3.1. Meng et al.'s depth-averaged equations

Let $oxyz$ be a terrain-following curvilinear coordinate system with the $x$ axis pointing down the chute, the $y$-axis pointing across the slope and the $z$-axis being the upward pointing normal, as illustrated in figure 2. The origin $o$ is placed at the top of the chute upstream of the release gate and the local angle of inclination of the chute $\zeta =\zeta (x)$. In these coordinates the chute base lies along $z=0$. The chute geometry and the initial conditions are independent of $y$. Moreover, figure 1(a), and movies 1 and 2, show that the flow front remains planar across the chute, without forming significant boundary layers adjacent to the sidewalls. The sidewall friction and lateral gradients of physical quantities are therefore neglected, and the whole flow is modelled independently of $y$.

In the curvilinear coordinate system the granular and phreatic (water) free surfaces lie at $z=h^g$ and $h^w$, respectively. When $h^g>h^w$ the flow is undersaturated, while if $h^g< h^w$ the flow is oversaturated. These flow configurations are shown schematically in figure 2 of Meng et al. (Reference Meng, Johnson and Gray2022). In the undersaturated regime there is a layer of dry grains above a saturated mixture of grains and water, and the associated volume fractions of the grains and the water are

(3.1a,b) \begin{equation} \phi^g=\phi^c, \quad 0\le z< h^g,\qquad \phi^w=\left\{\begin{array}{@{}ll} 0, & h^w\le z< h^g,\\ 1-\phi^c, & 0\le z < h^w,\end{array}\right. \end{equation}

respectively, where $\phi ^c$ is constant (Iverson & Denlinger Reference Iverson and Denlinger2001). In the oversaturated regime, there is a layer of water on top of the saturated mixture of grains and water. The volume fractions of the grains and the water are therefore

(3.2a,b) \begin{equation} \phi^g=\left\{\begin{array}{@{}ll} 0, & h^g\le z< h^w,\\ \phi^c, & 0\le z < h^g, \end{array}\right.\quad \phi^w=\left\{\begin{array}{@{}ll} 1, & h^g\le z< h^w,\\ 1-\phi^c, & 0\le z < h^g. \end{array}\right. \end{equation}

Note that the assumption that the volume fraction of the grains $\phi ^g$ equals $\phi ^c$ wherever grains are present, implies that Meng et al.'s (Reference Meng, Johnson and Gray2022) theory cannot resolve excess pore pressure effects that occur due to changes in volume fraction. Nevertheless, this paper shows that Meng et al.'s (Reference Meng, Johnson and Gray2022) theory is sufficient to quantitatively capture the experiments of Taylor-Noonan et al. (Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022).

The grains and the water are assumed to be incompressible with constant intrinsic densities $\varrho ^{g\star }$ and $\varrho ^{w\star }$, respectively. They also have independent velocity fields $\boldsymbol {u}^g=(u^g, w^g)$ and $\boldsymbol {u}^w=(u^w, w^w)$, where the components are measured in the downslope and normal directions. The depth-averaged velocities of the grains and the water, $\bar {u}^g$ and $\bar {u}^w$, in the downslope direction are defined as

(3.3a,b)\begin{equation} \bar{u}^g=\frac{\displaystyle\int_{0}^{h^g}\phi^g u^g\,{\rm d}z}{\displaystyle\int_{0}^{h^g}\phi^g\,{\rm d}z}=\frac{\overline{\phi^g u^g}}{\bar\phi^g}, \quad \bar{u}^w=\frac{\displaystyle\int_{0}^{h^w}\phi^w u^w\,{\rm d}z}{\displaystyle\int_{0}^{h^w}\phi^w\,{\rm d}z}=\frac{\overline{\phi^w u^w}}{\bar\phi^w}, \end{equation}

respectively. Meng et al. (Reference Meng, Johnson and Gray2022) defined the proportion of the flow height that is occupied by grains in the undersaturated and oversaturated regimes as

(3.4)\begin{equation} \mathcal{H}=\left\{\begin{array}{@{}ll} 1, & h^g> h^w,\quad (\text{undersaturated}),\\ h^g/h^w, & h^g\le h^w, \quad (\text{oversaturated}), \end{array} \right. \end{equation}

which is useful in defining a unified system of equations (their equations (4.68)–(4.77)) that are valid in both the undersaturated and oversaturated regimes. In the undersaturated regime Meng et al.'s (Reference Meng, Johnson and Gray2022) equations can be written in conservative form, which is convenient for the development of numerical methods (Kurganov & Tadmor Reference Kurganov and Tadmor2000). However, in the oversaturated regime the equations cannot be written in conservative form, due to the buoyancy terms in their equations (4.72) and (4.73), which makes the construction of numerical methods more complex (Maso, LeFloch & Murat Reference Maso, LeFloch and Murat1995; LeVeque Reference LeVeque2002; Parés Reference Parés2006; Diaz, Kurganov & de Luna Reference Diaz, Kurganov and de Luna2019).

Since Taylor-Noonan et al.'s (Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022) experiments are predominantly in the undersaturated regime, this paper makes the minimal possible modification to the equations (in the oversaturated regime) in order to put them in conservative form. Specifically the modified depth-averaged grain and water mass and momentum balance equations are assumed to be

(3.5)$$\begin{gather} \frac{\partial}{\partial t}(h^g\phi^c) + \frac{\partial}{\partial x}(h^g\phi^c\bar{u}^g) = 0, \end{gather}$$

(3.6)$$\begin{gather}\frac{\partial}{\partial t}(h^g\phi^c\bar{u}^g) + \frac{\partial}{\partial x}(\chi^g h^g\phi^c(\bar{u}^g)^2)+\frac{\partial}{\partial x}\left(\frac{1}{2}(h^g)^2\phi^cg\cos\zeta\right)=\mathcal{L}^g, \end{gather}$$

(3.7)$$\begin{gather}\frac{\partial}{\partial t}(h^w\bar\phi^w) + \frac{\partial}{\partial x}(h^w\bar\phi^w \bar{u}^w) = 0, \end{gather}$$

(3.8)$$\begin{gather}\frac{\partial}{\partial t}(h^w\bar\phi^w\bar{u}^w) + \frac{\partial}{\partial x}(\chi^w h^w\bar\phi^w (\bar{u}^w)^2)+\frac{\partial}{\partial x}\left(\frac{1}{2}(h^w)^2\bar\phi^w g\cos\zeta\right)=\mathcal{L}^w, \end{gather}$$

respectively, where $g$ is the constant of gravitational acceleration, $\mathcal {L}^g$ and $\mathcal {L}^w$ are source terms and $\bar \phi ^w$ is the depth-averaged water concentration. Using the assumed vertical water concentration distribution in the undersaturated and oversaturated regimes (3.1b)–(3.2b) as well as the definition (3.4) it follows that

(3.9)\begin{equation} \bar\phi^w=\frac{1}{h^w}\int_0^{h^w}\phi^w\,{\rm d}z=(1-\phi^c{\mathcal{H}}). \end{equation}

Equations (3.5)–(3.8) are identical to those of Meng et al. (Reference Meng, Johnson and Gray2022) in the undersaturated regime, while in the oversaturated regime the equations are asymptotically equivalent in the limit as the water thickness tends to the granular thickness from above, i.e. $h^w\rightarrow h^{g+}$. Moreover, the form has been chosen so that when $h^g=0$, (3.7)–(3.8) reduce to the shallow water equations. In the convective momentum terms in (3.6) and (3.8), $\chi ^\nu =\overline {(u^\nu )^2}/(\bar {u}^\nu )^2$ is the shape factor and its value deviates from unity for sheared velocity profiles. For non-unity values of the shape factor, the characteristic structure of the depth-averaged system changes, and leads to the formation of a thin precursor layer ahead of the granular front, which is unphysical (Saingier, Deboeuf & Lagrée Reference Saingier, Deboeuf and Lagrée2016). Saingier et al. (Reference Saingier, Deboeuf and Lagrée2016) also showed that at low Froude numbers, non-unity values of the shape factor make very little difference to the overall shape of the granular flow front for $h>0$. This paper therefore assumes that $\chi ^\nu =1$ for both species $\nu =g,w$ for simplicity, in line with virtually all other debris-flow models (Pitman & Le Reference Pitman and Le2005; Pelanti et al. Reference Pelanti, Bouchut and Mangeney2008; Pudasaini Reference Pudasaini2012; Bouchut et al. Reference Bouchut, Fernadez-Nieto, Mangeney and Narbona-Reina2016; Meng & Wang Reference Meng and Wang2018).

Since the buoyancy terms in equations (4.72) and (4.73) of Meng et al. (Reference Meng, Johnson and Gray2022) have been subsumed into the modified momentum balances (3.6) and (3.8), it follows that the leading-order source terms for the grains and the water are

(3.10)$$\begin{gather} \mathcal{L}^g =\underbrace{h^g\phi^cg\sin\zeta}_{\textit{Gravity}}- \underbrace{\frac{\bar{u}^g}{\vert\bar{u}^g\vert}\left(1-\gamma\mathcal{H}\frac{ h^w}{h^g}\right) \mu^bh^g\phi^c(g\cos\zeta+\kappa(\bar{u}^g)^2)}_{\textit{Basal friction}} \nonumber\\ +\underbrace{\frac{C^d}{\varrho^{g\star}} (\psi^w(s)-\psi^g(s))}_{\textit{Darcy drag}}, \end{gather}$$

(3.11)$$\begin{gather}\mathcal{L}^w =\underbrace{h^w\bar\phi^w g\sin\zeta}_{\textit{Gravity}} -\underbrace{C^w\bar{u}^w\vert\bar{u}^w\vert}_{\textit{Basal friction}} -\underbrace{\frac{C^d}{\varrho^{w\star}} (\psi^w(s)-\psi^g(s))}_{\textit{Darcy drag}}, \end{gather}$$

where $\gamma =\varrho ^{w\star }/\varrho ^{g\star }$ is the density ratio, $\mu ^b$ is a Coulomb-like basal friction coefficient for the grains and $\kappa =-\partial \zeta /\partial x$ is the curvature of the terrain-fitted coordinates, which modifies the basal pressure in the friction law (Savage & Hutter Reference Savage and Hutter1991; Gray et al. Reference Gray, Wieland and Hutter1999; Viroulet et al. Reference Viroulet, Baker, Edwards, Johnson, Gjaltema, Clavel and Gray2017). The water experiences a turbulent basal drag with the bed shear stress coefficient $C^w$, and the relative motion of the grains and the water within the mixture generates a Darcy drag, with Darcy drag coefficient $C^d$. The functional form of these coefficients, as well as the basal granular friction is discussed at greater length in § 3.2. The grain and water stream functions $\psi ^g$ and $\psi ^w$ in the Darcy drag terms are defined by

(3.12a,b)\begin{equation} \psi^g(s)=\int_0^{s}u^g \,{\rm d}z, \quad \psi^w(s)=\int_0^{s}u^w \,{\rm d}z, \end{equation}

where the internal height of the saturated grain–water mixture

(3.13)\begin{equation} s = \begin{cases} h^w , & {h^g > h^w}, \quad (\text{undersaturated}),\\ h^g, & {h^g \le h^w}, \quad (\text{oversaturated}). \end{cases} \end{equation}

The stream functions allow the grain and water downslope velocity profiles with $z$ to influence the relative motion of the two species. Specific profiles will be considered in § 3.3. It is these terms that are responsible for the deviation of Meng et al.'s (Reference Meng, Johnson and Gray2022) theory away from conventional debris-flow models that assume plug flow (Pitman & Le Reference Pitman and Le2005; Pelanti et al. Reference Pelanti, Bouchut and Mangeney2008; Iverson & George Reference Iverson and George2014; Bouchut et al. Reference Bouchut, Fernadez-Nieto, Mangeney and Narbona-Reina2016; Meng & Wang Reference Meng and Wang2016). Note that Meng et al.'s (Reference Meng, Johnson and Gray2022) depth-averaged viscous term for the water is neglected in (3.11), since it is not needed to model the experiments of Taylor-Noonan et al. (Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022). In addition, the use of the terrain-fitted coordinates implies that the basal topography gradient terms in equations (4.72) and (4.73) of Meng et al. (Reference Meng, Johnson and Gray2022) are zero, since $b=0$.

3.2. Granular friction, basal water drag and Darcy drag laws

Although the base of Taylor-Noonan et al.'s (Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022) chute is not roughened with particles, it is still rough enough that internal shear dominates over basal slip, as shown by the experimental peak flow velocity data in figure 5. It is therefore appropriate to apply the rough-bed basal friction law of Pouliquen & Forterre (Reference Pouliquen and Forterre2002), i.e.

(3.14)\begin{equation} \mu^b(Fr, h^g) = \mu_1 + \frac{\mu_2 - \mu_1}{1 + \displaystyle\dfrac{\beta h^g}{\mathscr{L}Fr}}, \end{equation}

where the Froude number $Fr$ is defined as

(3.15)\begin{equation} Fr=\frac{\bar{u}^g}{\sqrt{g h^g\cos\zeta}}. \end{equation}

The rate-dependent friction law therefore starts at $\mu _1=\tan \zeta _1$ at $Fr=0$ and asymptotes towards $\mu _2=\tan \zeta _2$ as $Fr/h^g$ tends towards infinity. The parameter $\beta$ is an empirical constant and $\mathscr {L}$ has the dimension of a length and is dependent on the properties of the particles and on the basal roughness. On a rough bed, the basal friction law (3.14) can be derived directly from the $\mu (I)$-rheology for granular flows (GDR-MiDi 2004; Jop, Forterre & Pouliquen Reference Jop, Forterre and Pouliquen2005; Gray & Edwards Reference Gray and Edwards2014). For flows on smoother slopes, Goujon, Thomas & Dalloz-Dubrujeaud (Reference Goujon, Thomas and Dalloz-Dubrujeaud2003) and Weinhart et al. (Reference Weinhart, Thornton, Luding and Bokhove2012) have shown that (3.14) still provides a good approximation for the friction, although the difference between $\zeta _2$ and $\zeta _1$ has to be reduced.

Figure 5. Non-dimensional downslope velocity profiles for the grains as a function of the non-dimensional depth, for the cubic ($m=0.5$), Bagnold ($m=2$) and linear shear with basal slip models ($\alpha ^g=0.6$). The blue shaded region represents the $\pm 1$ standard deviation about the downslope velocity in Taylor-Noonan et al.'s (Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022) 0.8 m$^3$ experiment. The measurements are made in a 0.02 s observation time window during peak flow (their figure 6c) at 0.5 m above the slope break at approximately $t=2.05$ s.

The bed shear stress for the water phase takes account of the turbulent friction arising from the bottom of the channel. In the Meng et al. (Reference Meng, Johnson and Gray2022) paper, $C^w$ was based on the Manning equation for open channel flows (Manning Reference Manning1891), and has the form

(3.16)\begin{equation} C^w=\bar\phi^w\frac{gn^2}{(h^w)^{{1}/{3}}}, \end{equation}

where $n$ is the Manning coefficient (Chertock et al. Reference Chertock, Cui, Kurganov and Wu2015). The factor $\bar \phi ^w$ ensures that it reduces to the classical Chézy coefficient for water in the absence of grains. However, there are other empirical relations in the literature. The coefficient $C^w$ can also be related to the Darcy–Weisbach friction factor $f^{DW}$ as

(3.17)\begin{equation} C^w=\bar{\phi}^w\frac{f^{DW}}{8}, \end{equation}

where the factor $\bar \phi ^w$ again ensures that in the absence of grains (3.17) reduces to the classical form of the pure water bed shear stress. A general and well-verified friction factor $f^{DW}$ is the White–Colebrook function corresponding to open channel flows (Silberman et al. Reference Silberman, Carter, Einstein, Hinds and Powell1963; Kleinhans Reference Kleinhans2005),

(3.18)\begin{equation} \sqrt{\frac{8}{f^{DW}}} = 5.74\log_{10}\left(\frac{h^w}{k_s}\right) + 6.24, \end{equation}

where $k_s$ is Nikuradse roughness length commonly related to some grain size percentile of the mass frequency distribution. Since Taylor-Noonan et al. (Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022) experiments are monodisperse, the hydraulic roughness length is chosen to be equal to the grain diameter, i.e. $k_s=d$ (Kleinhans Reference Kleinhans2005). For the simulations shown in this paper, $C^W$ is assumed to obey the Manning equation (3.16). However, all of the computations have also been done with the Darcy–Weisbach formulation (3.17)–(3.18), and the results are virtually identical. The only difference appears to be in the tail of the flow, where the roll waves have a different amplitude and wavelength. The fact that the results are not sensitive to the precise formulation of $C^W$ suggests that the buoyancy reduced granular friction (3.10) dominates the response of the system.

The form of the Darcy drag coefficient is

(3.19)\begin{equation} C^d = \mu^w\frac{(1-\phi^c)^2}{k}, \end{equation}

where $\mu ^w$ is the dynamic viscosity of water and the permeability $k = {(1-\phi ^c)^3d^2}/(180(\phi ^c)^2)$ is determined by Carman's formula for packing of spheres with diameter $d$ (Kozeny Reference Kozeny1927; Carman Reference Carman1937; Goharzadeh, Khalili & Jørgensen Reference Goharzadeh, Khalili and Jørgensen2005).

3.3. Assumed velocity profiles and associated stream functions

Vertical structure and velocity shear through the flow depth produce an important mechanism for differential species transport and the formation of large-rich and/or dry debris-flow fronts (Gray & Kokelaar Reference Gray and Kokelaar2010; Johnson et al. Reference Johnson, Kokelaar, Iverson, Logan, LaHusen and Gray2012; Baker et al. Reference Baker, Johnson and Gray2016; Gray Reference Gray2018; Viroulet et al. Reference Viroulet, Baker, Rocha, Johnson, Kokelaar and Gray2018). In Meng et al.'s (Reference Meng, Johnson and Gray2022) theory, shear-induced differential species transport is achieved by assuming downslope velocity profiles through the flow depth that deviate from plug flow.

In this paper three different granular velocity profiles are investigated that provide reasonable fits to Taylor-Noonan et al.'s (Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022) peak grain velocity data shown in figure 5. A general power-law downslope velocity, that satisfies the no-slip condition at the base and vanishing shear on the free surface, is given by

(3.20)\begin{equation} u^\nu(z) = \frac{1+2m}{1+m}\bar{u}^\nu\left[1-\left(1-\frac{z}{h^\nu}\right)^{({1+m})/{m}}\right], \end{equation}

where $\nu =g,w$. This tends to a linear profile as the parameter $m\rightarrow \infty$ and tends to a plug profile as $m\rightarrow 0$ (Ng & Mei Reference Ng and Mei1994). The case $m=2$ corresponds to a Bagnold profile, which arises naturally from the $\mu (I)$ rheology for dry granular flows (GDR-MiDi 2004; Jop et al. Reference Jop, Forterre and Pouliquen2006; Gray & Edwards Reference Gray and Edwards2014). This is consistent with the assumed basal granular friction law (3.14), and provides a reasonable fit to the experimental velocity data for the grains shown in figure 5. A better fit is obtained for the case $m=0.5$, which corresponds to a cubic profile. Rather than this indicating a deviation away from the $\mu (I)$ rheology, the apparently better fit is likely due to the grains slipping somewhat on the smooth aluminium chute base. One can also simply assume a linear velocity profile with basal slip to characterize either the grains or water velocity field

(3.21)\begin{equation} u^\nu(z) = \bar{u}^\nu\left(\alpha^\nu + 2(1-\alpha^\nu)\frac{z}{h^\nu}\right), \end{equation}

where for $\nu =g,w$ and the variable $\alpha ^\nu \in [0,1]$ controls the magnitude of the basal slip velocity (Gray & Thornton Reference Gray and Thornton2005; Gray & Ancey Reference Gray and Ancey2009). When $\alpha ^\nu =0$, (3.21) reduces to a linear profile with no slip at the base, and it reduces to plug flow, when $\alpha ^\nu =1$. For simplicity, the water is assumed to have a plug-flow profile, i.e. $u^w(z) = \bar {u}^w$.

The assumed grain and water velocity profiles enter Meng et al.'s (Reference Meng, Johnson and Gray2022) theory indirectly through the stream functions defined in (3.12). These are evaluated on the internal grain–water mixture surface $s$ defined in (3.13). It follows that for the power law velocity profile (3.20), the stream function is

(3.22)\begin{equation} \psi^\nu(s)=\frac{m}{1+m}h^\nu\bar{u}^\nu\left[\frac{1+2m}{m}\left(\frac{s}{h^\nu}\right)-1+ \left(1-\frac{s}{h^\nu}\right)^{({1+2m})/{m}}\right], \end{equation}

and for the linear profile (3.21) it is

(3.23)\begin{equation} \psi^\nu(s)=h^\nu\bar{u}^\nu\left[\alpha^\nu\left(\frac{s}{h^\nu}\right)+(1-\alpha^\nu)\left(\frac{s}{h^\nu}\right)^2\right]. \end{equation}

For the assumed plug-flow water velocity field it follows that

(3.24)\begin{equation} \psi^w(s)=s\bar u^w. \end{equation}

The stream functions are used in the Darcy-drag terms in the momentum sources (3.10) and (3.11). These control the relative depth-averaged motion of the grains and the water in the water-saturated granular region at the base of the flow.

4.1. Numerical method

The system of conservation laws (3.5)–(3.8) is solved numerically using the shock-capturing non-oscillatory central scheme of Kurganov & Tadmor (Reference Kurganov and Tadmor2000). This robust scheme has been used to successfully solve a number of closely related systems of conservation laws for dry granular flows (Edwards & Gray Reference Edwards and Gray2015; Baker et al. Reference Baker, Johnson and Gray2016; Rocha, Johnson & Gray Reference Rocha, Johnson and Gray2019), debris flows (Meng et al. Reference Meng, Wang, Feng, Wang and Zhou2018, Reference Meng, Wang, Chiou and Zhou2020) and submarine landslides (Sun et al. Reference Sun, Meng, Wang, Hsiau and You2023). The method is a semidiscrete Riemann-free solver that maintains the non-oscillatory property by using a flux limiter. Here, the weighted essentially non-oscillatory limiter detailed in Noelle (Reference Noelle2000) is chosen, as in Baker et al. (Reference Baker, Johnson and Gray2016). The non-oscillatory central scheme requires the system of (3.5)–(3.8) to be written in a vector form

(4.1)\begin{equation} \frac{\partial\boldsymbol{U}}{\partial t} + \frac{\partial\boldsymbol{F}(\boldsymbol{U})}{\partial x} = \boldsymbol{S}(\boldsymbol{U}), \end{equation}

where the vector of conservative fields is defined as

(4.2)\begin{equation} \boldsymbol{U}=(H^g,M^g,H^w,M^w)^{\rm T}=(h^g\phi^c, h^g\phi^c\bar{u}^g, h^w\bar\phi^w, h^w\bar\phi^w\bar{u}^w)^\text{T}. \end{equation}

Three field variables are easy to express in terms of the conservative variables

(4.3a–c)\begin{equation} h^g=\frac{H^g}{\phi^c},\quad \bar{u}^g=\frac{M^g}{H^g},\quad \bar{u}^w=\frac{M^w}{H^w}, \end{equation}

but the relation for $h^w$ is dependent on whether the flow is undersaturated or oversaturated. Using the definition of $\bar \phi ^w$ in (3.9), it follows that

(4.4)\begin{equation} h^w = \begin{cases} \displaystyle\frac{H^w}{1-\phi^c}, & (\text{undersaturated}),\\ H^w+H^g, & (\text{oversaturated}), \end{cases} \end{equation}

and hence the depth-averaged water concentration is

(4.5)\begin{equation} \bar\phi^w = \begin{cases} {1-\phi^c}, & (1-\phi^c)H^g > \phi^c H^w,\quad (\text{undersaturated}),\\ \displaystyle\frac{H^w}{H^w+H^g}, & (1-\phi^c)H^g \le \phi^c H^w,\quad (\text{oversaturated}). \end{cases} \end{equation}

In the undersaturated regime the flux vector $\boldsymbol {F}$ is

(4.6)\begin{equation} \boldsymbol{F}(\boldsymbol{U})= \left( \begin{array}{@{}c@{}} M^g \\ \displaystyle\dfrac{(M^g)^2}{H^g} +\dfrac{1}{2}\dfrac{(H^g)^2}{\phi^c}g\cos\zeta\\*[10pt] M^w \\ \displaystyle\dfrac{(M^w)^2}{H^w} +\dfrac{1}{2}\dfrac{(H^w)^2}{1-\phi^c} g\cos\zeta \end{array} \right),\quad (1-\phi^c)H^g > \phi^c H^w, \end{equation}

while in the oversaturated regime it is

(4.7)\begin{equation} \boldsymbol{F}(\boldsymbol{U})= \left( \begin{array}{@{}c@{}} M^g \\ \displaystyle\dfrac{(M^g)^2}{H^g} +\dfrac{1}{2}\dfrac{(H^g)^2}{\phi^c}g\cos\zeta\\ M^w \\ \displaystyle\dfrac{(M^w)^2}{H^w} +\dfrac{1}{2}(H^w+H^g)H^w g\cos\zeta \end{array} \right),\quad (1-\phi^c)H^g \le \phi^c H^w. \end{equation}

The characteristic wave speeds $\lambda _i$ ($i=1,\ldots,4$) of the system in both the undersaturated and oversaturated regimes are evaluated in Appendix A. In the undersaturated regime the structure of the system uncouples to generate two shallow-water-type systems (for the grains and for the water) that are weakly coupled through the momentum source terms. In the oversaturated regime there is stronger coupling through the flux vector. The system of equations is non-strictly hyperbolic in both undersaturated and oversaturated regimes, with explicit characteristic wave speeds given by (A4) and (A7).

The source vector $\boldsymbol {S}$ is

(4.8)\begin{equation} \boldsymbol{S}(\boldsymbol{U})= \left( \begin{array}{@{}c@{}} 0 \\ \mathcal{L}^g \\ 0 \\ \mathcal{L}^w \end{array} \right), \end{equation}

where $\mathcal {L}^g$ and $\mathcal {L}^w$ are defined in (3.10) and (3.11), respectively. The appearance of the water basal friction term in the form $-(1-\phi ^c\mathcal {H})gn^2u^w\vert {u\vert }^w/(h^w)^{{1}/{3}}$ is a potential challenge to the discretization of the source vector $\boldsymbol {S}(\boldsymbol {U})$, as the underlying semidiscrete system becomes stiff when the water thickness $h^w$ is small. In this case, small numerical oscillations might cause negative values of water thickness, which in turn would make it impossible to evaluate the characteristic wave speeds $\lambda _{3,4}$ in (A4b) and (A7b). To prevent the appearance of the negative thickness, this paper adopts the correction strategy of Chertock et al. (Reference Chertock, Cui, Kurganov and Wu2015) to correct the slopes of the conservative variables for small thicknesses during the reconstruction. Additionally, the water velocity at the centres of the cells, required to compute the water basal friction term, is computed at cell $j$ using the desingularization formula

(4.9)\begin{equation} \bar{u}_j^w = \frac{2H_j^w {M^w_j}}{(H_j^w)^2+\max((H_j^w)^2,\alpha^2)}, \end{equation}

where $\alpha =5\times 10^{-5}$ m. Equation (4.9) ensures that its influence on the water velocity is negligibly small for the thicknesses $h^w\ge 100\alpha /(1-\phi ^c)$, which are sufficiently small compared with the peak flow depth. Since Kurganov & Tadmor's (Reference Kurganov and Tadmor2000) semidiscrete approach is an explicit scheme, a small time step $\Delta t$ is needed to maintain stable computation. To avoid reducing the time step significantly and simultaneously preserve strong stability of the computation, a splitting technique is used to discretize the water and grain basal friction terms. This has proved to be effective in ensuring a stable computation without the expense of decreasing the time step significantly (see Liang & Marche Reference Liang and Marche2009). The time step $\Delta t$ is given by

(4.10)\begin{equation} \Delta t \le \frac{\Delta x}{\vert a\vert} CFL, \quad\text{where } a=\max\left\{\vert\lambda_{i}\vert\right\},\quad i=1,\ldots,4, \end{equation}

and the Courant–Friedrichs–Lewy number $CFL=0.1$ is chosen.

4.2. Computational geometry and boundary conditions

The computational domain is 20 m in length and the gate is located at $x_f={\rm \pi}$ m. In the experiments, the inclined plane is sharply connected to a horizontal plane. This causes the curvilinear coordinates $oxz$ to overlap for $z>0$ at the slope break. To avoid overlap in the flow thicknesses in the computations, a short smooth transition is therefore added between the inclined and horizontal planes. The inclination angle of the slope is

(4.11)\begin{equation} \zeta(x) = \begin{cases} \zeta_0, & {0\le x\le x_a},\\ \zeta_0\left(1-\displaystyle\frac{x-x_a}{x_b-x_a}\right), & {x_a < x < x_b},\\ 0^\circ, & {x \ge x_b}, \end{cases} \end{equation}

where $\zeta _0=30^\circ$, $x_a=x_f+6.73$ m, $x_b=x_f+7.13$ m. These values are summarized in table 1. The inclined section therefore extends 6.73 m downstream of the gate and the smooth transition is $0.4$ m in length. The smooth transition is sufficiently small compared with the overall chute length that its influence on the debris-flows dynamics is small. The centrifugal forces induced by the change in curvature $\kappa =-\partial \zeta /\partial x$ does, however, change the basal pressures in the transition region. This is accounted for by the curvature term in the granular source terms (3.10). Free outflow conditions are specified at either end of the domain, but in most simulations no material reaches either boundary. The computation domain is discretized into 4667 grid points, implying $\Delta x=0.0042857$ m.

Table 1. Computational chute geometry and physical parameters used to simulate Taylor-Noonan et al.'s (Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022) debris-flow experiments.

4.3. Initial conditions for the dry simulations

Dry grains are initially packed in a triangular region behind a vertical gate at $x=x_f$ and have a horizontal free surface. In the terrain-following curvilinear coordinates the initial height is therefore given by

(4.12)\begin{equation} h^g_{{ini}} = \begin{cases} \displaystyle (x_f-x)\cot\zeta_0, & { x_m \le x\le x_f},\\ \displaystyle (x-x_t)\tan\zeta_0, & {x_t < x < x_m},\\ \displaystyle 0, & \text{otherwise}, \end{cases} \end{equation}

where $x_m$ is the downslope position of the maximum depth (measured normal to the plane) and $x_t$ is the maximum upstream position of the triangular pile. The initial grain volume $V_{{ini}}$ is uniformly distributed across the chute, which is of width $W=2.09$ m. The two-dimensional projected area can therefore be used to determine $x_t$ and $x_m$, which are given by the expressions

(4.13a,b)\begin{equation} x_t=x_f - \sqrt{\frac{2V_{ini}}{W\sin\zeta_0\cos\zeta_0}}, \quad x_m=x_f - (\tan\zeta_0)\sqrt{\frac{2V_{{ini}}\sin\zeta_0\cos\zeta_0}{W}}, \end{equation}

respectively. Computations for the dry granular flows assume that $h^w=0$ and $C^w=0$, which implies that the depth-averaged water mass and momentum balances (3.7)–(3.8) are trivially satisfied. The depth-averaged granular mass and momentum balances (3.5)–(3.6) then reduce to those of Viroulet et al. (Reference Viroulet, Baker, Edwards, Johnson, Gjaltema, Clavel and Gray2017).

4.4. Initial conditions for wet simulations

In the wet flow experiments the initial volume of grains held behind the gate is in exactly the same configuration as the dry grains in § 4.3, and occupies a total volume $V_{ini}$. The only difference is that the interstices between the grains are now occupied by water rather than air. Since the grains are initially saturated, the intrinsic volumes of grains and water (i.e. the volumes of the pure phases) are therefore

(4.14a,b)\begin{equation} V^{g\star}_{ini}=V_{ini}\phi^g_{ini}, \quad V^{w\star}_{ini}=V_{ini}(1-\phi^g_{ini}), \end{equation}

where $\phi ^g_{ini}$ is the granular volume fraction in the initial deposit.

Meng et al.'s (Reference Meng, Johnson and Gray2022) theory uses a constant solids volume fraction $\phi ^g=\bar \phi ^g=\phi ^c$ that does not change during flow. The only way to model the initial dilatation of the granular matrix, observed in the Taylor-Noonan et al.'s (Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022)'s experiments, is therefore to change the assumed initial configuration to account for the subsequent change in $\phi ^g$ from $\phi ^g_{ini}=0.585$ to $\phi ^c=0.48$. For the grains, this is done by expanding the initial height, to account for the solids volume fraction, without changing the initial $x$ position, i.e.

(4.15)\begin{equation} h^g_{new}=\frac{\phi^g_{ini}}{\phi^c}h^g_{ini}, \end{equation}

where $h^g_{ini}(x)$ is defined in the same way as the initial dry height (4.13). In particular, the maximum height, which is still attained at $x=x_m$, is now

(4.16)\begin{equation} h^g_{max}=\frac{\phi^g_{ini}}{\phi^c}(x_f-x_m)\cot\zeta_0. \end{equation}

This expansion of the granular body in the normal direction to the chute, changes the assumed angle between the gate and the bed to

(4.17)\begin{equation} \theta = \arctan\left( \frac{h^g_{max}}{x_f-x_m} \right)=\arctan\left( \frac{\phi^g_{ini}}{\phi^c}\cot\zeta_0 \right), \end{equation}

as shown in the inset diagram in figure 2. This modifies the gate angle from $60^\circ$ in the dry case, to $\theta =63.67^\circ$ in the wet case, independently of the initial volume.

The total volume of the water $V^{w\star }_{ini}$ is also conserved, but it now occupies a smaller projected area in the $xz$ plane, because there is more pore space between the grains. For simplicity, it is assumed that the water remains in contact with the gate and occupies a triangular region defined by

(4.18)\begin{equation} h^w_{{ini}} = \begin{cases} \displaystyle h^w_{max}(x_f-x)/(x_f-x_w), & { x_w \le x\le x_f},\\ \displaystyle h^w_{max}(x-x_t)/(x_w-x_t), & {x_t < x < x_w},\\ \displaystyle 0, & \text{otherwise}, \end{cases} \end{equation}

where $x=x_w$ is the downslope position of the maximum water height $h^w_{max}$. Accounting for the dilatation of the grain matrix, conservation of $V_{ini}^{w\star }$ implies

(4.19)\begin{equation} h^w_{max}=\left(\frac{1-\phi^g_{ini}}{1-\phi^c}\right)(x_f-x_m)\cot\zeta_0. \end{equation}

This implies that in curvilinear coordinates the maximum initial height of the water lies at

(4.20)\begin{equation} x_w= x_f - h^w_{max}\cot\theta. \end{equation}

As can be seen in the inset of figure 2, the net effect of these modifications is to produce an initially saturated flow front that transitions to an undersaturated region in the tail, where the peak height occurs (in terrain-following coordinates).

4.5. Physical parameters used in the simulations

Taylor-Noonan et al. (Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022) provided the values of the slope angle $\zeta _0$, the grain and water densities $\varrho ^{g\star }$ and $\varrho ^{w\star }$, the grain diameter $d$ and the water viscosity $\mu ^w$. The base of the inclined plane is made of bare aluminium and therefore this paper assumes Manning's coefficient $n=0.016\ \text {s}\ \text {m}^{-1/3}$, which is within the commonly used range (see Chow Reference Chow1959). In the current theory, the solids volume fraction $\phi ^c$ is assumed to be constant; however, it varies with time and space in the experimental flows, as discussed in § 4.4. Taylor-Noonan et al.'s (Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022) wet experiment data (in their figure 9) suggests that $\phi ^c=0.48$ is a good approximate mean value. The initial solids volume fraction of $\phi ^g_{ini}=0.585$ is that of a very loose random packing. These experimental parameters and computational choices are summarized in table 1.

In addition, it is necessary to provide values of the parameters in the dynamic basal friction law (3.14) for the grains. The grains are of mean diameter $d=3.85$ mm. With such large grains direct measurements of the friction angles are difficult to obtain, because it is not easy to carry out the tilting table experiments to determine $h_{stop}(\zeta )$ as in Pouliquen & Forterre (Reference Pouliquen and Forterre2002). This paper therefore adopts the same values of $\mathscr {L}=1.65\times d$ and $\beta =0.135$ used by Meng et al. (Reference Meng, Johnson and Gray2022) to match Davies's (Reference Davies1988, Reference Davies1990) moving-bed-flume experiments with polyvinyl chloride (PVC) cut rod on a toothed belt.

To provide insights into the choice of angles $\zeta _1$ and $\zeta _2$, the parameters corresponding to the glass beads presented in Pouliquen (Reference Pouliquen1999) and the PVC rod presented in Meng et al. (Reference Meng, Johnson and Gray2022) are first used to conduct a series of dry flow calibration simulations. These are compared with the deposit in Taylor-Noonan et al.'s (Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022) experiment with an initial 0.8 m$^3$ volume of dry grains. Figure 6 shows that simulations with the PVC rod leave a significant deposit on the chute. This implies that $h_{stop}(\zeta _0)>0$ when $\zeta _2>\zeta _0$. This is not the case in the experiments, and hence the maximum friction angle $\zeta _2$ should be smaller than the slope angle $\zeta _0$. The simulations for glass beads show that the avalanche front, avalanche tail and the centre of mass of the deposit all lie too far upstream compared with Taylor-Noonan et al.'s (Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022) measured deposit. This is an indication that on average the friction is too high for the ceramic particles on the smooth aluminium bed. A good fit to Taylor-Noonan et al.'s (Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022) deposit is obtained by decreasing the value of $\zeta _2$ and increasing the value of $\zeta _1$ as shown in figure 6. All the computational results in the rest of this paper use the same frictional parameters, which are given in the last row of table 2.

Figure 6. Comparison between experimentally measured deposit morphology (red dash line) and theoretical predictions using the parameters for PVC rod (blue dotted line with the symbol ‘$\star$’), for glass beads (blue dash–dot line with the symbol ‘${\square }$’) and the parameters used in the paper (red solid line). The parameters are summarized in table 2. This plot is made using the Cartesian coordinate system $OXZ$ shown in figure 2.

Table 2. Material properties for (a) glass beads (Pouliquen Reference Pouliquen1999), (b) PVC rod on a rough bed (Davies Reference Davies1990; Meng et al. Reference Meng, Johnson and Gray2022) and (c) the present choice of parameters for Taylor-Noonan et al.'s (Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022) experiments with approximately spherical rough ceramic beads on a smooth aluminium chute.

6.1. Temporal and spatial evolution of an 0.8 m$^3$ saturated release

The simulated temporal and spatial evolution of the release of 0.8 m$^3$ of initially water saturated grains is shown in figure 9. As detailed in § 4.4, the dilation of the granular matrix on failure (and hence the dilated grain matrix during flow) are modelled by adjusting the initial granular and water heights to account for the dilatation. This results in the front of the flow being initially saturated, and the tail of the flow being undersaturated at $t=0^+$ (see the inset in figure 2). The initial collapse looks similar to that of the dry grains ($t\simeq 1$ s). However, the presence of the water in the main body of the flow reduces the basal granular friction in the source term (3.10), allowing the whole body to spread out faster than the purely dry flow. The tail of the grains therefore starts to move from its initial position by $t\simeq 1.82$ s.

Figure 9. Simulated spatial and temporal evolution of the grain and phreatic free surfaces after the 0.8 m$^3$ of water-saturated grains are released from the flume gate. The simulations assume the cubic velocity profile (3.20) with $m=0.5$, and the water has a plug-flow profile. Since the flow height is very small compared with the chute length, the flow thickness is multiplied by a factor of four and projected normally to the chute along the straight grey lines to aid visualization. An animation is included in the online supplementary material (movie 6). The inset depicts the roll waves developed in the tails at $t=5$ s in the range $x\in [8.26, 9.42]$ m and indicated by the dashed box.

By $t=1$ s a dry front has already formed. Small dry fronts can form in plug-flow simulations due to mobility differences between the grains and the fluid, as will be shown in § 7.1. Here, a cubic velocity profile (3.20) is assumed, and the increased size of the dry front is due to the combination of vertical structure and velocity shear. This allows the undersaturated flowing layer of dry grains near the free surface to be rapidly transported to the front, even though it started at the back of the initial release volume.

On the inclined section, the dry granular front is able to grow in size because dry grains are being continually supplied by velocity shear, and because the internal water front experiences high resistance. This increased resistance is due to the water friction term in (3.11) remaining finite as $h^w\rightarrow 0$, whereas all the other terms in the water momentum balance (3.8) tend to zero, i.e. the water friction generates singular behaviour. As the flow stretches out and the water layer becomes progressively thinner it experiences more basal resistance. In order to overcome this, an infinite gradient has to form at the water front for it to move.

The whole mixture accelerates more rapidly downslope than the purely dry flow. Indeed by $t=1.5$ s, the dry front has already reached the horizontal run-out plane, where it decelerates and tries to stop. It does not fully come to rest, but is pushed along by the pressure exerted by the continued supply of high-speed, low-friction undersaturated material from upstream. The dry front impedes the flow, and a bulbous head forms that is analogous to that developed in bidisperse particle flows (Denissen et al. Reference Denissen, Weinhart, Voortwis, Luding, Gray and Thornton2019). At the transition between the thinner more mobile upstream region and the thicker more slowly moving head, a shockwave forms. The shock is smaller than the purely dry case and does not immediately start travelling upstream. Instead, for $1.69\le t\le 3.64$ s the shock travels downslope at a slightly slower speed than both the granular and water fronts, which move downstream approximately in unison. For times $t\in [3.64,4.16]$ s, the shock briefly propagates a short distance upstream, before dissipating. During the whole of this time $t\in [1.69, 4.016]$ s the thicker undersaturated head region grows in size. The fact that granular and water fronts move at the same speed (at least for a brief period of time), suggests that the flow front is similar to the first type of travelling wave solutions found by Meng et al. (Reference Meng, Johnson and Gray2022). Meanwhile, in the tail, the grains experience less basal friction than the water, and they are able to flow faster downslope to leave a thin oversaturated region on the chute that takes a very long time to drain down.

At $t\simeq 4.29$ s, the granular front essentially stops, and there is some residual slow water percolation through the grains. A diffuse wave then travels upslope and almost all the grains come to rest by $t=6.24$ s. In the tail, the flow goes unstable and a close-up view shows that a series of roll waves form (see inset plot in figure 9). Similar waves can also be seen in the wet experiments in movie 2. At approximately $t=8$ s the grains are completely deposited on the horizontal run-out plane and the pore water drains slowly out of the granular matrix.

6.2. Comparison with Taylor-Noonan et al.'s (Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022) experimental time series

Taylor-Noonan et al. (Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022) photographed the flow with a high-speed camera that was located 0.5 m upstream of the slope break at $x=x_f+6.23$ m. They used these images to determine the approximate position of the granular and phreatic (water) free surfaces as a function of time (see also, e.g. figures 3 and 4). Taylor-Noonan et al.'s (Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022) experimental results are now compared with the simulated time series for each of the release volumes in figure 10(a–e). All the simulated flow profiles show that there is an initially dry front, that is followed by an undersaturated flow with a peak height that occurs within the first second after the initial front arrival. The arrival time of the wet front is broadly in line with that observed in experiment, although some of the fronts are highly undersaturated, but not completely dry. The main body of the flow travels past the camera within the first two seconds, when the flow is travelling very quickly. The peak height increases with increasing initial release volume, with the granular peak height occurring slightly before the phreatic peak height in all cases. Two seconds after the initial front arrival, all the flows are much thinner and slower, and there is a gradual transition to the oversaturated regime, where roll waves form both in the experiments and the numerical simulations. The simulated time series for the granular and phreatic free surfaces are in good quantitative agreement with measurement. This is remarkable, given that the basal granular friction law was calibrated by matching the deposit position of a single dry experiment. The results are also in broad qualitative agreement with some debris-flow types observed in the field (McArdell et al. Reference McArdell, Bartelt and Kowalski2007; Nagl, Hübl & Kaitna Reference Nagl, Hübl and Kaitna2020).

Figure 10. Comparison of the predicted and measured time-series of the flow depths at $x=x_f+6.23$ m for both (a–e) wet and (f–j) dry granular flows with source volumes of (a,f) 0.2, (b,g) 0.4, (c,h) 0.6, (d,i) 0.8 and (e,j) 1.0 m$^3$. The dashed red and blue lines show the experimental granular and phreatic (water) free surfaces determined by Taylor-Noonan et al. (Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022). The solid red line shows the grain thickness and the grey hashing shows the region occupied by grains. The blue solid line shows the position of the phreatic free surface, and the region beneath it is shaded blue to help visualize the water-saturated region. Taylor-Noonan et al.'s (Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022) high-speed camera data is available from https://doi.org/10.5683/SP3/1ZCUFY.

It is also interesting to compare the wet results with the equivalent time series for dry flows. These are shown in figure 10(f–j). For the smaller initial volumes $V_{ini}=0.2$ and $0.4$ m$^3$, high-speed images in Taylor-Noonan et al.'s (Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022) supplementary material show that the dry flows become dilute. This makes determining the free surface more difficult, and since the flow is expanded, the discrepancy between the theoretical and measured free-surface heights is significant. As the source volume of the dry flows increases, however, a larger portion of the flow depth exceeds 10 particle diameters and the grains behave more like a dense flow. The comparisons for the larger initial dry granular volumes are therefore very good. For all the volumes, the computed dry flows are slower than the wet flows, which leads to a wider time series, with a lower peak height, than the wet flows. In addition, roll waves do not emerge in the tail of the dry flows, which is also captured by the theory. For $V_{ini}= 0.8$ and $1.0$ m$^3$, the upslope propagating shockwaves (which deposit the grains) come into the camera field of view, and the data after approximately three seconds is cut (Taylor-Noonan et al. Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022). For the wet flows, it is evident that the fluid viscosity suppresses most of the saltation (regardless of the initial release volume), so the theoretical and experimental comparisons are better than for the dry flows.

6.3. Comparison with Taylor-Noonan et al.'s (Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022) final deposits

A LiDAR scanner was used to determine the deposit height of the grains after the water had fully drained out (at least a day after the experiment). In this final configuration the grains have a denser packing than during flow when shear induces dilation. In order to compare the simulations with the experimental data it is necessary to now contract the granular matrix. At each location the depth of the grains is therefore adjusted to account for the increase in solids volume fraction from its flowing value $\phi ^c$ to its deposited value, which is assumed to be the same initial solids volume fraction $\phi ^g_{{ini}}$,

(6.1)\begin{equation} h^g_{{dep}} = \left(\frac{\phi^c}{\phi^g_{{ini}}}\right) h^g. \end{equation}

The water height is then adjusted to conserve mass

(6.2)\begin{equation} h^w_{{dep}} = \begin{cases} \displaystyle\left(\frac{1-\phi^c}{1-\phi^g_{{ini}}}\right) h^w, & {h^g > h^w}, \\ h^w, & {h^g \le h^w}. \end{cases} \end{equation}

Figure 11 shows a comparison between the deposit thickness adjusted for the increase in volume and Taylor-Noonan et al.'s (Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022) measured deposit thickness for the five wet release volumes. As has been seen in the simulations of the 0.8 m$^3$ case (figure 9), the dry front tries to stop when it reaches the horizontal plane, but the high-mobility undersaturated flow behind pushes the front forwards. As a result the wet flow deposits all show markedly increased run-out distances compared with the dry deposits (figure 8). The larger wet releases go further than the smaller wet ones, which contrasts with the dry case, where the deposit centre of masses were all approximately at $X\simeq 1$ m. Also, rather than forming a large heap at the bottom of the chute, the wet deposits are thinner and spread out along the run-out pad. The thickest part of the deposit is in the vicinity of the front. Here there is a flat-topped deposit that is thicker and longer for the larger release volumes. The simulated deposit then connects to a thinner upstream section that did not pass through the shock that forms during the deposition process, shown in figure 9 and movie 6. The simulation and experimental deposits are in very good agreement with each other, both in terms of their overall morphology, their front positions and thicknesses. There are, however, some differences. The experimental deposits do not show evidence of a shockwave having formed, and so have a much more gradual transition between the thick frontal section and the thinner tail. This may be an indication that the viscosity of the granular material is able to diffuse the shock (Gray & Edwards Reference Gray and Edwards2014). In addition, a thin layer of granular material remains on the chute in the experiments, but the computations show that all the grains drain off. This may be an indication that the assumed friction in the tail is too low in the simulations.

Figure 11. Comparison of the simulated granular free surface adjusted for the decrease in volume (red solid line) with the experimental free surface (red dashed line) for initial release volumes of (a) 0.2, (b) 0.4, (c) 0.6, (d) 0.8 and (e) 1.0 m$^3$ of water-saturated grains (Taylor-Noonan et al. Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022). The predicted profiles assume a cubic velocity profile during flow and are taken at $t=8$ s when the grains (grey hashed region) have stopped moving and the pore water (blue shaded region) is draining out of the matrix. The black lines are the basal terrain, and the ‘$\star$’ symbols mark the start and end of the smooth transition.

7.1. Wet flow simulations assuming plug flow

The model in this paper differs from traditional debris-flow models by its inclusion of vertical flow structure and velocity shear, which provide an important mechanism for the differential transport of the water and the grains (Meng et al. Reference Meng, Johnson and Gray2022). In contrast, traditional debris-flow models usually assume plug flow of both the grains and the water through the depth of the flow, i.e.

(7.1a,b)\begin{equation} u^g = \bar{u}^g,\quad u^w = \bar{u}^w, \end{equation}

(Iverson & Denlinger Reference Iverson and Denlinger2001; Pitman & Le Reference Pitman and Le2005; Pelanti et al. Reference Pelanti, Bouchut and Mangeney2008; Pudasaini Reference Pudasaini2012; Iverson & George Reference Iverson and George2014; Bouchut et al. Reference Bouchut, Fernadez-Nieto, Mangeney and Narbona-Reina2016; Meng & Wang Reference Meng and Wang2018). In order to quantify the difference between the two approaches it is useful to solve the current model with the plug flow assumption (7.1). It follows from (3.12) that the grain and water stream functions are

(7.2a,b)\begin{equation} \psi^g(s) = s\bar{u}^g, \quad \psi^w(s) = s\bar{u}^w, \end{equation}

where recall that $s$ is defined as the height of the saturated grain–water layer in (3.13). These stream functions modify the Darcy drags in the source terms (3.10) and (3.11), and hence change the relative mobility of the grains and the water.

Figure 12 shows the computed temporal and spatial evolution of a 0.8 m$^3$ release of water-saturated grains assuming plug flow. Just as in the cubic velocity case, shown in figure 9, the initial release rapidly spreads out as it flows down the chute, and the flow front reaches the run-out pad at an almost identical time of $t=1.5$ s. The subtle difference is that the flow front only has a very small dry region. Since shear-induced transport has been eliminated in these simulations, the dry front that forms is purely due to the fact that the grains are more mobile than the water near the flow front. This mobility difference is a direct consequence of the degenerate nature of the basal friction law for the water, which requires an infinite gradient to form at the water front for it to propagate downslope. This mobility-induced relative motion of the water and grains near the front also implies that immediately behind the flow front the material is oversaturated (at $t=1.5$ s). This is not observed in Taylor-Noonan et al.'s (Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022) time series in figure 10(d), where the phreatic surface is below the grain surface in the vicinity of the front. In addition, now that shear-induced transport has been eliminated, the dry grains in the initially undersaturated region in the release (figure 2) cannot migrate forwards. The tail of the flow therefore remains undersaturated and does not develop roll waves, which also contradicts Taylor-Noonan et al.'s (Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022) observations.

Figure 12. Simulated spatial and temporal evolution of the grain and phreatic free surfaces when 0.8 m$^3$ of water-saturated grains is released from behind the flume gate. Both the grains and the water are assumed to have the plug-flow velocity profiles defined in (7.1). Since the flow height is very small compared with the chute length, the flow thickness is multiplied by a factor of four and projected normally to the chute along the straight grey lines to aid visualization. An animation is included in the online supplementary material (movie 7).

Although the small region of dry grains at the front may look like a small difference at $t=1.5$ s to the simulations in figure 9, it has a profound effect on the subsequent flow on the horizontal run-out pad. The increased amount of water close to the front dramatically increases its mobility, so it is easier to push it forwards. As a result, when the grains finally stop at approximately $t=7$ s the front lies at slightly over 18 m. This contrasts with a run out of approximately $16.4$ m when velocity shear is included (as seen in figure 9). Figure 13 shows that all the initial release volumes lead to extended run-out behaviour with the plug-flow assumption. Making a plug-flow assumption in debris-flow models may therefore lead to enhanced run-out predictions, dramatically different local flow compositions and the suppression of roll waves.

Figure 13. Comparison of the simulated granular free surface adjusted for the decrease in volume (red solid line) with the experimental free surface (red dashed line) for initial release volumes of (a) 0.2, (b) 0.4, (c) 0.6, (d) 0.8 and (e) 1.0 m$^3$ of water-saturated grains (Taylor-Noonan et al. Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022). The predicted profiles assume plug flow and are taken at $t=8$ s when the grains (grey hashed region) have stopped moving and the pore water (blue shaded region) is draining out of the matrix. The black lines are the basal terrain, and the ‘$\star$’ symbols mark the start and end of the smooth transition.

7.2. Influence of the assumed velocity profile on the final deposit morphology

The numerical simulations in § 7.1 of this paper showed that the run-out distance was overpredicted when plug-flow velocity profiles were assumed for the grains and the water. In contrast, the simulations in § 6 show that Meng et al.'s (Reference Meng, Johnson and Gray2022) model is able to make good quantitative predictions for the position of the Taylor-Noonan et al.'s (Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022) deposit, when a cubic granular velocity profile is assumed. Figure 5 shows that there is quite a bit of scatter in Taylor-Noonan et al.'s (Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022) peak-flow velocity data. A variety of functional forms could have been chosen to fit it. One may therefore legitimately ask how sensitive are the latter results to the precise parameterization of the velocity profile?

Figure 14 shows a comparison of the computed deposits for plug-flow (figure 14a), cubic (figure 14b), Bagnold (figure 14c) and linear profile with basal slip (figure 14d), velocity profiles, using the fitting parameters from § 3.3. The results in figure 14(b–d) use shear profiles that are broadly in agreement with one another, and all of them provide a good fit to Taylor-Noonan et al.'s (Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022) measured deposit height. This indicates that the precise velocity profile parameterization is not that important, provided that the overall shear rate is similar. In contrast, as discussed in § 7.1, plug flow eliminates shear-induced differential species transport, which reduces the size of the dry front and allows the debris-flow to run-out considerably further, as shown in figure 14(a).

Figure 14. Comparison of the simulated granular free surface adjusted for the decrease in volume (red solid line) with the experimental free surface (red dashed line) for an initial release volume of $0.8$ m$^3$ of water-saturated grains (Taylor-Noonan et al. Reference Taylor-Noonan, Bowman, McArdell, Kaitna, McElwaine and Take2022). The predicted profiles are taken at $t=8$ s when the grains (grey hashed region) have stopped moving and the pore water (blue shaded region) is draining out of the matrix. In the computations the velocity profile of the grains is assumed to be (a) plug-flow, (b) cubic, (c) Bagnold and (d) linear with basal slip. The black lines are the basal terrain, and the ‘$\star$’ symbols mark the start and end of the smooth transition.