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Flooding propagation is a crucial aspect of hydrological monitoring and forecasting. Previous studies have focused on hysteresis in the rating curve, caused by energy loss during flood propagation. However, the impact of tributary inflow on hysteresis downstream remains unclear, leading to inconsistent field observations on whether it strengthens or weakens hysteresis. In this study, we conducted flume experiments to identify the relationship between hysteresis in unsteady flow and the discharge magnitude of the tributary and the unsteady flow period in the mainstream. It was found that the discharge variations in the tributary had a larger influence on hysteresis compared to the periodical variations in the mainstream unsteady flow. Interestingly, the hysteresis of the unsteady flow had an initial strengthening followed by weakening as the tributary discharge increased. When the tributary inflow was low, the widening of the downstream cross-section sharpened the flood wave, increasing the hysteresis. However, as the tributary discharge increased to generate a backwater effect on the mainstream, the pressure gradient flattened flood waves, thereby weakening the hysteresis. This study improves our understanding of how tributary inflow affects flood propagation in the mainstream, offering new insights for flood prediction and control.

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  • The hysteresis of the unsteady flow exhibits a pattern of initially strengthening and then weakening as the discharge of the tributary increases.

  • The main effect of the change in hysteresis characteristics of unsteady flow was flow attenuation, which was generated by the dynamic factors of pressure gradient and frictional gradient-induced convective diffusion between the upstream and downstream flows.

channel confluence, hydrodynamics, hysteresis, rating curve, unsteady flow
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Unsteady flows are common in natural rivers and channels, especially during rapid flow changes (such as the operation of gates and pumps, flood events). Flood events often cause loss of human life and significant damage. Thus, it is crucial to predict flood wave propagation, which is a very complex hydrological and hydrodynamic phenomenon (Munoz & Constantinescu 2018). Previous studies show that during the passage of unsteady flow, discharge, velocity, and flow depth peak at different times, that is, they show a hysteresis (Nezu & Nakagawa 1991; Song & Graf 1996; Qu 2002). What is more, the inflow of tributaries could cause lateral turbulence of the mainstream (Alizadeh & Fernandes 2021) and increase the complexity of the flow at confluences (Constantinescu et al. 2011; Yuan et al. 2016; Tang et al. 2022). Previous studies on hysteresis of unsteady flow at confluences did not have a unified understanding, and opposing phenomena were even observed through field experiments (Meade et al. 1991; Guse et al. 2020). After analyzing field data from the Amazon basin, Meade et al. (1991) concluded that the backwater effect of the mainstream, caused by the inflow of tributaries, leads to an earlier peak-discharge in the mainstream, thereby reducing the hysteresis of floods. On the other hand, Guse et al. (2020), based on their study of confluences in four large river basins in Germany and Austria, found that the inflow of tributaries increases the hysteresis of the mainstream, which is due to the lower local soil saturation. However, neither of these studies thoroughly investigated the underlying dynamic mechanisms. Therefore, clarifying the role of tributaries in the hysteresis of the unsteady mainstream and elucidating the fundamental reasons for the changes in hysteresis at river confluences is essential for river basin flood control.

The hysteresis effect is the inherent characteristic of unsteady flow which has been widely reported in hydrology since the work of Haines (1930) and Richards (1931). Hysteresis is a relationship between two variables, that is, the lagging of an effect behind its cause marks a loop in their relationship. This definition indicates that there is causation in the relationship between the two variables. As shown on the left side of Figure 1, hysteretic behavior is observed in the ‘rating curve’ that characterizes the stage–discharge relationship in open channels (Miller & Lindner 2009; Muste et al. 2020). Another effect of the unsteady flows is the phase sequencing between the time series of the mean flow variables. To identify the underlying causes of these differences, previous studies refer to the Saint-Venant equations, which are commonly employed in hydraulic modeling to elucidate the propagation of unsteady flow in streams (Muste et al. 2020). While few field studies are available, several laboratory studies highlighted this essential characteristic of unsteady flow dynamics (Hunt 1997; Nezu et al. 1997). Laboratory studies also found that the differences between flow phases are more prominent for high flows and rate changes and the tributary inflow can effectively reduce the hysteresis value. In the above results, we found a link between hysteresis and energy change. For discussing these differences from the perspective of equations, the Saint-Venant equations that are often used in hydraulic modeling to describe unsteady flow propagation in streams were considered. The governing equation for a complete flood wave is defined by Equation (1):
(1)
Figure 1
Sketch of the hysteretic effect on the stage–discharge relationship: stage–discharge relationship (left) which is loop-rating curve, and non-overlapping hydrographs (right). U is the cross-sectional mean velocity, Q is the flow discharge, and H is the water level (Muste et al. 2022).
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Sketch of the hysteretic effect on the stage–discharge relationship: stage–discharge relationship (left) which is loop-rating curve, and non-overlapping hydrographs (right). U is the cross-sectional mean velocity, Q is the flow discharge, and H is the water level (Muste et al. 2022).

Figure 1
Sketch of the hysteretic effect on the stage–discharge relationship: stage–discharge relationship (left) which is loop-rating curve, and non-overlapping hydrographs (right). U is the cross-sectional mean velocity, Q is the flow discharge, and H is the water level (Muste et al. 2022).
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Sketch of the hysteretic effect on the stage–discharge relationship: stage–discharge relationship (left) which is loop-rating curve, and non-overlapping hydrographs (right). U is the cross-sectional mean velocity, Q is the flow discharge, and H is the water level (Muste et al. 2022).

Close modal
The momentum equation for a complete flood wave is defined by Equation (2):
(2)
where v represents the cross-sectional average velocity, C is the Chezy coefficient, R is the hydraulic radius, A is the cross-sectional area for flow, Q is the discharge, y is the water depth, t is time, x is the distance along the channel direction, and g is the acceleration due to gravity (Henderson 1966), and is the inertial term, is the additional friction term, i0 is the bed slope and is the resistance term. In the present context, for the convenience of analysis, the momentum equation is rewritten into the following equivalent form:
(3)

The type of dominant wave for a given situation is determined by the relative contribution of the terms in Equation (2). These terms continuously change their magnitude and signs (i.e., positive or negative) during the flood wave propagation, commensurate with the slope of the streambed at the site, the intensity of the propagating wave (i.e., its magnitude vs. duration), and the flood wave propagation phase (Henderson 1966). Previous use of Equation (2) for channel flow routing has shown that it is satisfactory for all flood wave propagation, regardless of the wave type: kinematic, diffusion, or full dynamic (Di Baldassarre & Montanari 2009).

To overcome the limitations of theoretical analysis, Mishra & Singh (1999) carried out the numerical simulation, and mentioned the influence of river contraction/expansion, bed roughness and river slope on the hysteresis between unsteady flow parameters, and made the change of hysteresis attributed to energy loss. However, in these formulaic analyses, they neglected the lateral inflow. To facilitate the analysis, the channel's configuration was simplified to a straight channel, thus neglecting the impact of lateral inflow on the unsteady flow of the mainstream. This simplification, in the presence of stage–discharge hysteresis, might result in substantial uncertainty in the estimation of discharge and discharge-related inferences such as model calibration to streamflow.

However, most of such relationships reported in the hydrological literature were described using descriptive indicators such as direction (clockwise or counterclockwise) and shape of the stage–discharge relationship curve, and how the shape and direction of the stage–discharge relationship curve evolves over time and space (Langlois et al. 2005; Lawler et al. 2006; Lloyd et al. 2016; Zuecco et al. 2016). These studies mainly focused on describing their observations in a single channel, which provided little insight into the fundamentals and underlying functioning of systems demonstrating those hysteretic behaviors. In particular, the previous research treated the hysteresis as a one-dimensional result in the model, lacking the analysis of a river system's physical process (Gharari & Razavi 2018). Therefore, it is very important to explain the hysteresis of unsteady flow at confluences from the perspective of the effect of flow hydrodynamics.

Due to the flow mixing, secondary circulation, post-confluence flow separation, contraction and backwater effects, the flow structure of open channel confluence stems is complex (Huang et al. 2002; Yuan et al. 2016; Xu et al. 2022). Many previous laboratory experiments (Best 1987), numerical studies (Shakibainia et al. 2010; Constantinescu et al. 2011; Jiang et al. 2022) and field investigations (Gualtieri et al. 2018; Li et al. 2022a, 2022b; Yuan et al. 2021, 2022a, 2022b, 2022c; Tang et al. 2022) have identified the structure of the flow at a confluence within the confluence hydrodynamic zone (CHZ), which includes: stagnation zone, separation zone, flow deflection zone, the shear layer, flow acceleration zone, and flow recovery zone of the CHZ. Among them, the separation zone is considered one of the primary mechanisms contributing to energy losses in the CHZ (Webber & Greated 1966; Arega et al. 2008; Ghostine et al. 2012; Mohammadiun et al. 2015; Schindfessel et al. 2015; Luo et al. 2018; Yuan et al. 2023). Hereby, the separation zone is particularly noteworthy in the study of the unsteady process at confluences, considering the relationship between the hysteresis of unsteady flow and energy loss (Mishra & Singh 1999).

We claim that previous studies on the hysteresis effect of unsteady flow have mainly focused on channels without lateral inflows, while for confluence channels with lateral inflows, only phenomenological descriptions have been made based on field observations. Due to the difficulty of describing channels with backwater effects using existing formulas (Kanda & Yanagiya 2008), this study investigates the unsteady flow in confluence channels by combining flume experiments with the theoretical foundation of previous research on hysteresis effect in straight channels. The objective of this study is to clarify how the tributary inflow affects the hysteresis of the unsteady mainstream flow. Hence, this study aims to investigate: (1) whether the inflow of tributary affects the hysteresis in the unsteady mainstream flow; (2) the intensity of the influence of the unsteady flow period of the mainstream and the inflow of the tributary on the hysteresis of the mainstream unsteady flow; (3) the driving mechanism of the hysteresis variations of mainstream unsteady flow at confluences.

The study is structured as follows. In Section 2, we introduce the experimental setup. In Section 3, we analyze the hysteresis variation of the unsteady flow and the variation of separation zone size in the laboratory experiment. In Section 4, we discuss the causes of hysteresis in confluence zones by employing the Saint-Venant equations and explain how a better understanding of the hysteresis of unsteady flow can improve the accuracy of hydrological forecasting.

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Experimental flume and instruments

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The experiments were performed in a laboratory-scale confluence (Figure 2), located at the National Key Laboratory of Water Disaster Prevention at Hohai University (Yuan et al. 2016). The straight upstream channels were concordant with a width of 0.3 m, a length of 3 m, and a height of 0.4 m, while the tributary channel was slightly wider (0.42 m) and longer (7 m) (Figure 2). The bed in the flume was flat and the junction angle was 60°. Flow was pumped from the downstream tank to the two upstream tanks through 110-mm-diameter PVC pipes, and the flow discharges were accurately controlled by the variable-frequency pump-valve system. The tail end of the water tank was provided with a tailgate to control the water depth.
Figure 2
Plainview of the laboratory-scale confluence and study areas.
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Plainview of the laboratory-scale confluence and study areas.

Figure 2
Plainview of the laboratory-scale confluence and study areas.
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Plainview of the laboratory-scale confluence and study areas.

Close modal
A particle image velocity measurement (PIV) system was used in the present study to capture the images in a plane for two-dimensional (2D) velocity components (Figure 3). The system included a double pulsed Nd: YAG laser (5 W) with a wavelength of 532 nm, a 2,560 × 2,048-pixel CMOS camera (JAI sp5000-cxp2) with a Kowa LM25HC 50 mm f/1.2 lens, PIV image processing software, image acquisition card and a high-performance computer. The flow was seeded with micro-spheres with a density of 1.04 kg/m3 and a mean diameter of 5 μm. The PIV procedure includes seeding of the flow, illumination of the measurement plane using laser and successive capturing of the illuminated plane using the cameras. The images were acquired on the PC using PIV image processing software (Liu et al. 2021).
Figure 3
Sketch of the PIV setup.
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Sketch of the PIV setup.

Figure 3
Sketch of the PIV setup.
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Sketch of the PIV setup.

Close modal
Three-dimensional flow velocities in the CHZ were measured using an acoustic Doppler velocimeter (ADV, NorTek Vectrino + , Oslo, Norway; maximum velocity: 4 m s–1; accuracy: ±10−3 m s−1) at a sampling frequency of 100 Hz over a period of 120 s (Yuan et al. 2016). Seeding materials consisting of neutrally buoyant hollow glass spheres (Potter Industries Sphericell, Valley Forge, USA) were added to provide acoustic scatting signals for ADV during the tests. Velocities were measured at points C and D perpendicular to the flume walls in each case (Figure 4). The probe was fixed at h = 13 cm (h represents elevation above channel bed) in the middle of the flume of all experimental runs and therefore the measurement range was h = 8 cm after subtraction of the 5 cm-long blind area (Figure 4).
Figure 4
Sketch of the ADV setup.
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Sketch of the ADV setup.

Figure 4
Sketch of the ADV setup.
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Sketch of the ADV setup.

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Experimental setting and data processing

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To better understand the effects of different inflow discharges and periods of unsteady flow on hydrodynamics and hysteretic properties, two types of unsteady flow with different periods and five different discharges of tributary were designed for the experiment (Table 1). The unsteady discharges in the mainstream, Qm, were varied between 5 and 12 L/s with 27 s periodic variations of the sine curve (Figure 5) (Lin et al. 2019). The steady discharges of the tributary, Qt, were different (Table 2). In particular, the experiment without a tributary was also set up to compare the effect of the tributary on hysteresis. The other experiments had the discharge with 54 s periodic variations. The water depth in the mainstream was from 19.5 to 20.5 cm and the water depth in the tributary channel was 20 cm (Figure 5).
Table 1

Experimental flow conditions

RunQm (L/s)Qt (L/s)ξ = Qt/(Qm + Qt)T (s)
0-27 5–12 – – 27 
3-27 3 0.2–0.375 
4--27 4 0.25–0.44 
5-27 5 0.29–0.50 
6-27 6 0.33–0.55 
0-54 5–12 – – 54 
3--54 3 0.2–0.375 
4-54 4 0.25–0.44 
5-54 5 0.29–0.50 
6-54 6 0.33–0.55 
RunQm (L/s)Qt (L/s)ξ = Qt/(Qm + Qt)T (s)
0-27 5–12 – – 27 
3-27 3 0.2–0.375 
4--27 4 0.25–0.44 
5-27 5 0.29–0.50 
6-27 6 0.33–0.55 
0-54 5–12 – – 54 
3--54 3 0.2–0.375 
4-54 4 0.25–0.44 
5-54 5 0.29–0.50 
6-54 6 0.33–0.55 

Note: Q is the discharge; T is the transformation period; The subscripts ‘m’ and ‘t’ represent the mainstream and the tributary respectively.

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Table 2

Nondimensional hysteresis, η

RunBefore the confluence zone η1After the confluence zone η2Increase rate Δη
0-27 0.28 0.42 0.50 
3-27 0.50 0.55 0.10 
4-27 0.51 0.45 −0.12 
5-27 0.56 0.45 −0.20 
6-27 0.56 0.35 −0.38 
0-54 0.32 0.39 0.22 
3-54 0.57 0.65 0.14 
4-54 0.57 0.37 −0.35 
5-54 0.60 0.33 −0.45 
6-54 0.60 0.16 −0.73 
RunBefore the confluence zone η1After the confluence zone η2Increase rate Δη
0-27 0.28 0.42 0.50 
3-27 0.50 0.55 0.10 
4-27 0.51 0.45 −0.12 
5-27 0.56 0.45 −0.20 
6-27 0.56 0.35 −0.38 
0-54 0.32 0.39 0.22 
3-54 0.57 0.65 0.14 
4-54 0.57 0.37 −0.35 
5-54 0.60 0.33 −0.45 
6-54 0.60 0.16 −0.73 
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Figure 5
Designed discharges of the flume experiments.
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Designed discharges of the flume experiments.

Figure 5
Designed discharges of the flume experiments.
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Designed discharges of the flume experiments.

Close modal

A dark room was arranged around the flume, a bracket was arranged inside the darkroom, a slide and a slider were arranged at the top of the bracket, and the CMOS camera was hung upside down on the bracket slider through the spherical head. The optical axis of the CMOS camera was vertically downward, perpendicular to the laser plane (i.e., the plane of measurement, which was a horizontal plane), and the positional relationship between the CMOS camera and the plane in which velocity was measured is as shown in Figure 3. PIV images were captured by the CMOS cameras which continuously transferred the data to the computer at a frame rate of 100 Hz and an imaging resolution of 3.75-4 pixels/mm, denoting 100 instantaneous vector fields per second. The raw data was saved to the main database in the PC for post-processing and further analysis.

Due to the limited measurement area of PIV, the experimental study area was divided into section 1 and section 2 to obtain the characteristics of the flow structure at the confluence. The x and y axes used in this study are shown in Figure 2. When viewed in the direction of flow, the 0 point coincides with the lower-left vertex of the study area. The velocity of flow components corresponding to the XY-axis directions is u and v, respectively. The velocity field was calculated by cross-correlating the particle displacements within matching interrogation regions in consecutive images. The velocity field was obtained using a multi-grid interrogation process. The interrogation area was 32 × 32 pixel2, the grid size was 16 × 16 pixel2, and the number of iterations was 5. The instantaneous velocity fields were calculated by the PIV processing software independently developed by Professor Wang Xingkui (Tsinghua University).

The velocity-area method was based on the empirical equation for the cross-sectional velocity distribution, where the velocity of the cross-section was measured and the water level at the cross-section was calculated to obtain the discharge (Doering and Hans 2001; Coz et al. 2012; Caissie 2021). The method results in a discharge overestimation of less than 4%, which falls within the acceptable error range for the experiment (ISO 748 2007). Firstly, the velocities at points C and D had to be measured with the ADV, while the corresponding water levels at the same time were measured with a water level meter (points A and B are the midpoints of their respective cross-sections), the velocities at the depth of 4, 8 and 12 cm were measured. Next, we used the PIV for sections A and B, measuring five layers (h = 4, 8, 12, 16, 18 cm) for each condition. This gives us both PIV and ADV measurements at points C and D in layers 4, 8 and 12 cm, and we should be able to compare these two sets of data to obtain the corresponding water levels at the time of PIV data collection. Finally, the discharge was obtained for cross-sections using the velocity-area method, which is used for discharge calculation in various industries (Bullard et al. 2007; Steinbock et al. 2016).

To calculate the value of the hysteresis, the dimensionless parameter, η, which represents the area of the loop in the nondimensional form of the curve and can be defined as a hysteresis parameter (Mishra & Seth 1996) was used. It can be expressed mathematically as:
(4)
where:
(5)
(6)

Here, T is the time period of the flood wave (i.e., the time of rise plus the time of recession); h is the dimensionless stage (a function of time); and Hmax, Hmin, and H are maximum, minimum, and time-varying depths, respectively, at a site of interest. Similarly, Q and q stand for discharge and dimensionless discharge, respectively.

The energy head or specific energy was defined by Chow (1959) as:
(7)
where y = flow depth, z = bed elevation and V = cross-sectional average velocity. In the present study, the total energy loss at a junction was evaluated in a global sense (Hsu et al. 1998a, 1998b; Lin & Soong 1979) as the difference between the entering and leaving energies per unit time as follows:
(8)
(9)
As Equation (8) is for a steady flow, Equation (10) has an integral form. By calculating the energy change at each moment in a period, the energy change in this period could be obtained by integrating:
(10)
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Rating curve of unsteady flow

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In order to illustrate the hysteresis behavior with experimental data, we plot in Figure 6(a)–(l) samples of directly measured stage and the discharge calculated by velocity-area method time series at the mainstream and tributary. Due to the limitation of the length of the article, we selected four periods of 54 seconds each as representative runs to study the impact of varying inflow discharge on the rating curve. We observe that the shape of the loops in Figure 6(a) and 6(e) and in Figure 6(b) and 6(f), respectively exhibit a similar pattern of variation, where the area under the loops of the mainstream appears to be smaller than that of the downstream. However, Figure 6(c) and 6(g) and Figure 6(d) and 6(h) present the opposite pattern. Although both Run 0-54 and Run 3-54 have enhanced unsteady hysteresis downstream, a comparison between Figure 6(i) and 6(j) reveals that in Run 3-54, the time hysteresis between maximum discharge and maximum depth of downstream is smaller than in Run 0-54 when compared to the mainstream. Second, Figure 6(i)–6(l) confirms the separation of the variable hydrographs hinted at in Figure 6, and their identical sequencing order.
Figure 6
Hysteresis effects on flow variables: (a–d) depth vs. discharge in the mainstream; (e–h) depth vs. discharge in the downstream; (i–l) relationships for pairs of variables measured during a flood wave propagating. The red arrow is the time of maximum depth reached in the mainstream, the green arrow is the time of maximum depth reached downstream, the black arrow is the time of maximum discharge reached in the mainstream, and the blue arrow is the time of maximum discharge reached downstream, respectively. Note: The subscripts ‘D’, and ‘M’ are downstream and mainstream, respectively.
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Hysteresis effects on flow variables: (a–d) depth vs. discharge in the mainstream; (e–h) depth vs. discharge in the downstream; (i–l) relationships for pairs of variables measured during a flood wave propagating. The red arrow is the time of maximum depth reached in the mainstream, the green arrow is the time of maximum depth reached downstream, the black arrow is the time of maximum discharge reached in the mainstream, and the blue arrow is the time of maximum discharge reached downstream, respectively. Note: The subscripts ‘D’, and ‘M’ are downstream and mainstream, respectively.

Figure 6
Hysteresis effects on flow variables: (a–d) depth vs. discharge in the mainstream; (e–h) depth vs. discharge in the downstream; (i–l) relationships for pairs of variables measured during a flood wave propagating. The red arrow is the time of maximum depth reached in the mainstream, the green arrow is the time of maximum depth reached downstream, the black arrow is the time of maximum discharge reached in the mainstream, and the blue arrow is the time of maximum discharge reached downstream, respectively. Note: The subscripts ‘D’, and ‘M’ are downstream and mainstream, respectively.
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Hysteresis effects on flow variables: (a–d) depth vs. discharge in the mainstream; (e–h) depth vs. discharge in the downstream; (i–l) relationships for pairs of variables measured during a flood wave propagating. The red arrow is the time of maximum depth reached in the mainstream, the green arrow is the time of maximum depth reached downstream, the black arrow is the time of maximum discharge reached in the mainstream, and the blue arrow is the time of maximum discharge reached downstream, respectively. Note: The subscripts ‘D’, and ‘M’ are downstream and mainstream, respectively.

Close modal

The influence of different factors on the hysteresis of unsteady flow

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To more directly demonstrate the variations in hysteresis, we utilized Equation (4) to calculate the hysteresis parameters of unsteady flow before and after passing through the confluence area under 10 runs, as shown in Table 2, and the variation of the value of hysteresis, Δη was calculated by Equation (11). Using Table 3, we analyzed the impact of different factors on hysteresis below:
(11)
Table 3

Energy change, ΔE (J)

RunEm + EtEdΔE
0-27 49.52 42.96 −6.56 
3-27 63.64 60.09 −3.55 
4-27 65.32 67.30 1.98 
5-27 69.44 71.98 2.54 
6-27 71.65 74.87 3.22 
0-54 94.23 87.29 −6.94 
3-54 131.07 128.59 −2.48 
4-54 136.10 137.38 1.28 
5-54 145.95 149.91 3.96 
6-54 154.57 159.49 4.92 
RunEm + EtEdΔE
0-27 49.52 42.96 −6.56 
3-27 63.64 60.09 −3.55 
4-27 65.32 67.30 1.98 
5-27 69.44 71.98 2.54 
6-27 71.65 74.87 3.22 
0-54 94.23 87.29 −6.94 
3-54 131.07 128.59 −2.48 
4-54 136.10 137.38 1.28 
5-54 145.95 149.91 3.96 
6-54 154.57 159.49 4.92 
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Effect of the cross-sectional expansion

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The effect of the tributary on the hysteresis variation along the confluence was evaluated by routing confluence flume design unsteady flow. Since the downstream width of the experimental flume was 1.4 times wider than that of the upstream, when the tributary was blocked (no tributary inflow), the flume became a straight channel with a width wider than that of the upstream. Comparing the runs without the tributary inflow and tributary inflow of 3 L/s, as evident from Table 2, the η-values for expansion downstream are generally larger than upstream. When the discharge of 3 L/s enters from the tributary at the cross-sectional expansion, the η-values in downstream are still larger than upstream. The η-values of Run 3-27 and 3-54 are from 0.5 to 0.55 and 0.57 to 0.65, separately, while the η-values of Run 0-27 and 0-54 are from 0.16 to 0.48 and 0.32 to 0.39, separately. It is obvious that when there is no inflow from the tributary, the larger the value of Δη, the more significant the hysteresis is in the downstream. Thus, the cross-sectional expansion can increase the hysteresis.

Effect of the tributary inflow

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The effect of inflow tributary on unsteady flow propagation is examined for eight runs, from Run 3-54 to Run 6-54 and from Run 3-27 to Run 6-27, respectively. In those eight runs with two different periods, the discharge of the tributary, Qt, is 3, 4, 5 and 6 L/s, separately. Table 2 lists η-values along the laboratory-scale confluence in the eight runs, indicating larger hysteresis in the downstream than those in the upstream except for Run 3-54 and Run 3-27 at the corresponding locations. This indicates that the hindrance effect of the tributary on the mainstream weakened the hysteresis downstream of the unsteady flows. For 3 L/s, the hysteresis in the downstream was stronger than that in the upstream. This is because the increase in hysteresis due to the downstream expansion is greater than the decrease in hysteresis caused by the 3 L/s inflow of the tributary.

Effect of the period of unsteady flow

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The effect of the period of unsteady flow is evaluated by comparing the two different periods. From run 1 to 5 the periods of the unsteady flow are all 27 s (Table 2), and from run 6 to 10 the periods of the unsteady flow are all 54 s (Table 2). The η-values in the upstream under 27 s conditions are lower than those for the 54 s conditions. In the downstream of the confluence, the reducing ratio of the 54 s conditions is larger than the 27 s conditions. Through the comparison of these two periods, it can be inferred that the hysteresis of long-period unsteady flow is more affected by the inflow of tributaries.

Relationship between hysteresis and energy loss

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As mentioned above, it has been suggested that hysteretic changes are associated with energy change (Ponce 1989; Mishra & Singh 1999). In addition, it was observed that the hysteresis increased with the expansion of the cross-section when the tributary inflow was zero, that is, the loss of energy increased hysteresis. When the tributary inflow was small, the hysteresis still increased after the unsteady flow through the confluence zone, which could be explained as only the increased energy from the tributary was not enough to resist the energy loss associated with the cross-sectional expansion. Therefore, the hysteresis began to decrease as the inflow flow increased.

We used Equation (10) to estimate the energy loss in the confluence and the results are presented in Table 3. It can be seen from Table 3 when the experiment without tributary inflow and the experiment with tributary flow is 3L/s, ΔE is negative, indicating that the total energy decreased in the downstream. On the contrary, ΔE is positive when tributary flows are higher than 4 L/s. Figure 7 shows the relationship between the change of energy loss and the change of hysteresis (except for the runs without tributary), Δη and ΔE are negatively correlated in the same cycle. This also confirms that the greater the energy loss, the greater the hysteresis. Meanwhile, compare the slope of the two trend lines of two periods in Figure 7, for the same energy loss, the larger the period, the larger the change of the hysteresis parameter, Δη, and the more obvious the hysteresis phenomenon, except for 3 L/s.
Figure 7
Hysteresis Δη vs. energy change ΔE. The red line shows the trend of Δη vs. ΔE in the runs with a period of 54 s, and the black line shows the trend of Δη vs. ΔE in the runs with a period of 27 s.
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Hysteresis Δη vs. energy change ΔE. The red line shows the trend of Δη vs. ΔE in the runs with a period of 54 s, and the black line shows the trend of Δη vs. ΔE in the runs with a period of 27 s.

Figure 7
Hysteresis Δη vs. energy change ΔE. The red line shows the trend of Δη vs. ΔE in the runs with a period of 54 s, and the black line shows the trend of Δη vs. ΔE in the runs with a period of 27 s.
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Hysteresis Δη vs. energy change ΔE. The red line shows the trend of Δη vs. ΔE in the runs with a period of 54 s, and the black line shows the trend of Δη vs. ΔE in the runs with a period of 27 s.

Close modal
Figure 8
Sketch of the CHZ with notation. B represents the width of the separation zone, L represents the length of the separation zone, Wd represents the width of the downstream, θ represents the intersection angle, and α represents the deflection angle of the shear layer (Best 1987).
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Sketch of the CHZ with notation. B represents the width of the separation zone, L represents the length of the separation zone, Wd represents the width of the downstream, θ represents the intersection angle, and α represents the deflection angle of the shear layer (Best 1987).

Figure 8
Sketch of the CHZ with notation. B represents the width of the separation zone, L represents the length of the separation zone, Wd represents the width of the downstream, θ represents the intersection angle, and α represents the deflection angle of the shear layer (Best 1987).
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Sketch of the CHZ with notation. B represents the width of the separation zone, L represents the length of the separation zone, Wd represents the width of the downstream, θ represents the intersection angle, and α represents the deflection angle of the shear layer (Best 1987).

Close modal

The variation of the separation zone size under unsteady flow

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The width of the downstream flow channel is relevant to flow energy loss. The separation zone often occupies 20–50% of the downstream flow channel width (Figure 8) (Best 1987). Therefore, the separation zone plays a crucial role in the calculation of energy loss in the confluence zone.

The flow structure in the separation zone is complex and it is difficult to accurately identify the size of the separation zone. In this study, the zero-velocity method (Yang et al. 2014) was adopted, which means that the contour line where the flow velocity was zero in the streamwise direction was taken as the boundary of the separation zone. The adopted method of measurement was implemented in a horizontal plane, and thus was 2D, whereas the separation zone was known to be 3D. To evaluate how these represent the depth-averaged flow modeled in theoretical models, the dimensions of the separation zone were determined on the depth of z/h = 0.4 flow field.

Figure 9 shows the instantaneous flow velocity distribution of the peak-discharge (Tp), medium-discharge (Tm) and valley-discharge (Tv) instantaneous moments under the Run 6-54 at z/h = 0.4. The black line represents the edge of the separation zone. Figure 9 indicates that the size of the separation zone is very small at Tp (Figure 9(a)), and its relative width is only about 0.1, accounting for 14% of the width downstream. The width of the separation zone does not change much at Tm, but the relative length increases from 1 at Tp to 2.8 at Tm. With the gradual decrease of the mainstream discharge, the width of the separation zone gradually increases, accounting for 28% of the mainstream at Tv. It follows that the dimension of the separation zone increases as the discharge of mainstream decreases, which is in agreement with the observations of Best & Reid (1984).
Figure 9
Plots of the separation zone under unsteady flow condition in the confluence at a different time. The black solid line represents the edge of the shear layer. Note: ‘Tp’, ‘Tm’ and ‘Tv’ are peak-discharge, medium-discharge and valley-discharge instantaneous moments, respectively.
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Plots of the separation zone under unsteady flow condition in the confluence at a different time. The black solid line represents the edge of the shear layer. Note: ‘Tp’, ‘Tm’ and ‘Tv’ are peak-discharge, medium-discharge and valley-discharge instantaneous moments, respectively.

Figure 9
Plots of the separation zone under unsteady flow condition in the confluence at a different time. The black solid line represents the edge of the shear layer. Note: ‘Tp’, ‘Tm’ and ‘Tv’ are peak-discharge, medium-discharge and valley-discharge instantaneous moments, respectively.
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Plots of the separation zone under unsteady flow condition in the confluence at a different time. The black solid line represents the edge of the shear layer. Note: ‘Tp’, ‘Tm’ and ‘Tv’ are peak-discharge, medium-discharge and valley-discharge instantaneous moments, respectively.

Close modal
The size of the separation zone for Run 6-54 and Run 6-27 are expressed in the rising and falling limbs with the change of the confluence ratio ξ (Figure 10). In Figure 10(a) and 10(b), as the discharge of the mainstream changed relative to total flow, the maximum relative width of the separation zone, B/Wb, increases following each of the least-squares regression equations (Best & Reid 1984) in both run 6-54 and run 6-27. However, as shown in  Figures 10(c) and 10(d), the variation of the length with the confluence ratio is different from the previous studies (Best & Reid 1984) for the steady flow. The L/Wb no longer satisfied the least-squares regression equations. The L/Wb on the rising limb of Run 6-27 begins to decrease after reaching 1.3 when the confluence ratio is 0.4, indicating that after the ξ is increased to 0.4, the increase of ξ would reduce the length of the separation zone which is somewhat inexplicable; it may be related to the oscillatory of the shear layer. On the falling limbs of the two runs, it can be clearly seen that the L/Wb of short-period (Run 6-27) and long-period (Run 6-54) unsteady flow are similar. There are two stages of L/Wb development with the increase of confluence ratio on the falling limbs (the blue dotted lines), which implies that the separation zone is split during the falling limbs, so the length of the separation zone has a process from increasing to decreasing and then increasing on the falling limbs. In addition, in both two runs, the L/Wb on the rising limbs is larger than the falling limbs, which may be due to the inhibition of the shear layer on the separation zone on the falling limbs.
Figure 10
(a) and (b) Relationship between the separation zone width, B/Wb, and the ratio of tributary to post-confluence total discharge, ξ; (c) and (d) relationship between separation zone length, L/Wb, and ξ, for Run 6-54 and Run 6-27. The blue dotted lines represent L/Wb for decreasing confluence ratio under two runs. It is divided into two stages under both runs. The orange dotted line represents L/Wb for increasing confluence ratio under Run 6--27.
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(a) and (b) Relationship between the separation zone width, B/Wb, and the ratio of tributary to post-confluence total discharge, ξ; (c) and (d) relationship between separation zone length, L/Wb, and ξ, for Run 6-54 and Run 6-27. The blue dotted lines represent L/Wb for decreasing confluence ratio under two runs. It is divided into two stages under both runs. The orange dotted line represents L/Wb for increasing confluence ratio under Run 6--27.

Figure 10
(a) and (b) Relationship between the separation zone width, B/Wb, and the ratio of tributary to post-confluence total discharge, ξ; (c) and (d) relationship between separation zone length, L/Wb, and ξ, for Run 6-54 and Run 6-27. The blue dotted lines represent L/Wb for decreasing confluence ratio under two runs. It is divided into two stages under both runs. The orange dotted line represents L/Wb for increasing confluence ratio under Run 6--27.
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(a) and (b) Relationship between the separation zone width, B/Wb, and the ratio of tributary to post-confluence total discharge, ξ; (c) and (d) relationship between separation zone length, L/Wb, and ξ, for Run 6-54 and Run 6-27. The blue dotted lines represent L/Wb for decreasing confluence ratio under two runs. It is divided into two stages under both runs. The orange dotted line represents L/Wb for increasing confluence ratio under Run 6--27.

Close modal
In order to more intuitively identify the relationship between the size of the separation zone and time, the time series of B/Wb, L/Wb and upstream discharge extracted from the PIV data for the separation zone shows the changes in size of separation zone with the variation of discharge (Figure 11). The size of the separation zone is basically negatively correlated with the discharge ratio, but there is a slight hysteresis between the maximum B/Wb, the maximum L/Wb and the minimum discharge in both two runs. Overall, although changes in the size of the separation zone are positively correlated with changes in the discharge ratio (Best & Reid 1984), there is a hysteresis between the size of the separation zone and the discharge, which may be due to the hysteresis between the discharge and the velocity, and changes of the velocity directly led to changes of the size of the separation zone. The blue part represents the process of L/Wb suddenly decreasing and then increasing again, which has no connection with the variation of discharge (this can be seen from the comparison of the three parameters in Figure 11), and this phenomenon exists in both run 6-54 and run 6-27. Obviously, the separation zone is affected by other factors except the confluence ratio, so the length of the separation zone decreases suddenly, but the width of the separation zone does not decrease.
Figure 11
Temporal evolution of dimensionless width of separation zone, B/Wd, and dimensionless length of separation zone, L/Wd, for the long-period (Run 6-54) and short-period (Run 6-27) unsteady flow. The red dashed vertical lines showed the corresponding time of the three parameters, and the blue parts showed the variation of L/Wd from a sudden decrease to increase.
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Temporal evolution of dimensionless width of separation zone, B/Wd, and dimensionless length of separation zone, L/Wd, for the long-period (Run 6-54) and short-period (Run 6-27) unsteady flow. The red dashed vertical lines showed the corresponding time of the three parameters, and the blue parts showed the variation of L/Wd from a sudden decrease to increase.

Figure 11
Temporal evolution of dimensionless width of separation zone, B/Wd, and dimensionless length of separation zone, L/Wd, for the long-period (Run 6-54) and short-period (Run 6-27) unsteady flow. The red dashed vertical lines showed the corresponding time of the three parameters, and the blue parts showed the variation of L/Wd from a sudden decrease to increase.
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Temporal evolution of dimensionless width of separation zone, B/Wd, and dimensionless length of separation zone, L/Wd, for the long-period (Run 6-54) and short-period (Run 6-27) unsteady flow. The red dashed vertical lines showed the corresponding time of the three parameters, and the blue parts showed the variation of L/Wd from a sudden decrease to increase.

Close modal
From Figure 12, we can see that the shear layer also has impacted the development of the separation zone. From T1 to T3, the deflective angle of the shear layer shifted from small to large. As the discharge of the main channel decreases, the tributary flow gradually enters the mainstream flow. In Run 6-54, a strong response relationship was observed between the angle of shear layer oscillation and the fluctuation in velocity. Specifically, when the mainstream velocity was high, the deflection angle of the shear layer was small, whereas when the mainstream velocity was low, the deflection angle of the shear layer was large (refer to Figure 12(c) and 12(d)). At the maximum flow velocity, the amplitude of the shear layer oscillation reached its minimum value, α ≈ 0°. At the minimum flow velocity, α peaked, at about 30°, which was half of the confluence angle θ. In Run 6-27, this kind of response relationship observed in Run 6-54 was not as good due to the quick variation of the discharge. Especially, as indicated in the blue circle, when the flow velocity decreases, the angle of shear layer oscillation changes very little and still remains on the right side of the channel. In this case, the shear layer could restrict the development of the separation zone.
Figure 12
The black dashed lines, red dashed lines, and blue dashed lines represent the positions of the shear layers representing time T1, T2, and T3, respectively. (a) and (b) indicated shear layer under unsteady flow condition in the confluence at different times. (c) and (d) indicated time series of velocity and θ at point L1 and L2, within the shear layer in the confluence for Run 6-54 and Run 6-27.
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The black dashed lines, red dashed lines, and blue dashed lines represent the positions of the shear layers representing time T1, T2, and T3, respectively. (a) and (b) indicated shear layer under unsteady flow condition in the confluence at different times. (c) and (d) indicated time series of velocity and θ at point L1 and L2, within the shear layer in the confluence for Run 6-54 and Run 6-27.

Figure 12
The black dashed lines, red dashed lines, and blue dashed lines represent the positions of the shear layers representing time T1, T2, and T3, respectively. (a) and (b) indicated shear layer under unsteady flow condition in the confluence at different times. (c) and (d) indicated time series of velocity and θ at point L1 and L2, within the shear layer in the confluence for Run 6-54 and Run 6-27.
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The black dashed lines, red dashed lines, and blue dashed lines represent the positions of the shear layers representing time T1, T2, and T3, respectively. (a) and (b) indicated shear layer under unsteady flow condition in the confluence at different times. (c) and (d) indicated time series of velocity and θ at point L1 and L2, within the shear layer in the confluence for Run 6-54 and Run 6-27.

Close modal
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The above results of this study indicate that the fluctuations in discharge within the tributary exerted a more pronounced influence on hysteresis compared to the periodic variations observed in the unsteady flow of the mainstream. This is somewhat different from the phenomenon observed by Mishra & Seth (1996). We will discuss those results through the calculation formula. According to previous calculation formulas, the flood wave in this flume experiment should be a diffusion wave. In the diffusive wave, ⁠, moreover, i0 ≈ 0, gravity gradient is not considered in this experiment, and the flow is primarily driven by an external force provided by the water pump. Therefore, Equation (3) can be simplified as:
(12)
where P represents the external force term, and the steady-uniform flow discharge ⁠, therefore ⁠. When the flood is rising, ⁠, and when the flood is falling, ⁠, so it is inevitable that during flood conditions ⁠, and thus ⁠; during recession conditions, ⁠, and thus ⁠. Therefore, Equation (12) shows that this curve is not a single water level-discharge relationship curve, but rather a hysteresis curve, and according to Equation (12), the size of the hysteresis curve is related to the additional friction term, the external force term, and the resistance term. When the resistance term is constant, the magnitude of the hysteresis curve is mainly determined by the external force term and the additional slope term. In this study, when there is no lateral inflow, the hysteresis is mainly determined by the external force and the additional friction, so the shorter the period, the smaller the hysteresis, because in conditions with a shorter period, the flow acceleration is greater, resulting in a larger external force.
When there is a tributary inflow, the momentum equation needs to include additional terms representing the momentum contributions from the inflow and outflow of the elemental volume. The additional momentum from inflow is ρqvqΔx, where vq represents the lateral inflow velocity component in the downstream direction. The outflow velocity of the elemental volume is equal to the flow velocity of the river, and its momentum is ρqvqΔx. Therefore, Equation (3) and (1) takes the following form:
(13)
(14)

Based on the analysis above, it can be concluded that in the case of lateral inflow, the hysteresis effect is mainly influenced by the external force term, the additional slope term and the additional momentum term. Therefore, when there is lateral inflow, the relationship between water level and discharge should also follow a hysteresis curve, which is consistent with the results of the experiment in this context. Moreover, in Equation (14), the resistance term plays an important role, which is related to the hydraulic radius. Thus, the width of the wetted area needs to be taken into account.

In fact, those terms are exclusively related to energy loss, and the intuitive manifestation of energy loss is the attenuation of the flood wave. Based on this, we infer that the magnitude of the hysteresis effect is primarily associated with the sharpness or attenuation of the flood wave. The main driving factors for the attenuation of the flood wave are the pressure gradient and friction gradient-induced convective diffusion between the upstream and downstream. When the tributary inflow is large (resulting in a higher water level in the mainstream), the pressure gradient dominates, leading to the attenuation of the downstream flood wave and a decrease in the hysteresis effect. However, when the tributary inflow is small (no significant increase in water level upstream), it is similar to an expansion of the downstream cross-section.

In this study, the discharge of the entire flume is controlled by the tailgate. Therefore, during the initial preparation of the experiment, the discharge of the mainstream is adjusted to a steady flow of 8.5 L/s, and then the water level in the flume is stabilized at 20 cm before proceeding with subsequent experiments. Since the inflow from the tributary is steady, it can be approximated as a channel that regulates the water level and discharge of the mainstream during the experiment. When the discharge of the mainstream reaches the minimum value (5 L/s), the tributary needs to supplement the mainstream to satisfy the condition of a total discharge greater than 8.5 L/s. At this point, the discharge of the tributary needs to be greater than 3.5 L/s. However, when the discharge of the tributary is less than 3.5 L/s, the overall discharge in the flume can be less than 8.5 L/s, and the upstream water level is lower than the water level downstream (Figure 13), which is equivalent to an increase in the wetted area of the downstream cross-section, resulting in increased energy loss and enhanced hysteresis effect. This can explain why, in this experiment, the hysteresis effect increases when the inflow from the tributary is less than or equal to 3 L/s, and decreases when the inflow from the tributary is greater than or equal to 4 L/s. Based on our inference, when the inflow from the tributary is 3.5 L/s, the hysteresis effect of unsteady flow between the upstream and downstream may not undergo any changes.
Figure 13
The upstream–downstream water level difference. As depicted in the figure, it represents the degree of attenuation of the flow through the confluence zone. This difference ΔH is calculated by subtracting the downstream peak from the upstream peak during a flood event.
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The upstream–downstream water level difference. As depicted in the figure, it represents the degree of attenuation of the flow through the confluence zone. This difference ΔH is calculated by subtracting the downstream peak from the upstream peak during a flood event.

Figure 13
The upstream–downstream water level difference. As depicted in the figure, it represents the degree of attenuation of the flow through the confluence zone. This difference ΔH is calculated by subtracting the downstream peak from the upstream peak during a flood event.
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The upstream–downstream water level difference. As depicted in the figure, it represents the degree of attenuation of the flow through the confluence zone. This difference ΔH is calculated by subtracting the downstream peak from the upstream peak during a flood event.

Close modal

The present study investigates the hysteresis characteristics of unsteady flow in a confluence zone through laboratory flume experiments. We observe the hysteresis phenomenon of unsteady flow and conduct a thorough analysis using Saint-Venant equations to validate the associated patterns. However, despite the reliability of laboratory flume experiments, we are unable to further validate these patterns using field data due to the limitations of field sampling conditions. To enhance the credibility of these patterns and facilitate their practical application, future studies should focus on validating the experimental results using field data.

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In this study, a PIV system was used to measure the unsteady flow field at the laboratory-scale confluence, and accurately measure the shape of the separation area and the plane two-dimensional instantaneous flow field of the confluence zone. Such a system, combined with ADV measurement and water level gauges, was applied to investigate the hysteresis, an important parameter describing the characteristics of unsteady flow. The experimental results revealed that the inflow of the tributary could effectively reduce the value of hysteresis and the expansion of the downstream section increased the hysteresis of the unsteady flow. Furthermore, under the condition of the same tributary inflow, the larger the period was, the more the energy increased in the same period, and the greater the hysteresis was. The variation of hysteresis was correlated to the variation of energy, the larger hysteresis implied larger energy loss. The main effect of the change in hysteresis characteristics of unsteady flow was flow attenuation, which was generated by the dynamic factors of pressure gradient and frictional gradient-induced convective diffusion between the upstream and downstream flows.

In addition, in terms of hydrodynamics, the study demonstrated that the development of the separation zone was divided into three phases: (1) when the discharge upstream increased, the length of the separation zone decreased, (2) when the discharge upstream was close to the peak value, the length of the separation zone suddenly peaked and then rapidly decreased to the minimum value, and (3) the discharge of upstream decreased, the length of the separation zone increased. During the three phases, the separation zone was divided into several parts which affect contaminants and sediment transport.

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This research was funded by the Fundamental Research Funds for the Central Universities (grant number B230201057). The authors also would like to thank Professor Bidya Sagar Pani of the Indian Institute of Technology-Bombay and Professor Bruce Melville of University of Auckland for help in revising this work. Thanks are also extended to Mengyi Wang, Supeng Wang, Hao Cao, Xiao Luo, Yunqiang Zhu, Guanghui Yan, Jiaming Liu, Jieqing Liu of Hohai University for their support in this research.

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All relevant data are included in the paper or its Supplementary Information.

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The authors declare there is no conflict.

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