Flow resistance in the channel-bar landscape of large alluvial rivers

Abstract

Accurate approaches for estimating flow resistance in large alluvial rivers are fundamental for simulating discharge, sediment transport, and flood routing. However, methods for estimating riverbed resistance and additional resistance in the channel-bar landscapes remain poorly investigated. In this study, we used in situ river bathymetry, sediment, and hydraulic data from the Shashi Reach in the Yangtze River to develop a semi-empirical approach for calculating flow resistance. Our method quantitatively separates flow resistance into riverbed resistance and additional resistance and shows high accuracy in terms of deviation ratio (~20%), root-mean-square error (~0.008), and geometric standard deviation (~3). Additional resistance plays a dominant role under low-flow conditions but a secondary role under high flows, primarily due to the reduction in momentum exchange in channel-bar regions as discharge increases. Riverbed resistance first decreases and then increases, which might be attributed to bedform changes in the lower and transitional flow regimes as flow velocity increases. Overall, our findings further the understanding of dynamic changes in flow resistance in the channel-bar landscapes of large river systems and have important implications for riverine ecology and flood management.

Graphical abstract

Keywords

flow resistance / channel-bar landscape / interaction region / large river / bedform

1 Introduction

Scientific investigations of the controls on river depth and velocity began in the 1700s and there is now a large and continually growing body of literature on the topic, including several reviews of flow resistance (Ferguson, 2013; Powell, 2014; Dai, 2021). Almost all the existing literature refers to the main channels of natural or engineered alluvial or bedrock rivers (Church and Ferguson, 2015; Ferguson et al., 2017; Mei et al., 2018). In contrast, very little is known about the bulk hydraulics of channel-bar interaction regions in large rivers, particularly their flow resistance behaviors (Mohanta et al., 2021; Ferguson et al., 2022).

Channel-bar interaction regions, which are the transition areas between main channels and high bars, occur in most large rivers worldwide and display complex flow movement (Dai et al., 2018; Mohanta et al., 2021). Previous experimental studies have observed lateral momentum transfer in these regions due to the channel-bar flow velocity difference (van Prooijen et al., 2005). In turn, the complicated flow and sediment regimes in interaction regions can influence the accuracy of discharge, sediment transport, and flood routing simulations (Chen et al., 2019a; Ferguson, 2021). Nevertheless, recent studies on flow resistance prediction have focused on laboratory-based methods (Knight et al., 2018; Chandra, 2019), and changes in individual resistance components under variable hydraulic geometry conditions are very rarely reported (Ferguson et al., 2019).

Currently, large rivers are subject to ongoing environmental change, including fluctuations in climate (Dai, 2021; Li et al., 2021) and human activity (Tellman et al., 2021; Lou et al., 2022; Zhou et al., 2022), and the responses of flow resistance to these changes have been widely noted (Church and Ferguson, 2015). However, flow and sediment parameters in channel-bar interaction regions remain poorly documented because of the practical challenges of measurement, especially at high flows (Chen et al., 2016, 2019b). Indeed, many previous studies have ignored lateral momentum transfer when estimating flow velocities in large rivers with channel-bar landscapes (Huthoff et al., 2008; Proust et al., 2009). Furthermore, commercial modeling tools, such as SOBEK, MIKE, and HEC-RAS, assume that no lateral momentum transfer occurs (Huthoff et al., 2008). Alternative approaches include the following: 1) continuing to ignore lateral momentum transfer while shifting the position and shape of interfaces (Patra and Kar, 2000; Cassells et al., 2001); 2) describing lateral momentum transfer by introducing the apparent shear stress (Christodoulou, 1992; Bousmar and Zech, 1999; Moreta and Martin-Vide, 2010; Chen et al., 2016; Chandra, 2019); and 3) resorting to a continuum model that resolves flow velocity as a function of the cross-channel coordinate (Shiono and Knight, 1991; van Prooijen et al., 2005). However, these methods are mainly based on laboratory experiments, and their applicability to large rivers is questionable (Chen et al., 2019a). Moreover, there is still a lack of attention on quantitatively assessing the relative contributions of different sources to flow resistance as well as evaluating the impacts of hydraulic geometry and flow regimes on flow resistance (Chandra, 2019).

In situ bathymetry, flow, and sedimentary data sets from 2004 to 2018 at the Shashi Reach, a representative channel-bar-type reach in the Yangtze River (Fig.1), provide a rare opportunity to quantify the flow resistance in the interaction regions of large rivers. The reliability and robustness of these data obtained under low- and high-flow conditions offer advantages over empirical or physical-based models, which are commonly associated with inherent uncertainties (Paarlberg et al., 2009; Ferguson et al., 2022). In particular, we were able to decompose total resistance into its component parts and evaluate their variations under different hydraulic geometry conditions. The specific objectives of this study were to 1) develop a semi-empirical approach for flow resistance calculation, 2) quantify the contributions of each resistance component to total resistance, and 3) clarify the change law of flow resistance in response to hydraulic geometry in the channel-bar-type landscapes of large alluvial rivers. The study findings enhance our understanding of the dynamic changes in flow resistance in channel-bar landscapes in large river systems and have important implications for riverine ecology and flood management.

2 Data and methods

2.1 Study area

The Yangtze River is 6300 km long and is usually divided into three parts—the upper Yangtze (source to Yichang), middle Yangtze (Yichang to Hukou), and lower Yangtze (Hukou to Datong) (Li et al., 2019; He et al., 2022). The middle Yangtze River (MYR) is characterized by a subtropical monsoon climate with an average annual temperature ranging from 15°C to 19°C and an average annual precipitation of 1000 mm (Yang et al., 2015). The Shashi Reach is located 170 km downstream of the Three Gorges Dam (TGD) in the MYR (Fig.1). The Shashi Reach is a part of the Upper Jingjiang Reach, which is one of the most dynamic reaches along the MYR (Xia et al., 2016).

The Shashi Reach displays a meandering and bar-braided channel pattern characteristic of the morphology of meandering channels (Hooke and Yorke, 2011; Wang et al., 2018). The sand-bed channels in the MYR begin in the Shashi Reach, covering approximately 18 km from Chenjiawan to Yuheping. Therefore, grain size and hydraulic parameters in this reach are sensitive to the operation of the Three Gorges Dam, and hence it is convenient to evaluate flow resistance response with in situ data. Additionally, it is a representative reach in the MYR with a channel-bar landscape. Several bars are located in this reach: the Taipingkou mid-channel bar upstream of Jingzhou Bridge, the Lalinzhou point bar on the convex side of the bend, and the Sanba mid-channel bar beneath the bridge. Based on the bathymetric elevation in the Shashi Reach, a high bar with an elevation greater than 38 m, and a channel-bar interaction region with an elevation between 32 and 38 m, were selected for this study (Fig.1).

2.2 Data collection

Hydrological, topographical, and riverbed sediment grain-size data for the Shashi Reach for August 2003, November 2003, August 2010, November 2010, August 2015, December 2015, March 2018, and August 2018 were obtained from the Changjiang Water Resources Commission. Data were processed according to a set of strict survey protocols issued by the Ministry of Water Resources (i.e., “Code for Liquid Flow Measurement in Open Channels” (GB-50179-2015)), ensuring an accuracy of > 95%. The topographical and sediment data were collected from the Changjiang Waterway Bureau (CWB), and the bathymetric survey followed the survey protocols of the “Specifications for Port and Waterway Engineering Survey” (JTS/T231-2021), which ensures a horizontal error of ≤ 1 m and vertical error of ≤ 0.1 m (Li et al., 2019). In total, approximately 5500 elevation points were obtained in the Shashi Reach (18 km in length and 1.5 km in width) with a spatial resolution of approximately 100 m × 40 m (river flow direction × river width direction), which is comparable to that of other large rivers including, the Rhine in Germany (Arnaud et al., 2019) and the Missouri in the USA (Skalak et al., 2013).

The hydraulic, topographic, and granulometric parameters of the channel-bar interaction regions of the Shashi Reach were divided into the eight groups summarized in Tab.1. Each set of data covered discharge (Q, m³/s), total channel width (B, m), main channel width (b, m), total water depth (h₁, m), channel-bar interaction region water depth (h₂, m), main channel water depth (h₃= h₁−h₂, m), average flow velocity (U, m/s), median grain size of bed material (D₅₀, mm), energy slope (J, -), and Froude number (Fr, -). Based on long-term statistics, the discharge of the Shashi Reach is approximately 10000 m³/s (low-flow channel) during the dry season and 35000 m³/s (high-flow channel) during the flood season.

2.3 Estimating flow resistance in the channel-bar interaction region

The flow resistance in the channel-bar interaction region mainly consists of form resistance (Ferguson et al., 2019), grain resistance (Powell, 2014), and additional resistance (Chen et al., 2019b). Based on the linear superposition approach initiated in Zurich, Switzerland (Yen, 2002), we followed the momentum concept, assuming that total resistance can be separated linearly as follows:

(1)\begin{array}{c}\tau ={\tau }^{\prime }+{\tau }^{″},\end{array}

where \tau {}^{\prime } is riverbed resistance (i.e., grain and form resistance) and \tau {}^{\prime }{}^{\prime } is additional resistance. The effect of friction is quantified through the widely used Manning’s roughness coefficient, which is sensitive to the resistance variation with flow and sediment regimes (Ferguson, 2010), thus

(2)\frac{n^{\prime }\sqrt{g}}{R^{1/6}}=\frac{\sqrt{{\tau }^{\prime }/\rho }}{U},

(3)\frac{n^{″}\sqrt{g}}{R^{1/6}}=\frac{\sqrt{{\tau }^{″}/\rho }}{U},

where n{}^{\prime } and n^{″} refer to the Manning’s roughness coefficients of the riverbed and additional resistance, respectively, \rho refers to the fluid density (kg/m³), g refers to the gravitational acceleration (m/s²), and R refers to the hydraulic radius (water depth, m). Hence, to satisfy Eq. (1), we obtain the following:

(4)\begin{array}{c}{n}^2={n^{\prime }}^2+{n^{″}}^2.\end{array}

With respect to riverbed resistance, a widely used empirical model introduced by van Rijn (1984) based on flume and field data can be applied to predict microtopography, including dune height \Delta and length \lambda, and is not, therefore, repeated here. The equivalent roughness height k_s in the van Rijn (1984) model can be computed as follows:

(5)\begin{array}{c}{k}_s=3D_{90}+1.1\Delta \left(1-{\mathrm{e}}^{-25\mathrm{\Delta }/\lambda }\right),\end{array}

where D_{90} is the surface grain diameter of which 90% of the material is finer, and D_{90}=1.6D_{50} according to local sand-bed sorting (Parker, 1991). Thus, riverbed resistance is derived as follows:

(6)n^{\prime }=\frac{R^{1/6}}{18\mathrm{l}\mathrm{g}\left({\displaystyle \frac{12R}{k_s}}\right)}.

With respect to additional resistance, an extension of the Moreta and Martin-Vide (2010) method, which considers the flow regime and channel geometry, is proposed, as follows:

(7){\tau }^{″}=\frac{1}{2}\rho C_{\mathrm{f}\mathrm{a}}{\left(\mathrm{\Delta }U\right)}^2,

(8)C_{\mathrm{f}\mathrm{a}}=K_{\mathrm{f}\mathrm{a}}{\left[\left(\frac{B}{b}\right){\left(\frac{h_2}{h_1}\right)}^{-\frac{1}{3}}{\left(\frac{h_1-h_2}{b}\right)}^{-\frac{1}{3}}\right]}^{\alpha }{\left(\frac{U}{\mathrm{\Delta }U}\right)}^{\beta },

where C_fa is a non-dimensional friction coefficient acting at the vertical channel-bar interface, ΔU ( = U₁−U₂) is the flow velocity difference, U₁ is the average flow velocity in the channel, U₂ is the average flow velocity in the interaction region, and the coefficients K_fa and the exponents α and β are used to determine the best fit for C_fa. Thus, additional resistance is described as follows:

(9)n^{″}=\frac{\left(\mathrm{\Delta }U\right)R^{1/6}}{U}\sqrt{\frac{C_{\mathrm{f}\mathrm{a}}}{2\mathrm{g}}}.

Subsequently, Eqs. (4), (6), and (9) were used to calculate the total, riverbed, and additional resistances, respectively.

3 Results

3.1 Calibration, verification, and evaluation

Based on the detailed hydrometric and bathymetric data, all Manning’s roughness coefficients (n_measured) were calibrated with water levels and flow velocities at corresponding discharges using HEC-RAS. After calibration through data sets 5–8 (951 low-flow sets and 3459 high-flow sets), pending parameters of 0.035 and 0.016, −0.22 and −0.528, and 1.788 and 2.109 were obtained for K_fa, α, and β, respectively (Eq. (8)) for low flows and high flows, with a coefficient of determination of 0.968 and 0.999, respectively. The proposed formula (Eq. (4)) was then verified using data sets 1–4 (501 low-flow sets and 2947 high-flow sets), and the results are shown in Tab.2. Our new predictors showed acceptable accuracy in channel-bar interaction regions, while the discrepancy ratio (DR) was 17.4% for low flows and 25.1% for high flows (Fig.2). However, a small difference in the mean prediction error does not necessarily indicate similar predicted flow resistance values between low and high flows. In fact, our proposed formula overestimated flow resistance under low-flow conditions but underestimated it under high-flow conditions.

To further evaluate our formula, two traditional models that are widely applied to the Yangtze River (Huang et al., 2004) were chosen for performance comparison. The first is the van Rijn (1984) model given by Eq. (6), although here, n\text{'} is equivalent to n and, therefore, it was not recalculated. The second model is the Peterson-Peterson model (Peterson and Peterson, 1988; Eq. (10)):

(10)\begin{array}{c}n=1.003R^{0.23}{J}^{0.224}{D}_{50}^{-0.017}.\end{array}

All of the models were assessed (using data sets 1–4) using statistical indices including the deviation ratio (DR), geometric standard deviation (GSD), and root-mean-square error (RMSE). Tab.2 shows that our proposed model (DR = 17.4%) outperformed the van Rijn (DR = 42.0%) and Peterson-Peterson (DR = 37.9%) models under low-flow conditions. In the case of high-flow conditions, our new model performed better (DR = 25.1%) than the van Rijn model (DR = 35.4%) but slightly worse than the Peterson-Peterson model (DR = 20.6%). Such a difference in model performance between low and high flows may be due to variations in compound hydraulic geometries, which are of great significance in channel-bar-type landscapes.

3.2 Variations in flow resistance with flow conditions and riverbed grain size

Both riverbed resistance and additional resistance changed significantly when the bar was inundated, with a mean increase in riverbed resistance of 22.9% (from 0.020 to 0.025) and a decrease in additional resistance of 43.6% (from 0.039 to 0.021) from low to high flows, respectively (Fig.3). Riverbed resistance was 0.5- and 1.2-times that of additional resistance at low and high flows, respectively, indicating a shift in the dominant role of total resistance during inundation. This may be attributed to variations in the mean water depth and flow velocity. When the water depth in the channel-bar interaction region is small, the flow velocity here is significantly less than that in the channel, indicating a large flow velocity difference and thus a big additional resistance. In contrast, when the water depth in the channel-bar interaction region is big, the flow velocity difference will be smaller and resulting in a small additional resistance.

Flow resistance showed strong relationships with mean water depths and flow velocities but a weak association with the median diameter of riverbed grains (Fig.3). Flow resistance increased significantly with increasing water depth, with a mean increase in riverbed resistance of 64.6% and 25.1% at low and high flows, and additional resistance of 90.6% and 115.8%, respectively. A minimum riverbed resistance (0.32 m/s) occurred with an increase above the mean flow velocity, suggesting a strong connection between riverbed resistance and flow regimes. Such a critical flow velocity was also observed by van Rijn (1984) and is likely attributable to form drag in the interaction region, contributing most to riverbed resistance. The transition of flow regimes also has significant impacts on bedforms, including plane beds, ripples, dunes, washed-out dunes, and anti-dunes (Fernandez et al., 2006). However, additional resistance gradually increased with an increase in the mean flow velocity. Furthermore, both riverbed resistance and additional resistance showed a weak relationship with riverbed grain size, indicating that grain resistance contributes little to total resistance in channel-bar interaction regions, which is consistent with sand-bed alluvial channels.

After the operation of the TGD, there are unavoidable impacts on the Shashi Reach, including its grain size and hydraulic parameters. The median grain size in the channel has increased from 0.019–0.283 mm in 2003 to 0.019–0.315 mm in 2018 (Tab.1). Meanwhile, median grain size in the bar has increased from 0.091–0.220 mm in 2003 to 0.096–0.290 mm in 2018 (Tab.1). This indicates a potential increase in the riverbed resistance in the channel-bar interaction region in response to the grain coarsening. In the meantime, hungry-water-driven channel erosion causes the amplification of the depth-width ratio (from 1 to 64 \times10⁻⁴ in 2003 to 8–84 \times10⁻⁴ in 2018; Tab.1), then an enhanced additional resistance in the interaction region may also occur. Therefore, the operation of the TGD may promote the increase of flow resistance in the interaction region of the Shashi Reach.

4 Discussion

4.1 Sensitivity analysis of the impact of hydraulic geometry on flow resistance

To test the performance of our proposed formula for predicting flow resistance in channel-bar interaction regions with variable hydraulic geometry, we selected one set of data from data set 4 to conduct a sensitivity analysis. For these data, h₁ = 6.18 m, h₂ = 3.07 m, B = 1525.01 m, b = 1416.77 m, U₁ = 0.68 m/s, U₂ = 0.37 m/s, and D₅₀ = 0.22 mm. The relationships between flow resistance and total width (B), main channel width (b), channel-bar interaction region water depth (h₂), main channel water depth (h₃), mean channel flow velocity (U₁), mean interaction region flow velocity (U₂), and median grain size of the bed material (D₅₀) are shown in Fig.4.

Riverbed resistance first decreases and then increases as water depth, mean flow velocity, and D₅₀ increase in the channel-bar interaction region, which might be attributed to the bedform variations under different flow regimes (Fig.4). For example, ripples and dunes may occur or wash out in lower and transitional flow regimes. However, neither changes in the hydraulic parameters of the main channel nor variations in the channel-bar-width ratio affected the riverbed resistance values in the interaction region. Furthermore, the changes in the riverbed Manning’s roughness coefficient were small (< 0.02) when the hydraulic parameters varied relative to additional resistance (< 0.05).

Additional resistance is sensitive to all hydraulic geometry parameters except D₅₀ (Fig.4), continuously increasing as B/b and h₂/h₁ decrease, h₃/b increases, or U/ΔU increases. For each individual parameter, additional resistance shows a positive correlation with b, h₂, h₃, and U₁, and a negative correlation with B and U₂. The relative influence of these factors on additional resistance, in descending order, is U₂ > h₂ > U₁ > b > B > h₃. These results indicate that the difference between the channel-bar flow velocities and total water fluxes in channel-bar interface regions are the two dominant factors controlling additional resistance.

4.2 Implications and further research

Previous studies have faced three main limitations when estimating flow resistance in the channel-bar interaction regions of large rivers (Chandra, 2019; Yang et al., 2022). First, calculation models are primarily proposed based on data from laboratories or small rivers (Shiono and Knight, 1991; Chen et al., 2019a). Second, many models are derived under the assumption that the Manning’s roughness coefficient in the channel and bar are known and remain constant; however, roughness always has complicated variations as flow and sediment regimes change (Moreta and Martin-Vide, 2010; Chen et al., 2016). Third, earlier models usually assumed that flow resistance in interaction regions is either riverbed resistance or additional resistance, forcing the simplification of complex dynamic changes between the two (Huthoff et al., 2008). Therefore, the resistance equation considering hydraulic geometries and flow regimes that can be applied in large alluvial rivers remains lacking. Importantly, our proposed formulae (Eqs. (4), (6), and (9)) overcome these limitations and, therefore, provide the first model to be successfully applied to the channel-bar-type landscape of a large river, the Shashi Reach of the Yangtze River. Such findings further the understanding of dynamic changes in flow resistance in the channel-bar landscapes of large river systems and have potentially important implications for riverine ecology and flood management.

However, our study remains preliminary with respect to the mechanisms controlling the geomorphological evolution of channel-bar interaction regions, and further work is needed (Church and Ferguson, 2015; Yamaguchi et al., 2019; Zhou et al., 2020; Ferguson et al., 2022). First, large rivers with wider flow regime ranges must be investigated (Fernandez et al., 2006). For instance, a transitional stage with washed-out dunes has been generated for a Froude number of approximately 0.6 in flume experiments (van Rijn, 1984) and 0.2–0.3 in the Missouri River, USA (Shen et al., 1978), yet the mean Froude number in our study was only 0.08. Second, logarithmic or exponential resistance formulas that consider large-scale roughness elements may be alternatives for riverbed resistance calculation where, for example, Eq. (6) cannot be applied to gravel or bedrock channels (Ferguson et al., 2017; Chen et al., 2020). Third, various turbulence parameters in channel-bar interaction regions should be considered when conducting field measurements because of the importance of turbulence information in quantifying velocity distributions (Knight and Shiono, 1990; Chandra, 2019). Fourth, studies concerning erosion and deposition, and sediment and pollutant transport within these interaction regions could be usefully carried out, particularly under various geometry and hydraulic conditions (Paarlberg et al., 2009). Fifth, an extension of our study to vegetated areas should also prove useful (Liu et al., 2020, 2021a, 2021b). Therefore, further application and extension of our proposed flow resistance equations to other global rivers with channel-bar landscapes will yield informative results.

5 Conclusions

We examined the response of riverbed resistance and additional resistance to hydrometric geometry in the channel-bar interaction region of large rivers based on in situ hydro-sediment-bathymetry data for the Shashi Reach of the Yangtze River. We developed a semi-empirical approach for predicting flow resistance that quantifies riverbed resistance and additional resistance separately. Our model outperforms conventional flow resistance models (e.g., the van Rijn model and the Peterson-Peterson model) by integrating dynamic changes into channel geometry, flow, and sediment regimes in the flow resistance calculation.

Overall, our study provides new understanding of a) the intensity of the momentum exchange in the channel-bar interaction regions of large rivers and its greater effects on additional resistance than on riverbed resistance; b) how additional resistance and riverbed resistance decrease and increase, respectively, during bar inundation; and c) the minimum value of riverbed resistance with increases in mean flow velocity. These findings have important implications for assessing fluvial responses to changes in flow and sediment regimes in channel-bar interaction regions and for evaluating associated impacts on hydropower development and operations, flooding, and aquatic ecosystems. Our insights also underline the need for more field-based turbulence and sediment observations across these interaction regions to better assess the impacts of changing channel-bar landscapes on fluvial systems in the context of climate change and human activity.

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