# Discharge distribution in open-channel T-shape bifurcations: effect of a reduced side branch width

Izihan Ibrahim, Nicolas Riviere, Ines Lebouteiller, Emmanuel Mignot

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## To cite this version:

Izihan Ibrahim, Nicolas Riviere, Ines Lebouteiller, Emmanuel Mignot. Discharge distribution in open-channel T-shape bifurcations: effect of a reduced side branch width. Journal of Hydraulic Engineering, 2022, 148 (9), 10.1061/(ASCE)HY.1943-7900.0002002. hal-05002570

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1 Discharge distribution in open-channel T-shape bifurcations: effect of a reduced side branch width

2 Izihan Ibrahim¹, Nicolas Riviere², Ines Lebouteiller³, Emmanuel Mignot⁴

3 1: PhD student, Department of Civil Engineering, International Islamic University Malaysia, 53100

4 Gombak, Selangor, Malaysia. izihan@iium.edu.my

5 2: Professor., LMFA – INSA Lyon, Université de Lyon, Ecole Centrale de Lyon, Université Claude

6 Bernard Lyon 1, CNRS, 20 avenue A. Einstein, 69621 Villeurbanne, France. Nicolas.Riviere@insa-lyon.fr

8 3: MSc student, LMFA – INSA Lyon, Université de Lyon, Ecole Centrale de Lyon, Université Claude

9 Bernard Lyon 1, CNRS, 20 avenue A. Einstein, 69621 Villeurbanne, France. ines.lebouteiller@insa-lyon.fr

11 4: Associate Professor., LMFA – INSA Lyon, Université de Lyon, Ecole Centrale de Lyon, Université

12 Claude Bernard Lyon 1, CNRS, 20 avenue A. Einstein, 69621 Villeurbanne, France.

13 Emmanuel.mignot@insa-lyon.fr

14 Abstract

15 The primary challenge for river managers in handling open-channel bifurcations is the prediction of
16 the discharge distribution to downstream branches. In this study, we aimed to summarize and
17 complete the available database and empirical correlations for T-shape bifurcations at 90°, both in
18 subcritical and transcritical flow regimes, in equal and reduced side-branch width geometries. An
19 experimental campaign generated a database of 668 new configurations in addition to 299 sets
20 collected from the literature. The existing correlation for predicting the discharge distribution in the
21 subcritical regime appears to be of limited agreement with these data, whereas a new empirical
22 relationship significantly increases the accuracy and is useful for a side-branch as narrow as a third of
23 the main branch width. Regarding the free-recirculation transcritical regime, the present data
24 validate the correlation proposed in 1967 for downstream branch ratios not tested by the authors.
25 An approach was proposed to predict the flow discharge distribution for configurations without a
26 *priori* knowledge of the flow regime.
## 27 Introduction

In this study, we investigated open-channel bifurcations, defined as geometrical singularities of a hydrographic network, where the inflow separates into two downstream branches. These singularities are encountered in several locations in the riverine environment, such as upstream from dams with a side flow bypassing the impoundment. Furthermore, they ensure a safe path for fish or intake structures located on the side of some rivers to extract raw water for human consumption.

Sediment transport in such bifurcations is a major issue in preventing extensive sediment discharge from the river from reaching the intake structure (Al-Zubaidy and Ismaeil 2022). Such bifurcations are also observed in sewer networks, where the combined sewer overflows evacuate, part of the exceeding sewer discharge capacity towards the surrounding environment. Finally, flooded streets reaching three-branch crossroads generate bifurcated channel flows. In all cases, the main flow continues straight ahead, while the side branch, generally narrower, makes an angle close to 90° with the main channel, forming a bifurcation known as “T-shape bifurcation”. Moreover, bifurcations are naturally present in river networks, mostly upstream from islands and in deltas, where the main flow does not necessarily continue in a straight direction but may form a Y-shaped bifurcation that is not symmetrical *a priori*, at any angle, and with any branch width ratio.

This study aims to summarize and improve the empirical discharge distribution correlations available in the literature. The remainder of this paper is organized as follows. The present introduction summarizes the available information regarding both the flow processes and empirical discharge distribution laws at bifurcations. The next section describes the experimental setup used in this study to complete the available dataset from the literature and test the existing discharge distribution correlations for both the subcritical and transcritical flow regimes in T-shaped bifurcations. The results are presented in the third Section, which summarizes the possible dimensional analyses of the discharge distribution coefficient. Part 1 of this section focuses on the subcritical regime by testing the applicability of Hager's (1991) correlation to bifurcations with various branch width ratios, and deriving a new empirical correlation based on field-adapted dimensionless parameters. Part 2
focuses on the free recirculation transcritical regime to test Lakshmana Rao and Sridharan (1967) discharge distribution correlation with present branch width ratios. The last part of Section 3 presents an integrated method for predicting the discharge distribution for any flow configuration. The applicability of this method, among other considerations, is discussed in Section 4.

## Flow description

Experimental works dedicated to better understand the flow patterns within open-channel bifurcations have mostly considered T-shaped junctions of simplified rectangular, smooth, and horizontal channels (Ibrahim et al. 2020). Neary and Odgaard (1993) and Neary et al. (1999) described the 3D flow structure within the bifurcation in the subcritical regime, including the recirculation zone (Kasthuri and Pundarikanthan 1987, Momplot et al. 2017) facing the accelerated zone in the side branch separated by a curved shear layer described by Mignot et al. (2014). In addition, Lakshmana Rao and Sridharan (1967) revealed different possible flow patterns in the “transcritical flow” regime, which were later explained by Rivière et al. (2014). When the flow is subcritical in the entire bifurcation, the maximum Froude number in the side branch is located just downstream from the intersection in the *vena contracta*, that is, the section with the maximum recirculation width.

If the water depth in the downstream channel increases or that in the side branch decreases (owing to a modification of the downstream controlling boundary conditions), the relative discharge entering the side branch increases. Consequently, the Froude number in the *vena contracta* (Fig. 1) increases, which eventually reaches 1 (choked flow section). Thereafter, the flow accelerates downstream from the critical section, as the flow section increases with decreasing recirculation zone width. This flow combining subcritical and supercritical flow areas was labeled as “transcritical” by Rivière *et al.* (2014). Depending on the downstream conditions and length of the branch channel, hydraulic jumps may form further downstream, leading to two characteristic configurations of the transcritical regime, as shown in Figure 1 of Rivière *et al.* (2014). First, for a water depth at the downstream end of the branch only slightly lower than for the appearance of the critical section, a
hydraulic jump appears in the diverging part of the side branch flow, namely in the region where the width of the recirculation decreases, that is between the *vena contracta* and the end of the recirculation zone. In such a case, the hydraulic jump does not cover the entire downstream channel width and is restricted to the main flow of the branch, that is, between the sidewall and shear layer bounding the recirculation zone. In this first configuration, named "drowned recirculation" by Rivière *et al.* (2014), the downstream boundary condition of the branch influences both the shape and depth of the recirculation. Second, as the downstream depth in the branch decreases further, the hydraulic jump moves downstream from the end of the recirculation zone, and then occupies the entire side branch width. In this second configuration, named "free recirculation," no downstream control occurs in the side branch because the flow in a certain channel section is in the supercritical regime over the entire width. The side branch discharge reaches its maximum value, and the discharge distribution becomes independent of the downstream boundary condition of this side branch. Conversely, in the drowned recirculation configuration described above, the supercritical flow region does not cover the entire side branch width, and a downstream control by the weir or fixed water depth at the downstream end of the side branch occurs via the recirculation zone. Therefore, the discharge distribution at the bifurcation is affected by the downstream boundary of the side branch.

A definite criterion or threshold for transition from the subcritical to transcritical regime has not yet been provided. Ramamurthy and Satish (1988) did consider, referring to data from Lakshmana Rao *et al.* (1968), that a choked flow occurs in the *vena contracta* when the downstream side-branch Froude number reaches 0.3 to 0.35, causing a transcritical flow. Later, Rivière *et al.* (2014) showed that the transition from drowned to free-recirculation transcritical subregimes occurs for a downstream side-branch Froude number reaching 0.65.

In addition, empirical correlations for the prediction of discharge distribution in the downstream branches have been derived by several authors for flows in T-bifurcations of 90° angle (among other angles), both in the subcritical and transcritical regimes. These are described below.
104 Discharge distribution correlations in subcritical regime

105 In previous investigations of subcritical flow regimes (also noted as “submerged” by several authors),
106 Lakshmana Rao et al. (1968), henceforth noted as LSY68, measured the discharge distribution in a
107 bifurcation with the side channel width half of the main branch, for a range of Froude numbers in the
108 downstream and side branches. The authors observed that the discharge distribution to the side
109 branch increases as the Froude number decreases in the main stream (owing to decreased
110 centrifugal forces applied on the flow in the intersection) or increases in the branch (owing to a lower
111 downstream control in this side branch). Hager (1991, see also 2010) proposed an empirical
112 correlation fitted to these data, based on governing non-dimensional parameters derived from a
113 dimensional analysis. However, this correlation was restricted to a branch-width ratio of 0.5.
114 Therefore, the first objective of the present work is to extend this discharge distribution database to
115 several width ratios lower than one and evaluate the applicability of the Hager (1991) fitting
116 correlation.

117 Rivière *et al.* (2007) conducted experiments for an equal-width T-shaped bifurcation (90° angle) in
118 subcritical flows. The authors proposed an empirical discharge distribution correlation that fits well
119 with their measurements, based on governing nondimensional parameters, but dissimilar to Hager
120 (1991). However, this correlation is restricted to bifurcations of equal widths. Moreover, the selected
121 parameters are constructed using the experimental weir crest heights as downstream boundary
122 conditions, whereas most bifurcations encountered in the riverine environment do not include such
123 downstream controls. This limits the applicability of Rivière *et al.* (2007) empirical correlation. The
124 second objective of the present work is to modify the nondimensional parameters and propose an
125 empirical discharge distribution correlation that is adapted to natural or man-made bifurcations,
126 based on downstream water depths in both divided channels. In addition, we aim to extend the
127 proposed correlation to several branch width ratios.
Note that other bifurcation data in subcritical regimes were provided in horizontal equal-width T-bifurcations, namely by Law (1965) when studying the flow pattern and discharge distribution, by Mignot et al. (2013) when investigating the impact of obstacles in the intersection on the discharge distribution, and by Li et al. (2021) when studying the effect of vertical distortion on bifurcation flow characteristics. This information is listed in Table 1 and included for the evaluation of the newly developed discharge distribution correlations.

### Discharge distribution correlations in transcritical regime

Lakshmana Rao and Sridharan (1967), referred to as LS67 in the sequel, intended to predict the discharge distribution in the “free-flow” condition, designated here as the free-recirculation transcritical regime. The authors studied five bifurcation angles and several ratios of side to main branch widths for each angle. The dataset of the 90° T-shape configuration considered four width ratio values: $R_b = ¼, ½, ¾$, and 1. They indicated that the discharge distribution to the side branch increased when the main channel Froude number decreased, thereby lowering the centrifugal force of the flow within the intersection. Moreover, the authors revealed that increasing the width ratio increased the discharge distribution to the side branch for all Froude number values in the main stream. They proposed a correlation for the discharge distribution as a function of the Froude number in the downstream branch based on two coefficients K and x, evolving as a 3rd order polynomial in relation to the branch width ratio. Rivière et al. (2014) tested the validity of this correlation for an equal-width T-shaped bifurcation and found that the coefficients recommended by LS67 are not optimal. Rivière et al. (2014) proposed to (i) slightly adjust the two coefficients of the correlation and (ii) change the location of the Froude number measurement in the downstream branch to a position one branch width downstream from the intersection instead of the entrance of the downstream branch (as proposed by LS67), where the flow is not fully one-dimensional. In addition, Rivière et al. (2014) reported that this correlation is adapted for both patterns of the transcritical regime: free flow and drowned recirculation. Therefore, the third objective of the
present work is to test the LS67 correlation for side to main branch width ratios differing from those
previously studied by the authors. In addition, we aim to verify whether the slight modification of the
coefficients by Rivière et al. (2014) improves the correlation accuracy.

Krishnappa and Seetharamiah (1963) also conducted an experimental campaign for bifurcations in
the transcritical regime with different branch width ratios and proposed an alternative discharge
distribution correlation. However, when tested against the data of Rivière et al. (2014) (in equal-
width configurations), the agreement was unsatisfactory. Rivière et al. (2014) suggested that this
may be attributed to the inclined side branch used in the author's laboratory configuration. The
correlation was evaluated based on the present experiments (not shown here), which confirmed the
discrepancy with the measured discharge distributions. Similarly, the model proposed by
Ramamurthy and Satish (1988) to predict the discharge distribution in the transcritical regime was
not considered here, which is consistent with the issues raised by Rivière et al. (2014). Later,
Ramamurthy et al. (1990) proposed a momentum-based theoretical model that was tested using
laboratory data for transcritical flow. However, the campaign, and the one proposed by Ramamurthy
et al. (2007), did not include measurements of water depths in the side branch; thus, these works are
not utilized here.

169 **Experimental approach**

170 *Experimental set-up*

The experiments were conducted in the same flume as in Rivière *et al.* (2007) and Mignot *et al.* (2013), with limited modifications. This system consists of two 0.3 m wide rectangular shape horizontal and smooth channels made of glass intersecting at a right angle at the mid-length of the main branch (Fig. 1). The main branch is 4.9 m long and side branch 2.6 m long. The improvement according to the Rivière *et al.* (2007) experimental set-up is the capacity to reduce the width of the side branch *b*<sub>y</sub>, while keeping a constant main branch width *b*<sub>x</sub> = 30 cm, where x and y indicate the main and side branch directions, respectively. This reduction was achieved by adding a wall parallel
178 to the sidewalls in the lateral branch and a second short wall in the continuity of the right wall of the
179 main branch to enclose the reduced area. Five branch width ratios were considered herein with R<sub>b</sub> =
180 b<sub>y</sub>/b<sub>x</sub> = 1, 26/30, 22/30, 18/30, and 10/30. The main tank is located downstream of the main branch.
181 The water is pumped towards the upstream tank with a valve plus flowmeter to adjust and register
182 the upstream discharge Q<sub>xi</sub> (ranging from 0.2 to 10 L/s with a typical uncertainty 0.05 L/s), where i
183 refers to inlet. A honeycomb screen was installed at the main branch inlet to stabilize and smoothen
184 inflow. An adjustable sharp-crested weir was installed at the outlets of both branches to control the
185 water depth, with C<sub>x</sub> and C<sub>y</sub> as the crest heights for the downstream and side channel weirs,
186 respectively. As the flow separates in the bifurcation, water leaving the side branch enters the side
187 reservoir, and is piped back to the main downstream reservoir via a flowmeter to measure Q<sub>yo</sub>
188 (typical uncertainty 0.05 L/s), where o refers to outlet. A point gauge is used to determine the main
189 as well as side channel water depths (h<sub>xo</sub> and h<sub>yo</sub> ranging between 4 and 10 cm, with a typical 0.5 mm
190 uncertainty) at their respective outlets, located 60 cm (i.e. two main channel width, 2b<sub>x</sub>) upstream
191 from the weirs (Fig. 1).

192 Dimensional analysis

193 Under the simplified geometry presented in Fig. 1 and steady conditions, the value of the discharge
194 Q<sub>yo</sub> reaching the side branch (or alternatively Q<sub>xo</sub> reaching the downstream branch) is dependent on
195 12 input dimensional variables as follows:

$$
196 \qquad Q_{yo} = f(g, \rho, \mu, \sigma, k_s, Q_{xi}, h_{xo}, h_{yo}, C_x, C_y, b_x, b_y) \tag{1}
$$

197 where g is the gravity, ρ and μ are the density and dynamic viscosity of water, σ is the air/water
198 surface tension, and k_s is the equivalent roughness elevation. By applying the VaschyBuckingham
199 theorem, the nondimensional discharge distribution correlation as a function of nine dimensionless
200 parameters is formulated as

$$
201 \quad R_q = \frac{Q_{yo}}{Q_{xi}} = f \left( Re_{bx} = \rho \frac{Q_{xi}}{b_x \mu}, We = \rho \frac{Q_{xi}^2}{b_h h_{xo}^2 \sigma}, \frac{k_s}{b_x}, F_{bx} = \frac{Q_{xi}}{b_x^{5/2} g^{1/2}}, \frac{h_{xo}}{b_x}, \frac{h_{yo}}{b_x}, \frac{c_x}{b_x}, \frac{c_y}{b_x}, R_b = \frac{b_y}{b_x} \right) \quad (2)
$$
However, considering the low roughness thickness of the glass material as well as the significantly high velocities and water depths, the flow remains in hydraulically smooth and fully turbulent regimes, with negligible effects of surface tension. Eq. (2) is reduced to

$$R_q = f\left(F_{bx}, \frac{h_{xo}}{b_x}, \frac{h_{yo}}{b_x}, \frac{c_x}{b_x}, \frac{c_y}{b_x}, R_b\right) \quad (3)$$

Considering the stage-discharge relationship at both weirs [$Q_o = f(C_{x,y}; h_o)$], the two respective parameters of Eq. (3), can be excluded. Among the possible combinations of dimensionless parameters, the two versions of this function are as follows:

$$R_q = \begin{cases} f\left(F_{Cx}, \frac{Q_{xi}}{b_x g^{1/2} c_x^{3/2}}, \frac{c_x}{b_x}, \frac{c_y}{c_x}, R_b\right) & (4) \\ f\left(F_{bx}, F_{xo} = \frac{Q_{xo}}{b_x g^{1/2} h_{xo}^{3/2}}, F_{yo} = \frac{Q_{yo}}{b_y g^{1/2} h_{yo}^{3/2}}, R_b\right) & (5) \end{cases}$$

In Eq. (4), following Rivière et al. (2007) for a T-bifurcation or Rivière et al. (2011) for a 4-branch intersection, the dimensionless parameters are based on the weir crest heights. Alternatively, Eq. (5) is based on water depths, which is valid regardless of the type of downstream control (weir, dam, lake, free-fall, etc.) that links the local water depth and discharge.

## Results

Six hundred and sixty-eight (668) configurations were measured in either the subcritical regime (everywhere, including the side branch) by sufficiently raising the side branch weir crest ($C_y$) or transcritical regimes when this weir crest was lowered (Table 1). In addition, the horizontal channel setup and minimum elevation of the downstream weir crest $C_x$ ensured that no flow fell into the supercritical regime in the upstream branch (Rivière and Perkins 2004), which is beyond the scope of the present work.

### Subcritical regime

In the subcritical regime, the two discharge distribution correlations available in the literature, proposed by Hager (1991 and 2010) and Rivière et al. (2007), are evaluated based on present and past measurements and discussed below. To the best of the authors' knowledge, the lowest
threshold value of the Froude number in the side branch reported in the literature to separate the subcritical and transcritical regimes is $F_{yo} = 0.3$. Thus, to ensure that all flows are in the subcritical regime, only experiments with $F_{yo} < 0.3$ are retained in the present section.

First, based on Eq. (5), Hager (1991) adjusted the measurements performed by LSY68 as follows:

$$R_q = 1 - \tanh[5(A - F_{yo})F_{xo}^{1/2}] \quad (6)$$

where $A = 0.56$ for $R_b = 0.5$, which is the only branch width ratio tested by LSY68. Note that Eq. (6) does not consider the effect of $F_{bx}$ on $R_q$. To extend the validity of Eq. (6) to $R_b$ values other than 0.5, the best fitted values of coefficient $A$ are obtained for the five $R_b$ runs of the present experimental campaign and plotted in Fig. 2 (insert), along with the value $A = 0.56$ for $R_b = 0.5$, as proposed by Hager (1991 and 2010). A monotonic decreasing trend is observed, but the value proposed by Hager (1991) departs slightly from the adjusted curve $A = 0.468R_b^{-0.306}$. This study assumes that this slight difference is owing to the measurement location of $F_{xo}$ at the entrance of the downstream branch, as performed by LSY68, whereas it is located further downstream in the present experiments (Fig. 1).

The discharge distribution $R_q$ predicted using Eq. (6) combined with the adjusted curve for $A$ values are compared in Fig. 2 with the measurements for all available flow configurations in the subcritical regime (ensured here by $F_{yo} < 0.3$). A relatively fair agreement was obtained for all $R_b$ values, with $R^2$ coefficients of determination exceeding 0.9. However, Fig. 2 indicates that the agreement is not perfect, as many points fall out of the +/- 10% range, exceeding +100% for $R_q < 0.15$, with the tendency of Eq. (6), to overestimate the lowest $R_q$ values and underestimate the largest values. Note that in this figure and the following figures, data from Law (1965) with $R_q = 0$ or $R_q = 1$ and two doubtful tests labeled #5 and #12 by these authors were excluded.

More recently, utilizing weir crest heights $C_x$ and $C_y$, Rivière et al. (2007) used Eq. (4) to propose the following empirical relationship for subcritical flow configurations (restricted to $R_b = 1$, neglecting the effect of $C_x/b_x$):

$$R_q = 0.5 - 0.04675F_{Cx} + 0.358 \left(1 - \frac{C_y}{C_x}\right) F_{Cx}^{-0.77} \quad (7)$$
The agreement with the experimental results of the authors exceeded that of Eq. (6) with $R^2 = 0.99$.

However, as mentioned above, this correlation is not adapted to field conditions where downstream weirs are not available for use, and does not include the effect of the branch width ratio. Another empirical correlation is sought, involving the dimensionless parameters in Eq. (5) instead of Eq. (4).

Because it is extremely difficult to independently adjust these parameters in the experiments, Eq.(7) is used to generate artificial data for various values of the triplet ($F_{bx}$, $F_{xo}$, $F_{yo}$). Several different series were generated, each with one of the three parameters varying, while the other two were maintained constant. This allows for the identification of the best functions that reproduce the influence of each parameter in Eq. (5). Consistent with Hager (1991), that is, Eq. (6), $F_{bx}$ has a limited influence on the discharge distribution (not shown here) and is discarded. The empirical coefficients are finely adjusted with the present experimental results using the least squares method, including the effect of the branch width ratio. The final empirical correlation in Eq. (8), provides the T-shaped bifurcated discharge distribution in the subcritical flow regime for $R_b$ ranging between 1/3 and 1.

$$R_q = a \ln \left[ \frac{b \cdot [\exp(1/a) - 1] \cdot \left\{ \frac{R_b F_{yo}}{F_{xo}} \right\}}{[\exp(1/a) - 1] + b \cdot \left\{ \frac{R_b F_{yo}}{F_{xo}} \right\}} + 1 \right], a = 1.012 \ \& \ b = 0.941 \quad (8)$$

where $R_q$ tends towards 0 (resp. 1) as $F_{yo}$ (resp. $F_{xo}$) tends towards 0. The major influence of the dimensionless parameter $R_b F_{yo}/F_{xo}$ is confirmed in Fig. 3a, with a significantly narrow dispersion of the measured discharge distributions. In addition, the validity of Eq. (8) is depicted in Fig. 3b, revealing better agreement with the measurements than Eq. (6) of Fig. 2, with correlation coefficients reaching 0.99 for all considered datasets and $R_b$ values.

### Free-recirculation transcritical regime

This sub-section aims at evaluating the discharge distribution correlations adapted to the “free recirculation” pattern of the transcritical regime (Rivière et al. 2014), also termed “free flow in the branch channel” by LSY68. In this regime, the downstream boundary of the side branch has no impact on the discharge distribution in the bifurcation. To ensure that all flows from the present
experiments belong to this regime, the following procedure is followed. For a fixed inflow $Q_{xi}$ and a fixed downstream weir crest height $C_x = 5$ cm, the height of the side branch weir crest $C_y$ is regularly lowered (thus increasing $R_q$ and the Froude number in the side branch $F_{yo}$) until the discharge distribution remains stable, as depicted in Fig. 4. All flow configurations on this plateau of constant $R_q$ (on the right side of the dashed lines in Fig. 4) are assumed to belong to the free-recirculation transcritical regime and are used hereafter.

Based on an extended experimental campaign, Lakshmana Rao and Sridharan (1967), hereafter referred to as LS67, proposed a discharge distribution correlation in this free-recirculation transcritical regime (with $C_y = 0$), based on Eq. (5) by neglecting the influence of $F_{bx}$ and $F_{yo$ as follows:

$$log_{10}(R_q) + K F_{xo}^{\alpha} = 0 \qquad (9)$$

where $K$ and $\alpha$ are constant coefficients that are dependent on the bifurcation angle and branch width ratio, respectively. For a T-shaped bifurcation (90° angle) and $R_b = 1$, LS67 proposed $K = 1.11$ and $\alpha = 0.995$. Based on their measurements, Rivière et al. (2014) proposed adjusting the coefficients to $K = 0.925$ and $\alpha = 0.98$ (motivated by the discrepancy between the purple square symbols and LS67's $y = x$ line in Fig. 5). However, the best agreement with the present data was obtained with LS67 coefficients for $R_b = 1$, which were thus retained in this analysis. Moreover, LS67 proposed fitting coefficients for three additional $R_b$ values: $K = 1.4$ and $\alpha = 0.672$ for $R_b = 0.25$; $K = 1.27$ and $\alpha = 0.86$ for $R_b = 0.5$ and $K = 1.16$ and $\alpha = 0.938$ for $R_b = 0.75$, which are used here to establish a discharge distribution law valid for a wide range of branch width ratios.

In addition, for another bifurcation angle (75°), LS67 proposed a 3rd order polynomial adjustment to obtain a correlation for $K$ and for $\alpha$. Applying the same approach to the authors' 90°-angle dataset results in the following equation, we get:

$$ \begin{aligned} K &= 0.43R_b^3 - 0.48R_b^2 - 0.35R_b + 1.51 \\ \alpha &= 0.95R_b^3 - 2.30R_b^2 + 2.06R_b + 0.29 \end{aligned} \qquad (10) $$

Fig. 5 validates the applicability of Eqs. (9) and (10) in predicting the discharge distribution of the present free-recirculation transcritical flows with other $R_b$ values than those used by the authors.
299 **All flow configurations**

300 As expressed above, the criterion $F_{yo} < 0.3$ considered in the first sub-section of the “Results” section 301 ensures that the flow configuration is in a subcritical regime everywhere (without a critical section at 302 the *vena contracta*). However, as expressed by Rivière et al. (2014): “Law and Reynolds (1966) found 303 that the critical Froude number for flow control was about $F_{yo} = 0.7$, which is coherent with our 304 observations, but Ramamurthy and Satish (1988), using the data of Lakshmana Rao et al. (1968) 305 proposed a value of 0.35 for this critical Froude number”. Rivière et al. (2014) proposed a limit of $F_{yo} = 306$ 0.65; however, the authors indicated it depends on the other boundary conditions, namely, $C_x$ (or $h_{xo}$) 307 at the outlet of the main branch, and $F_{xi}$ at the inlet. The measured outlet discharges and water 308 depths, as performed herein, do not permit a clear delimitation of subcritical and drowned- 309 recirculation transcritical regimes, as both exhibit downstream control by the outlet weir of the side 310 branch. Conversely, graphs, such as those in Fig. 4, allow an identification of the limit in transcritical 311 flows between drowned recirculation (with a critical section in the *vena contracta* but with a slight 312 downstream control by the side branch outlet) and free-recirculation (without downstream control 313 from the side branch). This limit (dashed lines in Fig. 4) appears to increase with decreasing $R_b$ for low 314 $F_{bx}$ values (i.e., from $F_{yo} = 0.35$ to 0.6 for $F_{bx} = 0.019$ as $R_b$ decreases from 1 to 1/3), as well as with 315 increasing $F_{bx}$ (or $Q_{xi}$). Thus, a dedicated analysis is required to derive a definitive correlation to limit 316 $F_{yo}$ as a function of $F_{bx}$ and $R_b$.

317 In practice, when facing a T-bifurcation flow, the flow regime (subcritical, drowned-recirculation 318 transcritical, or free-recirculation transcritical) is typically unknown *a-priori*. Nevertheless, a simple 319 methodology was proposed by Rivière et al. (2007) to predict the discharge distribution for any flow 320 condition by advocating to maintain the minimum $R_q$ value predicted by the correlations available in 321 subcritical and free-recirculation transcritical regimes. This method is replicated as follows:

322 $$R_q = \min [ \text{Eq. (8)}, \text{Eqs.(9-10)} ], \quad (11)$$

323 Fig. 6 confirms that this approach is efficient in predicting the discharge distribution for any flow 324 configuration (any boundary condition triplet of $Q_{xi}$, $h_{xo}$, $h_{yo}$). Note that this excludes the case of a
supercritical flow in the upstream branch, previously studied by Rivière and Perkins (2004), which can
occur only in a steep channel or downstream from a hydraulic structure, such as a sluice gate or a
weir. Note also that the validity of Eq. 11 can be affected by the location of the measurement of the
downstream water depths ($h_{xo}$, $h_{yo}$) if this occurs far downstream from the intersection. 1D
backwater computations will then have to be applied to relate the downstream water depths to the
water depths in the vicinity of the intersection at both outlets.

## Discussion

### Alternative approach based on measured water depths

The method proposed in Eq. (11) to predict the discharge distribution in subcritical (Eq. 8) and
transcritical (Eqs. 9 & 10) regimes is based on non-dimensional parameters resulting from the
dimensional analysis provided in Eq. (5), namely, the Froude numbers in the downstream branches
($F_{xo}$ and $F_{yo}$) and the branch width ratio $R_b$. Nevertheless, both Eq. (8) and Eq. (9) are implicit, and
must be solved iteratively, as Froude numbers are computed using the discharges in the downstream
branches ($Q_{xo}$ and $Q_{yo}$), which are determined by the discharge distribution $R_q$. Considering the
definitions of the downstream Froude numbers (see Eq. 5), Eq. (8) and Eq. (9) can alternatively be
expressed as

$$R_q = a \ln \left[ \frac{b \cdot [\exp(1/a) - 1] \cdot R_q}{[\exp(1/a) - 1] \cdot (1 - R_q) \cdot \left(\frac{h_{yo}}{h_{xo}}\right)^{3/2} + b \cdot R_q} + 1 \right], a = 1.012 \ \& \ b = 0.941 \quad (12)$$

for Eq. (8) and

$$R_q = 10^{-K \left[ \frac{Q_{xi}(1-R_q)}{b_x h_{xo}^{3/2} g^{1/2}} \right]^\alpha}, \quad \alpha \text{ and } K \text{ from Eq. (10)} \quad (13)$$

for Eq. (9).

The discharge distribution in Eq. (12) or Eq. (13) can be estimated based solely on the measured
boundary conditions, that is, the upstream discharge $Q_{xi}$ and downstream water depths $h_{xo}$ and $h_{yo}$.

Nevertheless, Fig. 7a shows that for subcritical flows, $R_q$ in Eq. (12) is sensitive to $(h_{yo}/h_{xo})^{3/2}$
experimental uncertainties. This can also be proved analytically using Eq. (12) (see Fig. 7a) as $R_q \approx -$
349

$13.6(h_{yo}/h_{xo})^{3/2}+13$, so that $dR_q \approx 3/2*13.6(h_{yo}/h_{xo})^{3/2}(dh/h_{yo} + dh/h_{xo})$. Considering $h_{xo} \approx h_{yo}$, $dR_q \approx 3*13.6(dh/h) \approx 40dh/h$. Assuming $dh/h = 1\%$ ($dh = 0.5$ mm and $h \approx 50$ mm), this yields $dR_q \approx 0.4$, a substantial value of uncertainty, thus explaining the large discrepancies between the measured data and Eq. (12), as shown in Fig. 7a. Note that Eq. (12) indicates that the subcritical regime will occur only if the side-to-downstream branch ratio of water depths, $h_{yo}/h_{xo}$, is in the range 0.917–0.968.

350

351

352

353

354

Conversely, for the flow conditions in transcritical conditions, Fig. 7b shows that Eq. (13) fairly estimates the discharge distribution based on the measured inflow discharge $Q_{xi}$ and outlet water depth in the main branch $h_{xo}$.

355

356

357

## Predictive approach

358

359

360

361

362

363

364

365

366

367

In most cases in the field, the downstream boundary conditions (at the outlets of both downstream branches) correspond to previously established stage discharge relationships rather than measured or fixed water depths, as in reservoirs. In such case, Eq. (8) or (12), along with the two downstream stage discharge relationships, form a set of three equations for three unknown variables: $R_q$, $F_{xo}$, and $F_{yo}$ in Eq. (8) (or $R_q$, $h_{xo}$, and $h_{yo}$ in Eq. (12)). Similarly, Eq. (9) and (10) (or 13 and 10), along with the downstream stage discharge relationship in the main branch, form a set of two equations for two unknown variables: $R_q$ and $F_{xo}$ in Eq. (9) (or $R_q$ and $h_{xo}$ in Eq. (12)). Thus, these sets of equations predict the discharge distribution in the bifurcation without any measurement in the downstream branches, knowing only the inlet discharge $Q_{xi}$, downstream stage discharge relationships, and bifurcation geometry ($R_b$ notably).

368

369

370

371

372

373

374

The approach used in Eq. (11) is applied. That is, for a particular $Q_{xi}$ and $R_b$, we maintain the minimum $R_q$ value predicted, on the one hand, by the set of three equations for the subcritical regime (Eq. (8) or (12) coupled with the two downstream stage discharge relationships), and on the other hand by the set of three equations for the transcritical regime (Eqs. (9 & 10), or Eqs. (13) and (10) coupled with the downstream main branch stage-discharge relationship). The comparison between the predicted and experimental values is limited to the present experiments only, as this test requires stage discharge relationships that are not available in the literature. The predicted discharge
distribution is of high quality, with an average error limited to $dR_q = 0.029$ compared to the
measurements, as plotted in Fig. 8. To the best of the authors' knowledge, this strategy is the most
adapted operational method for predicting the discharge distribution in a horizontal or limited-slope
T-bifurcation, both for subcritical and transcritical regimes, preventing a supercritical flow regime in
the main branch.

## Conclusion

The present work aims to summarize, validate, and extend the available discharge distribution
correlations for horizontal T-shaped bifurcations with simplified geometries in which the side branch
is narrower or of the same width as the main branch. The new empirical correlation for bifurcation in
the fully subcritical regime [Eq. (8)], compared to previous correlations from the literature, results in
substantial accuracy improvement and can be used regardless of the downstream conditions of the
bifurcation (mainly with fixed water depths or stage discharge relationships). Second, for flows in the
free-recirculation transcritical regime, the correlation proposed by Lakshmana Rao and Sridharan
(1967) [Eqs. (9-10)] agrees well with the present experimental datasets and those available in the
literature. A predictive method was derived for the discharge distribution based on the known
upstream discharge plus branch width ratio and stage discharge relationships in the downstream
branches. Note that the present discharge distribution prediction is accurate even without a priori
knowledge of the flow regime owing to the procedure proposed by Rivière et al. (2007) replicated in
Eq. (11).

Nevertheless, this study had several limitations. Initially, the side branch remained slightly wide, with
a minimum side to downstream branch width ratio $R_b = 0.33$, whereas Lakshmana Rao and Sridharan
(1967) considered a minimum ratio $R_b = 0.25$, and field conditions can exhibit significantly narrower
side branches. Thus, future work can be dedicated to bifurcations with significantly narrower side
branches. Moreover, the present configuration is restricted to right-angle T-shape bifurcations, and
although Lakshmana Rao and Sridharan (1967) extended their work to four additional angles, no
correlation is, to the best of the authors knowledge, available for subcritical flows in bifurcations with an angle departing from 90°. This can also constitute valuable future research.

402 **Data Availability Statement**

A table of the 668 measurement data points from the present experimental campaign is available
from the corresponding author upon reasonable request. This table comprises the width R<sub>b</sub>; three
discharges Q<sub>xi</sub>, Q<sub>xo</sub> and Q<sub>yo</sub>; corresponding water depths h<sub>xi</sub>, h<sub>xo</sub> and h<sub>yo</sub>; Froude numbers F<sub>xo</sub> and F<sub>yo</sub>.

406 Acknowledgments

This work was performed within the framework of EUR H2O'Lyon (ANR-17-EURE-0018) of Université
de Lyon (UdL), within the program "Investissements d'Avenir" operated by the French National
Research Agency (ANR). Part of the research was funded (for the first author) by the Ministry of
Higher Education (MoHE) Malaysia, that is, project no. FRGS19-130-0739.

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474 **Figure captions:**

475 **Fig. 1.** Sketch (a) and photograph (b) of the experimental set-up with indicated variables and
476 notations.

477 **Fig. 2.** Predicted discharge distribution in subcritical regime with Eq. (6) for all available flow
478 configurations with $F_{yo} < 0.3$; exact match ($y = x$) solid and +/- 10% dashed lines. Insert: best-fitted A
479 values for the 5 tested $R_b$ values of present experiments (blue) and A = 0.56 at $R_b$ = 0.5 (red) as
480 proposed by Hager (1991) when adjusting data from Lakshmana-Rao et al. (1968).

481 **Fig. 3.** Comparison of measured vs Eq. (8) predicted discharge distribution in subcritical regime for all
482 available flow configurations with $F_{yo} < 0.3$, (a) as a function of the dimensionless number $R_b \cdot F_{yo} / F_{xo}$,
483 and (b) with exact match ($y = x$) as solid and +/- 10% as dashed lines.

484 **Fig. 4.** Analysis of the subcritical-transcritical regimes transition for the 5 tested $R_b$ configurations. In
485 each series, $Q_{xi}$ and $C_x$ = 5 cm are maintained constant while lowering $C_y$, thus increasing $F_{yo}$. The right
486 most points for all series correspond to $C_y$ = 0. The dashed lines demarcate the assumed limits
487 between the free- and drowned-recirculation transcritical regimes.

488 **Fig. 5.** Evaluation of Eqs. (9) and (10) using present and literature data in free-recirculation
489 transcritical regime.

490 **Fig. 6.** Predicted discharge distribution with Eq. (11) for all 967 tested flow configurations (299 from
491 the literature and 668 from present work).

492 **Fig. 7.** (a) Effect of $(h_{yo}/h_{xo})^{3/2}$ on the measured discharge distribution (symbols) also compared with
493 Eq.(12) (red line). (b) Comparison of measured and predicted Eq.(13) discharge distributions for flows
494 in free-recirculation transcritical regime.

495 **Fig. 8.** Evaluation of the predicted discharge distribution using the minimum between Eq. (8) (or 12)
496 and Eq. (9) (or 13) coupled with downstream stage discharge relationships for a known upstream
497 discharge $Q_{xi}$ and a particular $R_b$ value.

498
<table><caption>**Table 1.** Flow configurations for present data and from the literature. *Number of data for which the result of Eq. (11) is so that the solution of Eqs. (9) & (10) is smaller than that of Eq. (8)</caption><thead><tr><th></th><th>R<sub>b</sub></th><th>Re<sub>bx</sub></th><th>F<sub>xo</sub></th><th>F<sub>yo</sub></th><th>Number of data</th><th>Eqs. (9) & (10) &lt; Eq. (8)*</th></tr></thead><tbody><tr><td></td><td>1</td><td>[5.6 – 30.8]x10<sup>3</sup></td><td>[0 – 0.38]</td><td>[0.02 – 0.73]</td><td><strong>132</strong></td><td>8</td></tr><tr><td></td><td>26/30</td><td>[7.9 – 30.0]x10<sup>3</sup></td><td>[0 – 0.37]</td><td>[0.01 – 0.71]</td><td><strong>126</strong></td><td>10</td></tr><tr><td rowspan="3">Present data</td><td>22/30</td><td>[7.8 – 32.1]x10<sup>3</sup></td><td>[0 – 0.40]</td><td>[0.01 – 0.68]</td><td><strong>132</strong></td><td>11</td></tr><tr><td>18/30</td><td>[8.6 – 30.7]x10<sup>3</sup></td><td>[0 – 0.35]</td><td>[0.02 – 0.68]</td><td><strong>142</strong></td><td>12</td></tr><tr><td>10/30</td><td>[8.8 – 31.4]x10<sup>3</sup></td><td>[0 – 0.36]</td><td>[0.03 – 0.65]</td><td><strong>136</strong></td><td>14</td></tr><tr><td>Law (1965)</td><td>1</td><td>[0.99 – 2.56]x10<sup>3</sup></td><td>[0.02 – 0.79]</td><td>[0.01 – 0.66]</td><td><strong>134</strong></td><td>79</td></tr><tr><td>Rivière (2007 & 2014)</td><td>1</td><td>[6.6 – 66.6]x10<sup>3</sup></td><td>[0.02 – 0.46]</td><td>[0.02 – 1.00]</td><td><strong>86</strong></td><td>40</td></tr><tr><td>Mignot (2013)</td><td>1</td><td>[5.53 – 23.3]x10<sup>3</sup></td><td>[0.09 – 0.41]</td><td>[0.09 – 0.50]</td><td><strong>14</strong></td><td>2</td></tr><tr><td>Li (2021)</td><td>1</td><td>[1.2 – 95.0]x10<sup>3</sup></td><td>[0.09 – 0.56]</td><td>[0.02 – 0.66]</td><td><strong>65</strong></td><td>35</td></tr></tbody></table>

501

502