Introduction
Flash floods, dam break floods, tsunami inundation, storm surge, tidal flow, and coastal swash can all develop hydraulic bores (also called shocks), characterized by steep fronts as flow depth rapidly increases (e.g.,
Reid et al. 1994;
Chaudhry 2008;
Chanson 2009;
Aleixo et al. 2019). Most of these phenomena can be destructive to infrastructure and to people, and all cause morphologic change to fluvial and coastal landscapes through sediment transport. Because of their high risk, rare occurrence and short duration, accurate flow data (e.g., depth, velocity, water surface slope, and other spatial and temporal derivatives) can be difficult or impossible to collect.
Bed shear stress (
) is the hydraulic parameter most commonly used to predict erosion, deposition, and sediment transport rates (e.g.,
Chaudhry 2008;
Dey 2014), but calculating it is particularly challenging for rapidly changing flows. The depth-averaged Saint-Venant equations are derived from Navier-Stokes equations, and assume that vertical velocity variations are much smaller than horizontal ones (
Chaudhry 2008). While the assumption limits the rigorous application of the Saint-Venant equations to gradually varying flows, they have generally been shown to be reliable for unsteady non-uniform flows such as flood waves (
Haizhou and Graf 1993;
Graf and Song 1995;
Nezu et al. 1997;
Rowinski et al. 2000;
Shen and Diplas 2010;
Mrokowska et al. 2015a,
b). Table
1 presents the Saint-Venant equation we used to calculate
from our experimental data. Mrokowska et al. (
2015b) compared the Saint-Venant equation with two simpler methods for flood waves, finding significant differences during times of more rapid change in discharge. In experiments of unsteady hydrographs without bores, Bombar (
2016) found reasonable correlations between five shear stress methods, including Saint-Venant. Mrokowska et al. (
2015a) evaluated shear stress in experimental hydrographs with bores, although they were unable to calculate stresses for several seconds after bore arrival. They detail challenges associated with obtaining flow data accurate enough to resolve the derivatives of depth (
,
) and velocity (
) required to solve the Saint Venant equation (Table
1; variables defined therein), even in their experimental case.
Other methods for calculating shear stresses require less comprehensive or different flow data than Saint-Venant, but have assumptions that are broken in strongly unsteady and non-uniform flows. Nonetheless, it is important to be able to quantify shear stresses and related uncertainties for unsteady open channel flows even in cases where the full Saint-Venant equation cannot be applied because of data limitations. We addressed this problem empirically, by comparing eight previously-proposed methods for calculating
to the Saint-Venant approach. We empirically determined the extent to which other methods capture shear stress magnitudes and trends during rapidly changing hydrographs. Table
1 summarizes equations and references for the methods we compare. Because previous studies have described simplifying assumptions for these methods, and our goal is to compare methods empirically rather than theoretically, Table
1 only includes a succinct overview of model simplifications and assumptions (see Supplemental Materials for additional model details).
After Saint-Venant, the next five methods (Table
1, Methods 2–6) use different combinations of hydraulic radius (
), depth-averaged velocity (
), water surface slope, and hydraulic roughness to calculate
. These methods can broadly be thought of as simplifications of Saint Venant that neglect time derivatives and assume quasi-steady flow. The
and
constant methods employ Manning’s
, which was originally derived for uniform flow (
Chow 1959). The
surface slope method assumes steady uniform flow, where acceleration and the hydrostatic pressure terms are neglected (
Chaudhry 2008). The Colebrook and Log-law methods assume fully developed boundary layer velocity profiles (
Colebrook 1939;
Swamee and Jain 1976;
O’Donoghue et al. 2016;
Moore et al. 2007). The last three methods (Table
1, Methods 7–9) use turbulent velocity fluctuations (
,
,
), scaled by coefficients calibrated based on steady flow conditions (
Dey et al. 2012;
Kim et al. 2000;
Soulsby 1983;
Stapleton and Huntley 1995;
Gross and Nowell 1985). Simplifying assumptions, such as quasi-steady flow, hydrostatic force balances, and boundary layer velocity profiles, are often applied to estimate
in slowly varying flows (e.g.,
Biron et al. 2004;
Dey 2014). Even for steady flows,
estimates often vary significantly when using different methods (e.g.,
Biron et al. 2004). Again, we fully acknowledge that theoretical assumptions are not met when applying these methods to rapidly varying hydrographs with bores, but argue that there is utility to measuring the accuracy of these methods under strongly unsteady conditions.
In summary, boundary shear stresses calculated using various methods may not match because they are based on different assumptions and require dissimilar data. It is generally unknown how consistent these methods are for very rapidly changing flows, where method assumptions may not be strictly valid. Our goals are (1) to empirically evaluate the accuracy and uncertainty of several shear stress methods that include quasi-steady approximations, for experimental cases of strongly unsteady flow in which data may not be sufficient to apply the Saint-Venant equation; and (2) to understand how bed shear stresses differ for flows with bores propagating over dry beds versus those over flowing water.
Discussion
Our experimental bores with very rapid changes in depth and velocity can be thought of as the unsteady end-member of flow conditions that are still relevant to natural flash floods and other extreme events. We do not have a method to independently confirm which shear stress methods are most accurate. Based on previous studies and force- and momentum-balance assumptions (e.g.,
Mrokowska et al. 2015a,
b;
Bombar 2016), we conclude that the Saint-Venant method is likely the most accurate measure of
for these rapidly changing hydrographs with bores.
The other methods are advantageous in requiring less data and being less dependent on calculated derivates (which can be noisy), but have assumptions likely broken during these flows. Nonetheless, Fig.
6 shows that these other methods are all nonlinearly correlated with
-Saint-Venant for dry beds, including the higher stress portions, during which flow was more rapidly changing. This result suggests, at least empirically, that these simpler methods may be adapted or calibrated to give reasonable shear stress estimates during flash floods. Fig.
7 shows similar power-law scaling for wet-bed bores between the various methods and the
shear stresses. At the same time, for dry beds the peak Saint-Venant based shear stress was roughly 10 to 20 times larger than peak
from other methods, and several times larger for wet bed conditions [Figs.
5(e and f)]. Methods that do not account for flow accelerations may underestimate stresses when flows are increasing rapidly.
For dry bed flows, velocity profiles in the lower 10.8 cm of the water column rapidly transitioned from highly variable with height above the bed to more typical boundary-layer velocity profiles within the first
, even as depth was still increasing and water surface slope was decreasing (Fig.
2). The assumption that a boundary layer has fully developed suggests that the Log-law method would be inappropriate for estimating shear stress for rapidly changing flows (
Kikkert et al. 2012,
2013;
O’Donoghue et al. 2016). However, in experiments with nonuniform flow accelerations, near-bed velocity profiles maintained their logarithmic form during both flow accelerations and decelerations (
Song and Chiew 2001). The rapid development of velocity profiles while the bulk flow is still changing suggests that relations relating depth-averaged velocity to shear stress may be reasonable over a range of unsteady flow conditions.
In field settings, data quality is likely to be a challenge with many of these methods, and with Saint-Venant in particular. Even with significant averaging of dry-bed runs to reduce variability, the calculated derivatives tend to make our Saint-Venant
calculations noisier than the methods that only use depth-averaged velocity [Figs.
4 and
5; Table
1; Eq. (
S6)]. Water surface slope requires accurate spatio-temporal measurements of small changes in water surface elevation. For this reason, methods using depth and surface velocity time series may be less noisy than slope-based methods.
Lags between flow acceleration and changes in turbulence may explain why the turbulence-based methods predict much smaller peak shear stresses during dry bed bore arrival, and no peak
for wet bed bore arrival. Comparing temporally accelerating pipe flow to steady pipe flow, He and Jackson (
2000) found delays in turbulence production and propagation from the flow boundary to the middle of the flow. Decelerating flow similarly had a time lag, where turbulent velocity fluctuations remained higher than predicted based on the slowing velocity alone. Song and Chiew (
2001) found similar lags for spatially accelerating and decelerating open channel flow. Relative to the other methods, these lags would predict reduced
at and immediately after bore arrival for turbulence-based methods, as well as higher
during recession. Although noisy, this pattern is generally observed [Figs.
5(e) and
6(f)]. For wet bed flows, we interpret that the lack of a turbulence-based peak
after bore arrival occurred because it took several seconds for enhanced turbulence caused by shearing within the fluid at the interface between bore and flowing water (pre-bore flow depth
) to fully propagate to the bed.
Many studies demonstrate that instantaneous stresses associated with turbulent velocity fluctuations play a large role in forces on grains and sediment transport (e.g.,
Sumer et al. 2003;
Schmeeckle et al. 2007). The lags embodied by the turbulence-derived shear stresses are physical. While turbulence contributes to the overall shear stress time series, we nonetheless interpret that these turbulence-based measures alone are incomplete representations of temporal shear stress changes. The empirical coefficients used in turbulence-based methods could vary with the degree and direction of unsteadiness (
Pope 2000). In addition, we acknowledge potentially significant instrumental and methodological limitations of our ADV data. Conducting more than three experimental runs at a given ADV height would have reduced uncertainties in our turbulence-based calculations. Turbulent velocities in the first several seconds are challenging to measure because of aerated flow immediately after bore arrival (especially in the dry bed case), uncertainty from despiking, and separating the mean velocity from turbulence when the mean and variability are simultaneously changing. In field settings and during extreme events it is often impossible to measure near-bed turbulence; most instruments suitable for field turbulence measurements need to be in contact with the flow. Our experimental analysis usefully shows that turbulence-based methods suggest lower peak shear stresses at bore arrival, compared to other methods.
A key motivation for improving shear stress calculations in natural flash floods is to quantitatively predict sediment transport rates and channel changes. Our shear stresses predict that bedload flux would be highest immediately after bore arrival, and also higher during bore passage over a dry bed than that over a wet bed. Rapid scour and deposition are also hypothesized to be caused by bore passage. In the current analysis, correlations between shear stress methods allow different stress estimates to be predicted from each other. However, these regressions are specific to these experiments. Future studies could evaluate how well unsteadiness metrics can improve shear stress estimates during bores and similar rapidly changing flows, especially for applications where it is impossible to measure flow accelerations with sufficient accuracy to apply Saint-Venant models.
Conclusions
The objectives of the present study were to better understand flash flood bores by (1) comparing approaches for estimating bed shear stress in flash floods that could feasibly be applied in field settings; and (2) comparing shear stresses for bores arriving over both dry beds and over flowing water. We conducted experiments comparing flows propagating over rough dry beds (immobile gravel and concrete) and over shallowly flowing water. Bed shear stresses were calculated by nine previously proposed methods using a variety of data, including water depth, slope, depth-averaged velocity, and turbulent velocity fluctuation.
We have two main conclusions. First, the most theoretically justifiable method we evaluated is a version of the Saint-Venant equations for depth-averaged flow, which includes local flow accelerations in its force balance. During and immediately after bore arrival this method predicted consistently higher shear stresses than the other eight methods. However, because the Saint-Venant approach assumes that flow only varies gradually, our shear stress calculations at and immediately after bore arrival are likely inaccurate. The other eight methods we evaluated are commonly used in cases where steady flow can be assumed, and so may not apply to rapidly changing hydrographs. Nonetheless, using power-law regressions we found that these methods were all moderately to highly correlated with the Saint-Venant approach. While our regressions are specific to these experiments, the correlations suggest that it may be possible to empirically calculate reasonably accurate shear stresses soon after bore arrival. Turbulence-based methods for calculating shear stresses gave the lowest peak shear stresses in response to bore arrival, which we interpret to primarily be caused by time lags between mean flow accelerations and turbulence generation. Challenges of measuring and separating turbulent fluctuations from mean velocities during bores may also contribute to lower peak shear stresses using turbulence-based methods.
Second, we conclude that bores propagating over relatively shallow flowing water ( to of ultimate water depth) tend to have considerably lower peak shear stresses from bore arrival than do bores moving over dry beds, indicating the extent to which shallow water can dampen and dissipate near-bed shear stresses, even when overlying bores cause extremely sudden changes in depth and velocity.