Drag Coefficients of Debris Accumulations

During flood events, floating wood debris are entrained in rivers and transported downstream (Keller and Swanson 1979; Diehl and Bryan 1993; Lyn et al. 2003; Schmocker and Hager 2011). Drift can accumulate at bridge piers or superstructure: the process is known as debris damming (Yeh et al. 2014; Stolle et al. 2018). Debris dams produce higher backwater rise (Schmockers and Hager 2013; Schalko et al. 2018), exacerbate the maximum scour depth at bridge piers (Melville and Dongol 1992; Pagliara and Carnacina 2011), and cause additional hydrodynamic forces on the structure; the combination of these effects may lead to bridge failure.

Estimation of drag forces in the presence of debris requires two modeling steps: (1) definition of the shape and extension of the accumulation in front of piers and/or deck of the bridge; and (2) calculation of the hydrodynamic force over such area. In this paper, we focus on the second step; for the accumulation geometry the reader can refer to Schmocker and Hager (2011), Gschnitzer et al. (2013), Schalko et al. (2019), Panici and de Almeida (2020), and De Cicco et al. (2020).

Some literature studies have been devoted to the determination of drag force on debris accumulations (Wellwood and Fenwick 1990; Parola et al. 2000; Panici and de Almeida 2018; Mauti et al. 2020). An operational synthesis of such studies is proposed by Cantero-Chinchilla et al. (2018). In most experimental campaigns, an idealized rigid obstacle was fixed to a bridge pier or superstructure model and only few authors used a woody debris accumulation in their test programs (Panici and de Almeida 2018; Parola et al. 2000). Most of the cited works propose to estimate forces through typical drag relations (see the next paragraph for details); drag coefficients ranging from 0.5 to 3 are indicated, but no explicit relations with control variables are provided.

In this work, we present results from 165 laboratory tests over wood accumulations aimed at defining a model for drag coefficient estimation as a function of the main geometric and kinematic control parameters. The model is validated through equivalent literature data (Panici and de Almeida 2018; Parola et al. 2000); for the sake of brevity in the following these two papers will be referred as PdA18 and Par00, respectively.

The classic (aeronautic) approach evaluates drag forces aswhere $F$ = force; $\rho$ = fluid density; $A$ = projection of the obstacle over a plane orthogonal to the flow direction; and $V$ = “undisturbed” reference flow velocity. For sufficiently high Reynolds numbers (defined by the velocity $V$ and some proper length scale of the obstacle), the drag coefficient, $C_D$, should only depend on the shape of the obstacle, including its roughness. In cases involving accumulations of wooden debris we may also expect some dependence on the porosity of the obstacle. For river hydraulic applications of Eq. (1) one must also consider the effects of gravity and of flow lateral confinement, through the Froude number $Fr=V/\sqrt{gh}$ (where $g$ is gravity and $h$ is the reference water depth) and the blockage ratio $BR=A/A_h$, where $A_h$ is the flow area related to water depth $h$ and $V=Q/A_h$, $Q$ being the flow rate. By increasing $Fr$ and/or $BR$ the difference between upstream and downstream water elevations also increases, affecting the corresponding pressure distribution on the obstacle, which, in turn, determines the drag force. It is well known [Chow 1973, Eqs. (17)–(27)] that ideal limiting combinations of $Fr$ and $BR$ are defined by

Eq. (2), holding for rectangular cross sections, identifies conditions for the restricted section to become a control for the flow, with a subcrical–supercritical transition at the obstacle.

From a different perspective, for low $Fr$ and $BR$ values the water elevations upstream and downstream the obstacle are very similar, so that depth and velocity values in either position can be equivalently assumed as reference condition in (1). However, for higher values of $Fr$ and $BR$ the choice of reference conditions becomes not obvious. In this work we only consider subcritical flows; consequently, we follow the approach of PdA18, and adopt the downstream values as reference conditions, as they do not depend on the presence of the obstacle (and represent values without the obstacle). The choice of upstream values as reference is also possible (see, for example, Par00 and Qi et al. 2014): however, application of consequent drag models requires the additional calculation of the backwater across the obstacle, which introduces additional uncertainties.

Finally, an alternative approach to Eq. (1) involves decomposing the force into two distinct components: a hydrostatic pressure force associated with streamwise variation of flow depth, and a dynamic force associated with flow velocity (Par00; Qi et al. 2014). We also tested such approach, but we do not consider it advantageous in terms of both accuracy and simplicity of use (see analyses in the Supplemental Materials).

The conceptual framework is synthetized in the following points.

The resulting dimensionless relation between the drag coefficient and other variables is, therefore

We organized the experimental campaign and comparisons with literature data along this conceptual model.

Experiments were conducted at the Fantoli Hydraulics Laboratory of Politecnico di Milano, in a 7.0 m long, 0.5 m wide, and 0.60 m deep plexiglass laboratory rectangular flume with zero bottom slope. Water discharge was measured by magnetic flowmeter. The downstream condition was controlled by movable bars at the end of the flume. Water depths were measured by 11 piezometers along the flume.

A rectangular frame with a nylon thread mesh inside was located at the centerline of the flume, 4.5 m downstream from the inlet section. The frame was connected with a dynamometer positioned above the flow, allowing for measurements of the horizontal component of the force on the frame. The dynamometer was calibrated at the beginning of each experimental series (details in the Supplemental Materials); the standard deviation around the calibration fitting is reported together with measured forces in the Supplemental Materials.

The net volume of the elements composing the accumulation was a control parameter measured prior to the experiment. The debris elements were introduced into the water upstream of the obstacle and allowed to accumulate against the frame (Fig. 1); any elements that escaped from the accumulation were retrieved downstream and subsequently reintroduced into the upstream flow. At first, force was measured over the empty frame, to estimate the fraction of the force due to the frame; in this way we could evaluate the net force over the accumulation only. The idea is that forces over debris accumulation do not depend on the shape of the underlying structure (pier, deck), while forces over the portion of the structure that are exposed to flow can be estimated separately.

Experimental tests were conducted using nonuniform natural debris elements, both with and without branches. Four types of debris elements were tested, with different densities, lengths, and shape, from almost cylindrical to mildly branched: such properties are indeed relevant control parameter for the formation of the accumulation (PdA18), and indeed our results suggest that they affect porosity $n$; however, we could not detect any correlation among $C_D$ and debris type: therefore, most of the experiments were carried out mixing all the debris elements. The length of sticks was between 7 and 13 cm, while their diameter varied from a few millimeters up to 10 mm.

Each test was conducted introducing debris elements into the flume at a distance 2.5 m upstream of the frame. After the debris dam had formed and the flow conditions had stabilized, water depths and drag force were recorded. The reference (downstream) water level was set as the average among three piezometers (P2-P3-P4 in the Supplemental Materials) placed within 2.05 and 1.45 m downstream of the obstacle.

Four Gopro Hero Black cameras were used to identify the geometry of the debris accumulation. Two frontal cameras were placed 1.30 m downstream of the obstacle, one above and one below the water surface. The third camera was installed looking vertically 80 cm above the debris dam. The last camera was positioned sideways, lined up with the debris jam, 80 cm away from the side of the flume. A remote control with wireless connection allowed simultaneous shooting. Prior to experiments, a squared mesh chessboard was put in water at the position of the frame for pixel-length conversion. The geometric parameters of the debris dam were computed by composing information from the four cameras. Details on image processing and consequent derivation of the geometry of the debris accumulations are provided in the Supplemental Materials. We here limit the explanation by saying that $a$ and $b$ (see Fig. 1) are evaluated as median values of the corresponding dimensions over the cross-stream projection of the submerged portion of the dam. Thickness $c$ is evaluated as the ratio of the gross submerged volume of the accumulation divided by the area of the cross-stream projection.

The experimental investigation consists of 165 tests carried out under steady and subcritical flow conditions. Tests were conducted maintaining the downstream water depth equal to approximately 0.25 m and varying the flow discharge from 15 to $110\mathrm{l}/\mathrm{s}$ and the net debris volume from 145 to $665{\mathrm{cm}}^3$. As mentioned above, the Froude number was calculated with the reference (downstream) flow velocity, $V$, and water depth, $h$: consequently, $Fr$ was defined a priori on the basis of the boundary conditions (range: $Fr=0.06\to 0.49$). Differently, ranges for the other dimensionless variables resulted from the actual shape of the self-formed accumulation ($BR=0.07\to 0.51$; $b/a=0.83\to 4.01$; $c/a=0.21\to 2.35$; $n=0.54\to 0.89$; $Re=2.1·{10}^4\to 2.2·{10}^5$), where the Reynolds number was calculated with the reference flow velocity, $V$, and $a$ as length scale. The complete data set is available in the Supplemental Materials. Fig. 2 shows present investigated condition in the $Fr–BR$ plane, together with literature data from PdA18 and Par00.

As explained, we could only fix $Fr$ among the control parameters in Eq. (3). Single-parameter analysis was, consequently, not possible. We perform multivariate regressions of the data, testing various regression function models. As a result of the exercise, we could not detect any dependence of $C_D$ on shape parameters $b/a$ and $c/a$. We also could not enhance any dependence on the Reynolds number, but this was expected due to its limited range of variation within our tests; however, we should anyway expect a limited effect of the viscosity for a turbulent flow around a blunt rough obstacle. Finally, we also did not detect any effect of density $n$. This result is not aligned with findings of Par00, who indicated that “porosity of the debris was found as one of the most relevant variables affecting the overload of bridge piers due to debris accumulations.” It should be, however, noticed that Par00 used thin porous plates as idealized debris accumulations: it is reasonable that effects of porosity diminish with the thickness of the obstacle, which finally behaves as an impervious body. Such behavior is also confirmed by Pagliara and Carnacina (2010) with respect to effects of accumulation porosity on scour around piers.

As expected, we detected a positive correlation among the drag coefficient and both the Froude number and the blocking ratio; a good interpolation formula of our data issuggesting an asymptotic value $C_{D0}=1.5$ for vanishing $Fr$ and $BR$, and increasing drag coefficient when $Fr$ and/or $BR$ increase. Fig. 3 shows the comparison between Eq. (4) and selected subsamples of data for nearly constant values of $BR$ [Fig. 3(a)] and $Fr$ [Fig. 3(b)]. The tendency of $C_D$ to increase for increasing value of $Fr$ can also be detected in PdA18. A direct comparison with the findings of Par00 is not possible, due to their different approach to the problem.

Fig. 4(a) shows the measured–calculated correlation for the whole experimental data set. Most of the data fall within the $\pm 20\%$ deviation from the perfect agreement line. Finally, Fig. 4(b) plots contour lines for $C_D$ according to Eq. (4) in the $Fr–BR$ plane. The figure allows us to observe the sensitivity of the model to the variables. Specifically, the region enclosed by the axes and the $C_D=1.8$ line can be regarded as an area of insensitivity concerning variations of $Fr$ and $BR$. This is because the drag coefficient remains within 20% of the asymptotic limit, $C_{D0}$, in this range.

Eq. (4) has been validated against available literature data for drag forces over wood debris accumulations (559 values from PdA18 and 21 values from Par00; corresponding ranges for $Fr$ and $BR$ in Fig. 2; complete data set in the Supplemental Materials). Fig. 5(a) shows the measured–calculated comparison for forces: overall the model correctly reproduces experimental data, with a tendency of underestimating the highest values from PdA18 and Par00. The dotted lines parallel to the perfect fit line contain 98% of the data, excluding only 1% of the point on each side. The solid line represents perfect agreement. Owing to the structure of the regression model, dashed lines would also correspond to lines of perfect agreement in plots with different values for $C_{D0}$. The dotted line corresponding to $C_{D0}=2.5$ can be viewed as a (safety side) envelope of experimental data. Notice that when present data only are considered, such an envelope can be restricted to $C_{D0}=1.9$.

Some statistics for the regression are reported in Table 1. Eq. (4) reproduces data with negligible distortion on the average. Due to its structure, the model cannot explain $C_D$ experimental values lower than 1.5; maximum calculated values also significantly differ from experimental ones. However, the mean percentage absolute error is relatively low (16%).

In addition to Eq. (4) we tested the performance of a zero-variate model, namely $C_D$ = constant = $C_{D0}=1.5$. This assumption neglects the amplification of the drag coefficient for increasing $Fr$ and $BR$. Results from the exercise can be appreciated in Fig. 5(b) and the last line of Table 1. Overall, the result is acceptable, with an obvious tendency to underestimate the largest values, which can be recognized both from the plot and from the average statistics in Table 1. The mean percentage absolute error is only 30% higher than that derived from Eq. (4). Having neglected the corrective effects of $Fr$ and $BR$, a value of $C_D$ as large as 3.1 is required for a 99% envelope of the experimental data.

From a quantitative point of view, a constant $C_D$ model, possibly slightly higher than 1.5, does not perform significantly worse than Eq. (4), being thus preferable for its simplicity. However, we claim that our regression model should be preferred, as it indicates the role of Froude number and blockage ratio as limiting conditions of Eq. (2) are approached.

From a different perspective it should be noticed that, in practical application, the proper definition of expected dimensions of the accumulation (and the related probability) is the most critical issue for design and/or verification scenarios (see introduction for references). It is reasonable to assume that uncertainties related to the characteristics of the accumulation are much larger than those related to the calculation of the drag force. Therefore, we propose to use Eq. (4) with $C_{D0}=1.5$, which represents a best fit estimate for forces, while safety factors may be introduced by assuming safety side evaluations of the dimension of the debris accumulation.

The aim of this work is to provide an operational tool to compute drag forces on (given) debris accumulations, typically over bridge piers and/or superstructures. We structured laboratory experiments within a conceptual framework, found a suitable regression of the data, and validated the result over two independent literature series. Tests were limited to subcritical flow conditions. Findings can be synthetized by the following points.

The main limitation of present study is that it does not provide indications for flows close to the transition from subcritical to supercritical conditions. In such a situation, however, the proposed conceptual model fails, as the flow is no longer controlled by downstream parameters. On the other side, it is not trivial to structure an experimental campaign with self-formed accumulations under such conditions.

All data, models, and code generated or used during the study appear in the published article.

Authors acknowledge the contributions of Marco Negri and Luca Martinelli for the support in designing and running the experimental campaign. This study was carried out within the RETURN Extended Partnership and received funding from the European Union Next-GenerationEU (National Recovery and Resilience Plan–NRRP, Mission 4, Component 2, Investment 1.3–D.D. 1243 2/8/2022, PE0000005).