# The advancing wave front on a sloping channel covered by a rod canopy
following an instantaneous dam break

Elia Buono<EMAIL_ADDRESS>Dipartimento di Ingegneria dell’Ambiente, del
Territorio e delle Infrastrutture, Politecnico di Torino, Torino, Italia
Gabriel G. Katul<EMAIL_ADDRESS>Department of Civil and Environmental
Engineering, Duke University, Durham, North Carolina, USA  Davide Poggi
<EMAIL_ADDRESS>Dipartimento di Ingegneria dell’Ambiente, del
Territorio e delle Infrastrutture, Politecnico di Torino, Torino, Italia

###### Abstract

The drag coefficient $C_{d}$ for a rigid and uniformly distributed rod canopy
covering a sloping channel following the instantaneous collapse of a dam was
examined using flume experiments. The measurements included space $x$ and time
$t$ high resolution images of the water surface $h(x,t)$ for multiple channel
bed slopes $S_{o}$ and water depths behind the dam $H_{o}$ along with drag
estimates provided by sequential load cells. Analysis of the Saint-Venant
Equation (SVE) for the front speed using the diffusive wave approximation lead
to a front velocity $U_{f}=\sqrt{\Gamma_{h}2g\phi_{v}^{\prime}/(C_{d}mD)}$,
where $\Gamma_{h}=-\partial h/\partial x$, $g$ is the gravitational
acceleration, $\phi_{v}^{\prime}=1-\phi_{v}$ is fluid volume fraction per
ground area, $\phi_{v}=m\pi D^{2}/4$ is the solid volume fraction per ground
area, $m$ is the number of rods per ground area, and $D$ is the rod diameter.
An inferred $C_{d}=0.4$ from the $h(x,t)$ data near the advancing front
region, also confirmed by load cell measurements, is much reduced relative to
its independently measured steady-uniform flow case. This finding suggests
that drag reduction mechanisms associated with transients and flow
disturbances are more likely to play a dominant role when compared to
conventional sheltering or blocking effects on $C_{d}$ examined in uniform
flow. The increased air volume entrained into the advancing wave front region
as determined from an inflow-outflow volume balance partly explains the
$C_{d}$ reduction from unity.

Keywords canopy drag; dam break; diffusive wave approximation; drag reduction;
Saint Venant Equation.

††preprint: AIP/123-QED

##  I Introduction

The sudden release of water following an instantaneous collapse of a dam has
received much research attention in hydrology (e.g. overland flow), ecology
(e.g. rapid inflow into wetlands or a marsh), hydraulics (e.g. flood routing),
and coastal engineering (tsunami on coastal plains) for well over a century
(Ritter, 1892; Whitham, 1955; Hunt, 1982; Hogg and Pritchard, 2004; Daly and
Porporato, 2004; Chanson, 2006; Begnudelli and Sanders, 2007; Ajayi, van de
Giesen, and Vlek, 2008; Carrivick, 2010; Chanson, 2009; Thompson  _et al._ ,
2011; Ma and Fu, 2012). The hydrodynamics describing the unsteady and shallow
nature of such flows are summarized by the Saint-Venant equation (SVE)
introduced in 1871 (de Saint-Venant, 1871; Hager, Castro-Orgaz, and Hutter,
2019). For a rectangular prismatic section of constant width $B$, the SVE in
their one-dimensional form are given by two partial differential equations:
the continuity and the area-averaged momentum balance. For the dam break
problem, the SVE is expressed as (Lighthill and Whitham, 1955; Whitham, 1955;
French, 1985; Hogg and Pritchard, 2004; Bellos and Sakkas, 1987)

$\frac{\partial h}{\partial t}+\frac{\partial Uh}{\partial x}=0,$ (1)

and

$\frac{\partial U}{\partial t}+U\frac{\partial U}{\partial
x}+g\left(\frac{\partial h}{\partial x}+S_{f}-S_{o}\right)=0,$ (2)

where $x$ is the longitudinal distance from the dam location with $x=0$ set at
the dam location, $t$ is time with $t=0$ set to the instant the dam is
removed, $h(x,t)$ is the water depth, $U(x,t)$ is the area-averaged or bulk
velocity, $g$ is the gravitational acceleration, $S_{o}$ is the bed slope, and
$S_{f}$ is the friction slope that is unknown in the SVE. It is the closure
model for $S_{f}$ in the SVE that frames the scope of the work. As early as
1892, analytical results for the dam-break problem were derived when $S_{f}=0$
and $S_{o}=0$ (Ritter, 1892; Ancey  _et al._ , 2008). The inclusion of finite
$S_{o}$ but keeping $S_{f}=0$ revises the classical Ritter solution to
(Chanson, 2009; LaRocque, Imran, and Chaudhry, 2012; Melis  _et al._ , 2019)

$U(x,t)=\frac{2}{3}\left(\frac{x}{t}+U_{o}+S_{o}gt\right),$ (3)

and

$h(x,t)=\frac{1}{9g}\left(2U_{o}-\frac{x}{t}+\frac{1}{2}S_{o}gt\right)^{2},$
(4)

where $U_{o}=\sqrt{gH_{o}}$ is the initial celerity speed. Here, the initial
conditions to the SVE are a dry channel bed. When $S_{o}=0$ and $t>0$,
equations 3 and 4 can be expressed in a dimensionless and compact form as
(Ritter, 1892)

$h_{n}=\frac{1}{9}\left(2-u_{n}\right)^{2},$ (5)

where $h_{n}=h/H_{o}$ is the dimensionless water depth,
$u_{n}=(x/t){(U_{o})}^{-1}$ is the dimensionless wave speed,
$t_{n}=t(H_{o}/g)^{-1/2}$ is dimensionless time, and $x_{n}=x/H_{o}$ is
dimensionless longitudinal position downstream from the dam. Revisions to
these results are numerous and include a gradual breaching of the dam (Capart,
2013), lateral contractions (Kocaman and Ozmen-Cagatay, 2012; Kocaman  _et
al._ , 2020a; Cozzolino  _et al._ , 2017), asymmetric geometry Ferdowsi,
Nemati, and Farzin (2021); Wang  _et al._ (2019, 2015a, 2020), steep $S_{o}$
(Ancey  _et al._ , 2008; Wang and Pan, 2015), and introduction of bends along
the channel (Frazão and Zech, 2002). Perhaps the most studied revision to the
inviscid solution is finite wall-friction (Dressler, 1952; Whitham, 1955;
Hunt, 1982, 1984a, 1984b; Chanson, 2009; Wang and Pan, 2015; Nielsen  _et al._
, 2022). In prior applications with finite wall friction, a resistance
formulation must be introduced to link $S_{f}$ to the variables being modeled
by the SVE ($h$ or $U$). The primary restriction imposed on such models are
recovering outcomes based on locally steady and uniform flow conditions. For
these idealized flow conditions, the most common formulation to parameterize
$S_{f}$ is Manning’s formula (Manning, 1891) that assumes a constant roughness
coefficient (=$n$) and links $S_{f}$ to the sought variables using

$S_{f}=\left[\frac{2gn^{2}}{R_{h}^{4/3}}\right]\frac{{U}^{2}}{2g},$ (6)

where $R_{h}=A_{c}/P_{w}$ is the hydraulic radius, $A_{c}=Bh$ is the cross-
sectional area of the flow, $P_{w}=B+2h$ is the wetted perimeter, and $n$ is
Manning’s roughness coefficient (in s m-1/3 when SI units are used). This
formulation is common given the accrued information on $n$ over the years for
different surface cover types (Chow, 1959; French, 1985). Also, theoretical
justification for equation 6 using turbulence theories and the energy cascade
are also emerging (Gioia and Bombardelli, 2001; Bonetti  _et al._ , 2017). In
some applications, the kinematic wave approximation is invoked whereby the
momentum balance is reduced to $S_{f}=S_{o}$ (Woolhiser and Liggett, 1967).
Invoking this approximation and inserting equation 6 into the continuity
equation, a broad class of self-similar solutions can be derived and
connections between the dam-break equations and the Fokker-Planck equations
have already been proposed (Daly and Porporato, 2004). However, extending such
solutions to the general SVE remains fraught with difficulties.

Controlled laboratory experiments on this topic remain also limited despite
the undisputed societal significance of the dam break problem (Chanson, 2009;
LaRocque, Imran, and Chaudhry, 2012; Ozmen-Cagatay and Kocaman, 2011). Some
laboratory studies considered (i) single isolated obstacles (Soares-Frazão and
Zech, 2008; Del Gaudio  _et al._ , 2022; Di Cristo  _et al._ , 2020; Kamra
_et al._ , 2019; Ansari, Khavasi, and Ghazanfarian, 2021; Vosoughi  _et al._ ,
2022; Kocaman  _et al._ , 2020b; Huo and Liu, 2023; Tan  _et al._ , 2023),
(ii) the initial stages of an instantaneous dam-break over smooth surfaces
(Stansby, Chegini, and Barnes, 1998; Ozmen-Cagatay and Kocaman, 2010; Espartel
and Manica, 2021), (iii) the use of polymer additives for inducing reductions
in $S_{f}$ (Jánosi  _et al._ , 2004), (iv) geometric alterations to the
channel section such as contractions, expansions, embankments and bends
(Frazão and Zech, 2002; Kocaman and Ozmen-Cagatay, 2012; Kocaman  _et al._ ,
2020a; Di Cristo  _et al._ , 2018), and (v) the role of sediments and movable
beds on $S_{f}$ (Abderrezzak, Paquier, and Gay, 2008; Zech  _et al._ , 2008;
Vosoughi  _et al._ , 2020; Biswal, Moharana, and Agrawal, 2018; Khosravi  _et
al._ , 2021; Fent, Zech, and Soares-Frazão, 2019; Qian  _et al._ , 2018).
However, the dam-break problem for channels covered by vegetation remains
under-studied with less than a handful being reported (Melis  _et al._ ,
2019). When the channel is vegetated, explicit inclusion of distributed drag
into the SVE is necessary as energy losses are no longer related to wall
friction (Nepf, 1999; Wu, Shen, and Chou, 1999; Lawrence, 2000; Green, 2005;
Huthoff, Augustijn, and Hulscher, 2007; Poggi, Krug, and Katul, 2009; Huai
_et al._ , 2009; Kothyari, Hayashi, and Hashimoto, 2009; Nepf, 2012; Etminan,
Lowe, and Ghisalberti, 2017).

The work here explores experimentally the effects of canopy drag on the
physics of the advancing front following the instantaneous removal of a dam
for varying static water depth $H_{o}$ behind the dam and $S_{o}$. The canopy
used is composed of staggered rigid cylinders covering the flume base
downstream from a dam where $S_{o}$ is varied from $S_{o}=0$ to 3$\%$.
Attention is drawn to the role of canopy drag reduction mechanisms as the
advancing front traverses the rod canopy. Thus, the two experimentally
controlled variables to be manipulated here are $S_{o}$ and $H_{o}$.
Comparison with a prior study (Melis  _et al._ , 2019) where the rod density
was much higher is also presented.

Figure 1: The experimental setup for the dam break problem. On the top left, a
picture of the channel downstream of the cofferdam and a picture of the wooden
cofferdam are featured. On the top right, three images from the early stages
of the dam break wave propagation ($H_{o}=$ 0.15m, $S_{o}=$ 0$\%$) are shown
along with the detected free surface (blue dots). On the bottom, side view of
the whole experimental facility (on the left) and longitudinal section view
(on the right) with the most meaningful dimensions

##  II Theory

The setup considered here is for an instantaneous removal of a dam that
results in a flood wave propagating downstream along a sloping rectangular
channel. The channel is covered by a uniform rigid cylindrical rod canopy that
acts to remove energy and momentum from the advancing flood wave. The
cylinders are staggered and presumed to have a uniform diameter $D$ and height
$h_{c}/h(x,t)>1$ after the dam break. The goal is to describe the front
position $x_{f}$ and front speed $U_{f}$ downstream from the dam for various
combinations of control variables $S_{o}$ and $H_{o}$. To arrive at an
expression for $S_{f}$ that accounts for the presence of cylinders to be used
in the SVE, a starting point is to consider a locally steady-uniform flow
within a canopy. The canopy is presumed to be sufficiently dense so that
ground and sidewall friction contributions to the total stress can be ignored.
Thus, a local balance between the gravitational contribution of the water
weight along $x$ and the drag resisting this motion results in

$\rho gS_{f}V_{w}=C_{d}A_{v}\rho\frac{U^{2}}{2g},$ (7)

where $\rho$ is the water density, $V_{w}$ is the water volume, $A_{v}$ is the
frontal area of the vegetation contained in $V_{w}$ and $C_{d}$ is the drag
coefficient. For the SVE, a force balance per unit ground area is preferred so
that $V_{w}=h(1-\phi_{v})$ and $A_{v}=mDh$, where $\phi_{v}$ is the solid
volume fraction per ground area determined by $\phi_{v}=m\pi D^{2}/4$, $m$ is
the rod density (i.e. number of rods per unit ground area). This force balance
leads to

$S_{f}=\left[\frac{(C_{d})~{}mD}{1-\phi_{v}}\right]\frac{{U}^{2}}{2g}.$ (8)

Equations 6 and 8 can be made equivalent when introducing a non-constant
Manning roughness given by

$n=\sqrt{\frac{C_{d}mD}{2g(1-\phi_{v})}}R_{h}^{2/3}.$ (9)

Setting $n$ to a constant value in models of $S_{f}$ cannot be reconciled with
a distributed drag formulation. The $C_{d}$, which frames the scope of the
work here, is influenced by numerous interactions between the canopy elements
and the moving water. In steady-uniform flows, $C_{d}$ is presumed to vary
with Reynolds number $Re=VL/\nu$, where $V$ and $L$ are characteristic
velocity and length scales respectively, and $\nu$ is the water kinematic
viscosity. A number of possibilities have been introduced in the literature to
define $L$ and $V$ in this context. Some set $L$ to be proportional to $D$,
rod spacing, or $R_{h}$. Likewise, $V$ was set to pore-scale velocity, the
constricted velocity, or a separation velocity (Etminan, Lowe, and
Ghisalberti, 2017). Corrections such as sheltering or blockage due to the
presence of an array of cylinders have also been studied for an isolated
cylinder and an array of cylinders (Tanino and Nepf, 2008; Stoesser, Kim, and
Diplas, 2010; Nepf, 1999; Lee  _et al._ , 2004; Etminan, Lowe, and
Ghisalberti, 2017; Cheng and Nguyen, 2010; Zdravkovich, 2000).

Returning to the water level description in $x$ and $t$ of an advancing
wavefront within a rod canopy, a number of simplifications have been adopted
to the SVE. Within the wavefront region, the front speed attains a quasi-
constant value so that the unsteady and inertial terms $\partial U/\partial
t+U\partial U/\partial x$ are small relative to remaining terms (Chanson,
2009). For these standard simplifications, the SVE and the continuity equation
become (Melis  _et al._ , 2019)

$g\left(\frac{\partial h}{\partial
x}+S_{f}-S_{o}\right)=0;~{}~{}\frac{\partial h}{\partial t}+U\frac{\partial
h}{\partial x}=0.$ (10)

At very high $Re$, $C_{d}$ may attain a quasi-constant value so that

$U=\sqrt{-A\left(\frac{\partial h}{\partial
x}-S_{o}\right)},~{}A=\frac{2g(1-\phi_{v})}{C_{d}mD}.$ (11)

Inserting $U$ into the approximated continuity equation and solving for
$h(x,t)$ results in

$h(x,t)=C_{1}+{C_{2}}t+\frac{x}{3}\left[S_{o}+E(S_{o},A)\right],$ (12)

where $E(S_{o},A)$ is given by

$\displaystyle E$
$\displaystyle=\frac{R_{1}}{A}+\frac{A{S_{o}}^{2}}{R_{1}};\rm{where}$ (13)
$\displaystyle R_{1}$
$\displaystyle=\left({A^{3}{S_{o}}^{3}-\frac{27A^{2}C_{2}^{2}}{2}+\frac{3\sqrt{3}}{2}\sqrt{27A^{4}C_{2}^{4}-4A^{5}C_{2}^{2}{S_{o}}^{3}}}\right)^{1/3}.$

Here, $C_{1}$, and $C_{2}$ are integration constants independent of $x$ or
$t$. Equation 13 applies when $S_{o}>0$. For $S_{o}=0$, the solution to the
simplified continuity and SVE system is (Melis  _et al._ , 2019)

$h(x,t)=C_{1}+{C_{2}}t-{\left[C_{2}{\sqrt{\frac{C_{d}mD}{2g(1-\phi_{v})}}}\right]}^{2/3}x.$
(14)

It is to be noted that equation 14 is not recovered from equation 12 when
setting $S_{o}=0$ as this condition resembles a singular limit (i.e. addition
of a new force). A near constant $C_{d}$ implies $h(x,t)$ is linear in $x$
(and $t$) at the advancing front region. The slope of this linear dependence
on $x$ varies with $C_{d}^{1/3}$ (i.e. sub-unity exponent).

Figure 2: A schematic of the instrument setup. 1) side view of the channel
with the dam released, 2) top view of the channel indicating the positions of
the measuring instrumentation, 3) detailed view of the advancing wave front
showing how the bulk velocity from imaged water depth was computed from water
level measurements at time $t$ and $t+dt$, 4) detailed view of the load cell
system illustrating the method for drag measurements including the assumption
of setting $F_{d}$ at $h/2$, 5) a sample drag time series record in raw (blue
line) and filtered (black line) form for $H_{o}=$ 0.15m, $S_{o}=$ 0$\%$,
$X_{lc}=$ 1.95m, $Y_{lc}=$ 0.05m, and 6) the computed bulk velocity (blue
line) for $H_{o}=$ 0.15m, $S_{o}=$ 0$\%$, $X_{lc}=$ 1.95m.

##  III Experiments

The flume facility has been described in prior studies (Melis  _et al._ ,
2019; Poggi  _et al._ , 2004) and will not be fully repeated. Briefly, the
tilting channel, the wooden cofferdam, the pneumatic pump release mechanism
for the dam removal (mimicking an instantaneous dam break), the rod canopy,
the water level imaging system, the load cells, the water level sensors, and
the data acquisition system are featured in Figure 1. The rectangular channel
shown in this figure has a length $L=11.6$m, a width $B=0.51$m, and a side
height $L_{s}=0.6$m. The channel sides are made of glass to allow imaging and
optical access. The $S_{o}$ was varied from 0% to 3%. The water behind the dam
was filled until the target $H_{o}$ is reached ($H_{o}=$ 0.15m, 0.20m, 0.25m,
0.30m). The quasi-instantaneous dam removal was carried out using a pneumatic
pump that pulls rapidly the cofferdam vertically upward. After the dam
removal, the water discharges from the end of the channel into a recirculating
tank while passing over a rectangular weir. Downstream from the dam, an array
of rigid cylinders of $D$=0.006 m and $h_{c}$=0.14 m were used to represent
the vegetation. The cylinders were fixed onto boards attached to the channel
bottom and cover an entire cross-section. A staggered rod configuration was
used for all runs with a constant density $m=194$ rods m-2.

To image $h(x,t)$ during and after the dam removal, four synchronized Sony
Handycam FDR-AX700 cameras were employed. The spatial resolution of each
camera was 1920 $\times$ 1080 pixels interrogated in time at 100 frames per
second. The cameras cover a total length of 4.2 m starting from 0.6 m behind
the cofferdam. Water was mixed with a Rhodamine dye and a green laser plane
was seated up parallel to the channel to enhance the automated detection of
$h(x,t)$ and the delineation of the water surface profile. MATLAB (Mathworks,
Natick, Massachusetts, USA) was used to analyze the movies by transforming the
detected $h(x,t)$ from pixel to metric coordinates as described elsewhere
(Melis  _et al._ , 2019). The duration of each experiment (i.e. a combination
of $H_{o}$ and $S_{o}$) ranged from 5 to 10 s.

Twelve load cells were used to record the drag force in $t$ at 1kHz on 3
downstream cross-sections away from the dam ($X_{lc}=$ 0.89m, 1.95m, 2.99m).
On each section, four load cells were placed with 0.1m spacing starting at
0.05m from the left side, according to the staggered canopy’s pattern. The
load cells used were eight Leane model DBBSM-1kg-003-000 along with four
Instrumentation Devices (model kD40s). The drag $F_{d}$ exerted on the
instrumented rod is transferred to the load cell through a rigid active beam
hinged on a fixed point as shown in Figure 2. The load cells were calibrated
by applying a known torque to the cell-beam system. During the experiments,
the torque onto the cell-beam system ($=M_{d}$) by $F_{d}$ was recorded. Then,
the actual $F_{d}$ was derived assuming $F_{d}$ is concentrated at half the
depth $F_{d}=M_{d}/(l_{b}-h/2)$ where $l_{b}$ is the distance from the channel
bottom to the hinge. Additionally, eight level sensors (Balloff model BUS004W)
were employed to record the water depth in $t$ at 1kHz behind the cofferdam.
These sensors were located at the cross-section center and spaced as $X_{ls}=$
-0.2m, -0.4m, -0.7m, -1m, -1.6m, -2m, -3m, -4m, where the free surface cannot
be detected with the imaging system. An acquisition card (National
Instruments, Austin, Texas, USA) was used for both load cells and level
sensors. LabVIEW (National Instruments, Austin, Texas, USA) was used to drive
data acquisition.

The $C_{d}$ was quantified from the measured $F_{d}$ using the quadratic drag-
law $F_{d}=(1/2)C_{d}(Dh)U^{2}$. This quantification requires the bulk
velocity on the instrumented cross-section $U_{lc}$ that was not directly
measured. For each instrumented cross section $X_{lc}$, the $U_{lc}$ was
computed from measured $h(x,t)$ using the continuity equation. Specifically,
for each $t$ the volume $V_{lc}$ forwarding $X_{lc}$ was computed by numerical
integration of $h(x,t)$ over $x$. The $V_{lc}$ was then numerically
differentiated to obtain the flow rate $Q_{lc}$ through the instrumented
section. The last step is to determine the bulk velocity from
$U_{lc}=Q_{lc}/(h(x=X_{lc},t)B)$. In Figure 2, an illustration of how the bulk
velocity was computed from this procedure is provided.

The dam break experiments reported here were compared to prior experiments
conducted in the same flume, same $S_{o}$-$H_{o}$ combinations, and for the
same staggered rod configuration (Melis  _et al._ , 2019). The main difference
between the present and the prior experiments is the rod density $m$. In the
prior experiments, $m=1206$ rods m-2 whereas here $m=194$ rods $m^{-2}$. These
prior experiments did not include load cell measurements or independent water
level measurements behind the dam. Hence, their effective $C_{d}$ was only
inferred by fitting a numerical solution of the SVE to imaged $h(x,t)$ for all
the $H_{o}$-$S_{o}$ combinations with assumptions about the inflow volume into
the channel following the dam break. The most pertinent finding from these
prior experiments was that a $C_{d}=0.4$ better describes the measured
$h(x,t)$ than the numerous models proposed in the literature (Melis  _et al._
, 2019).

A separate experiment was also conducted here in the same channel to determine
$C_{d}$ for the steady and uniform flow case and for $m=194$. In these
experiments, the staggered cylinder configuration was the same. The three
target $S_{o}>0$ values were also used. The $C_{d}$ for these experiments
determined from the load cells $C_{s,p}$ and separately from equation 6 when
setting $S_{f}=S_{o}$ are shown in Table 1 for completeness. In both cases,
$C_{d}$ was computed using the constricted cross-section velocity
$U_{c}=U/(1-\sqrt{2\phi_{v}/\pi})$ (Etminan, Lowe, and Ghisalberti, 2017).

Figure 3: The comparison between the drag coefficient $C_{d}$ derived for an isolated cylinder (Cheng, 2012), measured by the load cells $C_{d,p}$ and derived from friction slope $C_{d,f}$ using equation 7 \- all presented as a function of Reynolds number $Re_{d}$ for $S_{f}=S_{o}>0$ when steady and uniform flow conditions are attained. The agreement between $C_{d,p}$ and $C_{d,f}$ support the assumption that wall friction can be ignored relative to the drag for such a rod density. Table 1: Summary of the drag coefficient measurements from the load cells and from the friction slope $S_{f}$ for steady-uniform flow (i.e. $S_{f}=S_{o}$) for different $S_{o}$. Here, the $h$ is the mean water depth, $Q$ is the discharge, $U$ is the constricted cross-section velocity Etminan, Lowe, and Ghisalberti (2017), $F_{d}$ is the mean drag measured by the load cells, $C_{d,p}$ is the drag coefficient determined from the load cells, $C_{d,f}$ is the drag coefficient determined from the uniform flow result with $S_{f}=S_{o}$ (i.e. equation 7), and $Re_{d}$ is, as before, the Reynolds number. The overall agreement between the $C_{d}$ estimates by the two methods is better than $10\%$ on average. $\ S_{0}\ [\%]\ $ | $\ h\ [m]\ $ | $Q\ [m^{3}/s]\ $ | $\ U\ [m/s]\ $ | $\ F_{d}\ [N]\ $ | $\ C_{d,f}\ $ | $\ C_{d,p}\ $ | $\ Re_{d}\ $
---|---|---|---|---|---|---|---
1 | 0.08 | 0.016 | 0.42 | 0.037 | 1.00 | 0.87 | 2204
1 | 0.10 | 0.021 | 0.43 | 0.046 | 0.94 | 0.83 | 2266
1 | 0.12 | 0.026 | 0.45 | 0.069 | 0.85 | 0.93 | 2387
1 | 0.14 | 0.030 | 0.45 | 0.077 | 0.85 | 0.90 | 2380
2 | 0.06 | 0.017 | 0.60 | 0.061 | 0.96 | 0.95 | 3147
2 | 0.08 | 0.022 | 0.58 | 0.080 | 1.03 | 1.01 | 3038
2 | 0.10 | 0.029 | 0.60 | 0.105 | 0.94 | 0.97 | 3177
2 | 0.12 | 0.034 | 0.60 | 0.131 | 0.96 | 1.02 | 3142
2 | 0.14 | 0.040 | 0.59 | 0.156 | 0.99 | 1.07 | 3101
3 | 0.06 | 0.020 | 0.70 | 0.078 | 1.11 | 0.89 | 3672
3 | 0.08 | 0.028 | 0.73 | 0.113 | 1.01 | 0.88 | 3844
3 | 0.10 | 0.035 | 0.74 | 0.147 | 0.99 | 0.90 | 3882
3 | 0.12 | 0.043 | 0.75 | 0.184 | 0.97 | 0.92 | 3935
3 | 0.14 | 0.050 | 0.75 | 0.231 | 0.96 | 0.89 | 3943
Figure 4: Variations of the normalized drag, water level, velocity, and
$C_{d}$ as a function of time at the three load cell locations for all $H_{o}$
and $S_{o}$ conditions (columns). Here, $F_{o}$ is the normalizing force
computed using $C_{d}=1$, $U_{o}^{2}=gH_{o}$, and $\rho=1000$ kg m-3. Figure
5: The measured normalized water depth $h_{n}=h/H_{o}$ (ordinate) variations
with the normalized velocity $U_{n}=(x/t)/U_{o}$ (abscissa) along with the
Ritter solution (black solid line) for all four bed slopes $S_{o}$. In the top
panels, the blue lines represent the derived $h_{n}$ (solid line) and the 95%
confidence bound (dashed lines). The color scale is featured to show the
$log_{10}(N)$ data points used in the derivation of $h_{n}-U_{n}$ relation.
The bottom panels repeat the blue lines ($m=194$) from above and add earlier
experiments ($m=1206$) for the dense rod density case Melis  _et al._ (2019)
(red lines).

##  IV Results and Discussion

To address the study objectives, this section is organized as follows. A
comparison between measured $C_{d}$ for steady-uniform flow cases and dam-
break cases for all $S_{o}$ values is presented. The magnitude and controls on
$C_{d}$ for this rod configuration for steady-uniform flow can be used to
ascertain whether wall friction can be ignored relative to the canopy drag and
whether $C_{d}$ estimates from the load cells match the expectations from
equation 8. Next, the effects of $H_{o}$, $S_{o}$ and $S_{f}$ on the wavefront
are considered. These considerations are also used to summarize the data from
the experiments in a normalized manner. Once again, data from the prior study
($m=1206$) and the present experiments ($m=194$) are compared to assess the
effects of $C_{d}mD$ on $S_{f}$. To facilitate comparisons across the
$H_{o}-S_{o}$ cases and the two $m$ values, a single reference curve was
repeated in all of them based on equation 5. This ’base-line’ curve makes a
logical choice for a reference because it is derived for $S_{f}=0$ and
$S_{o}=0$. The physics of the advancing front wave is considered next. Two
regimes are shown to emerge when analyzing the measured front position $x_{f}$
against estimates from $U_{o}t$. The first is a rapid regime dominated by both
inertial and frictional effects, and a second regime trending towards the
diffusive wave approximation where the frictional effects experience a reduced
$C_{d}$. A discussion as to the possible causes of this drag reduction is then
offered. Throughout, the results in the figures are presented in dimensionless
form using the following: water depths are normalized by $H_{o}$, velocities
are normalized by $U_{o}$, time is normalized by $U_{o}/L_{o}$ where
$L_{o}=(C_{d}mD)^{-1}$ is a reference adjustment length (Belcher, Jerram, and
Hunt, 2003) taken at $C_{d}=1$ (i.e. steady-uniform flow case), forces are
normalized by $(1/2)C_{d}\rho DH_{o}U_{o}^{2}$ and horizontal distances are
normalized by $L_{o}$.

###  IV.1 Drag Coefficient from the Load Cells

For steady-uniform flow, the measured $C_{d}$ from the load cells $C_{d,p}$
and from friction slope $C_{d,f}$ shown in Figure 3 do not deviate
significantly from the accepted formulation for an isolated cylinder
$C_{d,iso}$ that is given by (Cheng, 2012; Wang  _et al._ , 2015b)

$C_{d,iso}=11(Re_{d})^{-0.75}+0.9{{\Gamma}_{1}}\left({{{Re}}_{d}}\right)+1.2{{\Gamma}_{2}}\left({{{Re}}_{d}}\right),$
(15)

where $Re_{d}=U_{c}D/\nu$ is the element Reynolds number and

$\displaystyle{{\Gamma}_{1}}\left({{{Re}}_{d}}\right)=$ $\displaystyle
1-\exp\left(-\frac{1000}{{{{Re}}_{d}}}\right),$
$\displaystyle{{\Gamma}_{2}}\left({{{Re}}_{d}}\right)=$ $\displaystyle
1-\exp\left[-{{\left(\frac{{{{Re}}_{d}}}{4500}\right)}^{0.7}}\right].$ (16)

This expression assumes that $Re_{d}<10^{4}$ (and is below the drag crisis
range for isolated cylinders) and that the drag from each cylinder operates in
isolation (i.e. no interference, sheltering, or blocking). The agreement
between $C_{d,p}$ and $C_{d,f}$ is also suggestive that wall friction that
impacts $C_{d,f}$ but not $C_{d,p}$ can be ignored relative to the canopy drag
for such a rod density. In the range of $Re_{d}$ considered here, a
$C_{d}=0.9-1.1$ appears to describe the steady-uniform flow data without any
significant dependency on $Re_{d}$ as shown in Figure 3. For this reason, the
reference drag $C_{d}=1$ is selected in the $L_{c}$ calculations used for
normalizing longitudinal distances. Returning to the dam break cases, the load
cell measured $F_{d}$ was used, together with the imaged water depth $h$, to
compute $C_{d}$ using $F_{d}=\left(1/2\right)C_{d}\rho DhU^{2}$. Here, the $U$
was computed from the continuity equation using the imaged water depth as
shown in Figure 4. The computed $C_{d}$ is well below unity as the wave front
passes. This reduction in $C_{d}$ is significant for all $S_{o}$ values. For
early times $t(U_{o}/L_{o})<1$, $C_{d}$ increases from some 0.2 to 0.6 as
shown in Figure 4 with a mean of about $C_{d}=0.4$. The mean value here is
consistent with the value inferred indirectly from fitting the SVE to measured
$h(x,t)$ in the prior study (Melis  _et al._ , 2019) despite the large
difference in $m$ between the two experiments. The agreement between the
reduced drag value ($C_{d}=0.4$) across the two experiments hints that
sheltering alone may not be the main mechanism responsible for drag reduction
as sheltering is expected to dependent on rod density.

Figure 6: The relation between wave front position $x_{f}$ and those
determined when assuming $x_{f}=U_{o}t$ from the top to the bottom: the time
$t_{f}$ multiplied by the characteristic velocity $U_{o}$, the normalized wave
front velocity $U_{f}/U_{o}$, the tip slope $\Gamma_{h}$ divided by
characteristic depth $H_{o}$ and the inferred drag coefficient $C_{d}$ using
the diffusive wave approximation.

###  IV.2 Bed Slope and Frictional Effects

To illustrate the simultaneous effects of $S_{o}$ and $H_{o}$ variations on
the depth-velocity relations, the experiments here ($m=194$) are summarized in
Figure 5 and then compared to equation 5 for $S_{o}=S_{f}=0$. The prior
experiments for $m=1206$ are also added for reference and are organized, as
before, by the two control variables $S_{o}$ and $H_{o}$. A number of comments
can be made about Figure 5:

  * •

Equation 5 over-predicts the advancing front wave speed (i.e. the $U_{n}$
associated with $h_{n}<0.05$) for all $S_{o}-H_{o}$ cases compared to their
frictional counterparts as expected;

  * •

The higher $m$ experiments result in higher water pile-up (i.e. higher
$h_{n}$) at the smaller $U_{n}<0.5$ values as expected;

  * •

With increasing $S_{o}$, the lower $m$ data approaches equation 5 (i.e. the
solution for $S_{o}=S_{f}=0$) except for the advancing wave front region. This
finding may be explained by the fact that increasing $S_{o}$ also increases
$S_{f}$ thereby diminishing their difference in the SVE ($S_{f}$ is always
finite and large in the presence of a canopy). The advancing front (i.e. the
region with $h_{n}<0.05$) always experiences a slow-down (lower $U_{n}$)
compared to equation 5 as noted earlier.

In sum, Figure 5 confirms all the logical expectations of the $u_{n}-h_{n}$
relations derived from the experiments for differing $m$ values, $S_{o}$, and
$H_{o}$. Common to all the cases is the slow-down for the advancing front
region when compared to the $S_{f}=0$ case due the canopy. In the absence of a
canopy, the Ritter solution reasonably describes the advancing wavefront for
the channel setup here as discussed elsewhere (Melis  _et al._ , 2019). The
advancing front region is now explored in detail.

###  IV.3 Two Dynamical Regimes

The dynamical regimes that introduce deviations from equation 5 are examined
along $x$. These regimes are identified by exploring how the normalized front
position $x_{f}/L_{o}$ varies against $tU_{o}/L_{o}$ as shown in Figure 6. Two
distinct regimes can be identified with a transition governed by the initial
$H_{o}$ for all $S_{o}$ (top panels). At small distances from the dam (first
regime, $x_{f}/L_{o}<$ 1), a robust $x_{f}=c_{f}U_{o}t$ can be seen from the
experiments where $c_{f}=1.26$ for $S_{o}=0$ and $c_{f}=1.33$ for $S_{f}=3\%$.
This implies that the advancing front velocity $U_{f}=c_{f}\sqrt{gH_{o}}$ is
roughly a constant and weakly dependent on $S_{o}$ given the small increase in
$c_{f}$ with increasing $S_{o}$. Setting $x/t=U$ in equation 5 leads to a
frictionless advancing front speed that is almost twice as fast (i.e.
$c_{f}=2$ for $S_{f}=S_{o}=0$) as discussed elsewhere (Hogg and Pritchard,
2004). Beyond a near constant $U_{f}$ for $x_{f}/L_{o}<1$, another dynamically
interesting feature of this regime is the rapid drop in measured $\partial
h/\partial x$ with increasing $x_{f}$ until the attainment of the second
regime $x_{f}/L_{o}>1$. The second regime marks a gradual slow-down in
$U_{f}/U_{o}$ compared to the near-constant velocity in the first regime (top
two rows in Figure 6), but trends towards a near constant
$\Gamma_{h}=-\partial h/\partial x$ with increasing $x_{f}/L_{o}$. Thus, while
the first regime experiences a near constant $U_{f}$ and a variable
$\Gamma_{h}$, the second regime is dynamically the opposite. The increase in
$H_{o}$ delays the onset of the slow down (or second) region with increased
$x_{f}$ (i.e. reduced $U_{f}$) for all $S_{o}$ values. It is expected that
with further increases in $x_{f}/L_{o}$, $U_{f}/U_{o}$,
$\Gamma_{h}L_{o}/H_{o}$, and likely $C_{d}$ all attain a constant value in
agreement with the diffusive wave approximation. The measurements reported in
Figure 6 seem to support this extrapolation for the low $H_{o}$ cases
considered.

The fact that $U_{f}$ varies may appear counter to the approximated physics in
equation 10. However, a detailed analysis (not shown here) conducted on the
data suggest that

$\frac{\partial Uh}{\partial x}\approx U_{f}\frac{\partial h}{\partial
x};\left|\frac{\partial U}{\partial t}+U\frac{\partial U}{\partial
x}\right|<<\left|\frac{\partial h}{\partial x}\right|.$ (17)

0Thus, the diffusive wave approximation remains plausible in the second regime
despite variations in $U_{f}$ with $x_{f}$.

Returning to the drag reduction issue, a $C_{d}$ was inferred from equation 11
using measured $U_{f}$ and measured $\partial h/\partial x$ for the
$S_{o}-H_{o}$ combinations. These computed $C_{d}$ values are shown in Figure
6. To be clear, the estimate of $C_{d}$ using a diffusive wave approximation
for the first region (i.e. $x_{f}/L_{o}<1$) cannot be correct. The computed
$C_{d}$ is only presented here to corroborate the transition zone from the
first to the second region around $x_{f}/L_{o}=1$, where the diffusive wave
approximation begins to apply for $x_{f}/L_{o}>1$. Beyond that $x_{f}$, a near
constant $C_{d}$ is attained from equation 11. The low $C_{d}$ values measured
by the load cells shown in Figure 4 agree with those computed from the
diffusive wave approximation using measured $h(x,t)$. This agreement in
reduced $C_{d}$ for the second region serves two purposes: (i) it shows that
the diffusive wave approximation is plausible for $x_{f}/L_{o}>1$, and (ii) it
confirms that a new drag reduction mechanism must be operating that is unique
to the dam break problem (i.e. it does not exist in the steady-uniform flow
cases).

###  IV.4 What Causes the Drag Reduction?

To recap the findings thus far, the $m=1206$ experiments yielded a $C_{d}=0.4$
inferred from $h(x,t)$ measurements by fitting an optimum $C_{d}$ to the SVE
so as to match $h(x,t)$. It was postulated that this fitted and reduced drag
is linked to the so-called drag crises (Melis  _et al._ , 2019). For an
isolated cylinder immersed in a steady and uniform background flow that
describes its far-field, the drag crisis occurs when well-organized vortex
shedding (e.g. Karman-vortex streets) are disrupted and transition to
randomized shedding with further increases in $Re_{d}$ (Singh and Mittal,
2005). This transition occurs at very large $Re_{d}(>10^{5})$ in the isolated
cylinder case and for steady-uniform flow as shown in Figure 7. Numerical
simulations confirm that the drag crisis commences when a critical Reynolds
number is reached where the boundary layer on the cylinders become turbulent
thereby maintaining attachment to the cylinder further downstream (Singh and
Mittal, 2005). The transition to a turbulent boundary layer and a sustained
attachment onto the cylinder has two effects. The first is that the pressure
differential between the front ($P_{i}$) and the back ($P_{s,t}$ for turbulent
versus $P_{s,v}$ for laminar) of the cylinder is reduced due to the partial
pressure ($P_{s,t}>P_{s,v}$) recovery following a longer downstream attachment
(Wen and Qiu, 2017) along the back of the cylinder as schematized in Figure 7.
Numerical simulations for an isolated cylinder with a uniform far field flow
have shown that this pressure recovery does occur because of delayed
separation on the back of the cylinder and can reduce the pressure
differential (referenced to $P_{i}$) by more than $25\%$. The second is that
the wake area is also reduced (red versus blue shades in Figure 7) as detailed
by simulations (Singh and Mittal, 2005; Wen and Qiu, 2017). The combined
effect of reduced pressure differential and reduced wake area leads to a
drastic reduction in $C_{d}$ (more than a factor of 2) - or the drag crisis.

It was conjectured in the prior study (Melis  _et al._ , 2019) that the dam
break problem leads to a disturbed and transient ’far-field’ background flow
region that enables the randomization of vortex shedding to be initiated and
persistent at much lower $Re_{d}$. This disturbed far-field state can enhance
momentum transport to the turbulent boundary layer region attached onto the
cylinder and allows the drag crisis to be maintained as schematized in Figure
7). A reduced $C_{d}$ is then to be sustained over an extended range of
$Re_{d}$ well below the critical value of $Re_{d}=10^{5}$ (where the laminar
boundary layer flips to a turbulent state for an undisturbed background
state).

Figure 7: Conceptual model for the boundary layer detachment from a cylinder
during the drag crisis (red wake, boundary layer is turbulent) compared with
pre-crisis (blue wake, boundary layer is laminar). The delayed separation
during the drag crisis has two effects: (i) it allows for a pressure recovery
(i.e. $P_{s,t}$ is less negative than $P_{s,v}$ shown at the dashed blue
center line of the cylinder), thereby reducing the pressure difference between
the front ($P_{i}$) and back of the cylinder (i.e. $P_{s,v}$ or $P_{s,t}$) and
(ii) it reduces the wake area behind the cylinder (red vs blue). Both effects
act in concert to reduce the form drag $C_{d}$ shown on top as a function of
$Re$. The dashed vertical lines show the $Re$ for the pre-crisis (blue) and
during the drag crisis (red).

The work here offers an amendment to this argument, which is the air
entrainment leading to density reductions at the advancing front in the second
region as shown in the photographs of Figure 1. Figure 8 presents a comparison
between the measured normalized volume of the inflow behind the dam using
water level sensors (ordinate) and the imaged volume of water as the front
progresses downstream from the dam for $x/t>0$ (abscissa). Noting that
$x_{f}\approx 1.3U_{o}t$ in Figure 6 and that the test-section region analyzed
in Figure 6 is for $x_{f}/L_{o}\leq 3$, it is clear that the imaged outflow
volume appears to be consistently larger by some 10-20$\%$ compared to the
inflow volume (i.e. volume behind the dam measured by detailed water level
measurements). From an experimental uncertainty point of view, the splashing
and breakup of water into fine droplets near the advancing tip region shown in
Figure 1 would have reduced, not increased, the imaged outflow volume. Water
droplets, which are not counted in the imaged $h(x,t)$ volume calculations,
will be missed. Since water mass is conserved, the imaged outflow volume
exceeding the inflow volume must then be associated with some air entrained at
the advancing front. With air volume entrained at the advancing wave front
region, the overall water density near the tip front must be substantially
reduced. A reduction in water density near the advancing front region leads to
a concomitant overall inferred $F_{d}$ and $C_{d}$ reductions in this
vicinity. This drag reduction mechanism may be acting in concert with the
randomization of vortex shedding associated with the earlier speculated drag
crisis (Melis  _et al._ , 2019). Thus, air entrainment, reduced water density
at the advancing front region, and a disturbed background state all conspire
to reduce $C_{d}$ by a factor of $2$.

Figure 8: A comparison between normalized water volume within the channel for
$t>$ 0 from the imaging system (ordinate) for $x>0$ and from level sensors
(abscissa) behind the dam $x<0$.

##  V Conclusions and Broader implications

The closure of the friction slope in the SVE continues to draw research
attention as it encodes all the solid-fluid interactions. In operational
models of flood waves, the $S_{f}$ is related to $U^{2}/(2g)$ using
conventional formulations that aim to recover the steady-uniform flow (e.g.
Manning’s formula). However, the presence of drag elements adds another layer
of complexity and that cannot be represented as an equivalent wall friction
derived using steady-uniform flow conditions. Those interactions are
represented by a drag coefficient $C_{d}$ that modifies $S_{f}$ as shown here.
As before, steady-uniform flow are assumed as base line state to link $C_{d}$
to $S_{f}$ in practice. This approach was deemed pragmatic in many dam break
and flood routing applications because there are numerous data sets and models
for $C_{d}$ when the drag elements are rod canopies (or other approximated
geometries). Much research has focused on corrections to $C_{d}$ from an
isolated cylinder and include sheltering, blockage, among others. The work
here suggests that $C_{d}$ from steady-uniform flow may be an overestimate by
a factor of 2. New physics at the advancing front related to air entrainment
and randomization of coherent vortex structures occur due to the unsteady and
disturbed nature of the flow away from the rod canopy. These two effects act
as drag reduction mechanisms that ’speed up’ the advancing front relative to
$C_{d}$ models derived from steady-uniform flow.

The work also identified two dynamically interesting regimes for the advancing
front velocity based on where the front location $x_{f}$ is relative to the
adjustment length scale $L_{o}=(C_{d}mD)^{-1}$. For $x_{f}/L_{o}<1$, the front
velocity is roughly constant and scales with $\sqrt{gH_{o}}$ and $\partial
h/\partial x$ rapidly declines in magnitude. A second regime for
$x_{f}/L_{o}>1$ emerges where the front velocity begins to decline but
$-\partial h/\partial x$ begins to attain its minimum. The diffusive wave
approximation further suggests that in the second regime, the drag coefficient
is reduced by a factor of 2 relative to its steady-uniform value.

This work must be viewed as an embryonic step so as to improve flood
forecasting in the future. Additionally, the present results offer benchmark
data for future numerical investigations given the increased focus on modeling
and simulating dam-break problems Bazilevs  _et al._ (2012); Dai and Jin
(2022); Wu, Shen, and Chou (1999); Ai  _et al._ (2022); Hernández-Dueñas,
Moreles, and González-Casanova (2023); Eames and Robinson (2022); Yan  _et
al._ (2023) over obstructions and vegetated bed scenarios He  _et al._ (2017);
Khoshkonesh  _et al._ (2022); Zhang  _et al._ (2016); Khoshkonesh  _et al._
(2023). The need to move beyond wall friction representation for energy losses
is recognized in large scale models but alternatives remain in short supply
(Melis  _et al._ , 2019). Progress on these alternatives using a quadratic
drag law is timely given the rapid advancements in water level measurements
from space (to within 10 cm) (Alsdorf, Rodríguez, and Lettenmaier, 2007) and
the wealth of research on drag coefficients for different geometries as
derived from steady-uniform flow. However, the disturbed and transient nature
of this flow was shown to lead to previously unexplored drag reduction
mechanisms (air entrainment, drag crisis). Future experiments should seek
novel methods to characterize the water density reductions and the
randomization of vortex shedding associated with the drag crisis for such
disturbed flows. From the simulation perspective, the results here hint that a
3-phase representation (solid, water, and air) may be needed to capture the
interplay between air entrainment and boundary layer separation at the solid
interface of the cylinders during the dam break.

###### Acknowledgements.

EB acknowledges Politechnico di Torino (Italy) for supporting the visit to
Duke University. GK acknowledges support from the U.S. National Science
Foundation (NSF-AGS-2028633) and the U.S. Department of Energy (DE-SC0022072).
DP acknowledges support from Fondo europeo di sviluppo regionale (FESR) for
project Bacini Ecologicamente sostenibili e sicuri, concepiti per
l’adattamento ai Cambiamenti ClimAtici (BECCA) in the context of Alpi Latine
COoperazione TRAnsfrontaliera (ALCOTRA) and project Nord Ovest Digitale e
Sostenibile - Digital innovation toward sustainable mountain (Nodes - 4).

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## Conflict of Interest Statement

The Authors have no conflicts to disclose.

## Data availability statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.