# Experimental Analysis of Velocity Distribution in a Coarse-Grained Debris Flow: A Modified Bagnold’s Equation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{Ba}, that is the ratio between the inertial grain stress and the viscous shear stress. The results of Bagnold’s measurements were then used to analyze different processes such as the grains sorting, the flows in gravel beds, and the critical conditions under which a particle remains suspended [18,19]. Takahashi [14] considered a uniform layer of granular loose material and applied Bagnold’s [16] constitutive equations to stony-type debris flow by integrating them under the assumption of constant grain concentration in the flow depth.

## 2. Material and Methods

#### 2.1. Pertinent Aspects of Bagnold’s Theory and Its Application to Debris Flow

_{Ba}) given by the ratio between the stresses due to the inertial forces and those due to the viscosity:

_{Ba}= (ρ

_{s}γd

_{p}

^{2}λ

^{1/2})/μ

_{f}

_{p}indicates the particle diameter, ρ

_{s}is the density of grains, μ

_{f}is the fluid’s dynamic viscosity, γ is the shear rate, and λ is the linear concentration of grains given by the following relation:

_{∗}is the maximum possible concentration (i.e., the so-called packing concentration), which can be obtained in static conditions where the friction between the sediment particles is of primary importance [47,48].

_{Ba}< 40), while the grain-inertia regime occurs for high values of the Bagnold number (i.e., for N

_{Ba}> 450). The intermediate-range of the Bagnold numbers indicates a transitional regime between the aforementioned ones. While in the macro-viscous regime, the stresses are determined by the interaction of the granular phase with the interstitial fluid, in the inertia regime, the inertia associated with the individual grains is more important.

_{i}is the so-called “friction coefficient” [16,50], and α is the collision angle. According to Bagnold’s experimental tests, it can be assumed a

_{i}= 0.04 and tanα = 0.75; Takahashi [14,49] suggested to assume a

_{i}= 0.42 for loose beds.

_{*}. While C

_{*}could be estimated by using either experimental data appositely collected or physically-based literature data, more investigations should be conducted to evaluate grain concentration C and its variation within the debris body.

#### 2.2. Experimental Apparatus and Procedure

_{50}= 3 mm. These experimental conditions were selected after preliminary setting runs and in accordance with the indications of previous literature works [33,39,51,52].

^{3}/s over a loose sediment layer of a thickness of around 0.10 m, which was previously saturated by slowly releasing a low water discharge Q

_{0}= 0.8 × 10

^{−3}m

^{3}/s. A permeable ground sill was positioned at the downstream end of the examined reach to avoid the degradation of the sediment layer.

_{,m}= 0.305.

_{50}. Within each sub-layer of thickness St, the grain’s instantaneous longitudinal velocity, u(t), was estimated as the ratio between the grain’s displacement for two consecutive frames (i.e., two consecutive times) and the time interval. In particular, by considering the orthogonal local reference system with an origin at the southwest corner of the frame, it was estimated u(t) = (s

_{i+}

_{1}− s

_{i})/(t

_{i+}

_{1}− t

_{i}) (where t

_{i+}

_{1}and t

_{i}are two consecutive times and s

_{i+}

_{1}and s

_{i}are the corresponding grain’s positions). Then, the time series of u(t), obtained for the grains within each sub-layer St by considering all the recorded frames, were used to determine both the corresponding time-averaged value u and the mean, $\overline{u}$, of the time-averaged velocities of the considered sub-layer St.

## 3. Results

#### 3.1. Grains Concentration Distribution

_{*}. The thickness of the static bed, z

_{o}, was estimated from the analysis of both of the recorded frames and of the measured velocity profiles.

^{2}= 0.96). The values of the grain concentration at the extreme points of the interpolating line are very close, respectively, to those of the surface concentration, Cs, and of the maximum possible concentration C

_{*}at the bed. Based on this, the following linear law is considered to approximate the distribution of the grain concentration

_{50}= 6 mm; St = 3D

_{50}= 9 mm) and the corresponding values of grain concentration C have also been estimated by applying the aforementioned four-step procedure. Then, the values of the grain concentration determined for each value of St were compared with the theoretical distribution given by Equation (5), as reported in Figure 6.

_{i,c}, and those measured, C

_{i,m}:

_{ts}is the number of the sub-layers of thickness St. It can be observed that σ assumes small and almost equal values for all the considered values of St. This suggests that the estimated grains concentration distribution does not depend on the selected value of St. Finally, a value of St = 2D

_{50}(6 mm) has been assumed for the subsequent analysis.

#### 3.2. Modified Bagnold’s Equation Applied to Debris Flow

_{Ba}has been rewritten as follows:

_{Ba}(z) = [ρ

_{s}γd

_{p}

^{2}λ(z)

^{1/2}]/μ

_{f}

_{Ba}-values obtained by using Equation (9), with the velocity and the grain concentration distributions respectively given by Equations (7) and (5), and the N

_{Ba}-values (hereon indicated as “experimental N

_{Ba}-values”) obtained by using the experimental velocity profile. Figure 8 also reports the N

_{Ba}-values obtained by using Equations (1) and (3). Figure 8 shows that while the distribution of the N

_{Ba}-values estimated by using Equation (9) is in agreement with the profile of the experimental N

_{Ba}-values, the distribution of the N

_{Ba}-values estimated by using Equation (1) deviates from it. In particular, the last one has N

_{Ba}-values greater than 1000, close to the bed, and a decreasing trend as one passes from the bed to the free surface. On the contrary, Equation (9) determines N

_{Ba}-values less than 1000 close to the bed and greater than 1000 close to the free surface. The latter trend of the N

_{Ba}-values is consistent with that observed in previous literature works [34,48,51], indicating the development of the frictional flow regime for N

_{Ba}-values <1000 and of the collisional-inertial flow regime for N

_{Ba}-values >1000.

_{Ba}-values >1000 and C ≅ Cs, the frictional regime in the intermediate zone with N

_{Ba}-values <1000 and C = C(z), and the static zone close to the bed with N

_{Ba}-values <<1000 and C ≅ C∗.

#### 3.3. Free Surface Grains Concentration and Procedure for Its Estimation

_{*}and Cs. While the parameter C∗ could be easily identified, as mentioned in Section 2.1, it is difficult to determine the free surface grain concentration, Cs.

_{m,i}and u

_{c,i}indicate, respectively, the measured and the calculated velocity values in the i-th sub-layer.

_{c}represents the critical friction angle.

_{c}could be assumed equal to the static friction angle of the material. This means that Equation (11) would allow us to obtain information on the grain concentration Cs, starting from the knowledge of the bed slope and of the static friction angle, which can be determined by simple shear tests.

_{c}was determined by using the direct cutting apparatus available at the Geotechnical Laboratory of the Engineering Department, University of Palermo. Such an apparatus is specially adapted to the present application because it is equipped by a large cutting box (30 × 30 × 20 cm

^{3}—see Figure 11), allowing us to test a sample, of adequate volume, of the material used for the experimental run.

_{n}, was obtained for each test. More details of the testing procedure can be found in Fichera [53]. From the cutting tests, it was also verified that all the examined samples assumed a very similar behavior. By considering the maximum value of the shear stress, τ

_{max}, of each test, the couples of values [τ

_{max}, σ

_{n}] were reported on a Cartesian plane. As Figure 12 shows, the coupled values [τ

_{max}, σ

_{n}] are arranged around an interpolating line, having an angular coefficient equal to φ

_{c,e}= 38°. Such an angular coefficient represents the “experimental” value of the static friction angle of the used granular material.

_{,m}= 0.305 and the inclination is β = 15°. Finally, a value φ

_{c}= 38° was obtained from Equation (11). It is clear that this value of φ

_{c}is equal to that obtained from the aforementioned direct cutting tests.

## 4. Discussion

#### 4.1. Velocity Profiles and Comparisons with Literature Data

#### 4.2. Main Aspects Derived by the Proposed Modified Expressions and Procedure

_{*}. This confirms that the assumption of uniform sediment distribution in the entire depth, which is generally used in analyzing the hydrodynamics of stony-debris flow, is not realistic.

_{*}, and the value of the grain concentration at the free surface, Cs.

_{Ba}and of the velocity for stony-type debris flows have been obtained.

_{Ba}), estimated by using the aforementioned modified expression, has highlighted that three different flow regimes can be identified within the debris body. In particular, the collisional regime is obtained close to the free surface, where the grain concentration is C = Cs and the N

_{Ba}-values are greater than 1000; the frictional regime is obtained in the intermediate zone, where C = C(z) and the N

_{Ba}-values are lower than 1000; close to the bed, where the concentration is C = C

_{*}and the N

_{Ba}-values are strongly lower than 1000, the static zone is established. The observed behavior is consistent with results obtained in other literature works (see in [34,51,53]), indicating the development of the frictional flow regime for N

_{Ba}-values <1000 and of the collisional-inertial flow regime for N

_{Ba}-values >1000.

_{*}, and free surface concentration, Cs. While parameter C

_{*}can be easily identified by using physically-based literature data, it is difficult to determine parameter Cs. Thus, first, the “optimal” values of parameter Cs were calculated by minimizing the mean square error between the velocity values estimated by using the proposed expression and the experimental ones taken from the literature. As is clear from the previous subsection, this analysis has allowed us to highlight the important role of parameter Cs in defining the velocity distribution within the debris body. The comparison between the velocity profiles estimated by considering the “optimal” values of Cs and the literature’s measured profiles (from [33,35,37,39]) has demonstrated that the proposed modified expression, which takes into account the variation of the grain concentration within the debris body, correctly interprets the measured velocities.

## 5. Conclusions

- (1)
- the distribution of the grain concentration can be interpreted by a linear law obtained between the value of the maximum package value, C
_{*}, at the bed and the value of the free surface concentration, Cs; - (2)
- by removing the hypothesis of uniform grain concentration along the entire depth, modified expressions of Bagnold’s number and of the longitudinal velocity, which take into account the variation of the grain concentration in the entire depth, are presented. The expression of the velocity profile includes two parameters: the maximum package value, C
_{*}, which could be determined by using either experimental data appositely collected or physically-based literature data, and the value of the free surface concentration, Cs; - (3)
- by using the modified expression of Bagnold’s number, it has been verified that a varying stress regime can develop within the debris flow. The N
_{Ba}-values are strongly lower than 1000 when close to the bed (frictional regime) and are greater than 1000 (collisional-inertial regime) when close to the free surface; - (4)
- it has been verified that, in the first approximation, surface concentration Cs can be estimated as a function of the static friction angle of the material, which can be determined by simple shear tests.

_{*}and Cs that can be easily identified through physically-based literature data and by simple shear tests, respectively. This result is of great importance, especially in numerical modeling of stony-type debris flows, which in nature especially occur in rapid valleys or in high-slope streams.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Plan view of the experimental apparatus; (

**b**) particular of the channel reach considered for the analysis (bed slope = 15°; assorted gravels with mean diameter D

_{50}= 3 mm; the numbers indicate the sections in which the channel was discretized).

**Figure 2.**Grain size distribution of the bed material (the x-axis indicates the grain diameter; the y-axis indicates the percent of finer by mass passing).

**Figure 4.**Variation of grains concentration in the depth of flow and particular of the grain configuration.

**Figure 5.**Measured grain concentrations C and interpolating law (Z indicates the direction orthogonal to the bed with an origin at the upper level of the static bed).

**Figure 6.**Comparison between the theoretical distribution of the grain concentration (Equation (5)) and the estimated values (C measured) by assuming different sub-layer thickness St: (

**a**) for St = D

_{50}; (

**b**) for St = 2D

_{50}; (

**c**) for St = 3D

_{50}. The x-axis indicates the grain concentration; the y-axis indicates the distance from the upper level of the static bed.

**Figure 7.**Comparison between the $\overline{u}$ profile (experimental profile) and the estimated ones by applying Equation (3) and by applying Equation (7). The dots represent the velocity u of selected grains (the x-axis indicates the longitudinal velocity; the y-axis indicates the distance from the upper level of the static bed).

**Figure 8.**Comparison between the N

_{Ba}-values by using the experimental velocity profile (experimental N

_{Ba}-values), the N

_{Ba}-values estimated by applying Equations (9) and (7) and the N

_{Ba}-values estimated by applying Equations (3) and (1). The dots represent the N

_{Ba}-values of selected grains; (the x-axis indicates the N

_{Ba}-values; the y-axis indicates the distance from the upper level of the static bed).

**Figure 10.**Comparison between the estimated “optimal” values of the parameter Cs and those measured and taken from the literature.

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Termini, D.; Fichera, A.
Experimental Analysis of Velocity Distribution in a Coarse-Grained Debris Flow: A Modified Bagnold’s Equation. *Water* **2020**, *12*, 1415.
https://doi.org/10.3390/w12051415

**AMA Style**

Termini D, Fichera A.
Experimental Analysis of Velocity Distribution in a Coarse-Grained Debris Flow: A Modified Bagnold’s Equation. *Water*. 2020; 12(5):1415.
https://doi.org/10.3390/w12051415

**Chicago/Turabian Style**

Termini, Donatella, and Antonio Fichera.
2020. "Experimental Analysis of Velocity Distribution in a Coarse-Grained Debris Flow: A Modified Bagnold’s Equation" *Water* 12, no. 5: 1415.
https://doi.org/10.3390/w12051415