Characteristics of wake morphology during debris flow when passing a cylindrical obstacle

Debris flows, which are prevalent geomorphic processes in mountainous regions, occur primarily due to the essential condition of steep slopes. Consequently, they are predominantly classified as supercritical gravity flows (Chen , 2015; Iverson, 1997; and Takahashi, 2009). When a debris flow meets an obstacle, its interaction resembles that of a supersonic gas flow confronting an obstacle in gas dynamics, given that the debris flow's velocity significantly surpasses the wave speed within the fluid. This interaction can lead to the emergence of either attached or detached shock waves (Amarouchene and Kellay, 2006). Earlier research on shock waves produced upstream of obstacles was concentrated in aerodynamics (Heil , 2004). However, with the broader application of relevant theories, the formation of shock waves during interactions between granular flows and supercritical open channel flows with obstacles has garnered increased scientific attention (Cui, 2019; Roux , 2022).

Savage (1979) first introduced the idea of shock waves in granular flows during his research on steady jumps upstream of a dividing plate. When a granular flow interacts with an obstacle, disturbances in the granular medium propagate through collisions between macroscopic particles. This implies that the sound speed within the granular flow is significantly lower in the air and less than the standard flow velocities of natural gravity flows. As a result, the shock waves' formation in granular flow–obstacle interactions is readily observable (Khan , 2020; Rericha , 2001; and Savage, 1988). Further studies have shown that a typical detached shock wave in granular flows exhibits parabolic shape, consisting of a central stagnant region surrounded by a moving grain envelope, resulting in nonlinear velocity increase in the radial direction (Amarouchene and Kellay, 2006).

Gray (2001, 2018) conducted a series of physical experiments on granular flows interacting with varied obstacles to produce shock waves. He adapted the analogous shallow water equations to simulate three-dimensional depth-averaged flows in complex terrains and subsequently compared the theoretical models with laboratory experiments. The outcomes confirmed the theory's capacity to accurately delineate the formation of normal shocks, oblique shocks, dead zones, and granular vacua, thereby establishing a comprehensive theory of shock waves in granular flows (Gray and Ancey, 2011; Gray and Cui, 2007; Gray and Edwards, 2014; and Gray , 2003, 2015).

When supercritical open channel flow encounters an impermeable barrier, observers typically note two flow patterns: a detached hydraulic jump or a wall-jet-like bow wave (Mignot and Riviere, 2010; Mignot , 2016; and Riviere , 2017). The transition between these patterns is governed by the ratio of incoming flow depth to the obstacle's characteristic size on its upstream face and by the Froude number (a dimensionless parameter that characterizes the relative magnitude of inertial force and gravity in a fluid) of the incoming flow. Riviere formulated a critical criterion for this mode transition, grounded in theoretical analysis and experimental observations (Riviere , 2017). Generally, for supercritical flows, when the obstacle is slender or the flow depth substantial, a wall-jet-like bow wave forms on the upstream face of the obstruction, directing the flow around the obstacle in a vertical jet pattern (Mignot and Riviere, 2010). Conversely, for wider and shallower flows, an independent hydraulic jump develops on the obstacle's upstream face (Roux , 2022).

Beyond granular and open channel flows, debris flows serve as typical examples of supercritical gravity flows in nature (Song , 2021, 2021b; Yan , 2023). The most frequent cylindrical obstructions that debris flows encounter include bridge piers and forests. Numerous investigations in the field and laboratories have delved into flow impact characteristics during debris flow passing a cylindrical obstacle (Cui , 2023; Wang , 2018a, 2018b, 2020). Nevertheless, research concerning the wake morphology of debris flow passing a cylindrical obstacle remains limited. The debris flows' intricate composition makes their interactions with obstacles more complex than those of granular or open channel flows. Thus, evaluating shock waves' formation characteristics during debris flow passing a cylindrical obstacle is vital for comprehending interactions between cylindrical obstructions, such as forests, pile-lined dams, bridge piers, and debris flows.

In order to quantitatively study the formation characteristics of wake morphology during the debris flows passing a cylindrical obstacle, this research employs a custom experimental apparatus: a cyclic flume designed for debris flows allows long-term stable circulation of debris flow through the use of a slurry pump and pipeline. This tool allows the debris flows to maintain a relatively stable motion, facilitating the quantitative measurement of pertinent parameters of wake morphology. The factors influencing the characteristics of cylindrical flow traces are quantitatively assessed, and theoretical derivations are utilized to introduce a predictive formula for wake morphology during debris flow passing a cylindrical obstacle. This formula is later verified using the experimental data.

The study investigated the characteristics of wake morphology during debris flow when passing a cylindrical obstacle. A cyclic flume designed for debris flows facilitated this examination [Fig. 1(a)]. The flume measured 6 m in length and 0.5 m in width. The cylindrical model was situated 4.5 m downstream from the upstream outlet, a point at which the debris flow had achieved a steady state. The design of the cyclic flume ensures a steady flow for an extended period, eliminating the need for repetitive tests in traditional water tanks to reduce errors. The width of the flume ensures that the sidewalls do not interfere with the formation of wake morphology, facilitating an unobstructed formation process. The stainless steel cylinder model was anchored at the flume's base and stood 15 cm tall. Three models were employed with 1, 2, and 3 cm diameter. An ultrasonic depth sensor (U-GAGE T30UX) was positioned upstream of the model to measure the flow's depth. A GoPro10 and two Sony cameras were strategically installed above the model, at the model's side and at the flume's tail end to document the wake morphology characteristics.

The Froude number was controlled within the range observed in natural field debris flows (Phillips and Davies, 1991), addressing the scaling issue between laboratory experiments and field conditions.

The material properties of debris flows constitute a second experimental similarity issue in addition to the previously mentioned scale issue. Natural debris flows belong to non-Newtonian fluids and exhibit yield stress, which differs from Newtonian fluids (Iverson, 1997). The used solid material was collected from a viscous debris flow channel in Pingwu County, Sichuan, China. The test cylinder's diameter and the slurry pump's bearing capacity are limited. Particles larger than 10 mm were removed using a sieve, and the remaining material was utilized for the experiments.

Two different solid volume fractions of debris flow were used in this experiment. The solid volume fraction is denoted as vs and represents the number of solid components in the total volume of the debris flow mixture. The liquid component of the mixture is characterized by the liquid density $ρ f$⁠, effective viscosity $μ$⁠, and liquid volume fraction $v f$⁠. It is widely accepted that both the solid and liquid components in debris flows are incompressible; therefore, $v s+ v f=1$⁠. Figure 2 depicts the grain size distribution curve and rheological characteristics of the debris flow samples.

The liquid phase of a debris flow consists of water and solid particles smaller than 0.0625 mm (Iverson, 1997). In this study, the proportion of solid particles smaller than 0.0625 mm was determined using the gradation curve. The bulk density of the liquid phase was calculated, and the corresponding debris flow liquid phase was prepared by adding water. Subsequently, the relationship between shear stress and shear rate in the liquid phase of the debris flow mixture was measured using a rheological test device. Figure 2(b) illustrates that the rheological characteristics of the liquid phase resemble those of a Newtonian fluid. To ensure more accurate determination of effective viscosity within the shear rate range relevant to debris flow, data points with shear rates greater than 50 were fitted. Rheological parameters were determined using a rheometer fitted with the Bingham model $τ= τ c+μ γ ̇$ to obtain the effective viscosity, μ (O'Brien and Julien, 1988; Wang , 2020).

Figure 3 illustrates the wake morphology of the debris flow surrounding a cylindrical obstacle. Experimental images reveal the absence of a distinct water leap phenomenon upstream of the obstacle. Upstream, a part of the flow ascends the obstacle, generating a rolling edge at the pinnacle and exhibiting reverse spillage. Beyond the wall jet, a different section flows parabolically around the cylinder's sides, eventually settling on the debris flow's surface and forming lateral jets. A side view reveals that the parabolic trajectory's highest point lies above the wall jet's fixed point. Moreover, lateral jets formed on either side of the cylinder appear nearly symmetrical. This flow pattern aligns well with the wall-jet-like bow wave as described by Riviere (2017).

A portion of the upstream flow climbs along the obstacle to form a wall jet, whose maximum height is represented by h_jet. In order to better investigate the factors influencing the climbing height and expand its applicability, h_jet is nondimensionalized using the momentum-based height. This is achieved by defining the nondimensional height as $h jet *= h jet/ v R 2 / 2 g$⁠. Simultaneously, two significant dimensionless numbers, Fr and h/R (the ratio of flow depth to obstacle diameter), are selected as influencing factors. Figure 4 reveals the relationship between these dimensionless numbers.

Figure 4(a) shows the relationship between Fr and $h jet *$⁠. An increase in Fr correlates with a decrease in $h jet *$⁠, indicating a consistent trend. However, when examining h/R in isolation, both high-density [Fig. 4(b), blue] and low-density debris flows [Fig. 4(b), red] display a decreasing relationship with h/R. This variation is potentially attributable to different fluid properties. Further discussion will probe the effects of the debris flow's internal viscous forces on its wake dynamics, particularly noting that as Fr and h/R rise, more material is expelled through the lateral jets during interactions with the obstacle.

The lateral jet, which serves as the upstream flow direction, is crucial in understanding the wake zone's extent behind cylindrical obstacles. Such knowledge is pivotal when assessing the protection and influence of clustered obstacles like forests, bridge piers, and pile structure arrays. Section II B introduced two key parameters, the wake's width W and the angle θ between the lateral jet and the central axis, to elucidate the lateral jet's characteristics. Figure 5 depicts the relationship between Fr and these parameters. As Fr increases, the angle θ between the lateral jet and the central axis starts to reduce [Fig. 5(a)]. Conversely, the wake width W displays an upward trend with a rising incoming Fr. Significantly, as observed in Fig. 5(b), similar Fr values coupled with a larger obstacle diameter lead to an expansion in the wake width W.

Figure 6 delineates the flow pattern classification diagram pertinent to the debris flow in the experiment. The red line represents Eq. (4). It can be seen that every experimental data point concerning the debris flow situates above this red line, indicating that they fall within the range of a wall-jet-like bow wave. This is consistent with the observed experimental phenomenon.

Figure 8 indicates the theoretical projections with the empirical values of θ and W. Due to certain simplifications applied during calculations, disparities exist between the theoretical and empirical values. Nevertheless, the results demonstrated that predicting the morphology of the debris flow cylinder is feasible. It is also evident that predictions are more accurate for low-density debris flows than for high-density ones.

Error assessments were conducted on three pivotal formulas related to the morphological prediction of the wall-jet-like bow wave, with the outcomes depicted in Fig. 9. As illustrated in Fig. 9, with an increase in Fr, the discrepancies in the formulas for $h jet *$⁠, θ, and W diminish. Figure 9(a) shows the calculation errors for $h jet *$⁠, with the MRE nearing zero. Figure 9(b) displays the nuanced calculation discrepancies for θ. As Fr increases, the RE gradually shifts from negative to zero and then to positive. For Fr = 6–8, the errors are mostly below 10%. Overall, the predicted values for θ tend to be underestimated, with an MRE of 2.98%. Although the error of W also decreases with the increase in Fr [Fig. 9(c)], the overall discrepancy remains positive, implying that calculated outcomes typically surpass measured ones. The most substantial errors are often related to high-density debris flow samples. The subsequent section will delve deeper into the influence of debris flow properties on these errors.

When $N Bag$ exceeds 200, the dominant stress in debris flow mixture is particle collision stress, and the influence of viscous shear stress increases with the decrease in $N Bag$ (Bagnold, 1954). Figure 10 shows the effect of viscous forces in debris flows on computational outcomes. For low-density debris flows with elevated N_Bag, the errors in the calculation formulas are primarily within 10%. The calculated values deviate from the actual values when N_Bag decreases, indicating an increase in the dominance of viscous stress within the debris flow. Specifically, for $h jet *$ and W, the calculated values tend to be overestimated. For θ, the calculated value is often underestimated at low volume and overestimated at high volume weight. Given the upswing in viscous forces within the debris flow and an increase in the solid volume fraction, resistance during the wall-jet ascent intensifies. This dynamic results in empirical outcomes that are lower than their computational counterparts. Similarly, heightened density and the resultant surge in viscous stress alter the rheological characteristics of the debris flow. A growth in yield stress culminates in variations in the gravity wave's propagation speed atop the debris flow, a factor not considered during calculations. Thus, it is imperative to factor in the influence of the rheological properties of debris flow on the critical wave speed to enhance the prediction accuracy of debris flow cylinder wake morphology. Future endeavors should address this concern more methodically, potentially through techniques such as numerical simulations. The impact of debris flow rheological properties on the calculation error of θ subsequently causes an increase in the calculation error of W under high-density conditions.

In conclusion, predicting the wave morphology caused by debris flows around cylindrical objects is crucial for disaster assessment and understanding the extent of these flows' impact on surrounding areas. This study, conducted through experiments in a recirculating flume, reveals that the flow pattern of debris flow around a cylindrical object resembles a wall-jet–like bow wave, without the formation of a hydraulic jump. The primary factors influencing this pattern are the Froude number and the ratio of flow depth to the cylinder's radius. The height of the wall jet decreases with increasing Froude number and depth-to-radius ratio. This relationship can be well approximated by the equation $h jet *=1.48 h R − 0.69F r 0.31 h / R − 1.47$⁠. Additionally, the morphology of the lateral jet is characterized by parameters θ and W, where θ decreases, and the scouring zone width increases with a rising Froude number. The relationship between these parameters can be accurately predicted using equations: $sin θ= g h v R$ and $W= 4 v x g v R 2 − v x 2 − F r 2 ⋅ v x 2$⁠. However, the accuracy of predicting the wall-jet-like bow wave morphology diminishes with increasing internal viscous stress of the debris flow, necessitating further exploration of the impact of the flow's rheological properties on its critical wave velocity. These predictions not only deepen our understanding of debris flow behaviors but also provide valuable guidance for the design and protection of infrastructural structures such as protective forests, pile forests, and bridge piers (Figs. 11 and 12).

This study was supported by the National Natural Science Foundation of China (Grant No. 41925030) and the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDA23090403).

The authors have no conflicts to disclose.

Wenrong Cui: Methodology (equal); Writing – original draft (equal). Jiangang Chen: Project administration (equal); Writing – review & editing (equal). Wanyu Zhao: Methodology (equal). Xiaoqing Chen: Funding acquisition (equal); Supervision (lead); Writing – review & editing (equal).

The data that support the findings of this study are openly available in Zenodo at https://doi.org/10.5281/zenodo.8125848, reference Cui (2023).

Side view of the cylindrical flow of debris flow (D and V refer to dilute and viscous, respectively, corresponding to bulk densities of 1680 and 1860; 10, 20, and 30 refer to the cylinder diameter in millimeters; and 1–5 refer to different groups of Fr values (Table I). Top view of the cylindrical flow of debris flow (D and V refer to dilute and viscous, respectively, corresponding to bulk densities of 1680 and 1860; 10, 20, and 30 refer to the cylinder diameter in millimeters; and 1–5 refer to different groups of Fr values (Table I).