Deciphering controls for the impact of geophysical flows on a flexible barrier: Insights from coupled CFD-DEM modeling

. Geophysical flows impacting a flexible barrier can create complex flows and solid-fluid-structure interactions, which are challenging to quantify and characterize towards a unified description. Here, we examine the common physical laws of multiphase, multiway interactions during debris flows, debris avalanches and rock avalanches against a flexible barrier system using a coupled computational fluid dynamics and discrete element (CFD-DEM) method. This model captures essential physics observed in experiments and fields. The bi-linear, positive correlations are found between peak impact load and Fr or maximum barrier deflection, with inflection points due to the transitions from trapezoid-to triangle-shaped dead zones. Our findings quantitatively elucidate how flow materials (wet versus dry) and impact dynamics (slow versus fast) control the patterns of the identified bi-linear correlations. This work offers a physics-based reference and insights for improving widely-used impact solutions for geophysical flows against flexible barriers.


Introduction
Flexible barriers are increasingly used to mitigate debris flows, debris/rock/snow avalanches, and rockfalls [1][2][3]. Determining the impact load exerted on a flexible barrier is a fundamental issue in hazard mitigation. Still, it has not been directly measurable in experiments or fields over the past three decades [4][5][6]. The difficulty is rooted in capturing the multiphase, multiway flowbarrier interactions, where many mechanisms can work simultaneously. Understanding such impact is thus of deep engineering and scientific importance.
The impact loads of geophysical flows on flexible barriers are typically estimated by simplified [2,3,7] and Froude-number-related analytical [8,9] solutions. Nonetheless, impact loads estimated by simplified solutions in physical tests lack scrutiny but are frequently referred to as reliable data for calibrating Frrelated analytical solutions [8,9]. Alternatively, many numerical methods, including continuum-based [10], discrete-based [11] and coupled frameworks [12,13], have been developed to explore the impacts on flexible barriers by geophysical flows. However, simplifications of the solid-liquid flow dynamics [10,11] and 3D nonuniform, permeable flexible barriers [12,13] have prevented a deeper understanding of the underlying relations and mechanisms behind the flow-barrier interactions.
Towards a unified description of the impacts when geophysical flows of variable natures against a flexible barrier system, this study scrutinizes underlying relations and mechanisms for widely-used solutions based on direct, numerical measures of flow-barrier forces and barrier load-deflection relations. * Corresponding author: mfguan@hku.hk

Methods and model setup
A coupled computational fluid dynamics and discrete element (CFD-DEM) method is employed to probe the dynamics during geophysical flows against a flexible barrier. A flexible barrier is modeled by DEM (Fig. 1a), while a debris flow is simulated as a mixture of discrete particles and a continuous slurry by DEM and CFD (Fig.  1d), respectively. A two-way coupling scheme offers a unified way to describe the solid-liquid interactions in a debris flow and between barrier components and debris liquid. The motions of a particle are governed by Newton's equations, and the fluid is controlled by the locally-averaged Navier-Stokes equation for each fluid cell with the finite-volume method. Further details can be found in our previous work [8,14]. This method has been benchmarked with classic geomechanics problems [14] and various engineering conditions [13,15,16,17].

Modeling a flexible ring net barrier
A flexible barrier typically consists of a ring net, brake elements, and cables ( Fig. 1a-upper). It is modeled by assembling the main ring net, ten brake elements, and five supporting cables ( Fig. 1a-lower). The bottom and lateral edges of the top and middle cables are fixed, mimicking the anchored boundaries. The Parallel Bond Model [18] implemented in DEM is employed to model all barrier components as connected nodal particles. For example, interlocking rings are idealized as connected nodal particles (Fig. 1b). Fig. 1c displays the local deformations of cable-ring-ring connections, consistent with field observations [3]. (c) present a comparison between field photos and numerical snapshots for the barrier, interlocking rings and local deformation characteristics, respectively. (d) shows a debris flow impacting the barrier, where split views display debris fluid and gap-graded particles (back half-space) as well as fluid streamlines and interparticle contacts (front half-space).

Model setup and simulated dynamics
We perform systematic simulations of Debris Flows (DF), Debris Avalanches (DA) and Rock Avalanches (RA) impacting a flexible barrier. A broad range of Fr (0.5 ~ 8.7) is produced with a pre-impact flow depth of ~ 0.3 m and varying initial velocities int = 0.5 m/s ~ 14 m/s. Fig. 1d shows a typical debris flow impacting a reduced-scale flexible barrier (0.9m-high, 1.8m-wide) constructed on an inclined channel with a slope of 20°, capturing critical physical processes, such as flow climbing, silting and retaining, the cable-ring-ring sliding, dewatering and small particles passing through.
Details of model geometry, debris-flow materials, barrier models and simulation conditions can be found in our newly published paper [8], which has complied a unified design diagram for flexible, slit and rigid barriers. By contrast, this work focuses on elucidating how flow materials and dynamics affect underlying relations for impacts of geophysical flows on a flexible barrier. Herein, the initial heights of viscous slurries in DF, DA and RA cases are set to 0.3 m, 0.15 m and 0 m, respectively. Clear differences in flow redirection, separation and overtopping dynamics, and barrier responses among DF, DA and RA cases can be observed from Videos S1, S2 and S3 that can be permanently archived at https://doi.org/10.5281/zenodo.6779488.  the total impact load b . This enables direct measure that delineates load components to b from individual debris-flow phases. Fig. 2d indicates that the peak value of the fluctuating solid-barrier contact forces (183.2 kN) is around nine times larger than that of the smooth fluidbarrier interaction forces (20.3 kN, see inset in Fig. 2d). Thus, s−b is the dominant debris-flow load contributor on a flexible barrier. Moreover, solid particles also trigger the first peak during the frontal impact (Fig. 2d), wherein the peak barrier load commonly occurs for rigid countermeasures [2,8]. In contrast, b Peak appears during overtopping for a flexible barrier (Fig. 2d), wherein flowing layer coexists with dead zones [16,19] (Fig. 2c). Therefore, b

Flow-barrier interactions and forces
Peak should be calculated as the sum of loads from the flowing layer and dead zone, especially in designing multi-level flexible barriers.   [8,9,15], with inflection points due to the shifts from trapezoid-to triangle-shaped dead zones (Figs. 3b ~ 3d). The boundaries of dead zones (runup surfaces) are roughly determined according to a velocity threshold [19], and details are presented in supplementary Fig. S1 (https://doi.org/10.5281/zenodo.6779488). We refer the shifts of dead zones to identify the slow-to-fast transitions of flow impact dynamics, which occurs at a higher Fr with a larger solid fraction of impinging flows. Further, the slope of Frb Peak relations in DF cases under fast impact dynamics is 7.7 times that under slow impact dynamics, whilst this ratio is around 3 for RA cases. Dry flows undergo a smaller bulk density and a longer energy-dissipative runup surface than wet flows (Figs. 3b and 3d). Further, grain shear stress is considered more effective in energy dissipation than fluid viscous shearing [20], and the viscous slurry may enhance flow velocity and impact pressures by decreasing the inter-particle friction [21] and dampening particle collisions [22]. Our results highlight that the discriminants of flow types (DF, DA or RA) and impact dynamics (slow or fast) are crucial for predicting Frb Peak relations, thereby enable potential improvements of the Fr-related analytical solutions.

Barrier load-deflection relations
Understanding barrier load-deflection phenomena is vital for evaluating peak impact, barrier deformation, and retainment capacity for practical designs. Notably, the spring solution given by b = b n h are investigated in the estimates of impact loads [3,7], where b n and h represent the equivalent barrier stiffness and maximum barrier deflection in the flow direction. We also present the complex bh relations in supplementary

Conclusions
This work presents systematic simulations of a flexible barrier system against geophysical flows of variable natures to identify underlying relations and mechanisms of the multiphase, multiway interactions. The employed fluid-solid coupling model captures essential physics observed in experiments and fields.
Physics-based numerical measures of flow-barrier forces and barrier load-deformation behaviors reveal the bi-linear, positive relations underpinning widely-used impact solutions, with inflection points caused by the transitions from trapezoid-to triangle-shaped dead zones. Specifically, the identified correlations relate the peak impact load to Froude-number or maximum barrier deformation. Our findings facilitate determining the impact loads of geophysical flows on flexible barriers based on flow materials (wet versus dry) and impact dynamics (slow versus fast).
Future work may explore the existence of similar relationships regarding the complexity of geophysical flows, including huge boulders [1], varying flow heights [3], and erosion [23].