# Shallow turbulent mixing layers in open-channel flows

Sébastien Proust, Bastien Cérino, Céline Berni, Vladimir I Nikora

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Sébastien Proust, Bastien Cérino, Céline Berni, Vladimir I Nikora. Shallow turbulent mixing layers in open-channel flows. Environmental Fluid Mechanics, 2025, 25 (1), pp.6. 10.1007/s10652-025-10013-9. hal-04890196

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# Shallow turbulent mixing layers in open-channel flows

Sébastien Proust<fnref n="1" /> · Bastien Cérino<fnref n="1" /> · Céline Berni<fnref n="1" /> · Vladimir I. Nikora<fnref n="2" />

Received: 19 September 2024 / Accepted: 7 December 2024

© The Author(s) 2025

## Abstract

A mixing layer (ML) forms when two streams of different speeds or densities merge. MLs are ubiquitous in nature and can be often observed in the atmosphere, ocean, rivers, canals, lakes and reservoirs. This review paper focuses on the turbulent MLs developing in open-channel flows when the vertical size of the ML is smaller than its streamwise and spanwise dimensions. Such MLs are referred to as shallow MLs (SMLs). The SMLs often involve large-scale features such as quasi-two-dimensional coherent structures with a bed-normal axis, streamwise-oriented vortices, secondary currents, gravity currents, and bed-induced turbulent structures such as large- and very-large-scale motions. Considering various types of SMLs, we distinguish SMLs driven by (i) spanwise inhomogeneity of hydrodynamic parameters, (ii) lateral changes in flow resistance, and (iii) spanwise heterogeneity in fluid density. As SMLs and associated flow structures largely control transverse exchanges of various substances (e.g., sediments, pollutants, nutrients) and heat, the mixing of substances and thermal mixing are also addressed. Then, commonalities and differences among the various types of SMLs are identified. The paper is concluded with suggestions on future research efforts for advancing the knowledge on SMLs and capabilities for their predictions and control.

**Keywords** Shear layer · Shear layer turbulence · Bed-induced turbulence · Coherent structures · Secondary flows · Gravity currents · Vegetated flows · Confluences · Sediment mixing · Thermal mixing

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<footnote>Sébastien Proust
sebastien.proust@inrae.fr</footnote>

<footnote>Vladimir I. Nikora
v.nikora@abdn.ac.uk</footnote>

<footnote>Bastien Cérino
bastien.cerino@inrae.fr</footnote>

<footnote>Céline Berni
celine.berni@inrae.fr</footnote>

<footnote n="1">¹ RiverLy, INRAE, 5 rue de la Doua CS 20244, 69625 Villeurbanne, France</footnote>

<footnote n="2">² School of Engineering, University of Aberdeen, Aberdeen AB24 3UE, UK</footnote>
# 1 Introduction

In the continental river networks as well as in estuarine and coastal areas, it is common to observe the merger of two currents of different speeds, which gives rise to a shear layer characterized by a significant jump in velocity occurring on a small length-scale [1], termed shear layer width ($\delta$). Shear layers can form e.g., at river confluences, in braided and anastomosing rivers, when a river flows into a lake or the sea, and at the interface between a free stream and a slower vegetated flow. Since shear layers are high velocity-gradient regions, where intense mixing can take place, they are often referred to as mixing layers (MLs) [2]. The MLs and associated flow structures largely control the spanwise and vertical exchanges of heat and matter such as contaminants, nutrients, and sediments. These exchanges can affect water quality, biodiversity, fish and invertebrate habitats, and endanger certain species (e.g. salmonids are very sensitive to changes in water temperature).

The present paper is a review article on turbulent MLs in open-channel flows (OCFs), which focuses on MLs whose vertical sizes are smaller than their streamwise and spanwise dimensions. Such MLs are referred to as shallow MLs (SMLs) [3]. In addition to this dimensional aspect, SMLs are vertically bounded by the channel bed and water surface, spanwise bounded by channel sidewalls, and experience vertical confinement and bed friction effects on the flow structure [e.g., 3–7].

The main objective of our analysis in this paper is to clearly define the specific features of SMLs in terms of time-averaged (mean) flow and turbulence (including relevant coherent structures), by considering the canonical two-dimensional (2D) plane free mixing layer (FML) as a reference. In addition, as in OCFs the occurrence of FMLs and transition from FML to SML remain unclear (despite their importance for understanding flow dynamics and improving predictive capabilities), these two points have been clarified. Last, we have determined the flow parameters that are (and are not) sensitive to the transition from FML regime to SML regime.

The (plane 2D) FMLs—which are spatially unbounded—have been extensively studied [8–29]. They are 2D in terms of time-averaged parameters but can be 3D in terms of instantaneous velocity field. A representative velocity scale of the FML is velocity difference $U_s = U_2 - U_1$, where $U_2$ and $U_1$ are characteristic mean streamwise velocities of the two ambient free streams (i.e. outside the ML). The velocities $U_1$ and $U_2$ are constant along the x-axis (streamwise direction) while the shear layer width $\delta$ and spanwise position $y_c$ of the average velocity $U_c = 1/2(U_2 + U_1)$ are linear functions of $x$ [8, 14, 19]. The growth rate of $\delta$ is controlled by the shear parameter $\lambda = (U_2 - U_1)/(U_2 + U_1)$ as $d\delta/dx = 2\sigma\lambda$, $\sigma$ being a constant in the range 0.06–0.11 [11, 24, 25]. The spanwise profiles of mean streamwise velocity $U$ are self-similar when plotting $U^* = (U - U_1)/U_s$ against dimensionless spanwise coordinate $\xi = (y - y_c)/\delta$ [9, 11, 12, 14, 22], as are self-similar the profiles of Reynolds stresses $-\overline{u'}\overline{v'}/U_s^2(\xi)$, $\overline{u'^2}/U_s^2(\xi)$, $\overline{v'^2}/U_s^2(\xi)$, and $\overline{w'^2}/U_s^2(\xi)$ [17, 22, 25], where $u'$, $v'$ and $w'$ are the velocity fluctuations along $x$, $y$, and $z$ (bed-normal) axes, and an overbar defines time-averaging. Note that self-similarity is observed in the fully developed turbulent mixing region but not in the transition from laminar to turbulent regime [8].

Within turbulent FMLs, besides ‘random’ turbulence, Dimotakis and Brown [13] noted that ‘there exist flow patterns and large-scale structures that appear to be dominant in determining the overall characteristics of such flows’. For instance, 2D large-scale vortices—termed Kelvin-Helmholtz coherent structures (KHCSs)—can be produced by the shear layer instability. The KHCSs were first visualized in FMLs by Brown and Roshko [11] for air flows, and by Winant and Brownand [12] and Dimotakis and Brown [13] for water

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<figure><img src="image_5.png" /><figcaption>**Fig. 1** Shallow mixing layers over smooth bed [7, 30]. Parameters at $x = 0.06$ m: **a** $\lambda_0 = (U_{2,0} - U_{1,0})/(U_{2,0} + U_{1,0}) = 0.22$, $h_{1,0} = h_{2,0} = 26$ mm, $Fr_{1,0} = U_{1,0}/\sqrt{gh_{1,0}} = 0.4$, $Fr_{2,0} = U_{2,0}/\sqrt{gh_{2,0}} = 0.6$, $Re_{1,0} = U_{1,0}h_{1,0}/\nu = 5474$, $Re_{2,0} = U_{2,0}h_{2,0}/\nu = 8374$, $c_{f1,0} = c_{f2,0} = 0.006$; **b** $\lambda_0 = 0.39$, $h_{1,0} = 25$ mm, $h_{2,0} = 24$ mm, $Fr_{1,0} = 0.3$, $Fr_{2,0} = 0.6$, $Re_{1,0} = 3470$, $Re_{2,0} = 7571$, $c_{f1,0} = c_{f2,0} = 0.006$; **c** $\lambda_0 = 0.81$, $h_{1,0} = 47$, $h_{2,0} = 57$ mm, $Fr_{1,0} = 0.2$, $Fr_{2,0} = 1.3$, $Re_{1,0} = 5016$, $Re_{2,0} = 57479$, $c_{f1,0} = 0.006$, $c_{f2,0} = 0.004$. Velocities $U_{2,0}$ and $U_{1,0}$ are measured at mid-depth. Dye injection at $x = 3.65$ m and $y = y_i$ (inflection point in the spanwise mean velocity profile)</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Fig. 2** SML over smooth bed with $\lambda_0 = 0.48$, $h_{1,0} = 56$ mm, $h_{2,0} = 55$ mm, $Fr_{1,0} = 0.4$, $Fr_{2,0} = 1.1$, $Re_{1,0} = 16602$, $Re_{2,0} = 45733$, $c_{f1,0} = 0.005$, $c_{f2,0} = 0.004$ [7, 30]: mean streamwise velocity $U^* = (U - U_1)/U_y$ (**a**), and normalized Reynolds stresses $-\overline{u'}\overline{v'}/U_s^2$ (**b**) and $\overline{v'^2}/U_s^2$ (**c**), versus spanwise coordinate $\xi = (y - y_c)/\delta$ at various streamwise positions $x$. The dashed lines indicate the maximum levels of the normalized Reynolds stresses for conventional plane FMLs [19]</figcaption></figure>

flows. KHCSs can form when the two incoming streams have equal densities ($\rho_1 = \rho_2$) [11, 12] or unequal densities [11], Figures 3 and 5 with [$\rho_1/\rho_2=1/7$ and 7]. In the latter case, the effect of the increase in density ratio by a factor 49 is to decrease the spreading angle of the vortices by approximately 2. The velocity fluctuations in FMLs can be predominantly caused by the KHCSs, as observed by Brown and Roshko [11], Winant and Brownand [12], and Dimotakis and Brown [13] for highly sheared flows with initial shear parameter $\lambda_0 = (U_{2,0} - U_{1,0})/(U_{2,0} + U_{1,0}) = 0.45-1$ (just behind the splitter plate). They can also be caused by 3D flow structures [16, 17], as 2D structures are not automatically present or dominant in FMLs, as observed by Pui and Gartshore [17] for weakly sheared flows with $\lambda_0 = 0.09-0.21$.

The turbulent SMLs have been firstly investigated by Chu and coworkers [3, 31–34]. In the simplest case of SMLs formed by two parallel incoming streams over a uniformly smooth bed (Figs. 1 and 2) or rough bed, SMLs feature a streamwise decrease in velocity $U_2$ combined with an increase in velocity $U_1$, while the evolutions of $\delta$ and $y_c$ are no more linear with $x$ [3, 4, 6, 7, 35, 36]. In addition, if the mean velocity profiles can remain self-similar over very large streamwise distances as illustrated in Fig. 2a, this is not the case for the Reynolds stress profiles (Fig. 2b and c) reflecting a lack of equilibrium in the shear layer turbulence. The latter is altered by the ambient-stream turbulence, namely the bed-induced
turbulence in upcoming OCFs [3, 4, 6, 7, 33]. This effect can be seen in Fig. 2c, as the distributions of spanwise velocity fluctuations are asymmetric (unlike FMLs) with higher values of $v^2/U_s^2$ on the high-speed stream side for $x \ge 1.65$ m.

Within turbulent SMLs, 2D large-scale structures can be observed for strongly sheared flows as illustrated in Fig. 1b ($\lambda_0 = 0.39$ at $x = 0.06$ m) and Fig. 1c ($\lambda_0 = 0.81$) or can be absent for a weakly sheared flow (Fig. 1a, $\lambda_0 = 0.22$) –as previously noticed by Pui and Gartshore [17] for FMLs. The available theoretical arguments (e.g., Lam et al. [37]) and experimental data (e.g., Proust and Nikora [38]) suggest that the emergence of the 2D coherent structures (2DCSs) in SMLs are triggered by hydrodynamic instabilities and then may develop along the flow following dynamics of two-dimensional turbulence. Therefore, the term ‘KHCSs’ for such 2DCSs may not be appropriate, strictly speaking. Therefore, for intercomparisons of FMLs and SMLs, we will use ‘KHCSs’ for the case of FMLs and 2DCSs when considering SMLs. The normalized predominant frequency (Strouhal number $S_t = f\theta/U_c$, where $\theta$ is momentum thickness) of the 2DCSs is in the range 0.032–0.044 (Table 5, [39–41], and Fig. 4) closely similar to the range of $S_t$ observed within FMLs (0.032–0.048, Table 5, [2, 12, 20, 42]). Like in FMLs [43], the shear layer instability in SMLs is manifested as a longitudinal oscillation of the vortex sheet that grows and leads to the formation of spiral-shape vortices [38]. Importantly, the data suggest that when the 2DCSs emerge in SMLs, their streamwise and spanwise dimensions are independent of the flow depth and bed-friction [7, 44]. However, the bottom-friction at the base of the vortical elements is the major mechanism responsible for the decay of the 2DCSs [5]. Last, surface waves and Froude numbers in the two streams do not affect the emergence of 2DCSs, their longitudinal development, and their decay (and subsequently the mixing processes when these structures are the predominant vortices populating the SML) [7, 44, 45].

The present review is about these turbulent SMLs. We focus on the key mechanisms controlling the SML structure and mixing processes for three major groups of SMLs most relevant for environmental applications:

1. Hydrodynamics-driven SMLs, including SMLs (i) with parallel incoming streams, (ii) with non-parallel incoming streams in idealized channel shapes, and (iii) at river confluences;

2. SMLs with two incoming streams of unequal densities;

3. Flow-resistance-driven SMLs, including SMLs induced by (i) lateral change in bed-roughness and (ii) in partly vegetated channel.

On the other hand, the review will not cover SMLs induced by vertical topographical forcings (two-stage channels also termed compound channels, and confluences with discordant beds). Nevertheless, we will not refrain from referring to some papers dealing with these types of SMLs if they exhibit features applicable to SMLs of types (1) to (3).

In this paper we attempted to synthesize findings from laboratory experiments (Table 1), numerical simulations and analytical equations (Table 2), and field observations (Table 3). Particular attention is paid to the parameters controlling the SML main features, considering both the mean flow and turbulence statistics (Table 4), including large-scale structures and their effects on the mixing of matter and heat (Table 5). As previously mentioned, we use conventional plane FMLs as a reference case (Tables 1, 2, 4 and 5). As shown in Table 1, the geometries and flow conditions can be highly simplified in the laboratory (compared to field conditions), with e.g., parallel incoming streams, rectangular cross-sections, horizontal flat smooth bed. This enables (i) to put aside the topographical and

<figure><img src="image_7.png" /></figure>
<table><caption>**Table 1** Some experimental studies on SMLs and plane 2D FMLs: test section length *L* and width *W*, hydraulic roughness HR, channel slope *S*₀, angle *α*₀ between incoming streams, initial shear parameter *λ*₀, initial density ratio *d*₀ and difference *Δρ*₀, and average flow depth *h*</caption><thead><tr><th>References<br>(Chronological order)</th><th>L × W<br>(m)×(m)</th><th>S<sub>0</sub><br>×10<sup>-3</sup></th><th>HR<sup>1</sup></th><th>α<sub>0</sub><br>(°)</th><th>λ<sub>0</sub><br>(–)</th><th>d<sub>0</sub>, Δρ<sub>0</sub><br>(–), (kg/m<sup>3</sup>)</th><th>h<br>(cm)</th></tr></thead><tbody><tr><td colspan="8"><strong>Shallow mixing layers (SMLs)</strong></td></tr><tr><td>Mosley [87]</td><td>2.8×1.3</td><td>16</td><td>r</td><td>0–180</td><td>n.a.<sup>2</sup></td><td>n.a.</td><td>n.a.</td></tr><tr><td>Ashmore and Parker [104]</td><td>10×2.3</td><td>10–15</td><td>r</td><td>20–110</td><td>n.a.</td><td>n.a.</td><td>n.a.</td></tr><tr><td>Best [86]</td><td>1 × 0.15</td><td>n.a.</td><td>r/s</td><td>15–105</td><td>n.a.</td><td>n.a.</td><td>n.a.</td></tr><tr><td>Chu et al. [3]</td><td>7×0.6</td><td>0</td><td>s</td><td>0</td><td>0.26–0.58</td><td>1</td><td>2–5</td></tr><tr><td>Babarutsi et al. [33]</td><td>7×0.6</td><td>0</td><td>s</td><td>0</td><td>1</td><td>1</td><td>1.4–8.2</td></tr><tr><td>Biron et al. [106]</td><td>3.5×0.3</td><td>0</td><td>s</td><td>30</td><td>0.29</td><td>1</td><td>16</td></tr><tr><td>Uijttewaal et al. [4]</td><td>20×3</td><td>0</td><td>s</td><td>0</td><td>0.36–0.40</td><td>1</td><td>4–7</td></tr><tr><td>Booij and Tukker [53]</td><td>20×3</td><td>0</td><td>s</td><td>0</td><td>0.34–0.63</td><td>1</td><td>5–10</td></tr><tr><td>Nezu and Onitsuka [131]</td><td>10×0.4</td><td>0.2–0.4</td><td>PVF</td><td>0</td><td>0</td><td>1</td><td>7</td></tr><tr><td>White and Nepf [41, 132]</td><td>13×1.2</td><td>0</td><td>PVF</td><td>0</td><td>0.78–0.96</td><td>1</td><td>5–14</td></tr><tr><td>Vermaas et al. [85]</td><td>12.5×2</td><td>0</td><td>BRT</td><td>0</td><td>0</td><td>1</td><td>8–22</td></tr><tr><td>Jesson et al. [126]</td><td>22×0.61</td><td>1.2</td><td>BRT</td><td>0</td><td>n.a.</td><td>1</td><td>12–16</td></tr><tr><td>Mignot et al. [171]</td><td>0.4×0.3</td><td>0</td><td>s</td><td>90</td><td>0.33</td><td>1</td><td>12</td></tr><tr><td>Schindfessel et al. [172]</td><td>1.3×4.4</td><td>0</td><td>r</td><td>90</td><td>n.a.</td><td>1</td><td>41</td></tr><tr><td>Mignot et al. [173]</td><td>0.3×0.6</td><td>0</td><td>s</td><td>0</td><td>1</td><td>1</td><td>7</td></tr><tr><td>Ismael et al. [117]</td><td>3.3×0.3</td><td>0</td><td>r</td><td>45</td><td>0</td><td>1.03–1.04</td><td>46</td></tr><tr><td>Yuan et al. [105]</td><td>0.6×0.4</td><td>0</td><td>s</td><td>90</td><td>0.21</td><td>1</td><td>16.3</td></tr><tr><td>Han et al. [174]</td><td>4.4×0.8</td><td>1.8</td><td>s</td><td>0</td><td>1</td><td>1</td><td>2–16</td></tr><tr><td>Caroppi et al. [39]</td><td>10×0.6</td><td>0.5–7</td><td>PVF</td><td>0</td><td>0.07–0.79</td><td>1</td><td>17</td></tr><tr><td>Akutina et al. [64]</td><td>26×1.1</td><td>3</td><td>BRT</td><td>0</td><td>0.15–0.41</td><td>1</td><td>3–6</td></tr><tr><td>Tinoco et al. [133]</td><td>1.2×0.8</td><td>0</td><td>PVF</td><td>0</td><td>0</td><td>n.a.</td><td>n.a.</td></tr><tr><td>Caroppi et al. [135]</td><td>10×0.6</td><td>4.8</td><td>PVF</td><td>0</td><td>0.41–0.80</td><td>1</td><td>8–17</td></tr><tr><td>Li et al. [67, 68]</td><td>8×1</td><td>0.1</td><td>PVF</td><td>0</td><td>0</td><td>1</td><td>13–14</td></tr><tr><td>Proust et al. [7, 30]</td><td>18×2</td><td>1.04</td><td>s</td><td>0</td><td>0–1</td><td>1</td><td>2–10</td></tr><tr><td>Táborda et al. [137]</td><td>6×0.8</td><td>0</td><td>PVF</td><td>0</td><td>0.73–0.85</td><td>1</td><td>10</td></tr><tr><td>Shi et al. [152]</td><td>8×2</td><td>1.4</td><td>s</td><td>0</td><td>0–1</td><td>1</td><td>2–10</td></tr><tr><td>Dupuis et al. [63]</td><td>26×1.1</td><td>3.1</td><td>BRT</td><td>0</td><td>0.45–0.84</td><td>1</td><td>3–8</td></tr><tr><td>Cerino et al. [44]</td><td>18×2</td><td>1.08</td><td>r</td><td>0</td><td>0.17–0.54</td><td>1</td><td>5–11</td></tr><tr><td>Villota et al. [138]</td><td>5.5×1.8</td><td>0</td><td>PVF</td><td>0</td><td>0</td><td>1</td><td>27</td></tr><tr><td>Duguay et al. [116]</td><td>1.5×0.5</td><td>0</td><td>s</td><td>60</td><td>0, 0.28</td><td>n.a., 0.33–0.66</td><td>7</td></tr><tr><td>Rivière et al. [175]</td><td>2.7×0.3</td><td>0</td><td>s</td><td>90</td><td>0.5</td><td>1</td><td>12</td></tr><tr><td colspan="8"><strong>2D plane free mixing layers (FMLs)</strong></td></tr><tr><td>Liepmann and Laufer [8]</td><td>1.52×0.17</td><td></td><td></td><td>0</td><td>1</td><td>1</td><td>n.a.</td></tr><tr><td>Wygnanski and Fiedler [9]</td><td>0.6×0.18</td><td></td><td></td><td>0</td><td>1</td><td>1</td><td>n.a.</td></tr><tr><td>Yule [10]</td><td>1.2×0.5</td><td></td><td></td><td>0</td><td>0.24, 0.54</td><td>1</td><td>28</td></tr><tr><td>Brown and Roshko [11]</td><td>0.17×0.05</td><td></td><td></td><td>0</td><td>0.45–1</td><td>1/7–7</td><td>n.a.</td></tr><tr><td>Winant and Browand [12]</td><td>1.5×0.10</td><td>0</td><td>s</td><td>0</td><td>0.48</td><td>1</td><td>10</td></tr><tr><td>Dimotakis and Brown [13]</td><td>2.4×0.5</td><td>0</td><td>s</td><td>0</td><td>0.66</td><td>1</td><td>50</td></tr><tr><td>Chandrsuda et al. [16]</td><td>1.2×0.6</td><td></td><td></td><td>0</td><td>1, n.a.</td><td>1</td><td>n.a.</td></tr><tr><td>Pui and Gartshore [17]</td><td>2.64×0.91</td><td></td><td></td><td>0</td><td>0.09–0.21</td><td>1</td><td>69</td></tr><tr><td>Breidenthal [18]</td><td>0.45×0.07</td><td>0</td><td>s</td><td>0</td><td>0.11–0.47</td><td>1</td><td>11</td></tr></tbody></table>
<table><caption>**Table 1 (continued)**</caption><thead><tr><th>References<br>(Chronological order)</th><th>L × W<br>(m)×(m)</th><th>S<sub>0</sub><br>×10<sup>-3</sup></th><th>HR<sup>1</sup></th><th>α<sub>0</sub><br>(°)</th><th>λ<sub>0</sub><br>(–)</th><th>d<sub>0</sub>, Δρ<sub>0</sub><br>(–), (kg/m<sup>3</sup>)</th><th>h<br>(cm)</th></tr></thead><tbody><tr><td>Oster and Wygnanski [19]</td><td>2.0×0.6</td><td></td><td></td><td>0</td><td>0.25–0.54</td><td>1</td><td>50</td></tr><tr><td>Ho and Huang [20]</td><td>1.8×0.1</td><td>0</td><td>s</td><td>0</td><td>0.31,0.2</td><td>1</td><td>10</td></tr><tr><td>Bernal and Roshko [21]</td><td>0.25×0.05</td><td></td><td></td><td>0</td><td>0.33–0.74</td><td>1–7</td><td>10</td></tr><tr><td>Bell and Mehta [22]</td><td>3.7×0.36</td><td></td><td></td><td>0</td><td>0.25</td><td>1</td><td>91</td></tr><tr><td>Mehta [23]</td><td>3.7×0.36</td><td></td><td></td><td>0</td><td>0.05–0.33</td><td>1</td><td>91</td></tr><tr><td>Loucks and Wallace [28]</td><td>1.2×0.6</td><td></td><td></td><td>0</td><td>0.34–0.44</td><td>1</td><td>n.a.</td></tr><tr><td>Balamurugan et al [29]</td><td>3×0.61</td><td></td><td></td><td>0</td><td>0.23–0.29</td><td>1</td><td>0.61</td></tr></tbody></table>

morphological effects (including movable beds) on the SMLs, and (ii) to isolate the effect of each key flow parameter (initial shear parameter, flow rate ratio, density ratio) or a geometrical parameter (junction angle, junction asymmetry, bed-roughness). The same simplifications are carried out in a big part of numerical studies (Table 2), which can be validated using laboratory studies. Last, the effect of complex geometries and large width-to-depth ratios on SMLs have been investigated using both numerical simulations and field measurements (as summarized in Tables 2 and 3).

Section 2 reports results on SMLs driven by heterogeneity in hydrodynamics, distinguishing SMLs with parallel and non-parallel incoming streams. Section 3 reports results on SMLs with two streams of unequal densities, which feature gravity currents (GCs). The main features of the SMLs driven by the spanwise heterogeneity in bed surface are presented in Sect. 4, where we investigate the effects of lateral bed-roughness changes and partly vegetated channels. The effect of the flow structures on the mixing of matter and heat is exposed in §5. The commonalities and differences among the SML types (1) to (3) are synthesized in §6. Last, we draw conclusions and present some perspectives for future research works on SMLs.

# 2 Hydrodynamics-driven shallow mixing layers

## 2.1 SMLs with parallel incoming streams

The studies of SMLs with parallel incoming streams have focused largely on the effects of bed-friction and vertical confinement on: (i) mean flow parameters ($U_1, U_2, U_c, U_s, \delta, d\delta/dx$, and $y_c$); (ii) statistics of velocity fluctuations (e.g., spanwise Reynolds stress $u'v'$); and (iii) emergence, development and decay of the 2DCSs—including their interplay with 3D structures.

### 2.1.1 Vertical flow confinement and bed-friction effects

The vertical confinement effect on MLs was quantified by flow depth $h$ [4, 6, 46], channel-width-to-depth ratio $W/h$ [47] or ML-width-to-depth ratio $\delta/h$ [7], while the bed-friction effects were quantified by the bed-friction length-scale $h/c_f$ [3], Eqs. 10–12] (see also Eq.

<footnote n="1">¹s, smooth; r, rough; BRT, bed-roughness transition; PVF, partially vegetated flow</footnote>

<footnote n="2">²n.a.: data not available</footnote>
<table><caption>**Table 2** Some numerical and analytical studies on SMLs and FMLS: direct numerical simulation (DNS), large-eddy simulation (LES), detached-eddy simulation (DES), stability analysis (SA), analytical developments (A), 2D shallow water equations (2D-h), $k-\epsilon$ turbulence model ($k-\epsilon$), Reynolds-averaged Navier-Stokes (RANS), and integral model (I)</caption><thead><tr><th>References<br>(Chronological order)</th><th>Model</th><th>Flow conditions</th><th>Validation<br>data</th></tr></thead><tbody><tr><td>Shallow mixing layers (SMLs)</td><td></td><td></td><td></td></tr><tr><td>Chu et al. [31]</td><td>S.A.</td><td>α₀ = 0°; V = 0; h = est.</td><td></td></tr><tr><td>Chu et al. [34]</td><td>S.A.</td><td>α₀ = 0°; c₇(y) or h(y); V = 0</td><td></td></tr><tr><td>Naot et al. [129]</td><td>k-ε</td><td>PVF: vegetated bank, corner, and floodplain</td><td>Exp. [176]</td></tr><tr><td>Babarutsi and Chu [56]</td><td>k-ε</td><td>Exp.: α₀ = 0°; λ₀ = 0.26 – 0.58; h = 2 – 5 cm. Field: river confluence</td><td>Exp. [3], Field.</td></tr><tr><td>Chen and Jirka [48]</td><td>S.A.</td><td>α₀ = 0°; λ₀ = 0 – 1; V = 0; h = est</td><td></td></tr><tr><td>Nadaoka and Yagi [130]</td><td>2D-LES</td><td>PVF: vegetated bank</td><td>Exp.</td></tr><tr><td>Booj and Tucker [53]</td><td>I</td><td>α₀ = 0°; λ₀ = 0.34 – 0.63, h = 5–10 cm</td><td>Exp. [53]</td></tr><tr><td>van Prooijen and Uijttewaal [50]</td><td>S.A.</td><td>α₀ = 0°; h = 42.67 mm; λ₀ = 0.39, 0.42</td><td>Exp.</td></tr><tr><td>Socolofsky et al. [51]</td><td>S.A</td><td>α₀ = 0°, λ₀ = -0.33-1.5, S=0–0.7</td><td></td></tr><tr><td>Cook and Richmond [112]</td><td>RANS</td><td>α₀ ≈ 80°, Qₜ = 0.9, 1.1; ΔT = 1.6°C, 9.6°C</td><td>Field</td></tr><tr><td>Constantinescu et al. [55, 99]</td><td>DES</td><td>α₀ = 60°; Mₜ = 1.03, 5.4; λ₀ = 0.05, 0.41; h = 0.23 m, 0.36 m</td><td>Field [107, 108, 177]</td></tr><tr><td>Kirkil [46]</td><td>DES</td><td>α₀ = 0°, λ₀ = 0.4, h = 6.7 cm, S₀ = 0, smooth or movable bed</td><td>Exp. [4]</td></tr><tr><td>Ramon et al. [115]</td><td>3D</td><td>α₀ ≈ 90°; Dₜ = 0.2×10⁻⁴; Qₜ = 0.13, 0.65, 1.53; Mₜ = 0.03, 0.72, 4.58</td><td></td></tr><tr><td>Constantinescu et al. [178]</td><td>DES</td><td>α₀ = 0°, 60°; λ₀ = -0.6, 0.01</td><td></td></tr><tr><td>Constantinescu et al. [168]</td><td>DES</td><td>α₀ = 0°, 60°; Mₜ = 1.04, 5.4</td><td></td></tr><tr><td>Cheng et al. [6]</td><td>DES</td><td>α₀ = 0°; λ₀ = 0.4, 0.5; h = 7, 3 cm.</td><td>Exp. [4]</td></tr><tr><td>Cushman-Roisin et al. [36]</td><td>A</td><td>α₀ = 0</td><td>Exp. [4]</td></tr><tr><td>Horna-Munoz et al. [100]</td><td>DES</td><td>α₀ = 60°, Mₜ = 2.1, 0.39; Fₐ = 4.9, 1.6; λ₀ = 0.22, 0.14</td><td>Field (T values)</td></tr><tr><td>Cheng et al. [101]</td><td>DES</td><td>α₀ = 0°, 60°; λ₀ = 0.4, 0.2;</td><td></td></tr><tr><td>Qu and Yu [140]</td><td>2D-h</td><td>PVF: α₀ = 0°; λ₀ = 0.24 – 0.47; h = 5.5 – 15 cm.</td><td>Exp. [41, 132]</td></tr><tr><td>Cheng et al. [118]</td><td>DES</td><td>α₀ = 0°, λ₀ = 0.37; Ri = 0, 0.03; h = 0.067 m;</td><td></td></tr><tr><td>Duquay et al. [97]</td><td>LES</td><td>α₀ = 120°; Mₜ = 0.91, 16.9</td><td>Field</td></tr><tr><td>Pandey and Mohapatra [102]</td><td>LES</td><td>α₀ = 90°; flat vs movable bed; small W/h. λ₀ = 0.1, h = 18 cm.</td><td>Exp.</td></tr></tbody></table>
<table><caption>**Table 2 (continued)**</caption><thead><tr><th>References (Chronological order)</th><th>Model</th><th>Flow conditions</th><th>Validation data</th></tr></thead><tbody><tr><td>Koken et al. [142]</td><td>DES</td><td>PVF; patch: L/h = 36, W/h = 1.6; λ₀ = 0</td><td></td></tr><tr><td>Jiang et al. [110]</td><td>DES</td><td>α₀ = 58°, λ₀ = 0.08–0.12, Ri = 0–0.037, |ΔT| = 2.5°C</td><td></td></tr><tr><td>Wang et al. [139]</td><td>2D-LES</td><td>PVF: S₀=0.00025–0.01; h = 1.5-14 cm</td><td></td></tr><tr><td>Free mixing layers (FMLS)</td><td></td><td></td><td></td></tr><tr><td>Rogers and Moser [25]</td><td>DNS</td><td>λ₀ = 0.25</td><td>Exp. [22]</td></tr><tr><td>Laizet [27]</td><td>DNS</td><td>λ₀ = 0.2</td><td></td></tr><tr><td>White and Helfrich [120]</td><td>ILES/DNS</td><td>U_c = 0; U₂ = −Uᵢ; ρ₂ > ρ₁; γ=0.05–1.67</td><td>Field</td></tr></tbody></table>

<figure><img src="image_8.png" /></figure>
<table><caption>**Table 3** Some field-based studies on SMLs</caption><thead><tr><th>References<br>(Chronological order)</th><th>Flow conditions</th><th>Focus on</th></tr></thead><tbody><tr><td>Rhoads and Kenworthy [95]</td><td>Small asym. confluence, α₀ = 60°, M<sub>r</sub> = 0.55 – 3.6, Q = 0.3–1.8 m<sup>3</sup> /s, ΔT = 2°C</td><td>SC1s, thermal mixing.</td></tr><tr><td>Kenworthy and Rhoads [88]</td><td>Small asym. confluence, α₀ = 60°, M<sub>r</sub> = 0.46–42, C<sub>r</sub>=0.1–23</td><td>suspended sediment mixing</td></tr><tr><td>Rhoads and Sukhodolov [107],<br>Sukhodolov and Rhoads [177]</td><td>Small sym. and asym. confluences, α₀ = 70°, 60°, 36°, ΔT = 1.5° – 3.5°C, M<sub>r</sub> = 0.35 – 1</td><td>SC1s, thermal mixing</td></tr><tr><td>Cook and Richmond [112]</td><td>Medium-sized asym. confluence, α₀ ≈ 80°, Q<sub>r</sub>=0.9, 1.1; ΔT=1.6°C, 9.6°C</td><td>GCs, thermal mixing</td></tr><tr><td>Lane et al. [167]</td><td>Large asym. confluence, α₀ ≈ 80°, M<sub>r</sub>=1.8, 3.6; C<sub>r</sub> = 0.036, 0.066;</td><td>suspended sediment mixing</td></tr><tr><td>Rhoads and Sukhodolov [108]</td><td>Small asym. confluence, α₀ = 60°, λ₀ = 0.41, M<sub>r</sub> = 5</td><td>lateral momentum flux</td></tr><tr><td>Laraque et al. [113]</td><td>Large asym. river confluence, α₀ ≈ 90°, λ₀ = 0.54; Q<sub>21</sub> = 170 kg s<sup>-1</sup>, Q<sub>31</sub> = 4740 kg s<sup>-1</sup></td><td>GCs, suspended sediment mixing</td></tr><tr><td>Sukhodolov et al. [35]</td><td>straight reach, α₀ = 0°, sandy bed; λ₀ = 0.5 – 1; h = 0.8 – 1 m</td><td>momentum fluxes, δ, U<sub>s</sub></td></tr><tr><td>Lewis and Rhoads [96]</td><td>Small asym. confluence, α₀ = 60°; M<sub>r</sub> = 0.3–18; D<sub>r</sub> = 0.1–1.7×10<sup>-3</sup>, Fr<sub>d</sub> = 2–23</td><td>GCs, SC1s, thermal mixing.</td></tr><tr><td>Rhoads and Johnson [90]</td><td>Large asym. confluence, α₀ = 85°; λ₀ = -0.05–0.71; M<sub>r</sub> = 0.01–0.34; Q<sub>r</sub>=0.06–0.34; h = 11–14 m</td><td>suspended sediment and thermal mixing, SC1 or SC2</td></tr><tr><td>Gualtieri et al. [103, 179]</td><td>Large asym. confluence, α₀ = 70°; λ<sub>0</sub> = 0.59, 0.63; M<sub>r</sub> = 0.07, 0.10; Q<sub>r</sub> = 0.32, 0.39; ΔT = 0.7°, 1°C; D<sub>r</sub> = 0.32–0.36×10<sup>-3</sup>, C<sub>r</sub> = 0.044, 0.038</td><td>suspended sediment, low and high flows</td></tr><tr><td>Sukhodolov and Sukhodolova [89]</td><td>Small sym. confluence, α₀ = 0°, 40°, 70°, W/h = 25; M<sub>r</sub> = 0.01 – 0.45; h = 0.3–0.5 m; λ₀ = 0.14–0.85</td><td>SC1s, SOVs</td></tr><tr><td>Pouchoulin et al. [121]</td><td>Medium-sized asym. confluence, α₀ = 0; λ₀ = 0.00–0.26; Q<sub>r</sub> =0.6–0.9; M<sub>r</sub>=0.6–1.5; ΔT = 0.3×2.5°C; D<sub>r</sub> = 0.5–3.2×10<sup>-4</sup></td><td>GCs, suspended sediment mixing</td></tr><tr><td>Yuan et al. [93]</td><td>Large asym. confluence, α₀ = 58°; λ₀ = 0.08, 0.53; M<sub>r</sub> = 0.06, 0.41</td><td>SC1, sediment mixing</td></tr><tr><td>Li et al. [91]</td><td>Large asym. confluence α₀ = 58°; λ₀ = 0.1–0.5; M<sub>r</sub> = 0.1–1.0; Q<sub>r</sub> = 0.2–0.8; Ri = 0.01–0.11; ΔT = 0.7°–2.5°C; D<sub>r</sub> = 1–3.8×10<sup>-4</sup>; C<sub>r</sub> = 0.2–2.5</td><td>SC1, kinetic energy and momentum coefficients, thermal mixing</td></tr></tbody></table>
<table><caption>Table 3 (continued)</caption><thead><tr><th>References<br>(Chronological order)</th><th>Flow conditions</th><th>Focus on</th></tr></thead><tbody><tr><td>Xu et al. [92]</td><td>Large asym. confluence, α<sub>0</sub> = 58°; λ<sub>0</sub> = -0.2-0.5; M<sub>r</sub> = 0.1-1.2; ΔT = 1.3°-2.9°C; F<sub>d</sub> = 2.2-6.2</td><td>GCs, SC1</td></tr><tr><td>Duguay et al. [97, 98]</td><td>Small river confluence, α<sub>0</sub> = 120°; λ<sub>0</sub> = 0.57, -0.1; Q<sub>r</sub> = 4.7, 1.1; M<sub>r</sub> = 17, 0.9; ΔT = 2-3°C</td><td>density-driven SOV cells, SC1s, thermal mixing and suspended sediment mixing</td></tr><tr><td>Yuan et al. [94]</td><td>Large asym. confluence, α<sub>0</sub> = 58°; λ<sub>0</sub> = 0.08-0.53; M<sub>r</sub> = 0.06-1.03</td><td>near-surface velocity dip, SC1s</td></tr><tr><td>Dureuil et al. [122]</td><td>Medium-size confluence, α<sub>0</sub> ≈ 0°; λ<sub>0</sub> = -0.34-0.74; M<sub>r</sub> = 1.31-0.01; γ = 0.77-106; D<sub>r</sub> = 0.7-9.1×10<sup>-4</sup></td><td>GCs, suspended sediment mixing</td></tr><tr><td>Xu et al. [180]</td><td>Large asym. confluence, α<sub>0</sub> = 58°; M<sub>r</sub> = 0.01, 1.12; V<sub>r</sub> = U<sub>2</sub>/U<sub>1</sub> = 0.12, 1.20; F<sub>d</sub> = 1.4, 2.5</td><td>KH vs. wake modes, unsteady SCs</td></tr></tbody></table>

Notations: momentum flux ratio $M_r = (ρ_2 Q_2 U_2)/(ρ_1 Q_1 U_1)$ where subscript 2 refers to the tributary and 1 to the main river; sediment concentration ratio $C_r = C_2/C_1$; discharge ratio $Q_r = Q_2/Q_1$; ΔT = $|T_2 - T_1|$; $D_r = |(ρ_1 - ρ_2)|/ρ_1$

<figure><img src="image_9.png" /></figure>
(3) in §2.1.2) –where $c_f$ is the quadratic law friction parameter averaged across the ML, with $c_f = f/4, f$ being Darcy-Weisbach friction coefficient– and the bed-friction number $S$ defined by Chu et al. [31] as

$$S = \frac{c_f \delta}{4h\lambda} \qquad (1)$$

with

$$\delta = U_s/(\partial U/\partial y)_{\max} \qquad (2)$$

being the vorticity thickness calculated at the inflection point of the mean velocity profile.

The $S$-number measures the stabilizing effect of the bed-friction over the destabilizing effect of the transverse shear on the large-scale horizontal turbulent motion [34]. This stability parameter emerges in the Orr-Sommerfeld equation [31, 32] or modified Orr-Sommerfeld equation for parallel flows [48, 49]. All hydrodynamic stability analyses of SMLs that have been conducted since the eighties for straight open channels [31, 34, 37, 48, 50–52] or compound open channels [32, 34, 49] relied on $S$-parameter alongside $\lambda$-parameter, as the emergence and development of quasi-2D CSs are largely dependent on these two parameters [37]. For instance, for hyperbolic tangent velocity profiles, the 2D structures cannot develop if $S \ge S_c$, with $S_c = 0.08$ to 0.12 for $\lambda_0 = 0.05$ to 0.83 (Chu et al. [34]), or $S_c = 0.09$ to 0.10 for $\lambda_0 = 0.001-1$ (Chen and Jirka [48]). A comparable $S_c$-value was observed in the experiments of Chu and Babarutsi [3] ($S_c = 0.09$ for $\lambda_0 = 0.26$ and 0.58), Uijttewaal and Booij [4] ($S_c = 0.08$ for $\lambda_0 = 0.36$ and 0.40), Proust et al. [7] ($S_c \approx 0.1$ for $\lambda_0 = 0.13–1.00$) and Cerino et al. [44] ($S_c \approx 0.1$ for $\lambda_0 = 0.17–0.54$).

## 2.1.2 Ambient stream velocities

Chu and Babarutsi [3] experimentally observed that for a SML formed by two parallel incoming streams with uniform bed-roughness, the relaxation towards uniformity across the channel (i.e., $U_s \approx 0$) was controlled by the bed-friction length-scale $h/c_f$. Relying on the momentum conservation equations for two ambient streams of equal cross-sections (outside the ML), and assuming that bed-friction coefficient and flow depth are constant across and along the flow, they proposed an exponential relationship for $U_s U_c = U_2^2 - U_1^2$:

$$(U_2^2 - U_1^2)/(U_{2,0}^2 - U_{1,0}^2) = \exp(-x^*) \qquad (3)$$

with $x^* = xc_f/h$, and subscript 0 refers to initial position $x^* = 0$. Equation (3) was found to be suitable for SMLs over a horizontal smooth bed and a constant flow depth, which were simulated by Cheng and Constantinescu [6] using 3D DES. In contrast, in the field experiments of Sukhodolov et al. [35] along a 45-m long river reach, a linear decrease in $U_s = U_{s,0} - \beta x$ gave a better fit to the measurements compared to Eq. (3).

In the experiments of Proust et al. [7] with SMLs over a smooth bed in a tilted flume, Eq. (3) was found to be applicable as long as mean spanwise velocity $V \ll U$. In the presence of a significant transverse flow ($V \approx U$), flow depth was not constant across the upstream part of the channel, friction coefficient was not constant along the flume, and across the flume at small $x$. This resulted in measured values of $(U_2^2 - U_1^2)/(U_{2,0}^2 - U_{1,0}^2)$ that deviate from $\exp(-x^*)$, though $U_s$ still $\to 0$ for 3-4 $h/c_f$ [7], Fig. 13]. With $c_f \sim 0.004–0.01$ [5, 7], this means that $U_s \approx 0$ for $x \sim 300h–1000h$ irrespective of the magnitude of $V/U$. When $V \approx U$, the ‘integral model of SMLs’ of Booij and Tukker [53] (which transversely
integrates the shallow water equations in the two ambient streams and ML) is of good help as it accurately predicts both velocities $U_c$ and $U_s$.

### 2.1.3 Mean velocity profiles

Like for the plane FMLs [8, 9, 14, 22, 23, 25], the mean streamwise velocity profiles for SMLs can be sigmoid and self-similar (i.e., the spanwise distribution of appropriately normalized velocity and spanwise coordinate is independent of $x$). This was observed for experimental (Fig. 2a, [4, 30, 40, 54]), field-based [35], and numerical [6] studies. However, the data [30] of Proust et al. [7] would suggest that self-similarity could be lost in the absence of 2DCSs (i.e. for small $\lambda_0$, see next §2.1.7). In addition, self-similarity is not met if a strong cross-flow ($V \approx U$) develops as experimentally observed by Proust et al. [54].

Self-similarity is also not fulfilled in the presence of a wake caused by a splitter plate or a confluence apex, which violates the classical sigmoid velocity profile. The wake originates from the no-slip boundary condition on the plate/apex surfaces resulting in two lateral boundary layers from either side of the plate/apex. This wake quickly disappears in the presence of intense lateral exchanges of mass and momentum for highly sheared flows [7]. In contrast, lasting wake effects were observed for small $\lambda_0$ for both FMLs [17, 20, 23] and SMLs [7, 38, 55] irrespective of flow depth and bed-friction [7]. Depending on the $\lambda_0$-value (and also on junction angle $\alpha_0$ inducing a stagnation zone and wake-like effect [47], see §2.2), the ML can be in a pure wake mode, a pure ML mode, or in an intermodal behaviour [47]. For the latter, Sukhodolov et al. [47] proposed the following model for a spanwise profile of depth-averaged velocity:

$$ (U - U_c)/U_s = 1/2\phi_m \tanh \xi - \phi_w^{1/2} \exp(-4 \ln 2\xi^2) \quad (4) $$

with $\phi_m + \phi_w = 1$, $\xi = 2(y - y_c)/\delta$, and $\phi_w$ relates to the ‘internal’ wake and $\phi_m$ to the external ML as follows: the velocity scale $\Delta U = \text{Max}(U) - \text{Min}(U)$ is the sum of the ML contribution $\phi_m \Delta U$ and the wake contribution $\phi_w \Delta U$ [47]. ‘The shape of the composite velocity profile is determined by the ratio of external and internal velocity scales $\phi_m/\phi_w$.’

### 2.1.4 Mixing layer width

Like for the plane FMLs [11, 12], the SML width $\delta$ is often quantified with the vorticity thickness (Eq. 2, [31]). By normalizing it as $\delta_1^* = c_f \delta U_{c,0}/(hU_{s,0}) = c_f \delta/(2h\lambda_0)$, Chu and Babarutsi [3] observed that the data on $\delta_1^*$ (five tests with $\lambda_0 \approx 0.3-0.6$) collapse along a single curve when plotted against $x^*$. A similar result was also obtained by Uijttewaal and Booij [4] (two tests with $\lambda_0 \approx 0.4$). When using the data [30] of 16 tests of Proust et al. [7] (Fig. 3a), we can see that the collapse is fairly good for $x^* \le 1$ when putting aside the ‘extreme’ spanwise-sheared flows, i.e. the weakly sheared flows with $\lambda_0 = 0.1-0.2$ and very highly sheared flows with $\lambda_0 = 0.8-1$. In addition, $\delta_1^*$ increases linearly with $x^*$ in Fig. 3a, which is equivalent to $\delta$ proportional to $\lambda_0 x$, as observed for the plane FMLs [11, 24, 25]. This is consistent with a FML behavior provided that: (i) KHCSs are present (i.e. if $\lambda_0 \gtrsim 0.3$, see §2.1.7), (ii) the wake effect is negligible (§2.1.2), and (iii) transverse flow is negligible compared to streamwise flow (not the case for $\lambda_0 = 0.8-1$).

To predict the streamwise evolution of $\delta_1^*$, Chu and Babarutsi [3] proposed a system of equations (partly based on Eq. (3)):
<figure><img src="image_10.png" /><figcaption>Fig. 3 Normalized mixing layer width $\delta^*$ as a function of $x^*$: a Chu and Babarutsi [3], with solution of Eqs. (5–7) assuming $U_c = U_{c,0}$ (continuous line), and with $\delta_{\max}^* = 0.065$ [56] (dashed line); b Cheng and Constantinescu [6], with Eq. (10) (continuous lines); c Van Prooijen and Uijttewaal [50] with Eq. (8) (continuous line). Data [30] of Ref. [7] with total discharges $Q = 14$ l/s (♦), 30 l/s (□), 60 l/s (○) and 120 l/s (△), and with $\lambda_0 = 0.1-1$ (marker color indicates $\lambda_0$-magnitude.)</figcaption></figure>

$$d\delta_1^*/dx^* = \beta U_{c,0}^2/U_c^2 \exp(-x^*) \quad (5)$$

$$\beta = \beta_0(1 - S/S_c) \quad \text{for } S < S_c \text{ and } \beta = 0 \quad \text{for } S \ge S_c \quad (6)$$

$$S = 0.5 \delta_1^* U_{s,0} U_c / (U_{c,0} U_s) \quad (7)$$

where $\beta = U_c/U_s d\delta/dx$ is the so-called spreading coefficient of the ML. Chu and Babarutsi [3] numerically solved Eqs. (5–7) assuming that $U_c = U_{c,0}$, $\beta_0 = 0.18$ and $S_c = 0.09$. With the same assumptions, we show in Fig. 3a the solution of Eqs. (5–7) (continuous line) against the data [30] of Ref. [7]. The agreement is poor for small $x^*$, while the maximum values of $\delta^*$ reached at high $x^*$ are comparable ($\delta_{\max}^* = 0.065$ [56]). An assumption that $U_c = U_{c,0}$ for all $x^*$ is not consistent with the measurements [7]. In addition, decreasing the coefficient $\beta_0$ to get a better fit with the data for small $x^*$ leads to underestimation of the $\delta_{\max}$-values, highlighting the limits of this approach. Last, additional measurements for $x^* \gtrsim 2$ (Fig. 3a) would be required to conclude on the existence of an asymptotic value $\delta_{\max}$.

Later, Van Prooijen and Uijttewaal [50] proposed the following simplification by assuming both $U_c = U_{c,0}$ and $\beta = \beta_0$ in (5):

$$\delta_3^* = \frac{c_f \delta U_{c,0}}{h U_{s,0}} = \beta_0(1 - \exp(-x^*)) + \delta_0^* \quad (8)$$

where $\delta_0^* = (c_f \delta_0 U_{c,0})/(h_0 U_{s,0})$ and $\delta_0$ is the thickness of the lateral boundary layers developing on both sides of the splitter plate, and assumed to equal the flow depth $h_0$. The relationship (8) is based on the data of Uijttewaal and Booij [4] and with $\beta_0 = 0.085$. Proust et al. [7] recently tested the relationship (8) against their own data [30]. The results are reported in Fig. 3c. Noticeable deviations of the measurements from Eq. (8) can be observed due to the streamwise changes in $U_c$ and high V-velocities –invalidating (3)– for more than half of the cases studied [7]. Sukhodolov et al. [35] proposed a modification of Eq. (8) based on the measurements along a straight river reach, which featured a linear decrease in velocity difference $U_s = U_{s,0} - \beta x$ for three various runs. This resulted in a parabolic function for $\delta$:
$$ \delta = \delta_0 + \beta_0 \frac{U_{s,0}}{U_{c,0}} x \left( 1 - \frac{B}{2U_{s,0}} x \right) \qquad (9) $$

where $\beta_0 = 0.11$, and $B$ is a parameter varying from one run to another, which limits the operability of Eq. (9).

Last, employing the definition of $\delta_1^*$ by Chu and Babaruti [3], but replacing $\lambda_0$ by the local value of $\lambda$, Cheng and Constantinescu [6] used in their simulations $\delta_2^* = c_f \delta / (2h\lambda)$, and found that

$$ \delta_2^* = 0.09 x^* \quad \text{for } x^* < 1.5 \quad \text{and} \quad \delta_2^* = 0.0364 x^*^3 \quad \text{for } x^* > 2 \qquad (10) $$

Proust et al. [7] applied relationships (10) to their data (Fig. 3b) and like in Fig. 3a, the linear relation between $\delta^*$ and $x^*$ (equivalent to $\delta \propto \lambda x$ similar to FMLs) is a good approximation for $x^* \lesssim 1$ putting aside the extreme spanwise sheared-flows. For high $x^*$, the proposed power function (10) seems to be a lower envelope curve for the measured data.

We can thus conclude that, the SML width $\delta$ starts increasing linearly with $x$ for a given $\lambda_0$-value (like for the plane FMLs) if: (i) 2DCSs are present (data [30] of Ref. [7]); (ii) wake behind a splitter plate or a confluence apex is weak; and (iii) a strong transverse flow is absent. For now, the upper limit of this linear part is not clearly known. There is currently no convincing relation that can accurately predict the evolution of $\delta$. In addition, the causes of the decrease in the growth rate $d\delta/dx$ are subject to continuing debate. The observed decrease could be due to a loss of coherence of the large-scale eddies [6, 46], or to a reduction of the transverse mean flow and associated lateral momentum transfer [7, 38]. The latter argument would be consistent with the fact that lateral momentum fluxes by the mean flow are always one or two orders of magnitude higher than the spanwise Reynolds stresses from the ML origin to the longitudinal position where the flow reaches spanwise uniformity ($U_s = 0$) [6, 7, 35, 38].

### 2.1.5 Initial growth rate of the mixing layer

For the plane FMLs, the initial growth rate of the ML can be specified as:

$$ d\delta/dx|_{x=x_0} = 2\lambda_0\sigma_0 \qquad (11) $$

with $\sigma_0$ ranging from 0.06 to 0.11 (Table 4, [11, 12, 24, 25]). The value $\sigma_0 = 0.0905$ of Brown and Roshko [11] is often taken as a reference as it is obtained by a best fit of various datasets [e.g. [8–11]]. It was found that in general the factor $\sigma_0$ varies with the density ratio $\rho_1/\rho_2$ [11] (where $\rho_1$ and $\rho_2$ are the densities of the two incoming flows), increases with the level of free-stream turbulence [17], and also depends on the state of the boundary layers on the splitter plate (turbulent vs. transition or laminar, tripped vs. untripped) [2, 22]. The parameter $\sigma_0$ may also vary in the case of the forced MLs, which are initially excited with periodic 2D oscillations [19]. The initial growth rate of the ML could also be sensitive to a longitudinal pressure gradient associated with acceleration or deceleration of the flow [57, 58].

For SMLs, the factor $\sigma_0$ in Eq. (11) ranges from 0.09 [4, 6, 7] to 0.18 [3], likely due to the same effects as listed for FMLs above. Booij et al [53] stated that ‘the spreading rate of a mixing layer is influenced by the turbulence outside the ML’ [53]. However, when looking at Fig. 3a or Fig. 3c, in the absence of a high transverse flow (i.e., at $\lambda_0 = 0.8-1$) or a wake (i.e., at $\lambda_0 = 0.1-0.2$), the data fit the growth rate reported by Brown
and Roshko [11]. There is therefore no indication that the initial growth rate for SMLs differ from that of the conventional plane FMLs.

Still, an open question is: over which streamwise distance the width $\delta$ is proportional to $\lambda_0 x$? The normalized distance $x^*$ is clearly not the appropriately scaled parameter, as $\delta^*$ is linear until $x^* = 0.2-0.3$ [4], $0.4-0.5$ [3], $1$ [7], $1.5$ [6] depending on the specific studies.

### 2.1.6 Mixing layer centerline

According to Proust et al. [7], four different definitions of the SML centreline can be considered. The ML centre can be firstly defined as ‘the spanwise location $y_m$ of the border between slow and fast flows that preserves initial discharges $Q_1$ and $Q_2$’. For instance, after van Prooijen and Uijttewaal [50], for a channel of width $W$, $y_m$ is solution of:

$$ \int_{y_m(x)}^{W} hU_d(x,y)dy = Q_2 \quad (12) $$

where integration is performed across the high-speed stream ($Q_2$), and subscript $d$ refers to depth-averaging. The use of (12) requires the knowledge of the depth-averaged velocity profiles. The $y_m$-values calculated from Eq. (12) [50] appeared to be essentially equal to the transverse coordinate $y_c$ where $U = U_c$ [4], which is another possible definition of the SML centre. Cushman-Roisin and Constantinescu [36] also considered $y_m(x)$ as a streamline separating two streams along which the spanwise exchange of mass is zero. They derived the relationship (13) from a mass conservation equation and Eq. (3):

$$ \frac{W_1^2}{y_m^2} - \left(\frac{U_{2,0}}{U_{1,0}}\right)^2 \frac{(W_2)^2}{(W - y_m)^2} = \left(1 - \left(\frac{U_{2,0}}{U_{1,0}}\right)^2\right) \exp(-x^*) \quad (13) $$

where $W_1$ and $W_2$ are the widths of the two streams at $x=0$, and with $W_1 + W_2 = W$. The values of $y_m$ predicted by (13) are in good agreement with those obtained in numerical simulations of mixing layers [59].

The SML centre can also be defined as the spanwise location $y_i$ of the inflection point in the $U$-distribution between fast and slow flows [3, 4, 7], which clearly differs from $y_c$ when the SML is asymmetrical [32, 38, 40]. Last, the SML centre can be defined as the location $y_t$ of the extreme in bulk turbulence statistics (e.g. in Reynolds stresses). For canonical plane FMLs, $y_i = y_c = y_t$ [26]. For SMLs, Proust et al. [7] experimentally observed that $y_i = y_c = y_t = y_m$ ($y_m$ calculated from Eq. 13) for weakly and moderately sheared flows (featuring $V \ll U$). For highly sheared flows (with $V \sim U$), some discrepancies appear between $y_m$ on the one hand, and $y_i$, $y_c$ and $y_t$ on the other hand, as $V$-velocity is maximum at the junction between streams, in contradiction with the streamline hypothesis for $y_m(x)$. However, for all sheared flows $y_i = y_c = y_t$ remains valid, suggesting that SMLs over a uniform bed-roughness are symmetrical. As for numerical simulations, the prediction of $y_c = y_i$ using the ‘integral model of SMLs’ developed by Booij and Tucker [53] is fairly good. A good agreement is also found between numerical (DES) simulations [6, 46] and experimental data, supporting $y_c = y_i$ [4]. We should also mention that Cheng and Constantinescu [6] proposed to normalize $y_c$ as $y_c^* = y_c U_{s,0}/(hU_{c,0})$ or $y_c^* = y_c U_s/(hU_c)$, to eliminate the effect of the initial velocity ratio, when plotting $y_c^*$ against $x^*$.
### 2.1.7 Two-dimensional coherent structures

The two-dimensional large-scale vortical structures with vertical axes were observed in both FMLs [11–13] and in SMLs [3, 7, 32, 38, 60, 61], as flow confinement does not prevent them from developing [7]. This is illustrated in Figs 1b–c, where the 2DCS spanwise length-scale $\delta_y^{CS} \sim 8-10h$ (can be detected from visualization). The 2D structures derive energy via velocity difference ($U_2 - U_1$) between the two streams [12]. For canonical plane FMLs, KHCSs were visualized/detected for $0.25 \le \lambda_0 \le 1$ [11–13, 18, 19, 62] (Table 5), while Pui and Garshore [17] noticed the absence of 2D structures for $\lambda_0 = 0.09-0.21$. In SMLs, 2DCSs were visualized/detected for $0.3 \le \lambda_0 \le 1$ [3, 6, 7, 35, 38, 39, 42, 44, 50, 63], while they were absent for $0.1 \le \lambda_0 \le 0.3$ [7, 38, 39, 44, 54, 61]. We can therefore suggest that, 2D structures can emerge if:

$$ \lambda_0 \gtrsim \lambda_c = 0.3 \qquad (14) $$

Relationship (14) was identified as a necessary condition for the emergence of 2DCSs in SMLs [7, 38, 44, 54]. Both FMLs and SMLs would therefore require a sufficiently high velocity shear for the development of 2D structures.

It is worth noting that condition $\lambda_0 \gtrsim 0.3$ is necessary but not sufficient for the 2DCSs to form in SMLs. Akutina et al. [64] have not observed the presence of 2DCSs for SMLs with a rough-to-rougher-bed variation at very low submergence $h/h_r = 1.5-3$ (where $h_r$ = roughness height) and for the entire studied range $0.15 \le \lambda_0 \le 0.41$. It is likely that the transverse shear layer turbulence could be hindered by a too high bed-induced turbulence production close to the bed. In addition, a ‘clear inflectional instability in the mean velocity profile’ [1] is another necessary condition for the emergence of 2D structures. When this condition is not fulfilled, even if $\lambda_0 \gtrsim 0.3$, 2DCSs cannot develop as observed by Proust et al. [54].

As illustrated in Fig 4a–c, 2DCSs have a spectral signature in the power density spectra of spanwise velocity component, i.e. a sharp bump at large scales, which is flanked by a short range of a $f^{-3}$ spectrum (on the right-hand side of the bump). A similar bump was observed in OCFs for various spanwise-sheared flow types, in which quasi-2D turbulence may co-exist with 3D turbulence [4, 6, 35, 38, 54, 65–68]. As shown by the co-spectra presented in Fig 4d–f, the 2DCSs are the major contributors to Reynolds stress $u'u'$.

<figure><img src="image_11.png" /><figcaption>**Fig. 4** Flow conditions of Fig. 1: a, d $\lambda_0 = 0.22$ and $h = 26$ mm; b, e $\lambda_0 = 0.39$ and $h = 26$ mm; c, f $\lambda_0 = 0.81$ and $h = 45$ mm [7, 30]. Measurements at $x = 3.65$ m and $y = y_i$ (inflection point). Power spectra (a–c) and co-spectra (d–f) of velocity fluctuations versus Strouhal number $St = f\theta/U$</figcaption></figure>
As for the downstream fate of the 2DCSs, the recent experiments of Proust et al. [7] and Cerino et al. [44] have confirmed that the bed-friction number S is involved in the development and decay of the 2DCSs (§2.1.1). The 2DCSs start being stretched longitudinally when

$$S \gtrsim S_{c1} = 0.01 \qquad (15)$$

and decay when

$$S \gtrsim S_{c2} = 0.1 \qquad (16)$$

in agreement with previous experimental and theoretical results [3, 4, 31, 34, 48]. Further downstream, the 2DCSs evolve in wave-like structures [6, 7], and eventually the large-scale transverse turbulent motion vanished [3, 7].

## 2.1.8 Large and very-large-scale motions

Large- (LSMs) and very-large-scale (VLSMs) motions have been observed in boundary layers [69], pipe flows [70] and OCFs [38, 45, 71–76]. LSMs in wall-bounded flows are believed to be packets of hairpin vortices generated in the velocity shear region near the bed [69]. VLSMs origin remains unclear. There are two main concepts that are currently under discussion: (1) VLSMs are result of a ‘coherent alignment’ of LSMs [70]; and (2) VLSMs are structures, independently formed via a self-sustained mechanism separate from LSMs [77]. The recent experimental data of Zampiron et al. [78], collected in two different facilities, suggest that VLSMs are a special type of coherent structures, different from LSMs or their arrangements. The streamwise wavelength of the LSMs $\Lambda_x \sim 2h-4h$ [45, 71–75, 79, 80], while for VLSMs $\Lambda_x \sim 20h-50h$ [45, 71–75, 80]. In uniform OCFs, the LSMs are fully developed for $x \gtrsim 50h-60h$, while VLSMs development requires much longer distances, i.e., $x \gtrsim 100h-150h$ [45, 71–75, 80]. For the case of fully established VLSMs, the key parameters potentially influencing their dynamics and characteristic lengths include width-to-depth ratio $W/h$, friction Reynolds number $Re_T = u^*h/v$, and, for the case of rough-bed flows, relative submergence $h/h_r$ [71, 73, 80, 81].

The studies of LSMs and VLSMs in the presence of SML are at their initial stage [7, 38, 45]. Investigating flows in a compound channel with a smooth main channel and two rough floodplains, Proust and Nikora [38] observed for three SMLs with $\lambda_0 \approx 0.2–0.7$ the spectral signatures of LSMs and VLSMs at $x \sim 30h$ in the main channel only, with $\Lambda_x \sim 2h$ for LSMs and $\Lambda_x \sim 20h$ for VLSMs. At larger distances from the flume entrance, LSMs and VLSMs quickly disappear along the flow under the combined effects of the transverse shear layer and the development of SCs and 2DCSs. It was also shown that LSMs and VLSMs were weakly altered by transverse currents. In the experiments of Proust et al. [7] on SMLs in a rectangular smooth-bed channel, the presence and development of LMSs/VLSMs were found to depend on the $\lambda_0$-value. For $\lambda_0 > 0.3$, i.e. in the presence of 2DCSs, LMSs and VLSMs were not detected, while for $\lambda_0 = 0.13–0.18$, LSMs and VLSMs were observed away from the splitter plate. Recently, Cerino et al. [45] conducted an experimental study on SMLs over a rough bed, yielding similar findings. They observed no spectral signature of VLSMs at measurement locations where strong 2DCSs, combined with transverse heterogeneity of the mean flow, inhibit the emergence of VLSMs. However, distinguishable VLSMs signature emerged once these 2DCSs transitioned from their vortex shape to a wave-like motion. Furthermore, Cerino et al. [45] noted that in parts of shallow shear flows where VLSMs did emerge, their contribution to velocity statistics $u'^2$ and
$\overline{-u'w'}$ were similar to those observed in uniform OCF (~40-50% for $\overline{u'^2}$ and ~30-50% for $\overline{-u'w'}$) [74, 82, 83].

### 2.1.9 Velocity fluctuations

For plane FMLs, the normalized primary Reynolds stress $-\overline{u'v'}/U_s^2$ is independent of velocity ratio $\lambda$ [17] if levels of free-stream turbulence are low [23], with $-\overline{u'v'}/U_s^2 \approx 0.01-0.02$ as reported in Table 4. In addition, Mehta [23] showed that when the mean velocity profile features a noticeable velocity deficit caused by the splitter plate (which is observed for small $\lambda_0$), the values of $-\overline{u'v'}/U_s^2$ are increased: from $\lambda_0 = 0.18$ to 0.05, Reynolds stress $-\overline{u'v'}/U_s^2$ rises from 0.017 to 0.1. For comparison, $-\overline{u'v'}/U_s^2 \approx 0.012$ for $\lambda_0 = 0.25-0.33$ in the absence of a velocity deficit.

For SMLs, the focus has been more on the dimensional stress $-\overline{u'v'}$ [3, 4] than on non-dimensional stress $-\overline{u'v'}/U_s^2$ as in SMLs $U_s$ tends towards 0 after a few length-scales $h/c_f$ (§2.1.1). However, it is interesting to examine the $-\overline{u'v'}/U_s^2$-values as long as $U_s$ is not negligible. Figure 5c shows $-\overline{u'v'}/U_s^2$ vs. local $\lambda(x)$ for the data [30] of Ref. [7] (along with the data of Mehta [23] for plane FMLs). For SMLs with initial shear $\lambda_0 \gtrsim 0.3$, $-\overline{u'v'}/U_s^2 \lesssim 0.02$ for a vast majority of data, as observed for FMLs in the absence of a measurable splitter-plate-induced velocity deficit. In contrast, for the weakly sheared flows with $\lambda_0 = 0.1-0.2$, a big part of the $-\overline{u'v'}/U_s^2$-values are higher than 0.02 in the range $\lambda =$

<figure><img src="image_12.png" /><figcaption>**Fig. 5** Data [30] of Ref. [7] from $x = 0.65$ m to 15.65 m, with $\lambda_0=0.13-1$ and $h \approx 2-10$ cm: peak correlation coefficient between spanwise and streamwise velocity fluctuations $R_{u',v'} = -\overline{u'v'}/(\sqrt{\overline{u'^2}}\sqrt{\overline{v'^2}})$ (a), and peak turbulence structure index $-\overline{u'v'}/\overline{u'^2}$ (b) (dashed lines indicate levels for FMLs [15]). Peak Reynolds stresses $-\overline{u'v'}/U_s^2$ compared with data of Mehta [23] for FMLs (c) and $-\overline{u'v'}/U_c^2$ (d) with $\lambda_c = 0.3$ in dashed lines, data with $Q = 141/s$ (◦), 301/s (□), 601/s (○) and 1201/s (△), marker color indicating the $\lambda_0$-magnitude</figcaption></figure>

<figure><img src="image_13.png" /></figure>
0.1–0.2 (i.e near the flume entrance) and comparable to Mehta’s values. This could be due to extra transverse turbulent fluxes caused by the splitter plate wake, which are of the same order of magnitude [84] as that of a ML.

Figure 5c shows that the presence or absence of 2DCSs is hardly detectable from the values of normalized Reynolds stresses-$\overline{u'v'}/U_s^2$, as observed by Pui and Garthmore [17] for plane FMLs. To see a possible effect of the 2DCSs, we have normalized $-\overline{u'v'}$ by $U_c$ in Fig. 5d. It appears that, in the presence of 2DCSs, i.e. with $\lambda_0 \gtrsim 0.3$, $-\overline{u'v'}/U_c^2$ can reach values far higher than in the absence of 2DCSs.

To relate velocity fluctuations to the turbulence structure more clearly, using the same dataset in Fig. 5a and b we have plotted the peak correlation coefficient $R_{u'v'}$ between $u'$ and $v'$ [8] and a peak turbulence structure index $-\overline{u'v'}/\overline{u'^2}$ [15]. For $\lambda_0 > 0.3$ and $0 \le S \lesssim S_{c1} = 0.01$, the coherence of 2DCSs measured by $R_{u'v'}$ (Fig. 5a) is high and the SMLs behave like FMLs with an index $-\overline{u'v'}/\overline{u'^2} \approx 0.38$ [15] (Fig. 5b). For $S_{c1} \lesssim S \lesssim S_{c2} = 0.1$, the 2DCSs are longitudinally stretched with a decrease in $-\overline{u'v'}/\overline{u'^2}$ with increasing S (Fig. 5b, [7]), combined with decreasing coherence in the structure (Fig. 5a). Last, if $S \gtrsim S_{c2}$, the transverse turbulent motion vanishes and $\overline{u'v'}$ tends towards very low values (i.e. that observed for uniform unsheared flows) (Fig 5a–b) as previously observed by several authors [3, 4, 31, 34, 48].

Last, it is important to compare the relative contributions of the velocity fluctuations and mean velocities to the lateral flux of streamwise momentum, i.e. $\overline{u'v'}/U_c^2$ and $(UV)/U_c^2$. In the experiments of Proust et al. [7] and in the numerical (DES) simulations of Cheng and Constantinescu [6], the fluxes $\overline{u'v'}/U_c^2$ are negligible compared to $(UV)/U_c^2$ over the whole dynamic range, from the ML origin to the relaxation towards uniformity ($U_s = 0$).

### 2.1.10 Link between S and velocity fluctuations

Relying on the kinetic energy equation of the large-scale motions, Chu et al. [34] linked the bed-friction number S to the velocity fluctuations as follows. Considering the production term of 2D structures $P_{2D} = -\overline{u'v'}/\partial U/\partial y$ and the bed-friction dissipation term $F = (2\overline{u'^2} + \overline{v'^2})c_f U/(2h)$ and assuming that the SML thickness $\delta = U_s/(\partial U/\partial y)_{\text{max}}$ (Eq. 2) and $U = U_c$, the ratio $F/P_{2D}$ can be expressed as:

$$ \frac{F}{P_{2D}} = \frac{c_f \delta}{4h\lambda} \frac{2\overline{u'^2} + \overline{v'^2}}{\overline{u'v'}} = S \frac{2\overline{u'^2} + \overline{v'^2}}{\overline{u'v'}} \approx 10S \quad (17) $$

also assuming that $\overline{u'^2} \approx \overline{v'^2}$ and $\overline{u'v'}/\overline{u'^2} \approx 0.3$ [15]. This implies that ‘the transverse motion would be suppressed if $F > P_{2D}$’ [34], which is equivalent to $S \gtrsim S_{c2} = 0.1$.

Using experimental measurements (within 16 various SMLs with $\lambda_0 = 0.13-1$) of S, $P_{2D}$, and 3D bed-induced turbulence production $P_{3D} = -\overline{u'w'}/\partial U/\partial z$, Proust et al. [7] also linked S to velocity fluctuations as follows:

$$ S = \frac{c_f \delta}{4h\lambda} \approx \lambda \gamma \frac{P_{3D}}{P_{2D}} \quad (18) $$

where $\gamma = 0.02$ is a dimensionless coefficient related to shear layer turbulence (i.e., $-\overline{u'v'} \approx \gamma U_s^2$). For $S \approx S_{c1} = 0.01$ (start of 2DCSs stretching), $P_{3D}/P_{2D} \approx 1-10$, and for $S \approx S_{c2} = 0.1$ (disappearance of the 2DCSs), $P_{3D}/P_{2D} \approx 100-500$.
### 2.1.11 Vertical variation in the SML structure

Owing to bed-friction, the structure of SMLs is non-uniform over the depth. This vertical non-uniformity was observed in detached-eddy simulations [6, 46], experiments in straight channels [4, 7, 40], and along a straight river reach [35]. The degree of vertical non-uniformity first increases downstream of the splitter plate [6, 7], and then decreases after the decay of 2DCSs [6]. In addition, vertical non-uniformity decreases with decrease in flow depth $h$ [6].

Regarding the SML key features, the width $\delta$ can increase from the channel bottom to the water surface [6, 7, 40, 46] by ~20-30% [7, 46], with the top of the ML on the low-velocity side slightly tilted toward the low-speed stream [6, 7, 46]. Spatial correlations of velocity fluctuations along the vertical direction show that the 2DCSs extend over the whole depth [4, 7]. The coherence of the large eddies decreases when approaching the channel bottom [46]. This is due to an increasing ratio $P_{3D}/P_{2D}$ [7], with the small-scale turbulence contributing to the diffusion of the 2D vorticity (similarly to viscosity), also resulting in a streamwise stretching of the 2DCSs being higher near the bed than near the water surface [7].

As for the lateral momentum transfer, the dispersive terms $[(UV)_d - U_dV_d]/U_c^2$ can be of the same order of magnitude as the turbulent fluxes $u'v'/U_c^2$ [6] as observed in compound channel flows [38, 40] or flows with a lateral change in bed-roughness [85].

## 2.2 SMLs with non-parallel incoming streams

When the two incoming streams are no longer parallel, the flow structure in the confluence hydrodynamic zone (CHZ) [86] was found to primarily depend on the confluence angle $\alpha_0$ [87], the degree of planform symmetry of the confluence [87], and the momentum flux ratio of the incoming streams $M_r = (\rho_2 Q_2 U_2)/(\rho_1 Q_1 U_1)$ (where 2 and 1 refer to tributary and main river) [88–92] or the discharge ratio $Q_r = Q_2/Q_1$ [87, 93, 94]. Besides the structures present in SMLs formed by parallel incoming streams (Sect. 2.1), the combined effects of streamline curvature and bed friction give rise to helical cells of Prandtl’s secondary current of the 1st kind (SC1) [89–98]. In addition, 3D turbulent coherent structures with a longitudinal axis can also be observed, which are termed streamwise-oriented-vortices (SOV) cells. For SMLs, SOV cells were typically observed in angled confluences [55, 89, 95, 97–102], and are associated with either SC1 cells or GCs. The complexity of the SML structure increases from idealized geometries to real river confluences (due to topographical forcings on the flow structures [103]), and from fixed bed to erodible bed.

### 2.2.1 Idealized geometries

Flow dynamics at angled confluences, and specifically in the CHZ, can be characterized by ‘six major regions of flow stagnation, flow deflection, flow separation, maximum velocity, flow recovery and distinct shear layers’ [86].

First, the extensions of these zones and associated flow structures are dependent on the magnitude of junction angle $\alpha_0$. Mosley [87] was first who analyzed the effect of $\alpha_0$ (Table 1), though indirectly, by experimentally exploring the scour pit formed in an erodible bed downstream a Y-shaped confluence. The scour hole depth $h_s$ in the CHZ increases rapidly as $\alpha_0$ increases from 15° to 90°, and more slowly up to 180°. As the confluence
is symmetrical, a pair of counter-rotating SC1 cells form with downflow between cells, which contributes to increasing bed shear stress. Flow acceleration, high turbulence level and SC1s are responsible for an increase in bed shear stress and thereby erosion in the CHZ. The increase in $\alpha_0$ gives rise to larger SC1s cells and higher $h_s$. The link between $\alpha_0$ and cell size can also be inferred from the laboratory (Table 1) and field data of Ashmore and Parker [104], as $h_s/h = 2.235+0.031\alpha_0$ where $h$ is the ambient average depth in the upstream branches. Another effect of $\alpha_0$ is the possible distortion of the ML, e.g., the ML can be distorted for $\alpha_0 = 90^\circ$ [105] but not for $\alpha_0 = 30^\circ$ [106].

Second, flow structure is dependent on the relative magnitudes of the upstream discharges $Q_1$ and $Q_2$, with total discharge $Q$ being fixed and stream cross-sections identical. In the experiments of Mosley [87], scour depth $h_s$ increases as $Q_1 - Q_2$ decreases and is maximum for $Q_1 = Q_2$, i.e., for $\lambda_0 = 0$. As total discharge $Q$ is constant in the CHZ, and 2DCSs cannot form for small $\lambda_0$ (Sect. 2.1.7), we can therefore deduce that the peak bed shear stress is predominantly influenced by the SC1 cells (compared to flow acceleration and 2D vertical vortices), and that the SC1 cell size increases with increasing $Q_r$ (from values lower than 1 to 1). Mosley [87] also found that ‘a relative increase in discharge in one upstream tributary caused the lower channel to swing into closer alignment with that tributary’.

Third, the flow structure was influenced by the momentum flux ratio $M_r = (\rho_2 Q_2 U_2)/(\rho_1 Q_1 U_1)$ [88–92], especially the path of the SML centerline. For a confluence with a main channel and a tributary, the momentum conservation equation [87, 88] gives

$$ \tan \phi = M_r \sin \alpha_0 / (1 + M_r \cos \alpha_0) \quad (19) $$

where $\phi$ is the angle between the SML centerline and the main channel direction in the vicinity of the junction (further downstream, the SML being curved adjusting to the prevailing flow direction). ‘As $M_r$ increases, greater deflection of the main stream flow by the tributary causes $\phi$ to increase’ [87].

The combined effects of a geometrical parameter ($\alpha_0$) and flow parameters (velocity ratio $U_2/U_1$, flow depth $h$) was numerically investigated by Cheng and Constantinescu [101]. They studied their effects on mean flow, turbulence structure and sediment entrainment mechanisms inside the SML and in its surroundings. For instance, SOV cells (that are effective in the sediment entrainment) form in the vicinity of the SML for $\alpha_0 = 60^\circ$ but not for $\alpha_0 = 0^\circ$, irrespective of $h$ and $U_2/U_1$.

Last, Sukohodolov et al. [47] recently found that, for predicting the decrease in velocity shear $U_s$ at angled flow configurations, it is required to consider a composite resistance factor $c_t = (c_f + c_a - c_p)$, where $c_a$ and $c_p$ are advective and pressure head coefficients, respectively. Then, relationship (3) becomes

$$ U_s = U_{s0} \exp(-(c_f + c_a - c_p)x/h) \quad (20) $$

assuming $U_c$ constant when moving downstream of the confluence [47], sup material. In this case, the calculation of $c_a$ and $c_p$ is not an easy task, as they are related to the lateral advective and turbulent fluxes of momentum, which vary with $\alpha_0$ and $M_r$ [47].

### 2.2.2 River confluences

Flow structure at river confluences was found to be dependent on the momentum flux ratio $M_r$ and total discharge $Q = Q_1 + Q_2$ [95]. With increase in the momentum flux ratio $M_r$,
the deflection of the mainstream flow by the tributary flow also increases. Correspondingly, the ML between the converging flows also shifts toward the outer channel bank [95].

The region of stagnated fluid near the upstream junction corner was observed by Rhoads et al. [107], which induces ‘a downstream velocity field characterized by two zones of maximum velocity separated by an intervening region of low velocity centered on the mixing interface’. For an asymmetrical river confluence with $α_0 = 60°$ (Table 3), Rhoads and Sukhodolov [108] observed that flow near the upstream junction corner has wake characteristics. When $M_r \sim 1$, the simulations of Constantinescu et al. [55] have shown that the ML is populated by quasi-2D successive eddies with opposite senses of rotation (von Karman vortex street), whose coherence is significantly less than that of 2DCSs. Flow dynamics is in the wake mode (§2.1.2), when $U_s$ is small and the stagnation zone induces an internal velocity deficit that acts like a wake behind an obstacle.

Like for ideal geometries, SC1 cells were identified in the field for angled river confluences. They were revealed by the presence of scour pits [109], or by (ADCP) measurements at a small stream confluence by Rhoads et al. [107] and Lewis and Rhoads [96]. At a small stream confluence, the channel-scale helical motion is a prominent feature of flow structure [90]. However, SC1 cells are also observed at a large asymmetrical river confluence [93], where one or two SC1 cells are observed depending on discharge ratio $Q_r$. The latter affects the helical motion which in turn affects sediment transport as well as the local bed morphology (large scour hole likely related to the downwelling and upwelling flows caused by SC1s). The simulations of Jiang et al. [110] also feature SC1 cells for a large river confluence with $α_0 = 58°$. Note that SC1 cells can be responsible for the existence of a velocity dip phenomenon in the near-surface layer, as observed by Yuan et al. [94] at a large confluence.

The effects of junction angle $α_0$ and momentum flux ratio $M_r$ on SC1s were examined by Sukhodolov and Sukhodolova [89] with a field-based experiment. While no SCs are detectable for $α_0 = 0$ irrespective of $M_r$-value, for $α_0 \neq 0$, they observed a pair of counter-rotating vortices or a single vortex depending on $M_r$-value. In addition, small-scale helical structures can develop in the immediate vicinity of the SML, with the same sense of rotation as the main cells but embedded within the large cells. Both $α_0$ and $M_r$ also define the lateral and streamwise extent of flow separation areas, which increase with increasing $α_0$ or decrease with increasing $M_r$.

At a small asymmetrical confluence, Constantinescu et al. [55] have observed two primary SOV cells (downstream of the junction) and two secondary cells (attached to the junction), which are adjacent to the shear layer. The diameter of SOV-like cells scales with flow depth. They have also shown that for natural confluences, the streamwise development of the SOV cells and of the mixing interface (see section 5) can be strongly affected by the presence of large-scale irregularities at the banks or on the channel bed. The simulation of Constantinescu et al. [99] additionally showed that SOV cells can develop in both ML and wake modes (§2.1.2), extracting mass and momentum from the SML. Dual counter-rotating SOV cells were also observed by Rhoads et al. [90]. In the field-based study of Sukhodolov and Sukhodolova [89], the SOV cells disappear as soon as the curvature of flow and SC1s vanish, and the SOV cell size scales with flow depth $h$ while SC1 cell size scales with $4-5h$. Interestingly, Jiang et al. [110] did not observe SOV cell in the vicinity of the SML on a large river confluence with $α_0 = 58°$ for the three hydrological scenarios investigated.

It is also worth mentioning the possible change in the free surface elevation in the CHZ, with a depression of the free surface in the separation zone [89], and with a superelevation of the free surface where the two converging flows meet [89], this superelevation increasing with $α_0$. The superelevation is due to an increase in pressure [55].

<figure><img src="image_14.png" /></figure>
Last, as observed with parallel streams by Proust et al. [7, 38] in the laboratory, or by Sukhodolov et al. [35] in a field-based experiment, Rhoads and Sukhodolov [108] found at a small stream confluence with $\alpha_0 = 60^\circ$ that the path of the SML was primarily influenced by the lateral fluxes of momentum (mean flow), the latter exceeding turbulence lateral momentum fluxes by one to two orders of magnitude. The reader is encouraged to check a comprehensive review of Constantinescu and Gualtieri [111] in this Special Issue, which is specifically focused on the river confluences.

### 3 Streams with unequal densities

A difference in density can be observed in natural flows when different temperatures, dissolved mineral or suspended sediment concentrations are present across the ML interface [97, 112–116]. In this review we will focus on the effect on a confluence dynamics of two streams with different densities but each of them well mixed when they met (see Fig. 6). We could imagine, especially for sand laden flows, a vertical stratification of one or two tributaries. It would then result in specific processes, such as the one explored by Ismail et al. [117] that will not be discussed here.

At river confluences, ‘the main effect of density contrast between the confluent flows is to generate lateral near-bed intrusions of heavier fluid and to strongly tilt the boundaries of the mixing interface’ [110]. Depending on the relative strength of buoyant and inertial forces, the ML can be: (i) fully stratified (and similar to a lock exchange); (ii) presenting density-driven secondary flows; or (iii) nearly vertical. The densimetric Froude number can be used to quantify this relative strength [119]. It is defined as

$$Fr_d = \frac{U}{(gh\Delta\rho/\rho)^{1/2}} \qquad (21)$$

<figure><img src="image_15.png" /><figcaption>**Fig. 6** Schematic diagram of a SML with two streams of unequal densities and $\rho_2 > \rho_1$. Inspired by Cheng and Constantinescu [118]</figcaption></figure>
where $U$ can be the mean streamwise velocity in the main channel [e.g., [119]], in the tributary [e.g., [96]] or downstream of the confluence apex [e.g., [110]] depending on the authors' perceptions or a problem at hand. Richardson number $Ri = 1/ Fr_d^2$ can be used equivalently [118]. However, these numbers take the same value whether the lighter flow is the faster or the slower (case depicted in Fig 6), even if the flow behavior was found to be different. They were then found to be not necessarily adapted to all cases [92, 98].

White and Helfrich [120] suggested another parameter to quantify this density gradient effect, i.e., $\gamma$-parameter, which is defined as the ratio of the time-scale of shear instabilities to the one relative to gravitational adjustment and can be written as

$$ \gamma = \frac{5L_u (gh\Delta\rho/\rho)^{1/2}}{h U_s} \qquad (22) $$

where $L_u$ is a characteristic length-scale for the horizontal shear, the ML width for example. Whereas $Fr_d$ is based on a mean velocity, the velocity scale considered in $\gamma$-parameter is the velocity difference $U_s$ between the two tributaries. Considering data available in the literature, it is difficult to identify the most efficient criterion to predict vertical stratification among $\gamma$ and $Fr_d$ [121] especially because in most of the natural streams, both criteria essentially coincide [122].

Even though these parameters do not fully describe the diversity of confluences with flows of unequal densities, they are still convenient to distinguish streams with large density differences from streams with lower density differences. For large density difference and low densimetric Froude number ($Fr_d < 2$ to $5$ [121]) the flow tends to evolve toward a stable stratified configuration at confluences [100]. The denser fluid plunged down the lighter fluid.

Small density differences ($Fr_d > 5$) can also have a dramatic effect on the behaviour of the ML [97, 112, 114, 118]. These confluences combine structures characteristic of shallow shear flow and gravity currents (GCs). The intrusion of dense fluid under light fluid can be limited transversely [98].

The comparison in the literature between confluences with and without density gradient has revealed the crucial role of it even for very small differences in densities, particularly considering the cases of concordant or discordant beds [123], for straight or bended downstream rivers [97, 110], and for confluences with large or small angles [103, 116]. Comparing the same flow with and without density gradient in the laboratory, we can see that the effect of density is crucial, but as it plays a role not so different of angle or bend, it can be hardly distinguished in natural rivers.

## 3.1 Time-averaged flow

A recirculation cell that occupies the whole channel depth develops for streams with unequal densities. The denser fluid goes towards lighter fluid near the bed and the lighter fluid towards denser fluid near the surface (lock exchange). This result in a tilt of the ML that can even become horizontal [121]. As a consequence, the definition of ambient streams velocities ($U_1$ or $U_2$) along the flow can be ambiguous both considering velocity profiles at a selected elevation or depth-averaged values. Considering the whole cross-section, Cheng and Constantinescu [118] showed that differences in density increase the distance needed to streamwise momentum redistribution between the two subsections.
In FMLs, the density gradients do not cause a tilting in the mixing interface and has much weaker effects on the flow characteristics than in SMLs. This results in a slightly smaller growth rate of the ML (visual growth or width of the intermediate density layer or vorticity thickness) when the faster flow is denser [11]. In contrast, a lighter faster flow increases the growth rate. The same statement seems to apply to SMLs [118], figure 6.

## 3.2 Turbulence statistics

In a confluence of two parallel flows (medium-size, with a width-to-depth ratio of 47), in the (DES) simulations of Cheng and Constantinescu [118], incoming streams with unequal densities generate larger values of TKE than streams with equal densities. They explained it as follows. With negligible density difference, high levels of turbulence mainly results from 2DCSs [7, 118] and are mainly confined to a small area in the ML, near the confluence apex. On the contrary, with strong density effects, the lock exchange flow induces oscillation of the front of intrusion of one fluid into the other, that generates high levels of TKE down to large distances from the origin of the ML. Cheng and Constantinescu observed a larger TKE amplification in the case of denser slower fluid than denser faster fluid [118].

Please note nevertheless that the simulation of Jiang et al. [110], based on the geometry of a large river confluence with $α_0 = 60°$, shows a decrease in turbulent kinetic energy at the free surface when the slower flow is denser, highlighting the difficulty in decoupling effects of geometry and density.

## 3.3 Coherent structures

Considering the strong effect of density variations on turbulent fluxes, in particular Reynolds stresses, we also expect effects on coherent structures [124].

For a SML with streams of unequal densities, the denser fluid plunged beneath the lighter fluid and, in the case of high velocity shear, the tilting of the ML stretches the 2DCSs [116], which causes a rapid loss of coherence [118]. This was corroborated by results of van Rooijen et al. [125] where 2DCS were absent for high $λ$ if difference of density was high. Whereas when equal density flows, the 2DCSs stretch, and then reach a quasi-equilibrium state with a wave-like structure, differences in densities prevent from reaching such equilibrium [118].

Density gradients induce secondary cross-flows that generate vortical cells orientated approximately along the streamwise direction at the interface between the two regions. Their length is ~10h. In their experiments, Duguay et al. [116] identified the density gradient as the main cause for this observed SCs, sometimes called SOVs. 'The coherence of the SOVs is altered by the momentum ratio of the confluence ($M_r$), with the SOVs being most coherent when a higher momentum denser channel collides with a lower momentum lighter channel' [116]. In the case of a confluence showing secondary currents induced by flow curvature (SC2s), the lock exchange flow induced by density difference can reinforce this secondary currents or act against them [92].
## 4 Flow-resistance-driven SMLs

SMLs can also be caused by a lateral change in flow resistance. Two flow configurations are often encountered: flows subject to a lateral bed-roughness transition (BRT) [63, 64, 85, 126–128]; and partially vegetated flows (PVFs), often observed in stream channels and riparian corridors [39, 41, 67, 68, 129–142]. Physical models of such configurations are shown in Fig 7a and b, respectively. When the lateral change in flow resistance does not vary in the streamwise direction, the spanwise velocity shear $U_s$ is self-sustained, which influences the streamwise evolution of all key parameters of the SML (width $\delta$, momentum thickness $\theta$, spanwise positions $y_c$ and $y_i$, large-scale eddy size, and turbulence statistics distributions).

<figure><img src="image_16.png" /><figcaption>**Fig. 7** a, c, e bed-roughness transition (BRT) glass - dense artificial grass; b, d, f partially vegetated flow (PVF) with emergent vegetation modelled by rigid stems set on the grass. Spanwise profiles of the mean streamwise velocity $U$ (c, d) and normalized Reynolds stress $-\overline{u'}\overline{v'}/U_s^2$ (e, f) for various inflow partition $Q_1 - Q_2$. New data collected at $x = 6$ m in an open-channel flume (18 m × 2 m) located at INRAE, Lyon, France</figcaption></figure>

<figure><img src="image_17.png" /></figure>
# 4.1 Bed-roughness transition

## 4.1.1 Mean flow

Vermaas et al. [85] were among the first who studied the effect of spanwise change in bed roughness on the structure of OCF. In their experiments, the flow is initially unsheared ($\lambda_0 = 0$ at $x = 0$, Table 1) and then enters a channel section with a lateral bed-roughness transition (BRT) (smooth to rough bed). This results in a transverse current and a net mass exchange from the decelerating flow over the rough bed to the accelerating flow over the smooth bed. Vermaas et al. [85] observed that, once scaled by the local velocity difference, the transverse velocity becomes independent of water depth. At the last measuring section ($x/h \sim 100$), the mean flow has not yet reached equilibrium. Along the flume, the spanwise position $y_i$ of the inflection point in the mean velocity profile remained located at the smooth-rough junction. The recent (preliminary) experiments conducted in the INRAE flume displayed in Fig. 7a suggest that $y_i$ can vary with the vertical confinement (i.e., with total discharge $Q = Q_1 + Q_2$ as the channel slope is constant) and/or the magnitude of the transverse mean flow. Indeed, in Fig. 7c, at $x = 6$ m, $y_i$ is shifted to the rough side for $Q_1 = 351$ l/s and $Q_2 = 851$ l/s, while for the three other cases $y_i$ remains located at the smooth-rough junction. The shift in $y_i$ is due to a transverse mean flow, whose magnitude increases with total discharge $Q$: the peak value of $V/U_c$ across the channel is 1%, 1.6%, 2.5%, and 4%, for $Q = 14$ l/s, 30 l/s, 60 l/s, and 120 l/s, respectively. Note that the transverse flow is a result of imposed upstream velocity difference, which was not the case in Vermaas et al. [85].

As for spanwise and vertical mean flow velocities, the presence of BRT induces Prandtl's secondary currents of the second kind (SC2) arising from turbulence anisotropy and flow heterogeneity caused by the transverse change in roughness. As shown in Fig. 7e, we can observe strong Reynolds-stress gradients from either side of the rough-smooth junction, which give birth to SC2 [72, 143, 144]. Hinze [145, 146] and Anderson et al. [144] demonstrated that these Reynolds-stress gradients 'cause a subsequent imbalance between production and dissipation of turbulent kinetic energy that necessitates secondary advective velocities' [144]. In the experiments of Akutina et al. [64], the SC2 cells (along with Reynolds stresses) are increasingly controlled by the roughness-scale as the flows become shallower.

In terms of mean spanwise velocity profile, the 'composite' channel (i.e., featuring a BRT) is comparable to a compound channel (with a step change in bed elevation across the flow). First, the mean velocity profile can be asymmetric (Fig. 7c) like in compound channel [32, 40, 147] with $y_i \neq y_c$. Second, within the ML, the lateral momentum exchange by the shear layer turbulence and the mean flow enables a part of the slow stream over the rough bed to be accelerated, while a part of the fast stream over the smooth bed is decelerated. The ability of the SML to accelerate the flow over the rough bed depends on the relative submergence over the rough bed, as shown in the experiments of Dupuis et al. [127]. In Fig. 7c, the larger the flow depth, the wider the extent of accelerated flow over the rough bed. Last, as for compound channel flows [148], the conveyance capacity of the channel is lowered by the lateral momentum exchange within the SML [85].

The mean velocity profile can exhibit a strong dependence on the relative submergence $h/h_r$ ($h_r$ being the height of the roughness elements), as shown in Fig. 7c in the case of artificial grass on the rough side, or when the BRT involves macro-roughness elements [63, 64, 127, 128], as demonstrated in the experiments of Akutina et al. [64] involving
a rough-to-rougher transition. Unlike in smooth-to-rough transitions, Akutina et al. [64] observed that in rough-to-rougher transitions, higher bulk velocities occur on the side with greater roughness. This difference increases as the flow becomes shallower, due to the increasing effective depth ratios. The effective depth ratio is defined as $d_{e,1}/d_{e,2}$, where $d_{e,1}$ and $d_{e,2}$ are the effective water depths accounting for the different porosities of the rough (index 2) and rougher (index 1) beds. An increase in $d_{e,1}/d_{e,2}$ leads to lower friction factor ratio $f_1/f_2$ (where f represents the Darcy friction factor). As a result, the rougher side, with its friction factor lower than that of the other side, experiences higher velocities. This trend suggests that decreasing relative submergence amplifies the impact of roughness differences between parallel flows, thereby increasing the velocity disparity between the two sides.

## 4.1.2 Turbulent statistics

For a BRT involving a rough bed made of a submerged cube array, Dupuis et al. [63] observed that the spanwise positions of the maxima differ for the longitudinal ($\rho u'^2$) and lateral ($\rho v'^2$) turbulent normal stresses. The maximum of $\rho v'^2$ is located directly above the outer cubes, whereas the maximum of $\rho u'^2$ is closer to the interface. Dupuis et al. [63] attributed this shift to the wakes generated by the outer cubes and the boundary layers on the upper faces of the cubes, which produce additional turbulence and likely shift the peak of $\rho v'^2$. On the high-speed side of the outer cube, Dupuis et al. [63] observed high longitudinal and vertical normal stresses, which are explained by the impingement of sweeps against this face (cross-flow toward the cube array) generated by the shear layer. For rough-to-rougher transitions, Akutina et al. [64] measured that the longitudinal-plane-averaged turbulence levels of the transverse velocity component are nearly as high ($v'/u* = 1.14 - 1.36$) as those of the streamwise component ($u'/u* = 1.52 - 1.70$). Additionally, the transverse turbulence intensity levels, normalized by the cross-section average velocity, increase with decreasing $h/h_r$, suggesting that turbulent diffusivity in the transverse direction increases in shallower flows.

In a BRT between a smooth bed and a rough bed composed of a submerged cube array, Dupuis et al. [63] observed a peak in the primary transverse Reynolds shear stress $-\rho u'v'$ occurring above the cubes of the rough bed. Akutina et al. [64] noted that $-\rho u'v'$ is not negligible in flow cases with high $h/h_r$ values. However, its magnitude decreases as $h/h_r$ decreases [64].

In rough-to-rougher transitions, Akutina et al. [64] observed that the $-\rho\overline{u'w'}$ flux is the strongest component of the turbulent momentum flux. The vertical profile of $-\rho\overline{u'w'}$, which exhibits a convex curvature, indicates the presence of a SC. In contrast, $-\rho v'\overline{w'}$ is almost negligible across all $h/h_r$ scenarios, indicating a minimal lateral transfer of vertical momentum by lateral velocity fluctuations [64].

## 4.1.3 Flow structures

The strong Reynolds stress gradients on either side of the rough-smooth interface give rise to SC2 [143–146]. In the experiments conducted by Akutina et al. [64], the SC2 cells (along with Reynolds stresses) are increasingly influenced by the roughness scale as the flows become shallower. In their shallowest case, SC2 cells were present only at the cube locations, unlike in the other flow cases where SC2 cells were independent of

<figure><img src="image_18.png" /></figure>
the roughness elements. This is attributed to the local roughness sublayer, which extends through the entire flow depth in the shallowest case. Vermaas et al. [85] observed SC2 cells primarily over the smooth part of their composite bed, extending across the entire water depth. For a lateral transition from rough to rougher, Akutina et al. [64] observed a similar large-scale secondary current at the interface between the two roughness types.

For flow over a BRT, the spanwise shear intensity increases downstream, allowing large-scale quasi-2D CSs to emerge at the roughness interface [63, 85]. Dupuis et al. [63] observed that 2DCSs were coherent across the water depth. In their study, the upper part of the 2DCSs exhibited a phase advance relative to the lower part. Consequently it was shown that 2DCSs were moving downstream with a vertical inclination in the longitudinal direction of approximately 20°-25°. The 2DCSs, as identified in Dupuis et al. [63], were found to contribute significantly to turbulence, accounting for approximately 35% of $\rho u'^2$ and 60% of $\rho v'^2$. Notably, Dupuis et al. [63] observed that 2DCSs continued to grow throughout the entire channel in their experimental setup. A significant difference of the findings of Akutina et al. [64], compared to other studies, is the absence of 2DCSs under very shallow conditions, despite high dimensionless shear with $\lambda > 0.3$. The presence of 2DCSs is often associated with SC2 cells, as both were observed to coexist [85, 149].

Additionally, Dupuis et al. [127] observed bed-induced turbulent structures near the roughness interface, where the flow is more turbulent and characterized by the presence of sweeps and ejections.

### 4.1.4 Mass and momentum transfers

Within the ML, the combined effects of lateral mass transfer and momentum exchange cause part of the slow stream over the rough bed to decelerate, while part of the fast stream over the smooth bed accelerates. Vermaas et al. [85] introduced a formulation to describe the mass exchange between parallel lanes with different roughness:

$$ \frac{\partial M_t}{\partial x} = \frac{4\rho h V_{100}}{B} \int_{x<<0}^{L} V_{100} dx \qquad (23) $$

where $V_{100}$ is the depth-averaged transverse velocity at the roughness interface, $x << 0$ defines a distance upstream of the BRT where flow is homogeneous, and $B$ is the width of the rough bed. The net mass exchange is observed to occur from the decelerating flow over the rough bed to the accelerating flow over the smooth bed [85]. As with the transverse velocity, Vermaas et al. [85] show that the mass transfers by the time-averaged flow is independent of the water depth once scaled by the local velocity difference.

Vermaas et al. [85] identified three main contributors to the lateral exchange of streamwise momentum: the transverse depth-averaged flow (lateral mass transfer), SC2 cells, and 2DCSs. This observation was later confirmed by Dupuis et al. [63].

Vermaas et al. [85] demonstrated that momentum transfer by 2DCSs was non-uniform with depth, with 2DCS-induced momentum transfer primarily occurring near the bed at the interface between different roughnesses. However, Proust et al. [7] later found that 2DCSs in smooth-bed OCF with a splitter plate decrease in intensity towards the bed, which physically seems to be more appealing. The maximum value of momentum transfers by 2DCSs is found to be located at the roughness interface. Over the depth, the region of high $u'v'$-values grows from bottom to surface and shifts to the rough bed, creating a tilting effect in the vertical distribution of the peak Reynolds stresses [85]. This tilting effect was attributed to potential interactions between 2DCSs and SC2s. For momentum transfers by 2DCSs,
Dupuis et al. [63] observed its maximum in the free flow above the cubes of their rough bed (where most of the lateral transfer of momentum occurs) with flux of momentum directed toward the rough bed (i.e., the low-speed section). A similar observation is made by Akutina et al. [64] who also observed turbulent momentum flux directed towards the rougher bed.

Vermaas et al. [85] observed the SC2 contribution to the lateral exchanges of momentum increasing as water depth increases. Akutina et al. [64] found that secondary currents participate in the momentum exchange from rough section to rougher section, similar to the fluxes observed by Vermaas et al. [85].

Comparing the relative contributions of the three mechanisms to lateral momentum exchange, Vermaas et al. [85] found that in a developed ML, the contributions from SC2 cells and 2DCSs are of the same order of magnitude. The relative contributions of these mechanisms were observed to be influenced by water depth. The same authors note that the contribution to momentum exchange by SC2s exceeds that of 2DCSs as the water depth increases. In contrast, Dupuis et al. [63] did not find similar contributions from 2DCSs and SC2 cells when conducting a momentum balance over the smooth section of their composite bed. In their study, lateral momentum flux occurred almost exclusively through turbulent shear stress, while SCs had a minor effect, and dispersive shear stress was negligible. Over their smooth section, the loss of momentum towards the rough section had similar order of magnitude as loss due to bed friction. The overall ability of the ML to exchange momentum between the two bed sections was shown to be dependent of $h/h_r$ [127]. These authors observe no mixing-layer induced acceleration of flow in the rough section for low $h/h_r$-value, while the opposite is observed for their test cases with greater relative submergence. A similar observation was made by Akutina et al. [64] who note that as $h/h_r$ decreases, the turbulent and secondary circulation transfer of momentum is increasingly inhibited. As the flow becomes shallower, meaning the relative submergence decreases, momentum transfer due to lateral dispersive stresses increases [64].

<figure><img src="image_19.png" /><figcaption>**Fig. 8** Spectrum of scenarios of spanwise 'roughness'-transitions in OCFs: a conventional smooth-to-rough ($h_r \ll h$) transition; b rough-to-rougher transition, both sides with $h_r \ll h$; c smooth/rough-to-submerged canopy transition, with $h_r \ll h$ on the left and $h_r \approx h$ on the right; d the same as in (c) but with 'extreme' canopy solidity ($1-\phi$) equal to 1.0, equivalent to the compound channel cross-section ($\phi$ is porosity); e smooth/rough-to-emergent canopy transition, with $h_r \ll h$ on the left and $h_r \ge h$ on the right; and f the same as in (e) but with 'extreme' canopy solidity equal to 1.0, equivalent to a conventional rectangular cross-section of OCF</figcaption></figure>

<figure><img src="image_20.png" /></figure>
## 4.2 Partially vegetated flows

### 4.2.1 Background comments

The previous section has focused on shallow flows over beds with transverse roughness change, either smooth-to-rough or rough-to-rougher (Fig. 8a, b). The studies of such shallow flows have covered a certain range of relative submergence $h/h_r$ and a variety of shapes and densities of roughness elements. They revealed that the flow may include a transient (development) section followed by an equilibrium part, within which the key mechanisms of transverse momentum exchange are in equilibrium. If a channel is straight and its bed roughness is uniform in the streamwise direction (but changing across the flow) then all turbulence parameters remain the same along the flow within the equilibrium section. These features apply to all BRT flows as well as to partially vegetated flows (Fig. 8c, e) which may be viewed, however, as a special subset of BRT flows.

There are several properties that differentiate partially vegetated flows (PVFs) from the BRT scenarios. First, the relative submergence on PVFs is typically comparable to the flow depth while for other rough-bed flows $h \gg h_r$ (although exceptions exist, e.g., [63]). Depending on $h/h_r$, PVFs can be further subdivided into two dynamically different flow types: (i) flows over submerged vegetation ($h \gtrsim h_r$, Fig. 8c); and (ii) flows with emergent vegetation ($h \le h_r$, Fig. 8e). Second, the presence of vegetation may change the nature of mixing layer development as it involves a vegetation-free flow and a porous media flow with a sharp transition from one to another. Unlike the free flow, a vegetation canopy induces a new type of fluid stress (known as dispersive) and the spatially distributed drag force that in turn are reflected in the emergence of additional terms in the appropriately averaged momentum, energy, and vorticity equations. In PVFs, the vegetation porosity $\phi$ emerges as an additional factor that can control transverse momentum exchanges and their mechanisms [41]. In extreme cases of zero porosities, PVFs become a conventional compound channel (Fig. 8d) or an OCF with the impermeable channel of the rectangular/cylindrical cross-section. Third, PVFs in most real-life situations involve flexible plants whose flexibility induces additional dynamic mechanisms such as, e.g., ‘monami’, or wave-like motions of the flow-vegetation interface(s). In the case of emergent vegetation, the monami may occur at the streamwise-vertical flow-vegetation interface. For the case of submerged vegetation, the flow-vegetation interactions are even more complex, involving simultaneous action of wave-like motion of both streamwise-vertical and streamwise-horizon tal flow-vegetation interfaces. If monami phenomena for fully vegetated beds has been already explored [150], there is no systematic information available yet in relation to the canopy-side interface. The vegetation flexibility is typically characterised by the flexural rigidity which often varies widely among plants and even within the same plant, presenting a significant difficulty in terms of its quantitative assessment, parameterisation, and model- ling. Fourth, vegetation in PVFs is not necessarily presented by long continuous vegetation regions of rectangular shape (as typically studied in laboratory conditions), but often con- sists of multi-scale patches of elongated shapes. Such vegetation patterns induce additional three-dimensionality effects in flow dynamics. Fifth, the streamwise extension of the real- life PVFs may be limited making it impossible to reach flow-vegetation equilibrium and highlighting the importance of the spatial transients in the flow structure over the whole canopy length. The combinations of the above factors may lead to some unique patterns of flow-vegetation interactions within PVFs, which may limit the potential for generalisations.
### 4.2.2 Mean flow: spanwise velocity distribution and secondary currents

Historically first systematic study of the PVF was completed, to the authors knowledge, by White and Nepf [41, 132] who considered the simplest case of PVFs with an emergent rigid canopy presented by the array of wooden circular cylinders (e.g., Fig. 8e). Their analysis employed the double-averaged hydrodynamic equations equally suitable for both vegetation-free and vegetated parts of the flow. Similar PVFs configurations have been explored in a number of follow up studies revealing additional interaction details specific to particular experimental setups, such as the effects of spatial density of cylinders within the canopy, their Reynolds number, patch aspect ratios, patch porosity, and other factors [e.g., [67, 137, 139, 142, 151]]. However, the initial large-scale picture, initially revealed by White and Nepf [41, 132], has been largely confirmed.

Specifically, the flow along the vegetation patch can be subdivided into two regions, the development region starting from the upstream canopy edge and the equilibrium region with spatially homogeneous distribution of flow characteristics along the flow. The spanwise distributions of mean velocities and turbulence parameters reflect a highly asymmetric ML (Fig. 7d, f), with mean velocity distribution in open-water part resembling that for the boundary layer while the streamwise velocity inside the canopy quickly decays following quasi-hyperbolic tangent profile and reaching an approximately constant value not far away from the interface. The inflection point in the spanwise velocity distribution is located within the open-flow region near the flow-vegetation interface. The total width of the ML consists of the width of the open-water part and an inside-canopy part, which is typically much smaller than the external open-water width of SML (Fig. 7d). The total SML width initially increases but then reaches a constant value within the equilibrium range. The above qualitative patterns may vary quantitatively depending on the combination of the PVF-controlling parameters mentioned above. E.g., comparing the PVFs with spatially homogeneous and heterogenous vegetation canopies, Li et al. [68] found that the heterogenous case requires a longer development region, also exhibiting some variations in streamwise distribution of the SML width, induced by fluctuations of the vegetation density in streamwise direction.

Increase in complexity of the emergent PVF configuration by making vegetation canopy flexible (e.g., flexible cylinders, flexible plant models or even real plants) brings the studied scenarios closer to the real-life situations. Interestingly, this enhanced configuration preserves, at least qualitatively, the overall flow features revealed in experiments with emergent rigid cylinders [e.g., [39, 134–136]]. Nevertheless, it has been observed that in the case of flexible vegetation the penetration of open-water flow inside the canopy has significantly increased compared to the case with rigid cylinders while spanwise distribution of turbulence parameters has been also affected. Furthermore, the plant flexibility alone may not be sufficient to describe specific interaction mechanisms as vegetation/plant morphology (e.g., presence or absence of leaves) may significantly influence the plant performance in terms of flow modification (e.g., the foliated conditions could enhance spanwise momentum exchange considerably [e.g., [39]]).

Moving from emergent vegetation to submerged canopies, it is instructive to start with its simplest case made of rigid cylinders. This scenario has been studied by Vilotota et al. [138], Shi et al. [152], Koken and Constantinescu [142], and Yan et al. [153], among others. The common outcome of these studies is that there are two interacting MLs in PVFs with submerged canopies: (i) ML forming at the horizontal flow-vegetation interface (at the canopy top); and (ii) ML forming at the vertical interface.

<figure><img src="image_21.png" /></figure>
Qualitatively, the features of these MLs resemble those reported for the fully vegetated beds and for the emergent canopies in PVFs, respectively. This complex structure highly depends on the controlling parameters such as relative submergence, canopy porosity, blockage factors and others. These parameters are influential for both the development region and the equilibrium region. More realistic cases of PVFs with submerged flexible canopies have not been studied yet, to the authors knowledge, representing the most challenging scenario with rich dynamics and an extensive parametric space.

Another important feature of the mean velocity field, in addition to the ML, is represented by the Prandtl’s secondary currents of the 2nd kind (SC2) in the vicinity of the vertical flow-vegetation interface, clearly observable in the crossflow planes. Depending on the combinations of controlling factors, the strength, size, location(s), and number of secondary flow cells may greatly vary. Nevertheless, the presence of secondary cells has been reported for all studied scenarios of PVFs, with emergent and submerged, rigid and flexible canopies [e.g., [137, 138, 142, 152, 153]].

### 4.2.3 Velocity fluctuations, secondary currents, and momentum transfer

The instantaneous velocity fields in PVFs involve a variety of multiscale turbulent motions that can be qualified as coherent features. The published work [e.g., [39, 42, 135, 137, 138, 142, 152, 153, 155]], complemented with the authors’ own studies [e.g., [40, 147, 151, 154, 156–159]], allow to summarise the wide family of coherent structures present in real-life vegetated flows as shown in Fig. 9. Although the contributions of specific scale-dependent coherent structures to bulk turbulence statistics highly depend on the combination of PVF-controlling factors, it could be safety claimed that the largest quasi-two-dimensional vortices formed at the vertical flow-canopy interface are among the major players in the spanwise momentum transfer (Fig. 7f). The role

Turbulence structure within, around, and above the plant canopy represents an interplay of multiple interacting mechanisms:

1. Boundary layer leaf-scale turbulence

2. Mixing layer leaf-scale turbulence

3. Stem-scale wake turbulence (von Karman vortices)

4. Leaf-scale wave-generated turbulence

5. Mixing layer canopy-scale turbulence

6. Boundary layer depth-scale turbulence

7. 3D and 2D turbulence associated with wakes and flow separation at a patch scale

8. 3D and 2D boundary layer and mixing layer turbulence at patch sides

<figure><img src="image_22.png" /><figcaption>**Fig. 9** Flow-vegetation interactions at multiple scales: a conceptual picture. Adapted from Nikora [154]</figcaption></figure>
of these vortices in the momentum exchanges was first highlighted in [41, 132]. Later it has been shown that they also ‘sponsor’ the generation of three-dimensional eddies and, even more, help to explain the emergence and/or the presence of SC2 cells in the averaged velocity fields. Detailed scenarios of potential mechanisms linking SC2s with large-scale horizontal eddies have been proposed in comprehensive numerical simulations of Koken and Constantinescu [142] and the extensive experimental study of Taborda et al. [137].

# 5 Mixing of heat and matter

The study of mixing processes in the river system is a key area of research, as the mixing of heat and matter (e.g., contaminants, nutrients, sediments) can affect water quality, biodiversity, fish and invertebrate habitats, and endanger certain species (e.g. salmonids are very sensitive to changes in temperature [112]). In the river system, confluences and partly-vegetated channels are two important locations where mixing interfaces (MIs) can form, and thereby, lateral exchanges of heat and matter can occur between two streams. A thermal MI is defined as a zone of strong lateral thermal gradient [96], i.e., a ‘narrow band of clustered isotherms within a cross-section’ [107], where the mixed fluid is defined e.g. as fluid with $0.2 < C < 0.8$ with $C = (T_1 - T)/(T_2 - T_1)$ [110], where T is temperature within the MI and $T_1$ and $T_2$ are the temperatures of the ambient streams outside the MI. In the same way, a MI for a transported substance is defined as a zone of high gradient in substance concentration. For instance, the MI of suspended sediments is the zone where highly turbid and less turbid waters interpenetrate [47]. Note that the ML can be viewed as a form of MI where intense mixing of momentum occurs.

When dealing with MIs of heat and matter, important issues are: (i) under what flow conditions are the two streams able to rapidly mix? (ii) what are the flow structures involved in the presence or absence (Fig. 10a) of a rapid mixing? We report here some answers from the literature on MIs in OCFs, which is mainly focused on thermal mixing and suspended sediment mixing.

<figure><img src="image_23.png" /><figcaption>**Fig. 10** **a** Flow with suspended load at Colorado-Green river confluence, © P. Novaresio. **b** Deposits of suspended sediments over a floodplain for a non-uniform flow in a compound channel flume [160] (looking downstream)</figcaption></figure>

<figure><img src="image_24.png" /></figure>
<table><caption>**Table 4** Mean flow parameters and turbulence statistics for plane 2D free mixing layers (FMLSs) and shallow mixing layers (SMLSs)</caption><thead><tr><th></th><th>FMLS</th><th>SMLS</th></tr></thead><tbody><tr><td>Mean flow</td><td>anti-symmetric & self-similar [8–10, 12, 14, 17, 22, 23, 25, 29]</td><td>anti-symmetric or asymmetric, self-similar or not, depending on: λ<sub>0</sub>, V-magnitude, lateral changes in HR, and x-position [4, 30, 35, 40, 46, 54, 60]</td></tr><tr><td>Mean velocity profile</td><td>for small λ<sub>0</sub> [17, 20, 23]</td><td>for small λ<sub>0</sub> [7, 38, 55]</td></tr><tr><td>Wake coexisting with ML</td><td>= 1 [26]</td><td>= exp(−x*) (Eq. 1) [3, 6, 7]</td></tr><tr><td>(U<sub>2</sub><sup>2</sup> − U<sub>s</sub><sup>2</sup>)/(U<sub>2</sub><sub>0</sub><sup>2</sup> − U<sub>s</sub><sub>0</sub><sup>2</sup>)</td><td>= C(x − x<sub>0</sub>) with C = C(λ<sub>0</sub>, d<sub>0</sub>) [11]</td><td>≠ exp(−x*) for V ~ U [7]</td></tr><tr><td>Width δ</td><td></td><td>≈ C(x − x<sub>0</sub>) [173]; solution of Eq. (8) [50], of Eq. (9) [35]</td></tr><tr><td>Plateau of δ</td><td></td><td>δ<sub>max</sub> = 0.13λ<sub>0</sub>/c<sub>f</sub> [3, 56]. Plateau of δ for PVF [40], and for σ<sub>0</sub> = 90° [171]</td></tr><tr><td>Width δ*</td><td>= 2λ<sub>0</sub>σ<sub>0</sub> with σ<sub>0</sub> = 0.06 [25], 0.08 [12], 0.09 [11], 0.11 [24]<br>= dδ/dx<sub>0</sub></td><td>= c<sub>f</sub>δ/(2hλ<sub>0</sub>), solution of Eqs. (5–7) [3]; = c<sub>f</sub>δ/(2hλ<sub>0</sub>), solution of Eq. (8) [50];<br>= c<sub>f</sub>δ/(2hλ), solution of Eq. (10) [6]<br>= 2λ<sub>0</sub>σ<sub>0</sub> with σ<sub>0</sub> = 0.18 [3], σ<sub>0</sub> = 0.09 [4]<br>= 2λδ with α = α(S) (Eq. 6) [3]</td></tr><tr><td>Initial growth rate dδ/dx<sub>0</sub></td><td>= 2λ<sub>0</sub>σ<sub>0</sub> with σ<sub>0</sub> = 0.06 [25], 0.08 [12], 0.09 [11], 0.11 [24]</td><td>Eq. (5) [3]</td></tr><tr><td>Growth rate dδ/dx</td><td></td><td>Eq. (12) [36]</td></tr><tr><td>Growth rate dδ*/dx*</td><td>linear function of x [8, 14, 19]</td><td>PVF: dθ/dx ≈ 0 for high x [41]</td></tr><tr><td>Centerline position y<sub>c</sub></td><td>dθ/dx = 0.01 [12, 29], θ ∝ λ<sub>0</sub>x [19], dθ/dx = 0.045λ<sub>0</sub> [2]</td><td></td></tr><tr><td>Momentum thickness θ</td><td>0.02–0.04 [29]</td><td></td></tr><tr><td>V/U<sub>s</sub></td><td></td><td>0.04–0.55 for λ<sub>0</sub> = 0.1 – 1 [7]</td></tr><tr><td>V/U<sub>c</sub></td><td></td><td>symmetric (u<sub>f</sub>v<sub>f</sub>/U<sub>s</sub><sup>2</sup>) or asymmetric (v<sub>f</sub><sup>2</sup>/U<sub>s</sub><sup>2</sup>) and not self-similar [30]. PVF: self-similar [40]</td></tr><tr><td>Velocity fluctuations</td><td></td><td>→ 0 if S ≈ 0.1 [3, 7, 31, 34]</td></tr><tr><td>Reynolds stress profiles</td><td>symmetric & self-similar [17, 29]</td><td>≈ 0.4 to 0 (↓ with S) [7]</td></tr><tr><td>Transverse motion</td><td></td><td>≈ 0.6–0.7 to 0 (↓ with S) [30]</td></tr><tr><td>−u<sub>f</sub>v<sub>f</sub>/u<sub>s</sub><sup>2</sup> max</td><td>≈ 0.4 [15]</td><td>≈ 4 for λ<sub>0</sub> = 0 (uniform flow) and ≈ 1.5 – 6 for λ<sub>0</sub> ≠ 0 [30]</td></tr><tr><td>−u<sub>f</sub>v<sub>f</sub>/(√u<sub>s</sub><sup>2</sup>√v<sub>s</sub><sup>2</sup>) max</td><td>0.5–0.6 [8, 15]</td><td>0.97 [26], 0.04 to 0.85 (↑ with h) [30]</td></tr><tr><td>u<sub>f</sub><sup>2</sup>/v<sub>s</sub><sup>2</sup> max</td><td>= 1.75 [15]</td><td>PVF: 0.04–0.05 [39, 40]; BRT: 0.03 [40]</td></tr><tr><td>v<sub>s</sub><sup>2</sup>/w<sub>s</sub><sup>2</sup> max</td><td>0.91 [15]</td><td></td></tr><tr><td>u<sub>s</sub><sup>2</sup>/max/U<sub>s</sub><sup>2</sup></td><td>0.01 [18], 0.03 [9, 10, 19, 29]</td><td></td></tr></tbody></table>

<figure><img src="image_25.png" /></figure>
<table><caption>**Table 4 (continued)**</caption><thead><tr><th></th><th>FMLS</th><th>SMLS</th></tr></thead><tbody><tr><td>$v^2_{\max}/U_s^2$</td><td>≈ 0.015–0.025 [9, 10, 19, 29, 181]</td><td>PVF: 0.016–0.026 [39], 0.017 [40], BRT: 0.012 [40]</td></tr><tr><td>$-u^2_{\max}/U_s^2$</td><td>≈ 0.01–0.02 [9, 10, 17, 19, 22, 23, 29, 181], ≈ 0.03–0.1 (wake) [23], 0.141d6/dx/λ [15]</td><td>0.00–0.01 [4], ≈ 0.01–0.02 [35, 47], 0.00–0.02 [30], 0.03–0.10 (wake) [30], PVF: 0.01–0.02 [41, 130, 137], ≈ 0.02 [39], ≈ 0.02 (new data, Fig. 7e), BRT: ≈ 0.01 (new data, Fig. 7e)</td></tr><tr><td>$-u^2_{\max}/U_c^2$</td><td></td><td>0–0.025 [30], 0–0.002 [6], 0–0.03 [38], PVF: 0.005–0.06 (t with α (density of vegetation) and Fr) [131] ≈ 0.07–0.09 for λ₀ = 0; ≈ 0.07–0.27 for λ₀ = 0.1–1 [7]</td></tr><tr><td>$\sqrt{u^2_{\max}/U_c^2}$</td><td></td><td></td></tr><tr><td>PVF: partially vegetated flow; BRT: bed-roughness transition</td><td></td><td></td></tr></tbody></table>

<figure><img src="image_26.png" /></figure>
<table><caption>**Table 5** Flow structures in plane 2D free mixing layers (FMLS) and shallow mixing layers (SMLS)</caption><thead><tr><th></th><th>FMLS</th><th>SMLS</th></tr></thead><tbody><tr><td>2D structures</td><td></td><td></td></tr><tr><td>Driving parameters</td><td>λ [11]</td><td>S [7, 31, 48], λ [7, 38, 54, 61], Re = U<sub>s</sub>δ/ν [48], δ/W [48], ρ<sub>1</sub>/ρ<sub>2</sub> [11]</td></tr><tr><td>Visualization/detection for</td><td>0.45 ≤ λ<sub>0</sub> ≤ 1 [11], λ<sub>0</sub> = 0.48 [12], λ<sub>0</sub> = 0.66 [13], λ<sub>0</sub> = 0.45 – 0.47 [18], λ<sub>0</sub> = 0.25 – 0.43 [19], λ<sub>0</sub> = 0.42 [62], λ<sub>0</sub> = 0.53 – 0.93 [42]</td><td>λ<sub>0</sub> ≈ 0.4 [13, 50], λ<sub>0</sub> = 0.47 – 1.13 [35], 0.3 ≲ λ<sub>0</sub> ≤ 1 [7], λ<sub>0</sub> = 0.37, 0.50 [61], λ<sub>0</sub> = 0.4 – 0.8 [39], λ<sub>0</sub> = 0.48 – 0.61 [63], λ<sub>0</sub> = 0.42 – 0.54 [44], λ ≥ 0.3 [38]</td></tr><tr><td>Development length</td><td>x ~ 2 cm [11], ~ 20 cm [12]</td><td>x ~ 10h–30h [6, 46], x ~ 65 cm/h [7]</td></tr><tr><td>Absence for</td><td>0.09 ≤ λ<sub>0</sub> ≤ 0.21 [17]</td><td>λ<sub>0</sub> ≲ 0.3 [7, 38, 44, 54, 61], λ<sub>0</sub> ≈ 0.25 – 0.26 [39], 0.15 ≤ λ<sub>0</sub> ≤ 0.41 [64], as long as λ ≤ 0.3 [38]</td></tr><tr><td>Streamwise stretching for</td><td></td><td>0.01 ≲ S ≲ 0.1 [7, 44], x/h ≳ 30 [6], ρ<sub>1</sub> ≠ ρ<sub>2</sub> [118]</td></tr><tr><td>Destruction for</td><td></td><td>S ≳ 0.1 [3, 4, 7, 31, 34, 44, 48], x/h ≳ 300 [6], x/h ≳ 100 [46]</td></tr><tr><td>Strouhal number S<sub>t</sub> = fθ/U<sub>c</sub></td><td>0.032 [20, 42], 0.044 [12], 0.044–0.048 [2]</td><td>0.032 [39, 41], 0.044 [40]</td></tr><tr><td>Growth rate dδ<sup>CS</sup><sub>y</sub>/dx</td><td>= 0.38λ [11]</td><td></td></tr><tr><td>Length-to-width ratio δ<sup>CS</sup><sub>x</sub>/δ<sup>CS</sup><sub>y</sub></td><td>≈ 3 [181]</td><td>≈ 1.25 – 3 [7]</td></tr><tr><td>Length δ<sup>CS</sup><sub>x</sub></td><td></td><td>≈ 3δ [46]</td></tr><tr><td>Width 2δ<sup>CS</sup><sub>y</sub></td><td></td><td>≈ 10–15h [46], 10–14h [7], ≈ 3–5 δ [38, 46]</td></tr><tr><td>Streamwise-oriented vortices (SOV)</td><td>2 counter-rotating cells [21]</td><td>2 counter-rotating cells for α<sub>0</sub> = 60° [55, 95, 99–101], 90° [102], 40° &amp; 70° [89], 120° [97, 98]</td></tr><tr><td>presence of SOV cells</td><td></td><td>for α<sub>0</sub> = 0° [101], for a large river confluence (W/h > 50) with α<sub>0</sub> = 58° [110]</td></tr><tr><td>absence of SOV cells</td><td></td><td>~ h [89]</td></tr><tr><td>SOV cell size</td><td></td><td></td></tr><tr><td>Secondary current Prandtl's first kind (SC1) (caused by flow curvature and bed-friction)</td><td></td><td>for α<sub>0</sub> = 60° [93, 95, 110]</td></tr><tr><td>presence of SC1 cell</td><td></td><td>↑ with increasing α<sub>0</sub> or decreasing Q<sub>1</sub> – Q<sub>2</sub> [87], variable with Q<sub>2</sub> [94], M<sub>r</sub> [95]</td></tr><tr><td>SC1 cell size</td><td></td><td></td></tr></tbody></table>
<table><caption>**Table 5 (continued)**</caption><thead><tr><th></th><th>FMLS</th><th>SMLS</th></tr></thead><tbody><tr><td>Number of cells</td><td></td><td>symmetrical junction: 2 [87]; asymmetrical junction: 1 or 2 depending on M<sub>r</sub> [89–92], Q or Q<sub>r</sub> [93]</td></tr><tr><td>Interplay with other structures</td><td></td><td>SCI position affected by GCs [110]</td></tr><tr><td>Secondary current Prandil's second kind (SC2) (caused by turbulence inhomogeneity and anisotropy)</td><td></td><td></td></tr><tr><td>Driving parameters</td><td></td><td>∂<sup>2</sup>/∂y∂z(v<sup>2</sup> - w<sup>2</sup>) [143]</td></tr><tr><td>Visualisation / detection</td><td></td><td>[38, 40, 63, 64, 85, 128]</td></tr><tr><td>Density effects inducing Gravity Current (GCs)</td><td></td><td></td></tr><tr><td>Driving parameters</td><td>ρ<sub>l</sub>/ρ<sub>2</sub> [11]</td><td>Fr<sub>d</sub> = U<sub>c</sub>/(g'h)<sup>0.5</sup> or Ri = 1/Fr<sub>d</sub><sup>2</sup> [100], α<sub>0</sub> [100], Q<sub>r</sub> [112], M<sub>r</sub> [167], γ [120]</td></tr><tr><td>Presence of GCs</td><td></td><td>for α<sub>0</sub> = 60° and Fr<sub>d</sub> = 1.6, 4.9 [100]; α<sub>0</sub> = 120°, M<sub>r</sub> = 16.9, Δρ = 0.5–2 kg/m<sup>3</sup> [97], Ri = 0.03 and λ<sub>0</sub> = 0.5 [118]</td></tr><tr><td>Large- and very-large-scale motions (LSMs & VLSMs) induced by bed and sidewalls</td><td></td><td>h: λ<sub>x</sub> ~ 2 h–4 h for λ<sub>0</sub> = 0 [45, 71, 75, 79, 80], λ<sub>x</sub> ~ 2 h for λ<sub>0</sub> ~ 0.2–0.7 [38]<br>h: λ<sub>x</sub> ~ 20 h–50 h for λ<sub>0</sub> = 0 [45, 71–75, 80], λ<sub>x</sub> ~ 20 h for λ<sub>0</sub> ~ 0.2–0.7 [38]; W/h, h/h<sub>r</sub>, and Re<sub>r</sub> = u<sup>*</sup>h/ν for λ<sub>0</sub> = 0 [71, 80]</td></tr><tr><td>LSMs driving parameters</td><td></td><td>40 h–50 h for LSM and 130 h–150 h for VLSM with λ<sub>0</sub> = 0 [45, 71, 72, 72–74, 80], 30 h for LSM and VLSM with λ<sub>0</sub> ~ 0.2–0.7 [38]</td></tr><tr><td>VLSMs driving parameters</td><td></td><td>2DCSs prevent VLSMs from emerging for λ<sub>0</sub> ~ 0.3 [7, 38, 45] except at small x [7, 38]</td></tr><tr><td>Development length x</td><td></td><td>SC2s prevent VLSMs from emerging for λ<sub>0</sub> = 0 [72], λ<sub>0</sub> = 0.2–0.7 [38]</td></tr><tr><td>Interplay 2DCS/VLSM</td><td></td><td>for small λ<sub>0</sub> (0.13, 0.18) [7]</td></tr><tr><td>Interplay SC2/VLSM</td><td></td><td></td></tr><tr><td>LSM/VLSM development</td><td></td><td></td></tr></tbody></table>

<figure><img src="image_27.png" /></figure>
## 5.1 Flow structures involved in mixing

In FMLs, molecular mixing was investigated based on chemical reaction between an acid and a base [13, 18, 62]. The pictures of Breidenthal [18] show that KHCSs dominate the mixing and the distribution of mixed fluid. Crowe et al. [161] also showed the role played by KHCSs in the dispersion of heavy particles, which are 'entrained in the rotating large-scale structures and may subsequently be centrifuged to the outer edges of the structures, giving rise to a lateral dispersion exceeding that of a fluid'. Their experimental and numerical results also indicated that particles accumulate at the periphery of the KHCSs.

Large-scale structures play also an important role in mixing and dispersion in shallow water flows [162, 163]. For instance, 2DCSs are responsible for the lateral transfer of fine sediments between river main channel and floodplain, and thereby, the sedimentation and formation of a fluvial levee over the floodplain [160, 164, 165], as illustrated in Fig. 10b (levee being formed in the downstream part of the flume, where flow reached streamwise uniformity and lateral advection is negligible [160]). For SMLs with parallel streams, the simulations of Kirkil [46] have also shown that the transverse mixing of scalar concentration is higher near the surface than near the bed, which is consistent with the increasing size of the 2DCSs from bottom to surface [7], Figure 23a].

Cross-channel mixing can also be ascribed to the transverse advective transport by SC1s, which form e.g. downstream of angled river confluences [88, 107]. In the simulations of Jiang et al. [110], mixing is enhanced by a SC1 cell that extends inside of the MI. Kwon et al. [166] have recently shown that SC1 cells highly contribute to the mixing of suspended sediments. The SC1 cells can also play an opposite role, as observed by Yuan et al. [93] at a large river confluence, where the two SC1 cells 'restricted the core size of high sediment concentration and downwelling flows acted as a barrier hindering the exchange of sediment between the two rivers'. The same observation was made by Xu et al. [92].

SOV cells can also participate to the mixing at angled confluences [89, 97–99, 101]. In the simulations of Cheng and Constantinescu [101] of a small ideal confluence with $α_0 = 60°$, the SOV cells play a major role in enhancing mixing between the two streams (besides inducing the largest bed shear stresses beneath them, and thereby, sediment entrainment). However, according to Jiang et al. [110], the increase in the scale of the confluence (e.g., aspect ratio of the main channel) impedes the formation of the SOV cells. The latter are also impacted by the presence of GCs [98, 100], the interplay between GCs and SOV cells being dependent on the magnitude and direction of density difference $ρ_2 – ρ_1$ [98].

GCs are also important flow structures involved in the mixing of upcoming streams at river confluences [96, 100, 112, 113, 121, 167], and more specifically in the CHZ, where the waters of one tributary slides beneath that of the other tributary. This can lead to a constant temperature across the channel in the near-surface layer as observed by Cook et al. [112] at an asymmetrical confluence with $α_0 = 80°$ and $ΔT = 9.6°C$ although waters are not fully mixed, showing strong vertical stratification. The near-surface transverse mixing is actually dependent on the momentum ratio $M_r$ as observed by Lane et al. [167] at a large river confluence (postjunction width greater than 1 km), where a channel-scale flow circulation enhancing mixing occurred for high $M_r$ only. Downstream of the CHZ, the flow remains stratified vertically as observed e.g. by Pou-choulin et al. [121] along the Rhône river. The vertical stratified flow can generate
additional turbulence caused by the horizontal shearing between the two liquid masses, which contributes to mixing as observed by Laraque et al. [113] along the Amazon river. Last, the simulations of a small river confluence by Horna-Munoz et al. [100] or the measurements at a medium-sized confluence by Pouchoulin et al. [121] have shown that GCs increased mixing in the CHZ compared with corresponding cases without GCs, and ‘regardless of whether the high-momentum stream contains the higher-density fluid or the lower-density fluid’ [100]. However, ‘when density effects are weak, GCs can reinforce or weaken the mixing associated with SC1 cells depending on which tributary contains the denser fluid’ [100]. Xu et al. [92] also observed at a large confluence that GCs can work together or against the SC1 cells, enhancing or reducing mixing.

## 5.2 Mixing efficiency

For a given geometrical configuration, the mixing efficiency can highly vary at different times depending on the flow parameters of the incoming streams, and on the resulting flow structures in the CHZ and farther downstream. It was shown by Lane et al. [167] for the large river confluence of Río Paraná and Río Paraguay, as depending on specific flow configurations the two rivers can mix within the range from 8 km to 400 km downstream of the junction. The momentum ratio $M_r$ and junction angle $\alpha_0$ are two key parameters as they partially affect the position of the ML and MI (Eq. 19), the existence and size of the SC1 cells, SOV cells and GCs [97, 110, 168].

The initial velocity shear $\lambda_0$ is also a key parameter as it drives the emergence of 2DCSs (§2.1.7 and Table 5). However, if 2DCSs within a vertical MI can play a major role in the lateral transfer of momentum, matter and heat, for small streams or flows in open-channel flumes (as the transverse width of the 2DCSs can be of the same order of magnitude as the channel width [7, 38]), 2DCSs may not be effective to rapidly mix two streams for intermediate-scale or large rivers, as observed by Pouchoulin et al. [121]. In addition, as 2DCSs are driven by the velocity shear $U_s$ that rapidly decreases downstream the junction of two streams (see §2.1.2 [6, 7]), these structures cannot participate to transverse mixing beyond a longitudinal distance exceeding several $h/c_f$, i.e. several hundreds of h assuming that $c_f \sim 0.01$ [5].

As previously seen, there is no doubt that the GCs can enhance the mixing in the CHZ [110, 112, 118]. However, if a vertical stratification is very stable then it will slow down the mixing farther downstream (note that for FMLs, vertical stratification was found to reduce rates of mixing [11]). This can be dramatic, e.g. along the Amazon river where complete mixing requires more than 100 km from the confluence between the rivers Negro and Solimões, due to higher densities of the waters of the Solimões River [113]. Last, it should be mentioned that Ramón et al. [119, 169] found that GCs seem to primarily influence the pattern rather than the rate of mixing.

The mixing processes are also sensitive to flow dimensions [96] and combined effects of flow magnitude and momentum flux ratio [95], i.e., mixing may be enhanced at low flows with high $M_r$-values. The aspect ratio of the postjunction channel $W/h$ can also be an important factor. High $W/h$ ratios enhance mixing by the bed-induced turbulence, and enhance the effect of topographical forcings on the mixing patterns [103, 167]). For instance, Gualtieri et al. [103] found that near-bed GCs can interact with the bathymetry, resulting in strong upwelling and downwelling flows. All that said, one can easily understand that the interplay between the various mixing processes makes difficult to predict the

<figure><img src="image_28.png" /></figure>
mixing length for a wide range of variable hydrological conditions. In any case, the estimate of mixing should not be made based on the free surface pattern of mixing [110, 121].

# 6 Commonalities and differences among ML types

The consideration of different types of SMLs in the preceding sections demonstrated that depending on the parameters of flow, channel geometry, and bed surface roughness the manifestation of SMLs can be different and not easily predictable. In this section we will briefly discuss common features across SML types and their variations.

The SMLs exhibit spanwise mean velocity profiles that are anti-symmetrical over a uniform bed-roughness (Fig. 2a, [4, 30]) or asymmetrical with a bed-roughness transition (BRT) or in a partly vegetated flow (PVF), with the asymmetry being more noticeable for PVF than for BRT (Fig. 7c–d, [38, 40, 41, 139]). This asymmetry does not prevent the presence of self-similarity of the velocity profiles when using two different length-scales and velocity scales from either side of the interface between adjacent bed surfaces [32, 40]. However, self-similarity requires that turbulence is fully developed and 2DCSs are present [7, 30]. With these conditions, self-similarity can be observed over very large distances ($x/h \sim 270$ in Fig. 2a). Like FMLs [23], SML velocity profiles feature a velocity deficit on the low speed side for small $\lambda_0$ caused by the wake induced by the lateral boundary layers on the junction apex or splitter plate.

For parallel incoming streams, the initial growth rate of the SML is proportional to the shear parameter $\lambda$ like for plane 2D free mixing layer (FML), i.e., essentially controlled by the mean flow. Further downstream, the decrease in the growth rate is not easily predicted, as potentially controlled by both large-scale coherent structures and lateral transfers of mass and momentum by the mean flow. For both parallel incoming streams and angled flow configurations, the spanwise momentum fluxes by the mean flow can greatly exceed turbulent fluxes [7, 35, 47, 108]. As a result, SML width $\delta$ could be essentially controlled by the mean flow from the SML origin to its decay.

The 2DCSs can be observed in all types of MLs (Table 5). A high initial shear parameter $\lambda_0$ is required so that 2DCSs can develop, i.e., $\lambda_0 \ge 0.25$ for FMLs [11–13, 18, 19, 42, 62] and $\lambda_0 \gtrsim 0.3$ for SMLs [3, 6, 7, 35, 38, 39, 44, 50, 54, 63] irrespective of flow sub-mergence, aspect ratio $W/h$, and bed-friction coefficient $c_f$. The normalized most-amplified frequency $S_i$ (Strouhal number) for SMLs is in the range 0.032–0.045 [39–42], comparable to that for FMLs (0.032–0.048, [2, 12, 20, 42]), with no noted influence of $\lambda_0$, $h$, $c_f$, and of the presence of an initial wake (caused by a splitter plate or a confluence junction). As observed by Chandrsuda et al. [16] in FMLs, the 2DCSs cannot persist within SMLs if the ambient-stream turbulence —namely the bed-induced turbulence— is too strong compared to the shear-layer-induced turbulence. This can be quantified by the bed-friction number $S$: 2DCSs cannot persist in the flow if $S \gtrsim 0.1$ [3, 4, 7, 31, 34, 44, 48]. It was also found that the stopping of vortex pairing and simultaneous longitudinal stretching of the 2DCSs occurs when $S \gtrsim 0.01$ [7, 44]. The stretching of the 2DCSs can also occur in the presence of GCs [118]. Last, 2DCSs prevent VLSMs from emerging for $\lambda_0 \gtrsim 0.3$ [7, 38, 45] except at small $x$ [7, 38].

The streamwise-oriented-vortex (SOV) cells were observed in SMLs essentially for angled flow configurations [55, 89, 101, 102] (while they were observed in FMLs for parallel incoming streams by Bernal and Roshko [21]). They are located from either side of the mixing interface (MI), and could be associated with the presence of the SC1 cells that are
induced by flow curvature and bed-friction [89]. The size of SC1 cells increases with the junction angle $\alpha_0$ [87], and the number of cells (one or two) is dependent on the momentum flux ratio $M_r$ [95].

The various turbulent coherent structures induce velocity fluctuations, whose transverse distributions are not self-similar over large distances (Fig. 2b-c) unlike spanwise distribution of mean velocities (Fig. 2a). For SMLs, self-similarity of the velocity fluctuation profiles is perturbed by bed-induced turbulence, as that for FMLs was perturbed by free-stream turbulence [16, 23]. The magnitude of the peak of the normalized Reynolds stress $-\overline{u'v'}_{\max}/U_s^2$ does not differ between FMLs and SMLs and between SML types at small $\lambda_0$, either in the absence of initial wake ($-\overline{u'v'}_{\max}/U_s^2 \approx 0.01-0.02$, see Table 4) or in the presence of wake ($\approx 0.03-0.1$). For medium and high values of $\lambda$, like for FMLs [17], the absence of two-dimensionality in the large-scales does not influence the Reynolds stresses significantly. The effect of 2D structures is hardly visible on the values of $-\overline{u'v'}_{\max}/U_s^2$. In contrast, the emergence of 2DCS leads to an increase in $-\overline{u'v'}_{\max}/U_c^2$ (Fig. 5d). Correlation coefficient $R_{u'u'}$ and the Townsend index $-\overline{u'v'}/\overline{u'u'}$ are key parameters to detect the zone of FML-behavior in SML, with respective values of 0.5-0.6 and 0.38, but also to determine the state of the shear layer turbulence as these parameters decrease during the stretching and decay of the 2D structures [7, 44].

The common features and differences across SML types, mentioned above, are not exhaustive. The SMLs at some combination of parameters are not yet systematically studied, and at the moment they are represented by one or two papers (sometimes reporting contrasting findings). Nevertheless, we consider the available information as a sufficient base for the forthcoming generalization and theoretical conceptualizations.

# 7 Perspectives

Shallow mixing layers are ubiquitous in nature and in man-made structures, and therefore the advanced knowledge on SML dynamics and scaling features are required for their quantitative assessment, reliable prediction, and effective control. In this paper we subdivided the known forms of SMLs into three major groups: (1) hydrodynamics-driven SMLs, (2) SMLs affected by density stratification, and (3) SMLs induced by spanwise heterogeneity in flow resistance. Each of these groups have been further subdivided depending on specific combinations of boundary conditions and external parameters. Although 'topography-driven' SMLs like those in compound channels [e.g., 32, 38, 40, 49, 54, 61, 66, 147, 148, 164, 165]] have not been reviewed in this paper, they can be fairly considered as an intermediate form between first and third groups of SMLs. In this respect we would like to note that our subdivision of SMLs into distinct groups should be viewed as preliminary. It is likely that the whole spectrum of possible SMLs is continuous and may exhibit much greater variety of SMLs than have been reviewed in this paper. Nevertheless, at this stage of SML knowledge it is useful to introduce at least a preliminary SML classification to underpin follow-up studies.

Most of cases considered in this review relate to the idealised situations of homogeneous boundary conditions in the streamwise direction. The exploration of such SMLs needs more efforts, particularly in relation to the SML development to equilibrium, emergence of 2DCSs and associated critical value(s) of the shear parameter $\lambda$ (e.g., is $\lambda_0 = 0.3$ universal across SML scenarios?), density effects on SMLs, and other issues highlighted in
this review. Although studying idealised SMLs scenarios is an important step towards better understanding the real-life manifestations of SMLs, the expansion in this research field ought to cover more realistic patterns and scenarios. Some of them are listed below.

1. *SMLs as elements of more complex flow patterns.* In many situations, SMLs are not standalone fluid phenomena but an element(s) of more complex flow patterns. Typical examples are shallow jets and wakes that include SMLs as their in-build components. The basic knowledge on these flow patterns has been under development since the 90s of past century [e.g., [5, 163, 170]], although specific details on their associated SMLs in such flow types are not fully explored and likely exhibit special features differentiating them from the idealised hydrodynamics-driven SMLs discussed in this review. Exploring SMLs in such configurations would be necessary for building up a more comprehensive knowledge based on SMLs.

2. *Multi-scale SMLs.* Real-life SMLs are not single-scale phenomenon, often involving several scales at which various-scale SMLs evolve. Such multi-scale SMLs can be observed in rivers where large-scale SMLs (e.g., at confluences) can co-exist and interact with smaller scale SMLs generated, e.g., due to riverbank irregularities. This multi-scale nature of SMLs, if present, can introduce additional complexity in their quantitative description and prediction.

3. *Curved SMLs.* The centrelines of river SMLs are often curved (e.g., at confluences) introducing an additional mechanism of moment transfer which needs to be accounted for in theoretical considerations and in quantitative assessments. An approach based on the employment of the Serret-Frenet frame-axes to account for the local velocity direction, proposed for this problem by Mignot et al. [171], is a potentially fruitful research direction that needs to be explored in full. This relates to both standalone SMLs and also to SMLs associated with curved shallow jets which have not been systematically studied yet in spite of their important practical relevance.

4. *SMLs with streamwise-inhomogeneous boundary conditions.* The studies with homogeneous streamwise boundary conditions need to be supplemented with inhomogeneous streamwise conditions which better resemble real-life flow scenarios. Typical examples include formation of SMLs along finite-size patches of instream vegetation and bed sediment patches of different roughness compared to surrounding bed surface.

We consider the exploration of the above topics as an important research work that should secure a coherent knowledge based on SMLs and thus create a better justified foundation for practical applications.

**Acknowledgements** We are grateful to Professor George Constantinescu, Professor Carlo Gualtieri, and Etienne Dureuil for fruitful discussions on SMLs. The work has been partially supported by the EPSRC/ UK grant 'Secondary currents in turbulent flows over rough walls' (EP/V002414/1), and by the RiverLy research unit of INRAE.

**Author Contributions** SP coordinated the writing of the review. SP, BC, CB and VN equally prepared an initial draft. All authors contributed equally to editing the manuscript.

**Data availability** No datasets were generated or analysed during the current study.

## Declarations

**Conflict of interest** The authors declare no conflict of interest.
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