15$ , the difference in simulation comes from the limitation of the streamwise length of the computational domain. For particle flow, the decrease of $\xi _{sp}$ corresponds to the enhancement of the particle settling effect, indicated by an increase in $\tilde S$ . This results in the front leaving the constant-velocity slumping stage earlier, which is consistent with previous observations (Gladstone & Woods Reference Gladstone and Woods2000; Gadal et al. Reference Gadal, Mercier, Rastello and Lacaze2023). The front position for $\tilde S=0.033$ simulated in this work is basically enveloped by the front curves of $\tilde S=0.076$ and $\tilde S=0.026$ from Gadal et al. (Reference Gadal, Mercier, Rastello and Lacaze2023). All of the above show that our simulation results have good credibility.

Figure 5.

Temporal evolution of ( $a$ ) front position $\tilde x_{\textit{front}} - \tilde L_g$ and ( $b$ ) front velocity $\tilde u_{\textit{front}}$ . The lines indicate the front position and velocity given by the salinity contour, and the data marked in symbols is obtained from the most downstream particle’s position. The inset in panel ( $b$ ) shows the duration of slumping stage $\tilde t_{sp}$ at different $\tilde S$ s.

The salinity distribution and the particle dataset yield similar front position and velocity values in the dual-driven current, showing minimal differences. This suggests a high degree of coordination between salinity and particle diffusion and propagation under strong fluid–particle interactions. During the acceleration stage, the front velocity in figure 5( $b$ ) is independent of $\xi _{sp}$ , consistent with the findings in Gladstone & Woods (Reference Gladstone and Woods2000). The duration of the slumping stage ( $\tilde t_{sp}$ ) is gradually shortened as $\tilde S$ increases corresponding to a decrease in $\xi _{sp}$ . The reason for the short slumping stage at large $\xi _{sp}$ is that the low particle settling velocity enhances the backflow’s propagation (Huang et al. Reference Huang, Imran and Pirmez2008), enabling the reflected wave to catch up with the front more quickly and alter the front dynamics. When $\xi _{sp}$ is small, the stronger particle settling effect is primarily responsible for the earlier end time of the slumping stage.

Figure 6.

Spatial distribution of spanwise-averaged relative volume fraction of saline fluid $\langle \alpha _s \rangle$ and particle flow profile. The heat map shows the relative volume fraction of the saline fluid, and the black line indicates the outer contour of the particle flow. Here ( $a$ , $d$ , $g$ , $j$ ) $\xi _{sp} = 6.0$ , ( $b$ , $e$ , $h$ , $k$ ) $\xi _{sp} = 1.0$ and ( $c$ , $f$ , $i$ , $l$ ) $\xi _{sp} = 0.17$ ; ( $a$ – $c$ ) $\tilde t = 3$ , ( $d$ – $\!f$ ) $\tilde t = 6$ , ( $g$ – $i$ ) $\tilde t = 9$ and ( $j$ – $l$ ) $\tilde t = 12$ .

The salinity and particle distribution of the dual-driven current at various $\xi _{sp}$ s are depicted in figure 6. It can be seen that the salinity propagation is closely connected to the particle movement in the dual-driven current. A reduction in $\xi _{sp}$ , which decreases the interstitial fluid density and thereby increases particle settling, results in a slower advancement velocity, reduced ability to maintain head shape (Huang et al. Reference Huang, Imran and Pirmez2008) and a lower relative saline fluid fraction $\alpha _s$ near the head. Compared with the salinity-dominated current (figure 6 $a$ ), the particle-dominated current (figure 6 $c$ ) is more prone to interfacial instability at early stages, leading to the formation of KH billows. The interfacial instability structures in the salinity-dominated current (figure 6 $a$ , $d$ , $g$ , $j$ ) are generally confined to the region where $\tilde z\lt 1$ . The excited interfacial vortex propagates downstream with the current, rapidly expands in size, and then slowly dissipates as salinity diffuses. In contrast, the interfacial vortex in particle-dominated current essentially does not move streamwise (figure 6 $\!f\!$ , $i$ , $l$ ). The vortex continues to grow spatially throughout the evolution. Such pronounced vortices have also been reported in previous studies on PD gravity currents (Nasr-Azadani & Meiburg Reference Nasr-Azadani and Meiburg2014; Lu et al. Reference Lu, Liu, Xie, Sun, Yang and Guo2024). These vortices form due to the interaction between wall-reflected bores and vortices induced by the settling of particle groups. As the strength of vortex structure progressively weakens, the reduced fluid forces are insufficient to retain the particles within the vortex, especially at small $\xi _{sp}$ . Thus, particles are displaced by centrifugal force, leading to an expansion in the vortex’s spatial range.

From the above analysis, the key phenomena and transition regimes from SD to PD current can be summarized as follows.

1. (i) Material transport: analysing the variations in transport capacity for salinity and particles is crucial for comprehending SPDD current behaviour.

2. (ii) Interfacial instability: this phenomenon is linked to the entrainment regime and thus the density dilution of interstitial fluid, significantly altering current dynamics.

3.2. Salinity convection–diffusion and particle transport

The material transport capacity can be reflected by the equivalent material height $\tilde h$ and current height $\tilde h_t$ which includes the transporting material covering transported particle part $\tilde h_{p,t}$ and saline fluid part $\tilde h_{s}$ . The difference between the two is that the latter isolates the contribution of the deposited particles $\tilde h_{p,d}$ , yielding $\tilde h = \tilde h_t + \tilde h_{p,d} = ( \tilde h_{p,t} +\tilde h_{s}) + \tilde h_{p,d}$ . The equivalent material height $\tilde h$ can be given by referring to Cantero et al. (Reference Cantero, Lee, Balachandar and Garcia2007, Reference Cantero, Balachandar and García2008), Shin, Dalziel & Linden (Reference Shin, Dalziel and Linden2004) and Birman, Martin & Meiburg (Reference Birman, Martin and Meiburg2005) as

(3.1) \begin{equation} \tilde h = \int _0 ^{\tilde L_z} \langle \tilde \rho _m \rangle \mathrm{d} \tilde z = \int _0 ^{\tilde L_z} \frac { \langle \alpha _p \rangle \left (\rho _p - \rho _w \right)}{\rho _{m0} - \rho _w} \mathrm{d} \tilde z + \int _0 ^{\tilde L_z} \frac {\langle \alpha _{\!f\!} \rangle (\langle \alpha _s \rangle \rho _s + \langle \alpha _w \rangle \rho _w - \rho _w)}{\rho _{m0} - \rho _w} \mathrm{d} \tilde z. \end{equation}

The first and second terms on the right-hand side represent the height contributed from particles and salinity denoted as $\tilde h_p =\tilde h_{p,t} + \tilde h_{p,d}$ and $\tilde h_s$ , respectively. The deposited particle height $\tilde h_{p,d}$ is calculated by

(3.2) \begin{equation} \tilde h_{p,d} = \frac {\rho _p - \rho _w }{\rho _{m0} - \rho _w} \int _0 ^{\tilde L_z} \langle \alpha _{p,d} \rangle \mathrm{d} \tilde z, \end{equation}

where $\langle \alpha _{p,d} \rangle$ denotes the spanwise-averaged volume fraction of deposited particles, which is given by

(3.3) \begin{equation} \langle \alpha _{p,d}(x) \rangle = N_{p,d}^l(x) V_p N_x / L_x L_y, \end{equation}

in which $N_{p,d}^l(x)$ is the deposited particle number in the region $\mathcal{R} = (x-({L_x}/{2N_x}),x+({L_x}/{2N_x}))$ . The deposited particles here refer to the particles that are in contact with the bottom wall and have a velocity $|\boldsymbol{\tilde u}_p|$ less than $10^{-2}$ . It is important to note that both the equivalent material height $\tilde h$ and current height $\tilde h_t$ account for the material transported by large-sized vortices in the flow field, aligning with the approach in previous studies (Cantero et al. 2007, 2008). This is primarily because these vortices are a significant feature generated by the current and exert a non-negligible influence on its dynamics.

Figure 7.

Spatiotemporal evolution of material heights for various cases: ( $a$ – $c$ ) $\xi _{sp} = 6.0$ , ( $d$ – $f$ ) $\xi _{sp} = 1.0$ and ( $g$ – $i$ ) $\xi _{sp} = 0.17$ ; ( $a$ , $d$ , $g$ ) relative salinity height $\tilde h_s / \tilde h$ , ( $b$ , $e$ , $h$ ) relative particle height $\tilde h_p / \tilde h$ and ( $c$ , $f$ , $i$ ) equivalent material height $\tilde h$ .

Figure 7 illustrates the spatiotemporal evolution of three material heights ( $\tilde h_s / \tilde h$ , $\tilde h_p / \tilde h$ and $\tilde h$ ) at varying $\xi _{sp}$ s. Regions with locally large salinity height $\tilde h_s$ often corresponds to regions with locally large $\tilde h$ , indicating that the interstitial fluid is essential to the material transport. The relative salinity/particle height ( $\tilde h_s / \tilde h$ , $\tilde h_p / \tilde h$ ) in the head essentially stays constant in the slumping stage. It suggests that the composition ratio of salinity to particle in the head region remains in equilibrium. This balance is crucial for maintaining the constant front velocity regime of the slumping stage. During the slumping stage, the equivalent material height $\tilde h$ near the head is approximately unity, which corresponds to half the initial height of the fluid–particle column. This observation aligns with previous studies (Necker et al. Reference Necker, Härtel, Kleiser and Meiburg2002; Cantero et al. Reference Cantero, Lee, Balachandar and Garcia2007; Bonometti & Balachandar Reference Bonometti and Balachandar2008; Meiburg et al. Reference Meiburg, Radhakrishnan and Nasr-Azadani2015). The position of the maximum $\tilde h$ progressively shifts away from the head as $\xi _{sp}$ decreases after entering the deceleration period (approximately $\tilde t\gt 10$ ) in figure 7( $c$ , $f\!$ , $i$ ). When $\xi _{sp}$ is small, the inherent sedimentation properties of the particles cause the density difference between the gravity current and the ambient fluid to decay more. It corresponds to a faster decay of the forward ability (also see in figure 5), making it increasingly difficult for the particles to be transported downstream by the current.

Figure 8.

Spatial distribution of the ratio of transporting particle height to particle height $\tilde h_{p,t} / \tilde h_p$ : ( $a$ ) $\tilde t = 3$ , ( $b$ ) $\tilde t = 6$ , ( $c$ ) $\tilde t = 9$ , ( $d$ ) $\tilde t = 12$ , ( $e$ ) $\tilde t = 15$ and ( $f$ ) $\tilde t = 18$ .

We then discuss the ratio of transporting particles to deposited particles at different locations. Figure 8 illustrates the ratio of transporting particle height relative to total particle height, $\tilde h_{p,t} / \tilde h_p$ , at different times and varying values of $\xi _{sp}$ . During the slumping stage, $\xi _{sp}$ has minimal influence on the ratio of transporting particle height, indicating that particle transport capacity and sedimentation are largely independent of $\xi _{sp}$ . In this phase, the maximum $\tilde h_{p,t} / \tilde h_p$ near the front remains at unity, meaning the particles here are primarily transported in suspension. Particles in the upstream gradually settle, leading to a decrease in $\tilde h_{p,t} / \tilde h_p$ . In the deceleration stage, gravity currents with larger $\xi _{sp}$ exhibit higher $\tilde h_{p,t} / \tilde h_p$ values, particularly in terms of maximum $\tilde h_{p,t} / \tilde h_p$ near the front. This implies a great capacity of particle suspension at larger $\xi _{sp}$ , corresponding to the greater forward velocity observed in figure 5. Upstream, after the current passes, $\tilde h_{p,t} / \tilde h_p$ stabilizes at approximately 0.3, indicating that most particles are deposited. This low $\tilde h_{p,t} / \tilde h_p$ results from the movement of particles within the near-wall region under the influence of residual flow.

Figure 9.

Spatiotemporal evolution of current heights for various cases: ( $a$ – $c$ ) $\xi _{sp} = 6.0$ , ( $d$ – $f$ ) $\xi _{sp} = 1.0$ and ( $g$ – $i$ ) $\xi _{sp} = 0.17$ ; ( $a$ , $d$ , $g$ ) relative salinity height $\tilde h_s / \tilde h_t$ , ( $b$ , $e$ , $h$ ) relative transporting particle height $\tilde h_{p,t} / \tilde h_t$ and ( $c$ , $f$ , $i$ ) current height $\tilde h_t$ .

Figure 9 depicts the spatiotemporal evolution of the proportions of transporting particle height ( $\tilde h_{p,t}$ ) and salinity height ( $\tilde h_s$ ) in the current height ( $\tilde h_t$ ), and the evolution of the current height ( $\tilde h_t$ ). The distribution patterns are qualitatively similar to those in figure 7. At larger $\xi _{sp}$ , the particle effect on the current is weak, resulting in heights in figure 9( $a$ – $c$ ) comparable to the heights observed in figure 7( $a$ – $c$ ). Conversely, when $\xi _{sp}$ is small, particle deposition reduces transporting material heights by $\tilde h_{p,d}$ , causing the values in figure 9( $h$ – $i$ ) to be lower than those in figure 7( $h$ – $i$ ).

Figure 10.

Time evolution of ( $a$ ) transporting particle quantity $N_{p,t}$ and ( $b$ ) particle deposition rate $\mathrm{d} N_{p,d} / \mathrm{d} \tilde t$ , both of which are normalized by the total particle quantity $N_p$ . The inset in panel ( $a$ ) shows the transporting particle quantity at varying $\xi _{sp}$ s at $\tilde t = 12$ .

A statistical analysis of the number of deposited particles is crucial for understanding the deposition process in gravity current. Figure 10 illustrates the time evolution of normalized transporting particle number $N_{p,t}/N_p$ and the corresponding deposition rate $\mathrm{d} (N_{p,d}/N_p) / \mathrm{d} \tilde t$ under different values of $\xi _{sp}$ . During the early stage of the current ( $\tilde t\lt 10$ ), the deposition process is independent of $\xi _{sp}$ , and the deposition rate scales with $\tilde t^{\;0.5}$ , which is consistent with previous findings (Necker et al. Reference Necker, Härtel, Kleiser and Meiburg2002; Espath et al. Reference Espath, Pinto, Laizet and Silvestrini2014; Francisco et al. Reference Francisco, Espath and Silvestrini2017). As the current experiences a short stage ( $8\lt \tilde t\lt 10$ ) developing from the slumping stage to the deceleration stage, the deposition rate decreases following a scaling law of $\tilde t^{-2.25}$ . For $\tilde t\gt 10$ , a decrease in $\xi _{sp}$ leads to a lower deposition rate. It is attributed to the large-sized vortex structures in the flow field (Necker et al. Reference Necker, Härtel, Kleiser and Meiburg2002) as seen in figure 6, which trap some particles and hinder sedimentation. This phenomenon is evident from the higher number of transporting particles at very small $\xi _{sp}$ , as shown in the inset of figure 10( $a$ ). At this stage, the deposition rate scales approximately with $\tilde t^{-6}$ , slightly higher than the earlier decay rate (Necker et al. Reference Necker, Härtel, Kleiser and Meiburg2002; Espath et al. Reference Espath, Pinto, Laizet and Silvestrini2015). This difference is attributed to the higher particle settling velocity $\tilde \gamma _t$ in this study (Necker et al. Reference Necker, Härtel, Kleiser and Meiburg2005; Espath et al. Reference Espath, Pinto, Laizet and Silvestrini2015; Francisco et al. Reference Francisco, Espath and Silvestrini2017).

We compute the average vertical predominant forces ( $\boldsymbol{\tilde {F}}^{\boldsymbol{\nabla} p}_z$ , $\boldsymbol{\tilde {F}}^{d}_z$ , and $\boldsymbol{\tilde {F}}^{c}_z$ ) acting on the transported particles (Xie et al. 2023a) to elucidate the effect of salinity–particle interaction on particle suspension, as illustrated in figure 11. Changes in $\xi _{sp}$ have a minimal effect on the dominant drag force (figure 11 $a$ ) and pressure gradient forces (figure 11 $b$ ) before the slumping stage. At this stage, the primary pressure gradient force initially rises and then falls, while the secondary drag force mostly stays constant. After that, both ( $\boldsymbol{\tilde {F}}^{\boldsymbol{\nabla} p}_z$ and $\boldsymbol{\tilde {F}}^{d}_z$ ) gradually diminish. As the particle concentration increases, the contact force takes on greater significance in maintaining particle suspension (figure 11 $c$ ). With $\xi _{sp}$ decreasing, and given that the dimensionless deposition rates are comparable (figure 10), a larger number of particles are deposited, further reducing the mixture density of the gravity current. This reduction in density weakens the flow dynamics, significantly decreasing the pressure gradient force and drag force, as shown in figure 11( $a$ , $b$ ). Consequently, the particle suspension ability declines and the particle settling is enhanced, as illustrated in figure 8. Additionally, with particles gradually settling, the increased particle concentration near the wall leads to a corresponding rise in the contact force, as depicted in figure 11( $c$ ).

Figure 11.

Average vertical forces acting on transported particles over time. ( $a$ ) Pressure gradient force $\boldsymbol{\tilde {F}}^{\boldsymbol{\nabla} p}_z$ , ( $b$ ) drag force $\boldsymbol{\tilde {F}}^{d}_z$ and ( $c$ ) contact force $\boldsymbol{\tilde {F}}^{c}_z$ .

3.3. Entrainment and detrainment

One of the key mechanisms to entrainment is the formation of a series of billows resulting from interfacial shear instability, which alters the density difference with the ambient fluid by affecting the material concentration of the current (Wells & Dorrell Reference Wells and Dorrell2021). Here we focus on current entrainment during the slumping phase characterized by a constant velocity. Figure 12( $a$ – $\!f\!$ ) displays the spanwise-averaged flow profiles averaged over $\tilde x = 1.4 \!\sim\! 1.8$ at $\tilde t =2.5$ (in the slumping stage), including fluid streamwise velocity $\langle \tilde u_{\!f\!} \rangle$ , fluid vertical velocity $\langle \tilde w_{\!f\!} \rangle$ , fluid density $\langle \tilde \rho _{\!f\!} \rangle$ , relative particle volume fraction $\langle \alpha _p \rangle /\alpha _{p0}$ , mixture density $\langle \tilde \rho _m \rangle$ and gradient Richardson number $\langle Ri_g \rangle$ . The spatial distribution of the vertical total force $\boldsymbol{F}^T_z$ on the particles in the slumping stage for various cases is shown in figure 12( $g$ – $i$ ).

Figure 12.

Spanwise-averaged current profiles at $\tilde t =2.5$ averaged over $\tilde x = 1.4{\sim}1.8$ : ( $a$ ) fluid streamwise velocity $\langle \tilde u_{\!f\!} \rangle$ ; ( $b$ ) fluid vertical velocity $\langle \tilde w_{\!f\!} \rangle$ ; ( $c$ ) fluid density $\langle \tilde \rho _{\!f\!} \rangle$ ; ( $d$ ) relative particle volume fraction $\langle \alpha _p \rangle / \alpha _{p0}$ ; ( $e$ ) mixture density $\langle \tilde \rho _m \rangle$ ; ( $f$ ) gradient Richardson number $\langle Ri_g \rangle$ . The insets in panels ( $a$ ) and ( $f\!$ ) plot $\langle \tilde u_{\!f\!} \rangle$ and $\langle Ri_g \rangle$ , respectively, at current height $\tilde h_t$ which is given in table 2 at different $\xi _{sp}$ . The inset in panel ( $b$ ) shows the relationship between mean fluid vertical velocity $\langle \tilde w_{\!f\!} \rangle$ near the wall ( $\tilde z = 0 {\sim} 0.05$ ) and $\xi _{sp}$ . Spatial distribution of vertical total force $\boldsymbol{\tilde F}^T_z$ acting on particles at $\tilde t = 2.5$ for ( $g$ ) $\xi _{sp} = 6.0$ , ( $h$ ) $\xi _{sp} = 1.0$ and ( $i$ ) $\xi _{sp} = 0.17$ . The higher density of the granular medium arises from the projection of three-dimensional Lagrangian particle points onto a two-dimensional plane.

Table 2.

Average current interface elevations over $\tilde {x} = 1.4\!-\!1.8$ at $\tilde{t} =2.5$ .

For the sake of interface definition, the current height $\tilde h_t$ is considered as the interface elevation. The average interface elevations over $\tilde x = 1.4{\sim}1.8$ at $\tilde t =2.5$ are presented in table 2. The downward sedimentation of particles inevitably causes the current interface to descend, and the degree of descent increases with the decrease of $\xi _{sp}$ . With regard to $\langle \tilde u_{\!f\!} \rangle$ profiles (figure 12 $a$ ), the decrease of $\xi _{sp}$ reduces $\langle \tilde u_{\!f\!} \rangle$ at the interface but increases $\partial \langle \tilde u_{\!f\!} \rangle / \partial \tilde z$ here. It increases $\langle \tilde u_{\!f\!} \rangle$ below the interface, which is consistent with the results of Shringarpure et al. (Reference Shringarpure, Cantero and Balachandar2015). Figure 12( $b$ ) illustrates that, in most regions (excluding the range $\tilde z = 0.4 \!\sim\! 0.7$ ), a decrease in $\xi _{sp}$ results in a leftward shift of the $\langle \tilde w_{\!f\!} \rangle$ profile. This shift indicates a reduction in upward vertical velocity (at $\tilde z \gtrapprox 0.7$ and the very near the wall as shown in the inset) and an increase in downward vertical velocity (at $\tilde z \lessapprox 0.4$ ). This phenomenon is closely associated with particle sedimentation. Conversely, within the range $\tilde z = 0.4 \!\sim\! 0.7$ , smaller $\xi _{sp}$ values cause the profile to shift to the right, a behaviour attributed to enhanced interface instability. Simultaneously, the interface fluid stratification is reduced with $\xi _{sp}$ decreasing, as evidenced by the decrease in $\partial \langle \tilde \rho _{\!f\!} \rangle / \partial \tilde z$ in figure 12( $c$ ), and the fluid–particle mixture stratification also diminishes (figure 12 $e$ ) under the impact of the particle spatial distribution. Additionally, as shown in figure 12( $f$ ), $Ri_g$ near the interface is positively correlated with $\xi _{sp}$ and clearly falls below the classical shear instability threshold of 1/4 (Nasr-Azadani et al. Reference Nasr-Azadani, Meiburg and Kneller2018; Caulfield Reference Caulfield2021). This explains why, during the slumping stage, currents with low $\xi _{sp}$ are more prone to become unstable and form interfacial billows. Correspondingly, the particles in the low- $\xi _{sp}$ regime are more easily rolled up by the stronger fluid forces provided by the vortices as shown in figure 12( $g$ – $i$ ).

Figure 13.

Time evolution of entrainment coefficients for ( $a$ ) saline flow $\mathcal{\tilde E}_{s}$ and ( $b$ ) particle flow $\mathcal{\tilde E}_{p}$ .

By setting thresholds on the volume fraction to capture the envelope ( $\langle \alpha _s \rangle = 10^{-3}$ and $\langle \alpha _p \rangle / \alpha _{p0} = 10^{-2}$ ) which is similar to He et al. (Reference He, Zhao, Hu, Yu and Lin2018), we determine the corresponding current volumes per unit width, $\tilde V_{F,s}$ and $\tilde V_{F,p}$ . Then, we can get the current volume change rate per unit width by $\tilde Q_{F,s(p)} = \mathrm{d} \tilde V_{F,s(p)} / \mathrm{d} \tilde t$ . The entrainment velocity ( $\tilde W_{s(p)}$ ) and entrainment coefficient ( $\mathcal{\tilde E}_{s(p)}$ ) can be given by (Ottolenghi et al. Reference Ottolenghi, Adduce, Inghilesi, Roman and Armenio2016)

(3.4) \begin{equation} \tilde W_{s(p)} = \frac {\tilde Q_{F,s(p)}}{\tilde L_{F,s(p)}}, \;\;\;\; \mathcal{\tilde E}_{s(p)} = \frac {\tilde W_{s(p)}}{\tilde u_{front,s(p)}}, \end{equation}

where $\tilde L_{F,s(p)}$ denotes the length of the upper interface. Given that the deposited particles form only a very thin layer and often coexist in thin layers with near-wall moving particles, their inclusion in the interface identification using volume fraction has a negligible effect on the captured volume and can therefore be ignored. Figure 13 depicts the time evolution of entrainment coefficients ( $\mathcal{\tilde E}_{s(p)}$ ). The entrainment coefficients for particle flow and saline flow are distinct: $\mathcal{\tilde E}_{s}\gt 0$ in figure 13( $a$ ) and $\mathcal{\tilde E}_{p}\lt 0$ in figure 13( $b$ ), respectively. This indicates that the saline flow is always entraining the fluid, while the particle flow is always detraining the fluid due to particle gravity. Here $\mathcal{\tilde E}_{s}\gt |\mathcal{\tilde E}_{p} |$ , meaning that in the dual-driven current, the entrainment from salinity convection–diffusion is always greater than the detrainment from particle flow sedimentation. Both $\mathcal{\tilde E}_{s}$ and $\mathcal{\tilde E}_{p}$ undergo the initial fast decay, governed by complex vertical and horizontal dynamics. After that (in the slumping stage), $\mathcal{\tilde E}_{s(p)}$ decreases slightly or stabilizes, aligning with He et al. (Reference He, Zhao, Hu, Yu and Lin2018). Finally, $\mathcal{\tilde E}_{s(p)}$ at large $\xi _{sp}$ exhibits a significant rise as a result of impact on the downstream wall. The entrainment/detrainment of saline/particle flow is independent of $\xi _{sp}$ in the early stage ( $\tilde t\lt 1$ ), with $\mathcal{\tilde E}_{s}$ decaying faster ( ${\sim}\tilde t ^{-1.25}$ ) than $\mathcal{\tilde E}_{p}$ ( ${\sim}\tilde t ^{-0.75}$ ). Then, the particle sedimentation effect impacts both entrainment and detrainment. The entrainment coefficient of saline flow is approximately 0.1 as close to the previous reports (Van den Bremer & Hunt Reference van den Bremer and Hunt2014; He et al. Reference He, Zhao, Hu, Yu and Lin2018; Parker et al. Reference Parker, Burridge, Partridge and Linden2020; Richardson & Hunt Reference Richardson and Hunt2022; Salizzoni et al. Reference Salizzoni, Vaux, Creyssels, Craske and van Reeuwijk2024), and is facilitated by the decrease in $\xi _{sp}$ , particularly during the deceleration phase. This is because the presence of particles promotes interface instability in figure 6. This explains also why, as $\xi _{sp}$ decreases, the particle flow detrainment weakens. In the slumping stage, when $\xi _{sp} = 0$ , $\mathcal{\tilde E}_{p}$ decays at a rate of $\tilde t^{-0.05}$ , whereas $\mathcal{\tilde E}_{p}$ is essentially constant when $\xi _{sp} = 6.0$ . The high- $\xi _{sp}$ flow exhibits a weaker detrainment during the deceleration period (figure 13 $b$ ), primarily because the interfacial vortices are continuously excited (figure 6).

Figure 14.

Time evolution of entrainment coefficients for ( $a$ , $c$ , $e$ ) saline flow $\mathcal{\tilde E}_{s}$ and ( $b$ , $d$ , $f$ ) particle flow $\mathcal{\tilde E}_{p}$ with different interface definition thresholds: ( $a$ , $b$ ) $\xi _{sp} = 6.0$ , ( $c$ , $d$ ) $\xi _{sp} = 1.0$ and ( $e$ , $f$ ) $\xi _{sp} = 0.17$ .

Figure 14 illustrates the time evolution of the entrainment and detrainment coefficients ( $\mathcal{\tilde E}_{s}$ and $\mathcal{\tilde E}_{p}$ ) under different current interface definition thresholds ( $\langle \alpha _s \rangle$ for saline flow and $\langle \alpha _{p} \rangle / \alpha _{p0}$ for particle flow). Within the range of $\langle \alpha _s \rangle$ and $\langle \alpha _{p} \rangle / \alpha _{p0}$ values shown in the figure, changes in the threshold values do not qualitatively alter the temporal trends of $\mathcal{\tilde E}_{s}$ and $\mathcal{\tilde E}_{p}$ . Specifically, increasing the $\langle \alpha _s \rangle$ threshold slightly reduces the entrainment coefficient, while increasing the $\langle \alpha _{p} \rangle / \alpha _{p0}$ threshold slightly enhances the detrainment coefficient. This behaviour arises because larger threshold values encompass smaller captured current volumes. Considering that interface vortices can contain substantial amounts of matter and influence current dynamics below, we here treat vortices near the interface as part of the gravity current. Accordingly, adopting a relatively small threshold value captures most of the saline and particle flow contours, thereby providing a more comprehensive depiction of material entrainment and detrainment.