2.1. General framework equations in 2-D geometry

We consider a steady incoming free-surface flow of thickness h, mean velocity u and mean density p down a flat slope with inclination 9, as shown in Figure 1. A nearly triangular stagnant zone is formed upstream of the dam, as shown in Figure 1. Momentum conservation over a control volume, V ₀, upstream of the dam (pink area in Fig. 1) allows us to show that, in a steady-flow regime, the resulting force, F, normal to the dam is the sum of four contributions:

(1)

Fig. 1. Sketch of the flow and control volume, V0 (pink area). H is the obstacle height; α_zm is the mean angle of the dead zone; α_sl is the mean angle of the free surface above the dead zone; a is the angle between the velocity, , and the ground; h and u are the thickness and the depth-averaged velocity outside the influence area of the obstacle; h* and u* are the thickness and the depth-averaged velocity of the overflow at the top of the dam (in the x _* direction); and L is the length of the influence zone upstream of the obstacle, assumed to be close to the length of the dead zone.

V ₀ is a volume per unit width and we consider forces per unit width. F_u is a purely dynamic force resulting from the momentum variation between sections S ¹ and S2, defined in Figure 1: F_u = α(δ_u cos α)ρu ² h where 5_u is the velocity ratio, u*/u, u* is the velocity at the top of the dam, as defined in Figure 1, and a is the deflection angle (i.e. the angle between u* and the ground). The coefficient f3 depends on the shape of the velocity profile and is defined by .The relative velocity reduction, (u - u*)/u, is simply assumed to be proportional to the deflection angle, a, which gives:

(2)

where K is a velocity-reduction coefficient. Fh is a purely hydrostatic contribution due to the incoming flow undisturbed by the obstacle, , where k is the earth pressure coefficient classically introduced for gravity-driven flows of granular materials (Reference Savage and HutterSavage and Hutter, 1989) or snow (Reference McClung and MearsMcClung and Mears, 1995; Reference Bartelt, Salm and GruberBartelt and others, 1999). F_w is the x-axis component of the weight of the control volume, V0: F_w = ρgV0 sin θ. Ff is the basal friction force assumed to be proportional to the y-axis component of the weight of the control volume, V0 (Coulomb friction): Ff = μ_zmρgV0 cosθ, where μ_zm is the friction coefficient between the dead zone and the ground. We discuss μ_zm further below.

The volume, V₀, can be calculated from Figure 1 as follows:

(3)

where h* is the thickness of the outcoming flow defined in Figure 1, δ _h is the depth ratio, h*/h, H is the obstacle height and L is the length of the influence zone upstream of the obstacle. The zone of influence upstream from the dam is defined as the length of the disturbed flow upstream of the dam (flow depth and velocity are not equal to the incoming flow depth, h, and velocity, u) and can be approximated by tan(α_zm) = H/L By mass flow rate conservation, the flow depth ratio, δ _h, is equal to 1/δ_u, if we assume that the density is unchanged. The angle a is equal to (α_sl + α_zm)/2 where α_sl is the angle of the free surface inside the control volume, V₀. We detail below how we can estimate the angles a_zm and α_sl.

If we neglect the second-order term in Equation (3), then Equation (1) can be synthesized in terms of the normalized force, F/(0.5ρu₂h), versus the incoming Froude number, :

(4)

The second-order term is strictly negligible when h* ≈ L and h* w h, which is almost true when the Froude number is not too high (typically less than 5–10). When the Froude number is more than 5–10, the contribution due to the volume, V₀, in the total force is so weak (purely dynamic force) that an error in V₀ has no effect on the total resulting force. This analytical model has been initially developed for granular flows (Reference Faug, Beguin and ChanutFaug and others, 2009) for which the parameters (k, β, μ_zm, K) have been determined, as well as expressions to calculate the angles a_zm and α_sl, which give the deflection angle, a. Results for granular flows are briefly reported in the next subsection.

2.2. Validation of the analytical model for 2 -D granular flows

We performed discrete-particle simulations using a linear damped spring law between spherical particles with a Coulomb failure criterion, in order to simulate 2-D steady granular flows down an inclined slope, as shown in Figure 1 (see details in Reference Faug, Beguin and ChanutFaug and others, 2009). These discrete numerical simulations were based on the molecular-dynamics method introduced by Reference Cundall and StrackCundall and Strack (1979) and widely used to simulate dense granular flows (Reference Ertas, Grest, Halsey, Levine and SilbertErtas and others, 2001; Reference Silbert, Ertas, Grest, Halsey, Levine and PlimptonSilbert and others, 2001; Reference Da Cruz, Emam, Prochnow, Roux and Chevoirda Cruz and others, 2005). The following microscopic parameters were needed to describe the contacts between grains: the normal and tangential stiffnesses, k_n and k _t; the restitution coefficient, e (linked to the damping coefficient) and the interparticle friction coefficient, p. The influence of these parameters was discussed by Reference Silbert, Ertas, Grest, Halsey, Levine and PlimptonSilbert and others (2001): (1) k_n and k_t have no effect in the limit of rigid grains (overlap between particles < 1/1000d, where d is the particle diameter); (2) e has little effect except for extreme values e = 0 and e = 1; (3) μ has a greater effect on the results, but its influence becomes weak at low values, typically <0.5. In our simulations, we used the following values: k_n = 10⁴ N m∼¹, k_t = 1/2 k _n, e = 0.5 and μ = 0.5 (corresponding to a typical value of the internal friction angle of granular materials). Here we used the commercial code PFC2D (Itasca Consulting: http://www.itasca.com/pfc/index.php). These numerical simulations allowed us to estimate the macroscopic empirical laws to close Equation (4) for the case of granular materials. A first result of the numerical simulations is that the basal friction, μ_zm, was shown to be constant, equal to tan 9_{m in}, where 9_{m in} is the minimal angle below which no steady flow is possible (stopping of the flow) (Reference PouliquenPouliquen, 1999; GDR Reference MiDiMiDi, 2004). Second, the mean angle of the dead zone with the horizontal, θ – α_zm, was shown to be equal to 9_{m in} for all slope inclinations, 9. The length, L, of the influence zone of the obstacle was then assumed to be equal to the length of the stagnant zone and accordingly was defined as: tan(θ – θ_min) = H/L. Third, the free-surface angle, α_sl, was shown to be a simple linear function of the slope inclination, 9: α_sl = aθ + b, where a and b are coefficients depending on the incoming flow regime. We defined 9_max as the maximum angle above which uniform flows were not possible. For uniform flows (#_min < 9 < Omax), the angle α_sl was expressed as:

(5)

For non-uniform flows (θ > θ _max), the angle αsl was expressed as:

(6)

Equation (4) was successfully tested on data from discrete numerical simulations using the following values (Reference Faug, Beguin and ChanutFaug and others, 2009): k = 1, β = 5/4 (for a Bagnold-like velocity profile (GDR Reference MiDiMiDi, 2004)), κ = (1 – e)/(π/2) (where e = 0.5 is the restitution coefficient), 9_{m in} = 14° and θmax = 24° (typical values for 2 -D granular flows (GDR Reference MiDiMiDi, 2004)). The value of K is simply estimated from a purely collisional regime assuming that u*/u scales as e when α = π/2 which gives e = 1 – κ(π/2) according to Equation (2). Figure 2 shows predictions of the analytical model compared to discrete simulations. The predictions of the model are in very good agreement with the discrete simulations, without having introduced any fitting parameter. Note that Equation (4) predicts a force ratio F/(0.5ρu₂h) = 2β[1 – (1 – κα) cos a] at high Froude numbers, which gives a value of 2 for a = π/2, compatible with the drag coefficient classically given for a flat obstacle in the inertial regime (high Froude number). Figure 2 also reports the force normalized by the purely hydrostatic force, ¹² pgh₂ cos 9. It is interesting to consider the asymptotic prediction for this ratio when 9 tends towards 9_{m in} (i.e. the Froude number, Fr, tends to zero). According to the analytical model, this ratio should scale as κ + (H/h) (2 + H/h) (1/ cos 9_{m in} ). With κ = 1 and H/h = 1, this gives a value of 1 + 3/cosθ_min. The ratio is then close to 4 (cosθ_{m in} ≈ 1), as found in discrete simulations and shown in Figure 2. In section 2.3, we use the analytical model for dry snow and we provide the model parameters adapted to dry snow.

Fig. 2. Granular flows: prediction of the model for the rescaled force, F/(0.5ρu2h) and F/(0.5ρgh2 cosθ), vs slope, compared to data from numerical discrete simulations. The following parameters were used: β = 5/4, k = 1, κ = 0.31 (e = 0.5), θ_{m in} = 14°, θmax = 24° (Reference Faug, Beguin and ChanutFaug and others, 2009).

2.3. Outlook for 2-D snow flows

Here we use Equation (4) for dry snow flows and we describe the parameters characterizing the behaviour of dry snow. Cemagref has designed a 10 m long and 20 cm wide channel at Col du Lac Blanc, Alpe d’Huez, France, described in detail by Reference Bouchet, Naaim, Ousset, Bellot and CauvardBouchet and others (2003, Reference Bouchet, Naaim, Bellot and Ousset2004). Recent investigations on flows of dense and dry snow (for T < 0°C) down this 10 m long flume showed that snow exhibits some properties similar to those of granular flows (Reference RognonRognon, 2006; Reference Rognon, Chevoir, Bellot, Ousset, Naaim and CoussotRognon and others, 2008). Dry and dense snow is a polydisperse granular material. Similarly to granular flows, there exists a minimum angle below which the flow is stopped, and a maximum angle above which flows accelerate along the channel. Steady and uniform flows are only possible between these two angles. Typical values were derived from experimental investigations at Col du Lac Blanc for dense and dry snow (Reference Rognon, Chevoir, Bellot, Ousset, Naaim and CoussotRognon and others, 2008): 9_{m in} = 33° and θ_max = 42°. These angles were obtained in a narrow channel with typical flow depths ∼10cm. Even if these angles are likely to be influenced by wall effects (whose extent is still under discussion), we will use these angles for snow in the following. The flow is divided into two layers: a highly sheared layer of snow grains (typically 1 mm in size) at the base, surmounted by a low shear layer of aggregates (with a maximum size close to the flow depth). This flow configuration results in a typical velocity profile close to velocity profiles obtained from discrete numerical simulations on bi-disperse granular flows. A value of β close to 1 is then reasonable for snow flows.

The friction law for snow flows is still an open question. However, an effective friction law corresponding to a Voellmy model could be fitted to data from the Lac Blanc chute (Reference RognonRognon, 2006): μ = μ_s + (g/ξ)Fr². The Voellmy model is classically used in snow-avalanche engineering applications (Reference Bartelt, Salm and GruberBartelt and others, 1999); μ_s is a dry friction coefficient and ξ is a turbulent coefficient. Both parameters (μ_s, ξ) were fitted corresponding to various choices. The best fit was obtained for μ = 0.61, i.e. θ _s = arctan(/μ_s) = 31.2°, and ξ = 1050 m s∼², which are values encountered for snow avalanches (Reference Salm, Burkard and GublerSalm and others, 1990). With respect to the limit angle, 9_{m in} = 33°, which would imply μ = tan 9_{m in}, a value of ξ = 1400 m s∼² was obtained.

We assume that the empirical laws derived from investigations on granular flows to estimate the free-surface angle, α_sl, the dead-zone angle, α_zm, and consequently the deflection angle, a, as well as the friction, μ_zm, still hold for our granular-like snow flows. A value of k = 1 is chosen. Here, K is calculated using a very low value for e that is more compatible with the properties of snow material: e = 0.1 gives K = 0.57.

The angle a is assumed to be equal to:

(7)

where α_sl is defined by Equations (5) and (6).

In steady and uniform flow conditions, or within a flow regime for which the effect of acceleration terms (time-derivative terms in momentum conservation) can be neglected, we have the following equation corresponding to equilibrium between gravity and friction forces (tan 9 = μ):

(8)

Equation (8) allows us to eliminate the slope, 9, in Equations (7) and (4) and to express the rescaled force, F/(0.5ρu2h), as a function of the incoming Froude number, Fr, without the prior knowledge of the slope (information which is included in the Froude number). This raises the question whether we may extrapolate the predictions of our analytical model to slope inclinations less than 0_{m in} in transient conditions towards stopping. The answer to this question is positive if the system is able to reach an equilibrium state for which the assumption tan θ = μ still holds. This assumption seems reasonable in the case of decelerating flows evolving towards stopping (tails of avalanches), for which the effect of time-derivative terms is expected to be negligible. We use this assumption in the following when applying equations established for the steady regime to full-scale avalanche flows.

It is important to note that the rescaled force, F/(0.5ρu2h), depends on the ratio H/h. Indeed, attempts to find a relation between this ratio and Fr are meaningful only if H/h remains constant. Figure 3 gives the prediction of the analytical model in terms of the rescaled force vs Fr for different values of H/h, 9_{m in} and θmax, keeping θ_max - θmin constant. The curves show that the results are not very sensitive to the value of 9_min. The results are, rather, influenced by the geometry corresponding to a varying ratio, H/h. We also show the rescaled force, F/(0.5ρgh2 cos 9), for the same set of parameters to give the prediction of the analytical model at very low Fr in order to highlight the transition towards the hydrostatic regime.

Fig. 3. Snow flows. Prediction of the analytical model for the rescaled forces (a) F/(0.5ρu2h) and (b) F/(0.5ρgh2 cosθ) vs the Froude number at different ratios, H/h (0.1, 1 and 5). The following parameters were used:β = 1, k = 1, κ = 0.57 (e = 0.1) and ξ = 1000 ms-². Predictions are given for two pairs (θ_{m in} ,θ_max) keeping θ_max – θ_min = 9°: [33°; 42°] (p_s = 0.65) and [10°; 19°] (p_s = 0.18).

2.4. Outlook for 3 -D effects with lateral fluxes

2.4.1. General framework equations in 3-D geometry

We propose a modification of Equation (4), taking into account 3-D effects corresponding to lateral fluxes and to the modification of the shape of the dead zone by these lateral fluxes. The resulting analytical model is a simplified model that does not take into account the entire effect of flow spreading, which would require fully 3-D numerical models (e.g. Reference Naaim, Naaim-Bouvet, Faug and BouchetNaaim and others, 2004). Here, we consider the thickness, h_L, and the depth-averaged velocity, u_L, that correspond to the lateral fluxes around the obstacle. Figure 4a gives a top view of the flow configuration in 3-D geometry. A dead zone is formed upstream of the obstacle. The base of the dead zone in the plane (x, y) is triangular and is characterized by angle γ.

Fig. 4. (a) Top view of the flow and dead zone (pink area) in 3-D geometry. 7 is the mean angle of the dead zone in the plane (x, y). h and u are the thickness and the depth-averaged velocity outside the zone of influence of the obstacle. ξ_a is the width of the incoming flow. h* and u* are the thickness and the depth-averaged velocity at the top of the dam. ξ is the width of the obstacle. h ^L and u ^L are the mean thickness and the depth-averaged velocity in the flow branch corresponding to lateral fluxes. L is the length of the dead zone at the centre (in y-axis direction) of the obstacle, and L(y) is the length of the dead zone at a given position, y. Note that h*, u*, h ^L and u ^L are mean values in sections S* (overflow) and S ^L (lateral fluxes). Due to the symmetry of the problem, we only show one lateral flux. (b) Side view at a position y of the flow overflowing the obstacle. We use a notation similar to that of the 2-D configuration given in Figure 1, but here the variables depend on the position, y.

Compared to the 2-D geometry, the angles α_zm, α_sl and a depend on the transverse position, y, along the width, £, of the obstacle as depicted in Figure 4b. As a first approximation, and in the absence of well-documented experimental evidence of the shape of the dead zone, we assume a simple triangular shape for the dead zone in a plane normal to the z axis (Fig. 4a), and the length, L(y), at location y can be expressed as L(y) = c + dy. The boundary conditions require 0 = c + d(ℓ/2) (edge of the obstacle) and H/tan(θ - θ_min) = c (centre of the obstacle), which leads to:

(9)

The dead-zone angle at position y is α_zm(y) = arctan[H/L(y)). The free-surface angle is assumed to vary linearly between π/2 (edge of the obstacle) and the value of α_sl in 2-D given by Equations (5) and (6), which gives α_sl(y) = 2(π/2 - α_sl)(y/ℓ)+ α_sl. The mean angle of deflection is then given by α(y) = (α_zm(y) + α_sl(y))/2. The dependence of these angles on y complicates the calculation of the momentum balance (3-D integrals).

We assume that the basal friction is proportional to the weight in the z-axis direction. The total basal force is caused by the basal friction below the dead zone and the sum of the weights of the dead zone and the fluid above the dead zone: . Here is the control volume in 3-D geometry (between sections S, S* and S ^L in Fig. 4):

(10)

Equation (10) can be solved numerically and is work in progress. In section 2.4.2 we propose some rough approximations to simplify the model in order to be able to provide an analytical solution under restrictive conditions.

2.4.2. Simplified analytical model for 3-D snow flows

For the sake of simplicity and to provide an analytical model, we here consider the mean values over the obstacle width:

(11)

(12)

(13)

One parameter is introduced: r = ℓ_a/ℓ is the ratio between the width of the incoming avalanche flow, ℓ_a, and the width of the obstacle, £. Similarly to the 2-D case, we can apply the momentum balance over the control volume, . We simplify the calculation by taking the mean values over the obstacle width (given by Equations (11–13)) out of the 3-D integrals, which gives:

(14)

(15)

We split the resulting ratio, (F/ℓ_a)/(0.5ρu₂h), into two parts: Equation (14) is the contribution due to the dynamic force and Equation (15) is the contribution of the sum of the incoming pressure force, the weight and the basal friction force. Three parameters have to be quantified: the mean ratio of velocities at the centre, the mean ratio of the flow depths, and the mean ratio of velocities on both lateral sides, The mean lateral flow depths ratio, where h ^L is the thickness of lateral fluxes, is determined by the conservation of the mass flow rate: .

We use the following assumptions to derive simple empirical laws for these quantities. The mean ratio of velocities at the centre is calculated from the mean angle, similarly to the 2-D geometry. The mean ratio of flow depths at the centre is assumed to be ∼1, which corresponds to the assumption that the typical size of the overflowing flow is close to the typical size of the undisturbed flow. This is almost true for the case of 2-D granular flows (Reference Faug, Beguin and ChanutFaug and others, 2009) but it remains an assumption for 3-D flows, in the absence of well-documented experimental data. The mean ratio of velocities on both lateral sides is assumed to depend on the mean angle, γ (by analogy with the definition of according to where we simply assume a value of κ _L equal to K = (1 - e)/(π/2). This latter assumption is argued by the fact that the restitution coefficients are similar for collisions in the planes (x, z) and (x, y). The angle 7 is defined as 7 = arctan [(ℓ/2)/L] (Fig. 5a).

Fig. 5. Avalanche flows at Lautaret: predictions of the analytical model compared to the measured data (full black curve). The following parameters were used: β = 1, k = 1, H/h = 1, θ_{m in} = 33°, 9_max = 42°, ξ = 1000 ms– ₂ and κ = κ _L = (1 – e)/(π/2) with e = 0.1. The grey dashed curve shows the contribution from the sum of the hydrostatic force, the gravity force and the basal friction force (Equation (1 5)). The black dashed curve shows the contribution from the incoming dynamic force (Equation (14)). (a) 1 5 February 2007 avalanche (Reference Thibert, Baroudi, Limam and Berthet-RambaudThibert and others, 2008) with r = 7. (b) 26 March 2008 avalanche (Baroudi and Thibert, in press) with r = 3.