1 Introduction

Discharge from high dams can cause downstream total dissolved gas supersaturation, leading to an increased incidence of gas bubble disease and fish mortality in the river reach downstream of the dam [1]. This issue is widely recognized as a significant negative environmental consequence resulting from numerous dams, including those with only low-to-moderate head discharge [2, 3]. Recent research has shown that the convergence of rivers contributes to the outflux of saturated dissolved gas in the post-confluence region, creating a temporary refuge with lower total dissolved gas saturation [4,5,6,7]. This provides a new framework to alleviate the gas supersaturation problem caused by dams. However, a deeper understanding of the mechanism underlying the rapid outflux of supersaturated gas at confluences is needed.

These processes are associated with the gas exchange between two incoming streams and across the air–water interface. The latter is more efficient, and its flux can be parameterized as the product of the gas transfer velocity and the difference between surface water gas concentration and the equilibrium concentration with the atmosphere [8, 9]. While the concentration difference in the field is fairly easy to measure in almost any type of aquatic system [10, 11], the gas transfer velocity, which is the key parameter regulating such a gas exchange, is primarily controlled by turbulence on the water side of the interface [12] and it cannot be easily estimated. The surface renewal theory suggests that the gas transfer velocity is correlated with the dissipation rate of turbulent kinetic energy near the water surface [11, 13,14,15,16]. Thus, information on the turbulent dissipation rate near the free surface at a confluence could provide an indirect measure of the gas transfer velocity and be used to predict the dissolved oxygen transport downstream of the confluence apex.

Many laboratory experiments have been performed to investigate the time-averaged flow structures at river confluences [17, 18]. Best [19, 20] proposed a general conceptual model which consists of six hydrodynamic regions including the flow stagnation zone, the shear layer, the flow deflection zone, the flow separation zone, the maximum velocity zone, and the flow recovery zone (Fig. 1). Of particular importance are the shear layer or mixing interface [21,22,23] and the flow separation zone [24, 25]. The mixing interface zone typically resembles a shallow mixing layer for momentum ratios far from unity and is characterized by a large mean velocity gradient in the spanwise direction and the formation of co-rotating vertically-orientated vortices (VOVs). Such high velocity gradients and the advection of strongly coherent vortices are critical for pollutant transport downstream of the confluence apex. Downstream of the confluence apex, a separation zone generally forms when the incoming flow in the higher angle tributary has sufficient momentum and inertia to detach from the lateral channel wall [25, 26]. The size of the separation zone is a function of the angles between each tributary and the main channel downstream of the confluence, the momentum ratio and the flow ratio [24, 27]. Moreover, the occurrence of large-scale coherent structures within the mean flow, particularly of secondary flow helical cells and of streamwise-orientated vortices (SOVs) near the mixing interface are common hydrodynamic features observed at concordant confluences [28,29,30]. Twin surface-convergent helical cells generally form on both sides of the mixing interface in the vertical plane and can rapidly evolve into a single, channel-width circulation cell due to the difference in the curvature of the two incoming flows [31,32,33,34].

Fig. 1
figure 1

Time-averaged flow structures at an asymmetrical confluence

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Most of the previous studies of the turbulent flow structure at concordant bed confluences have focused on the shear layer/mixing interface and/or the separation zone [27,28,29, 35,36,37,38,39]. High turbulence intensity and turbulent kinetic energy are generally observed inside the mixing interface region [35,36,37]. Moreover, if the mixing interface is distorted, then the turbulent kinetic energy, Reynolds shear stress and the occurrence probability of ejection and sweep events within the shear layer are further amplified [38]. Two different mechanisms were identified by Constantinescu [35] to be responsible for the formation of VOVs. The wake mode of the mixing interface dominates when the momentum flux ratio is close to unity and it is characterized by counter-rotating vortices, similar to the von Kármán vortex street forming in the wake of bluff bodies. The Kelvin–Helmholtz (K-H) mode is characterized by the formation of co-rotating vortices that merge as they are advected inside the shear layer, similar to a classical mixing layer. This mode dominates when the momentum ratio is far from unity, meaning strong transverse shear is present across the shear layer [21, 28]. Recent studies have proposed an intermodal behavior of mixing at angled confluences, showing that although the momentum flux ratio is not unity the mixing interface still evolves in a wake mode at a short distance from stagnation zone, and the switch from wake mode to mixing-layer mode occurs when the velocity deficit associated with the wake becomes small with increasing distance from the apex [23]. The separation zone is characterized by a complex three-dimensional shape, with high turbulence kinetic energy and low pressure [27]. A high vorticity zone develops around the boundary with the faster moving fluid from the corresponding tributary [40].

Despite the numerous investigations of the complex flow structure at (concordant) river confluences, information on the near-surface turbulence dissipation at confluences is very scarce and incomplete. Such information is essential for further research on gas exchange. Previous experimental studies on turbulence at confluences have generally used Acoustic Doppler Velocimeters (ADVs) to collect velocity data [25, 27, 38, 41, 42]. Such instruments cannot provide accurate measurements near the water surface. Furthermore, the K-H and wake vortices inside the mixing interface have different growth rates [23], leading to different turbulence characteristics. Thus, one needs to quantify their respective effect on the dissipation rate. Such effects have not been studied so far.

The main goal of the present study is to investigate the distribution of the turbulence dissipation near the water surface at concordant-bed channel confluences, which should also provide information on the gas-transfer velocity distribution at such confluences. Experiments in a laboratory-scale confluence are conducted for two different values of the momentum flux ratio where the K-H mode and respectively the wake mode dominate. Measurements were performed using a Particle Image Velocimetry (PIV) system that was used to analyze the horizontal velocity distribution near the water surface. The objectives of this study are: (1) to investigate the spatial distribution of the turbulent dissipation rate and the mechanisms responsible for its amplification at a laboratory-scale concordant bed confluence; (2) to identify the effect of the two different mixing interface modes on that spatial distribution of the turbulence dissipation rate.

2 Methodology

2.1 Experimental setup and instrumentation

The laboratory-scale confluence adopted in this study is shown in Fig. 2. The flume had a rectangular cross-section, while the junction angle between the two tributaries was 60°. Both tributaries were 3 m long and 32 cm wide, while the post-confluence channel was slightly longer (7 m) and wider (42 cm). These geometric relationships seem to be common in river confluences [38, 43]. Water was pumped from the downstream tank to two upstream tanks through two independent PVC pipes. The flow was controlled by a computer and closely regulated using two ultrasonic flowmeters and two pump–valve systems (accuracy ± 0.01L/s). Honeycombs at the inlet, followed by sufficiently long channels, were used to ensure fully developed flows on reaching the confluence. The regulation of the water level in the flume was realized by operating sluice gates at the outlet section.

Fig. 2
figure 2

Sketch of the experimental setup showing the confluence flume with a 60-degree junction angle and the PIV measurement locations

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Two different discharge ratios were selected in this study to obtain two different characteristics of the shear layer as well as two different sizes of the separation zone. In the first case, the discharges of the tributary and that in the main channel Q1, and Q2 were 6.0 and 9.0 L/s, respectively. The water depth in the post-confluence channel was 20 cm (accuracy ± 0.1 cm). Thus, the discharge ratio (Qr = Q1/Q2) was about 0.67 and the momentum flux ratio (\(M_{{\text{r}}} = Q_{1} U_{1} /Q_{2} U_{2}\)) was about 0.45, which are expected to generate K-H vortices. In the second case, Q1 and Q2 were both set to 7.5 L/s, and thus the momentum flux ratio was 1 resulting in counter-rotating vortices inside the mixing interface. The total discharge and the water depth in the post-confluence channel were equal in both cases. Details regarding the hydraulic parameters are listed in Table 1.

Table 1 Hydraulic parameters for the two experimental cases
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To estimate the dissipation rate of the turbulent kinetic energy, the flow velocity near the water surface was measured using a particle image velocimetry (PIV) system. The system consisted of a 532 nm wavelength YAG laser with a power of 10 W, a high-speed CMOS camera with a resolution of 2560 × 2048 pixels, and a computer equipped with PIV software. To visualize the turbulent flow field, PIV seeding particles (silver-coated hollow glass spheres) having a diameter of 50 µm and a density of 1100 kg L−3 were added to the water in the confluence flume before starting the experiments. The experiments were conducted without any external light and the particles were visualized only under the illumination of the YAG laser. The laser power output was regulated by a laser control system, and the pulse duration between lasers was adjusted to optimize the laser intensity and to obtain a uniform illumination of the flow plane. In addition, the CMOS camera mounted above the flume was used to view and record the sequential images of the flow fields in horizontal planes.

2.2 Data collection and post-processing

The coordinate origin and the orientations are shown in Fig. 2. The streamwise, spanwise, and vertical coordinates in the confluent channel are respectively denoted as x, y, as well as z, and the coordinate origin is set at the bottom of the channel, at the corner of the right bank. The instantaneous velocity components in the x, y, and z directions are denoted as u, v, and w, respectively, while time-averaged velocity components are U, V, and W. The corresponding turbulent fluctuations are u′, v′, and w′.

The measurement area is shown in Fig. 2. Area A was located at the junction of the tributary channel and Area M was located in the main channel, upstream of A. Areas B, C, and D are the first, second, and third square-sized areas, respectively, placed downstream of A, each having a length equal to the width of the post-confluence channel. To avoid the effect of waves, the horizontal measurement planes were located 1 cm below the free surface. In area B, multiple horizontal planes located from 1 cm above the bed to 1 cm below the water surface, with a spacing of 1 cm, were used for taking measurements. In addition, in Case 1, the dissipation rate of the near-surface turbulence, following flow velocity recovery, was measured 1 cm below the free surface. The measurements spanned over a horizontal plane placed at a downstream distance of ten times the post-confluence width. PIV measurements were also performed in an x–z plane, by projecting the YAG laser light along the centerlines of the two upstream channels, to understand how the dissipation rate is distributed in the vertical direction.

PIV images were captured for 100 s using the camera which continuously transferred data to a computer at a frame rate of 100 Hz in all planes. The raw data were post-processed and further analyzed in PIVLAB. The used PIV algorithm was FFT window deformation, which has proven to be a precise algorithm in previous studies [44,45,46]. Furthermore, the sizes of the interrogation areas for the three passes were 128·128, 64·64 and 32·32 pixels respectively, with 50% overlap. After background subtraction, the synthetic images contained on average 10 particles per interrogation area (32·× 32 pixels). Finally, the 2D velocity field with a spatial resolution of 4 mm·4 mm was obtained. A stationarity analysis of the velocity data was performed to test the appropriate choice of the PIV measurement duration. The variance of the flow velocity within the mixing interface was found to converge well in 60 s (Fig. 3) suggesting that the recording time of 100 s was sufficient to capture the major sources of variation in the velocity field (for more details see Yuan et al. [38, 47]).

Fig. 3
figure 3

Examples of stationarity analysis of u and v velocities within the shear layer/mixing interface

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In this study, the dissipation rate was estimated using the inertial dissipation method, namely inertial subrange fitting, which assumes isotropic turbulence and is based on the theory of energy cascade from the larger to the smaller scales [48].

$$E(k_{w} ) = A\alpha_{k} \varepsilon^{\frac{2}{3}} k_{w}^{{ - \frac{5}{3}}}$$
(1)

where, \(\varepsilon\)[W kg−1] is the dissipation rates of turbulent kinetic energy, \(E(k_{w} )\) [m3 s−2 rad−1] is the one-dimensional wave number spectrum of turbulence fluctuations, \(k_{w}\) is the wave number [rad m−1], \(\alpha_{k} = 1.5\) [–] is the Kolmogorov constant, and \(A\) is a constant which depends on the direction of turbulence fluctuations: for the streamwise component \(A = \frac{18}{{55}}\) [–]. For the spanwise and vertical components \(A = \frac{4}{3} \times \frac{18}{{55}}\) [–] [48].

We used the streamwise velocity component to calculate the dissipation rate [49, 50] due to the damping of vertical velocity fluctuations near the free surface [51]. The frequency spectrum in Fig. 4 was calculated using a fast Fourier transformation of the covariance function with a Parzen window and was then converted to a wave number spectrum using Taylor’s frozen turbulence hypothesis. The validity of Taylor’s frozen turbulence hypothesis was tested based on \(\left( {\overline{{u^{^{\prime}2} }} } \right)^{\frac{1}{2}} /U_{{{\text{flow}}}} \le 0.15\) criterion [52], and the following transformation was applied:

$$E(k_{w} ) = \left( {\frac{{U_{flow} }}{2\pi }} \right)S(f)\quad {\text{and}}\quad k_{w} = \frac{2\pi f}{{U_{flow} }}$$
(2)

where \(S(f)\) is the longitudinal frequency spectrum, f is the frequency, and \(U_{{{\text{flow}}}} = (U^{2} + V^{2} )^{\frac{1}{2}}\) is the horizontal relative velocity.

Fig. 4
figure 4

Examples of power spectra of u velocity of water body within the shear layer, main channel flow and separation zone boundary near the free surface

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Thus, the dissipation rate was estimated as:

$$\varepsilon = 2\pi U_{{_{flow} }}^{ - 1} \left( {\frac{s(f)}{{A\alpha_{k} f^{{ - \frac{5}{3}}} }}} \right)^{\frac{3}{2}}$$
(3)

3 Results and discussion

3.1 Flow structure inside the confluence hydrodynamic zone

The portion of the river system affected by the combination of flows at a confluence is known as the Confluence Hydrodynamic Zone (CHZ) [53]. Two-dimensional time-averaged horizontal velocity field of CHZ, 1 cm below the water surface, is shown in Fig. 5. The general conceptual model of confluent flows proposed by Best [19] was observed clearly in this Fig. 5. The zone with the velocity close to 0 was the stagnation zone. This zone got deflected towards the tributary in Case 1 (Mr = 0.45; Fig. 5a), while it tended to be in the middle of two flows in Case 2 (Mr = 1.00; Fig. 5b).

Fig. 5
figure 5

Time-averaged horizontal relative velocity fields inside the CHZ, at 1 cm below the water surface (z/H = 0.95) in Case 1 (a) and Case 2 (b)

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Downstream of the stagnation zone, a mixing interface formed. Similarly, its centerline position was also related to Mr. In Case 1, it was closer to the tributary from y/W = 0.5 to 0.55, while in Case 2 it was at about y/W = 0.6. Figure 6 shows the results of vortex extraction using the instantaneous flow fields (for more details see Chen et al. [54]). In Case 1, the mixing interface was in the KH mode since vortices were co-rotating (Fig. 6a). The initial vortices produced behind the stagnation zone grew along the interface and vortex pairing resulted in the formation of larger KH vortices. In Case 2, the mixing interface was in the wake mode in which successive vortices were shed via the interactions of the separated shear layers on the two sides of the confluence apex with opposite rotational directions (Fig. 6b).

Fig. 6
figure 6

Visualization of vortical structures in an instantaneous flow field for Case 1 (a) and Case 2 (b). Vectors denote velocity fields, and the color contours represent the swirling strength within the vortex cores, where blue and red indicate clockwise and counterclockwise vortex filaments, respectively

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At the downstream corner of the right bank, a separation zone developed. The width of the separation zone expanded downstream and was related to the tributary inflow. In this study, the velocity isolines were used to define the size of the separation zone [25, 47]. In Case 1, the width of the separation zone (Ws) was about 0.14W, and its length (Ls) was approximately 1.55W, while in Case 2 its scale got larger such that the ratio Ws/W was close to 0.24 and the Ls/W ratio was about 1.80, based on measurements conducted at z/H = 0.95 (Fig. 5). Across the separation zone boundary, a large velocity gradient was found, leading to flow shearing.

In the main channel, the average flow velocities near the water surface were approximately 0.13 m/s in Case 1 and 0.09 m/s in Case 2, with slightly lower velocities on the sides influenced by the flume wall (Fig. 5). The velocity increased as the flow entered the post-confluence region. In Case 1, in the post-confluence channel, the flow originating from the main channel accelerated from 0.13 m/s to 0.24 m/s (Fig. 5a); while the flow originating from the tributary was sharply deflected and its velocity increased more, from 0.09 m/s to 0.24 m/s, due to the combined effects of mutual flow interference and expansion of the separation zone. In Case 2, the velocity accelerated from 0.11 m/s to 0.23 m/s for the flow originating in the main channel and from 0.11 m/s to 0.26 m/s for the flow originating in the tributary channel (Fig. 5b).

3.2 Vertical distribution of turbulent dissipation rates inside the CHZ

We use the inertial dissipation model to analyze the spatial distribution of turbulent dissipation inside the confluence hydrodynamics zone. This model is consistent with Taylor’s frozen turbulence hypothesis. This hypothesis is valid in confluent shear layers as suggested by Rhoads and Sukhodolov [21]. However, Lin [55] suggested that the recirculating fluid was hardly consistent with this hypothesis, and hence the dissipation rate inside the separation zone was not estimated using the inertial dissipation model. Moreover, inside the boundary of the separation zone, high turbulence intensity led to large values of \(u^{\prime}/U_{flow}\), which is also inconsistent with that hypothesis, as suggested by Bluteau et al. [52].

Figure 7 shows the spanwise and vertical distributions of the turbulent dissipation rates at x/W = 0.6 in the post confluence region. The dissipation rates \(\varepsilon H/U_{m}^{3}\) near the water surface varied from 10–5 to 10–3 in the post-confluence channel, while they were about 2.2 × 10–5 in the upstream tributary and 5.6 × 10–5 in the main channel. These results show large variations over the confluence. The spanwise distribution of the dissipation rate mainly peaked near the shear layer/mixing interface as well as inside the separation zone boundary (Fig. 7a). In addition, due to the boundary layer generated by the friction between the flow and the flume wall, slightly higher dissipation rates were observed at the flume sidewall.

Fig. 7
figure 7

Distribution of turbulent dissipation rate at section M in Case 1, in the spanwise direction (a) and the vertical direction (b). Um is the mean flow velocity of the post-confluence channel section. y/W = 0.50 to y/W = 0.57 represents the area where the shear layer is located, and y/W = 0.81 to y/W = 0.86 is the central position of the main channel side of the post-confluence channel. The black solid stippled line represents the vertical turbulent dissipation rate in the center of upstream with Q = 9 L/s, and the black solid line represents its fitted curve through the law of wall

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At z/H = 0.95, the turbulent dissipation rate \(\varepsilon H/U_{m}^{3}\) inside the shear layer (~ 7.7 × 10–4) was about 20 and, respectively, 5 times larger than that measured along the centerline of the flow on the main channel side (~ 3.5 × 10–5) and on the tributary side of the post confluence channel (~ 1.4 × 10–4), respectively (Fig. 7a). Moreover, the dissipation rate close to the tributary-side of the post-confluence channel was one order of magnitude higher than that inside the upstream tributary channel. This can be attributed to the combined effect of the separation zone boundary and the shear layer, which also explains why the dissipation rate is higher compared to the main channel side of the post-confluence channel.

At z/H = 0.95, the spanwise velocity gradient on the boundary of the separation zone is larger than inside the shear layer/mixing interface (Fig. 5a), causing stronger shearing. In addition, the planar turbulent kinetic energy at the free surface (\(k = \frac{1}{2}(u^{{\prime}{2}} + v^{{\prime}{2}} )\)) near the separation zone boundary was about 1.7 × 10–3 m2s−2, much higher than that measured inside the shear layer (about 1.0 × 10–4 m2s−2). Probably, due to these differences in the turbulent kinetic energy at z/H = 0.95, \(\varepsilon H/U_{m}^{3}\) near separation zone boundary (about 7.0 × 10–3) was an order of magnitude larger than that inside the shear layer (Fig. 7a).

Next, the vertical distribution of dissipation rate at x/W = 0.6 is analyzed. We mainly focus on its values inside and outside of the shear layer. Inside the shear layer, where K-H vortices are present (Fig. 6a), the dissipation rate varied little over the water depth (Fig. 7b). Near the water surface, the dissipation rate across the shear layer was about twenty times larger than that in the main channel side of the post confluence channel. Close to the channel bed, the difference between its values inside and outside of the shear layer was relatively small. The main reason for this finding is that the contribution of bottom shear to generation of turbulence increased and was larger than flow mixing at the bed.

Outside the shear layer, the dissipation rate in the central area of the main channel side of the post-confluence channel was nearly the same as the one inside the main channel. Assuming the shear stress remains constant over the water column and equal to the bed shear stress, we used the law of the wall [56], namely \(\varepsilon = \frac{{u_{*}^{3} }}{kz}\), where \(u_{*}\) is the bed friction velocity and \(k = 0.41\) is the Karman constant, to fit the dissipation rate curve. The \(u_{*}\) obtained from the fit curve was estimated to be 0.55 cm/s, and the ratio of it to the average flow velocity inside the upstream main channel was 3.9%. Thus, far away from the shear layer, the turbulent dissipation still depended on near-bed friction effects. Furthermore, at each water depth, the dissipation rates near the separation zone boundary were of the same order of magnitude. All these values were one order of magnitude larger than the averaged values measured inside the shear layer. This was expected, as the lateral shearing and the turbulent kinetic energy near the separation zone boundary were stronger compared to those recorded inside the shear layer.

3.3 Streamwise and spanwise dissipation rate distribution near the free surface

Figure 8 shows the distribution of the turbulent energy dissipation rate at 1 cm below the water surface for Case 1 inside Areas A, B and D. Inside Area A, the incoming flows started to merge, resulting in a shear layer without the formation of a separation zone. Near the shear layer, the dissipation rate was about 1 to 2 orders of magnitude greater than that in the surrounding regions. Along the shear layer, the peak of the curve moved laterally, and the maximum dissipation rate decayed longitudinally (Fig. 8a). Additionally, the influence of the VOVs expanded as the mixing interface gradually increased in width with increasing distance from the confluence apex.

Fig. 8
figure 8

Spanwise distribution of the turbulent dissipation rate at z/H = 0.95 for Case 1. Figures (a), (b), (c) and (d) show the distribution of the dissipation rate inside area A, area B, area C and the area situated downstream of the maximum velocity zone

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Inside Area B, higher dissipation rates \(\varepsilon H/U_{m}^{3}\) ware estimated near the separation zone boundary (10–3) (Fig. 8b). The shear layer inside Area B was still characterized by fairly large dissipation rates (10–4). These values gradually decreased along shear layer. Meanwhile, the curve peak also moved laterally at a slow rate. Under the combined effects of the separation zone boundary and the shear layer, the dissipation rate inside the tributary-side of the post-confluence channel gradually increased in the longitudinal direction (Fig. 8b and c). However, the longitudinal variation of the dissipation rate along the main-channel side of the post-confluence channel was very small, indicating that flow acceleration has a minor effect on the dissipation rate at confluences.

Downstream of the maximum velocity zone, the velocity gradually recovered. However, a large spanwise velocity gradient was found inside area D, with low velocities near the right bank where the separation zone was located (Fig. 5a). The dissipation rate decreased as the lateral distance from the right bank increased (Fig. 8d). This suggests that the effect of the separation zone was also non-negligible at the flow recovery zone (Fig. 9).

Fig. 9
figure 9

Comparison of dissipation rates at z/H = 0.95 in the two cases inside area A (a-c), area B (d-h) and area C to D (i-l)

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At x/W = 10.7, the flow velocity difference between the right bank and the left bank was less than 0.04 m/s (Fig. 10f), thereby it could be considered that the flow velocity has almost recovered. The dissipation rate \(\varepsilon H/U_{m}^{3}\) was evenly distributed in the spanwise direction, with values between 3.5 × 10–4 and 7 × 10–4 (Fig. 8d). These values were an order of magnitude higher than the value inside the upstream main channel (5.6 × 10–5).

Fig. 10
figure 10

Spanwise distribution of production and dissipation of turbulent kinetic energy at z/h = 0.95, at x/W = −0.48 (a), x/W = 0.60 (b), x/W = 1.55 (c), x/W = 2.98 (d), x/W = 10.70 (e) and their U velocity distribution in spanwise (f) in Case 1

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3.4 Comparison between the dissipation rate distributions for Case 1 and Case 2

Comparatively, the distributions of the dissipation rate near the free surface for Case 1 and Case 2 were quite similar. Inside area A, although the turbulent motions produced inside the mixing interface by the two momentum flux ratios were different (Fig. 6), the turbulent dissipation rates near the mixing interface were of the same order in the two cases (Fig. 9a–c). The turbulent dissipation rate inside the mixing interface in Case 2 was a little lower than that in Case 1, specifically inside Area A of the post-confluence channel, due to the smaller velocity difference across the mixing interface in Case 2. The dissipation rate on the tributary side of post-confluence in Case 2 was higher than that in Case 1. Also, the position of the curve-peak, roughly representing the centerline of the mixing interface, was closer to the left bank in Case 1, due to the higher momentum ratio. In addition, in Case 2 the smaller discharge flow on the main channel side, had a lower dissipation rate.

Compared with Case 1, the dissipation rate in Case 2 was larger on the tributary-side of the post-confluence channel, relatively speaking, and quickly increased to attain the same order of magnitude as that inside the mixing interface along the downstream direction (Fig. 9d–h).Moreover, the dissipation rate in Case 2 was higher inside the shear layer region of the flow recovery zone (Fig. 9i–l), which should be due to the larger separation zone forming in Case 2.

3.5 Production and dissipation of turbulent kinetic energy

The turbulent energy production near the water surface in Case 1 is analyzed to better understand the change in the turbulent dissipation rate and the transport of turbulent kinetic energy. In the post confluence region, the spanwise velocity gradient near the water surface was dominant compared to the velocity gradients in the streamwise and the vertical directions. Hence, we estimated the turbulent production rate using the product of the Reynolds shear stress and the lateral velocity gradients, namely \(P = - \overline{u^{\prime}v^{\prime}} (\frac{\partial U}{{\partial y}} + \frac{\partial V}{{\partial x}})\) [57].

Figure 10a shows the spanwise distribution of the turbulent energy production and dissipation rate near the water surface inside Area A. Within the shear layer, the turbulence production \(PH/U_{m}^{3}\) was about 5.3 × 10–3 and it was an order magnitude larger than the local dissipation. On the main channel side of the post-confluence, near the shear layer, the local production was slightly larger than the local dissipation \(\varepsilon H/U_{m}^{3}\) (between y/W = 0.4 and 0.65), while some distance away (between y/W = 0.65 and 0.85) the production by mean shear was an order of magnitude smaller. In the center of the main channel and on the tributary side of the post-confluence, the turbulence production by mean shear was relatively low (less than 10–6) compared to other regions (Fig. 10a). Hence, near the water surface the shear layer was one of the dominant sources for turbulent kinetic energy production. Although most of the turbulent kinetic energy was dissipated inside the shear layer, the excess turbulent energy was transported to other regions such as the center of the main channel. On the tributary-side, near the shear layer (lying between y/W = 0.1 and 0.25), the local energy production rate \(PH/U_{m}^{3}\) was about one to two orders of magnitude smaller than its dissipation rate \(\varepsilon H/U_{m}^{3}\). Its magnitude (10–8–10–6) was smaller than that on the main channel side (10–7–10–6) due to the lower \(\overline{u^{\prime}v^{\prime}}\) resulting from smaller flow discharge in the tributary compared to the main channel.

In the post-confluence channel, the absolute maximum of the turbulent production rate \(PH/U_{m}^{3}\) was found to occur at the separation zone boundary. Its magnitude decreased as one goes along this boundary (from 10–2 at x/W = 0.6 to 10–3 at x/W = 2.98) (Fig. 10b–d). The local turbulent production rate near the separation zone boundary was an order of magnitude larger than its dissipation (Fig. 10b and c), suggesting that flow separation, as well as the large velocity gradient across the separation zone, were the main supply source of turbulent kinetic energy. A relative maximum of the production rate \(PH/U_{m}^{3}\) (about 1.1 × 104) was found inside the shear layer, but its value was an order of magnitude lower than the local dissipation due to the decrease of the velocity gradient across the shear layer. This suggests that part of its turbulent kinetic energy is supplied from upstream of the shear layer or the separation zone.

At the downstream of the maximum velocity zone, the flow gradually recovered Although the velocity difference in the shear layer gradually decreased, a large spanwise velocity gradient was still found at x/W = 1.55 and x/W = 2.98 due to the presence of flow with lower velocity near the right bank of the post-confluence channel (Fig. 10f). This probably explains why, in Fig. 10c, the production rate \(PH/U_{m}^{3}\) in shear layer slightly increased compared to that in Fig. 10b (e.g., from 1.1 × 10–4 to 2.5 × 10–4). Furthermore, at x/W = 2.98, inside most regions situated near the right bank, the local turbulent energy production was an order of magnitude larger than its dissipation rate (Fig. 10d). This effect is associated with the presence of the upstream separation zone at that bank.

At x/W = 10.7, the velocity difference between the two streams was lower than 0.04 m/s as the flow recovered. The magnitude of the local turbulent energy production \(PH/U_{m}^{3}\) near the right bank decreased to 10–4, but was still slightly greater than the dissipation rate at the same locations. At other locations situated away from the right bank, the local dissipation rate was greater than the turbulence production (Fig. 10e). This shows that the upstream separation zone is an important source of supply of turbulent energy. Therefore, at these locations, most of the energy deficit may be supplied by the excess energy generated inside the right bank, upstream area.

3.6 Discussion

3.6.1 Near-surface turbulent dissipation at confluence

The distribution of the dissipation rate of turbulent kinetic energy at the free surface of a laboratory-scale confluence was analyzed in the previous sections. Figure 11 shows the conceptual map of the dissipation rate of near-surface turbulence at a confluence according to the observations of this study. The dissipation rate near the free surface increases inside the CHZ with higher dissipation rates mainly present inside the shear layer/mixing interface and near the main separation zone. On the mainstream-side of the post-confluence channel, the dissipation rate remained low; while on the tributary-side, the dissipation rate gradually increased between the shear layer/mixing interface and the separation zone. Inside the flow recovery zone, relatively high and low dissipation rates were found on the right and left banks, respectively. Further downstream these differences gradually decayed. Evidently, the momentum flux ratio, which can affect both the mixing interface and the size of the flow separation region, is an important factor affecting the distributions of the dissipation downstream of the confluence apex. In this section, we compare these results with previous studies reporting free-surface dissipation rates (Table 2).

Fig. 11
figure 11

Conceptual map of the dissipation rate of near-surface turbulence at a laboratory-scale asymmetric confluence

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Table 2 Comparison of free-surface dissipation rates reported in the literature
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The turbulent dissipation rate inside the upstream channel was mostly due to bottom shear. Previous studies have found that near-surface dissipation rate in open channels is controlled by bottom roughness, bed slope, Froude number and hydraulic diameter [58,59,60,61,62]. For instance, in a tilting flume with a highly rough bed [58], the dissipation rates near the free surface \(\varepsilon H/U^{3}\) increased from 1 × 10–3 to 1.5 × 10–3. as the slope increased from 0.25% to 1%. In this study, the dissipation rate \(\varepsilon H/U^{3}\) estimated near the free surface of the upstream main channel (~ 1.2 × 10–4) was relatively low due to the flat channel bed and the relatively low Froude number.

Previous studies have investigated the turbulent structure of mixing interfaces at a confluence [36,37,38,39]. It is well known that mean flow shear can increase the turbulent kinetic energy and Reynolds stress near the mixing interface. But the distribution of the free-surface turbulent dissipation rate inside the mixing interface region, especially sustaining different mixing modes (i.e. K-H mode and wake mode), was not previously studied. Our study has found that mean flow shear inside the mixing interface can increase the near-surface dissipation rate by one to two orders of magnitude compared to that inside the surrounding regions containing flow originating in the two incoming streams. Moreover, even though the dynamics of the mixing interface vortices is different for the KH and wake modes, the effects of these vortices on the near-surface dissipation rates were comparable.

The flow at the boundary and downstream of the main separation zone also was characterized by high dissipation rates due to large mean velocity gradients. Liu et al. [63] estimated the turbulence dissipation rate inside the recirculating flow region at a slot fishway and found that its value decreased with increasing distance from the jet center. Although the turbulent dissipation inside the separation zone was not available in this study, one can infer that, similar with the recirculating flow at the slot fishway, the dissipation rate inside the separation zone at a confluence shall gradually decrease with increasing distance from the separation-zone boundary.

Our study showed that the dissipation rates inside the main channel change little through the flow acceleration zone (Fig. 8b). Although flow velocity increased gradually inside the zone, the turbulent intensity change was small (i.e. for y/W = 0.81, the turbulent intensity was about 0.032 at x/W = −0.48, and about 0.028 at x/W = 0.83). Therefore, the flow acceleration should have a minor effect on the free-surface dissipation rate. This observation is consistent with Kokic et al. [60] who suggested that more than half of the variance in the dissipation rates could not be attributed to the flow speed. Natchimuthu et al. [64] also suggested that the bivariate relationships between gas transfer velocity and water velocity should not be strong for flatter-stream reaches. The other flow acceleration zone situated on the tributary side was affected by both the shear layer/mixing interface and by the separation zone. This explained the occurrence of relatively high dissipation rates inside this flow acceleration zone.

Both the momentum flux ratio and Froude number significantly affect near-surface turbulent dissipation at the confluence. The momentum flux ratio is an important factor controlling the distributions of the dissipation downstream of the confluence apex. A higher momentum flux ratio biases the shear layer more toward the main channel, reducing the size of the low dissipation rate region. It can also result in the formation of a larger separation zone, accelerating the dissipation rate of the flow from the tributary and enhancing turbulent dissipation downstream. In this study, as the Froude number of the post-confluence channel was constant in both cases, its direct impact on the post-confluence dissipation rate was not investigated. However, in Case 1, the dissipation near the water surface in the main channel, with a larger Froude number, exceeded that at the tributary, which suggests that the Froude number can increase the post-confluence dissipation rate without impacting its distribution. At large-scale River confluences, complex hydrodynamics and regions of high flow shear can occur [65,66,67,68,69]. The free-surface turbulence dissipation can be affected by many factors such as wind shear, bottom shear, flow shear, geomorphological heterogeneity [56, 70,71,72,73]. Future studies of gas exchanges conducted at large-scale confluences should focus on these aspects.

3.6.2 Implications for gas transfer at river confluence

Total dissolved gas (TDG) supersaturation frequently is observed downstream of these dams and persists for long distances despite the engineering measures implemented to mitigate their effect. As a result, TDG supersaturation affect the migration and spawning of finless porpoises and fish, and was also shown to adversely affect the aquatic ecology in the Yangtze River [74, 75]. Downstream of the dam, river confluences naturally reduce TDG supersaturation. Previous studies have indicated that confluences promote TDG supersaturation dissipation, leading to a decrease in the dissolved gas concentration in the post-confluence channel [4, 6, 74]. Near-surface turbulence is the primary control on the gas transfer velocity at the air–water interface. In this section, the dissipation rate near water surface at the laboratory-scale confluence was used to indirectly analyze the gas transfer velocity.

Using the surface renewal model, the gas transfer velocity at the confluence can be estimated from the turbulent dissipation rate calculated in this study, i.e., \(K_{L} = \alpha (\varepsilon \nu )^{\frac{1}{4}} Sc^{{ - \frac{1}{2}}}\), where α is an experimental constant and Sc is the Schmidt number [11, 14]. Spatially, it was found that the gas transfer velocity at the confluence was characterized by considerable heterogeneity, with the shear layer and the separation zone boundary characterized by K-H and wake type vortices exhibiting 2–4 times higher gas transfer velocities compared to the flow inside the remaining parts of the main channel. After the flow entered the confluence channel from the tributary, its gas transfer velocity increased by 1.8 times. Furthermore, high gas transport velocity (twice as fast as upstream) in most regions of the post-confluence channel persisted for considerable distances downstream of the confluence apex.

Surface divergence has also been shown to correlate with the gas transfer velocity at the unsheared free water surface, known as surface divergence mode, which identifies the root mean square divergence \(\beta ^{\prime}\) as the central parameter [76, 77]. Previous studies have found that it can successfully predict the gas transfer velocity using the flow structure at a free air–water interface evaluated by PIV measurements [78]. Also, the combination of these two models is more accurate than either one alone [76, 79]. Future study will also focus on the surface velocity divergence intensity and combine them for evaluating gas transfer velocity.

4 Conclusions

Laboratory experiments based on PIV measurements in horizontal planes were conducted to investigate the turbulence characteristics near the free surface of a confluence for two discharge ratios. Based on the spatial distribution of the turbulent dissipation/production rate, the following conclusions were reached:

  1. (1)

    The turbulent energy dissipation rate near the water surface increases significantly in the post-confluence channel, presenting considerable spatial heterogeneity. The largest dissipation rate of the near-surface turbulence was observed mainly within the shear layer/mixing layer and the separation zone boundary, where its value is about two orders of magnitude greater than the near-surface dissipation rate in the incoming flows upstream of the confluence apex.

  2. (2)

    The separation zone and the shear layer/mixing interface are the dominant flow features that affect turbulence near the free surface of a confluence. At the right bank, the near-surface turbulence behind the corner is affected by both the separation zone and the shear layer, resulting in a dissipation rate on the tributary-side that surpasses that observed on the opposite side by approximately one order of magnitude. Furthermore, within the flow recovery zone, the relative contribution of the separation zone to near-surface turbulence gradually amplifies.

  3. (3)

    As the distance from the channel bottom increases, the dissipation rates inside and outside the mixing interface both gradually decrease due to the weakening of bottom shear. However, the K-H or wake vortices have a compensatory effect on the flow near the shear layer/mixing layer, resulting in only minor changes in the dissipation rate. Furthermore, within the mixing interface, there was no significant distinction in the near-surface dissipation rates between the dominant K-H mode and wake mode.

  4. (4)

    Even after the flow recovers almost fully in the post-confluence channel, the dissipation rate remains consistently one order of magnitude higher than that observed in the upstream channel. Within the shear layer/mixing interface and the separation zone boundary, the local turbulent energy production exceeds the dissipation rate. On the contrary, near the water surface at downstream locations, the dissipation rate surpasses the local production rate. These observed trends provide valuable insights into the turbulent energy transport and balance mechanisms taking place in both the spanwise and downstream directions at the confluence.

  5. (5)

    The surface renewal model indicates that gas transfer velocity had significant spatial heterogeneity at confluences, with K-H and wake type mixing interface vortices having a velocity 2–4 times higher than in the main channel. Downstream of the apex, the post-confluence flow was characterized by higher gas transfer velocity, about twice as fast as those in the upstream flow. This region of high gas transfer velocity extended for a distance is > 10W.

As large-scale confluences hydrodynamics is more complex due to the impact of floodplains, and the effects of atmospheric forcing via winds, heat fluxes, etc., future studies should focus on these aspects while investigating turbulence characteristics.