Investigating the effects of clay/sand content on depositional mechanisms of submarine debris flows through physical and numerical modeling

Abstract

In this study, we carried out a series of physical experiments using a submersed flume model to investigate how sand/clay content influences the depositional mechanism of submarine debris flows. A three-dimensional biphasic numerical model, with a Herschel-Bulkley rheology, was used to back-analyze the physical experiments. The calibrated numerical model was then used in a back-calculation to investigate the effects of viscosity on the deposition process. The results show that as submarine debris flows mix with water during the deposition process, shear stress at the slurry-water interface generates a vortex that leads to a swirl-wedge front head. High-viscosity slurry flows with a swirl-wedge front head travel at a higher aspect ratio and with a greater radius of rotation. Hydroplaning was observed when the front head was lifted by water during flow. The lifting height increased with flow depth fluctuation. Higher viscosity slurry was found to lift more rapidly due to its larger vortex and the decrease in density at the front head over time, both of which promote fluctuation. Although a high-density slurry has a greater lifting height, the ratio of lifting height to front head height is lower, indicating a smaller lifting force influence. Lower density flows have higher kinetic energy as the transfer of potential energy into kinetic energy is more efficient. Kinetic energy dissipation comprises three stages: (1) gravity-dominated coherent flow and hydroplaning lead to a rapid increase in kinetic energy; (2) sharp reduction in kinetic energy as slurry mixes with water, coherence and hydroplaning are reduced, and the influence of the shear stress increases; (3) slurry mixed very well with water, turbidity current dominates the kinetic energy dissipation. High-density slurry dissipates quicker in the last two stages. In stage 3, which dominates the temporal evolution of the debris flow, the Froude number is lower than 1, the flow thins and elongates, and the deposition process of submarine debris flows is dominated by gravity, and the difference of morphology of the different cases become clear.

Appendix Refraction analysis for the flume tests

Refraction analyses were carried out for the physical flume tests to ensure the measurement accuracy. The distance between the image plane and the objective plane was firstly measured, followed by the calculation of the percentage difference in the measured velocity (Patil and Liburdy 2012; Patil and Liburdy 2013). The distance refraction error can be calculated from the geometry using Snell’s law (Born and Wolf 2013). As shown in Figure 13, when light passes from one medium to another, the angle of incidence θ₁ changes to the angle of refraction θ₂, the observation error D can be calculated from.

$$ D=B\left(\mathit{\tan}{\theta}_1-\mathit{\tan}{\theta}_2\right) $$

(A1)

For the current test set up, a total of three refractions happened from the camera to the observed slurry, as shown in Figure 14. In this process, the light wave from the high-speed camera passes 4 mediums including air, tempered glass, water, and perspex. Each observation error can be calculated in each refraction, with the final distance error given by the sum of each observation error, as.

$$ \varDelta D={D}_1+{D}_2+{D}_3 $$

(A2)

where D₁ = B₂(tanθ₁ −  tan θ₂), D₂ = B₂(tanθ₁ −  tan θ₃), and D₃ = B₄(tanθ₁ −  tan θ₄) are distance errors caused by interfaces of air-tempered glass, tempered glass-water, and water-perspex. B₁ is the vertical distance from the camera to the outer flume wall, B₂ is the thickness of the outer flume, B₃ is the vertical distance between the outer and inner flume wall, and B₄ is the width of perspex. The angles of refraction are calculated as (Table 5):

$$ {\displaystyle \begin{array}{l}{\theta}_1= ta{n}^{-1}\left(\frac{L}{B_1+{B}_2+{B}_3+{B}_4}\right)\\ {}{\theta}_2= si{n}^{-1}\left(\frac{R{n}_1}{R{n}_2}\mathit{\sin}{\theta}_1\right)\\ {}{\theta}_3= si{n}^{-1}\left(\frac{R{n}_2}{R{n}_3}\mathit{\sin}{\theta}_2\right)\\ {}{\theta}_4= si{n}^{-1}\left(\frac{R{n}_3}{R{n}_4}\mathit{\sin}{\theta}_3\right)\end{array}} $$

(A3)

where θ₁ is the angle of incidence, θ₂ the angle of refraction from air to tempered glass, θ₃ is the angle of refraction from tempered glass to water; and θ₄ is the angle of refraction from water to perspex.

For debris slurries moving within a time interval of Δt, the accuracy of velocity is considered in two directions, along and perpendicular to the flow direction (v₁, v₂), as shown in Figure 15. The observed velocity perpendicular to the dimension from the center to the object plane can be calculated from $\frac{l_2}{\varDelta t}$, and the real velocity v_r is $\frac{l_1}{\varDelta t}$. According to the geometry, the ratio of the observed velocity over the real velocity can be expressed in terms of the distance as $\frac{L_2}{L_1}$. The observed velocity in the direction from the center to the object plane is ${v}_1=\frac{BD}{\varDelta t}$, the real velocity in this direction is ${v}_r=\frac{AC}{\varDelta t}$, and the error ratio is equal to $\frac{BD}{AC}$. Therefore, the velocity error in both directions can be calculated from:

$$ {\displaystyle \begin{array}{l}{R}_1=\frac{L_2-{L}_1}{L_1}\\ {}{R}_2=\frac{BD- AC}{AC}=\frac{v_1}{\varDelta D\left({L}_2+ v\varDelta t\right)-\varDelta D\left({L}_2\right)}-1\end{array}} $$

(A4)

where ΔD is a calculation formula of based on Eq. (A2) and serve as a function of L₂.

The farthest distance from the object plane (high-speed camera) to the center of the frame is 455 mm (frame’s center point to frame’s edge, the farthest point on the frame), and the error rate in the range of 0 to 455 mm is calculated from 0.66 to 0.81% and from 0.66 to 1.1%. This is less than 1.5% which is acceptable.

The error rate of the flume experiment is below 1% by narrowing the distance (B₃ in Figure 16) between the inner and outer flume.

Table 5 List of refractive indices of the different mediums at room temperature (20 °C) (Born and Wolf 2013)

Fig. 13

Refraction of light at the interface between two media of different refractive indices

Fig. 14

Plan view of the experimental setup for the refraction calculation

Fig. 15

Calculation of the velocity error in two directions during the debris slurry movement

Fig. 16

Calculated error rate of R₁ and R₂