# Comparison between 2D Shallow-Water Simulations and Energy-Momentum Computations for Transcritical Flow Past Channel Contractions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Steady-State Water Surface Profiles for Gradually Varied Open Channel Flow

#### 2.2. Solution of the 2D Depth-Integrated Shallow-Water Equations (SWE)

#### 2.3. Model Verification Using Subcritical Flow: Simple Backwater Curves in Expanding/Contracting Channels

#### 2.4. Construction of Water Surface Profiles for Transcritical Flow past a Contraction

## 3. Results: Model Validation Using Supercritical and Transcritical Flow in Laboratory-Scale Flumes

#### 3.1. Supercritical Flow Past a Channel Contraction

#### 3.2. Gradually Varied Transcritical Flow in a Parshall Flume

#### 3.3. Gradually Varied Transcritical Flow in a Khafagi Flume

#### 3.4. Predicting the Hydraulic Jump Position in the Khafagi’s Venturi Flume Experiments

## 4. Results: Transcritical Flow in Long Channels

#### 4.1. Predicting the Onset of Transcritical Flow and Jump Position in a Long Channel with a Linear Contraction

#### 4.2. The Role of Grid Refinement on Capturing 2D Flow Features

#### 4.3. Influence of the Contraction Geometry on the Discrepancy between 1D and 2D Models

## 5. Discussion and Conclusions

- For transcritical flow past short, horizontal channels with relatively smooth contractions and negligible flow separation (e.g., for the experimental cases considered in the validation section), the deviations between models and experiments seem larger than among models. The discrepancy between models and experimental data is consistent with the well known limitations of the depth-averaged shallow-water model, in particular the impact of non-hydrostatic pressures and streamline curvature on the flow. The standard 1D theory shares this limitation.
- Considering its simplicity and negligible computational cost, the classical 1D theory performs remarkably well for a wide range of flow conditions and relatively smooth channel contractions. In particular, the 1D model yields a good prediction of the transition to supercritical flow at the contraction. Perhaps more importantly, the 1D model is more conservative, in the sense that it predicts an earlier onset of critical flow at the contraction as the tailwater depth decreases.
- The grid resolution used in the 2D SWE simulations plays an important role in capturing the spatial flow patterns, so that coarse grid 2D simulations provide essentially the same information as 1D ones. The implication of this observation is that the discrepancies among various 2D models with different spatial grid resolution may be as large as those between the 2D models and a classical 1D energy-momentum calculation. The impact of grid resolution on the agreement between 1D and 2D models is relevant in practice, as field-scale hydrodynamic models in fluvial dynamics rely on the available topography, whose spatial resolution is often limited to the meter scale.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**We study free surface flow in long channels, where tranquil flow under mild slope conditions is altered by a symmetric contraction. Depending on channel geometry and flow parameters, the contraction may induce a transition from subcritical to supercritical regime at the narrowest throat, and then back to subcritical flow downstream of it. Transcritical flow is revealed by distinctive two-dimensional flow patterns, both in terms of water surface morphology and spatial structure of the velocity field. Typical flow features associated with transcritical flow past a contraction include oblique standing waves and flow separation that disturb the flow over long distances, promoting flow focusing, the appearance of stagnation regions and the loss of flow symmetry.

**Figure 2.**Model verification for subcritical flow in converging (

**a**) and expanding (

**b**) channels. The channel slope is $S=0.002$, the total discharge is $Q=500$ m${}^{3}$/s and Manning’s friction coefficient is $n=0.04$. The boundary conditions in the 2D model are consistent with mild slope and subcritical flow: known total discharge, Q, at the upstream left boundary, and critical flow at the downstream right boundary. For the 1D theory we simply integrate (1)–(2) backwards with final condition $h\left(L\right)={h}_{c}$. We observe a very good agreement between the 1D and 2D theories.

**Figure 3.**Comparison between 2D shallow-water simulations and 1D backwater curves for subcritical flow past a symmetric contraction in a long channel. (

**a**) Schematic plot of the channel geometry, and full-channel water surface profile computed using the 1D theory. (

**b**) Sample water surface profile along the center of the channel, for ${L}_{u}=2000$ m and ${L}_{d}=4000$ m, and ${L}_{e}=100$ m. (

**c**) Comparison of water surface profiles predicted by the 1D and 2D models around the contraction, when the length of the expansion zone is ${L}_{e}=100$ m. (

**d**) Comparison for ${L}_{e}=50$ m. The 1D prediction is more conservative: it yields a lower water depth at the narrowest section, therefore suggesting an earlier onset of critical flow conditions at the contraction.

**Figure 4.**Construction of water surface profiles for transcritical flow past a channel contraction using the 1D energy-momentum theory. Blue arrows indicate hydraulic control under mild slope conditions, and direction of integration of Equations (1) and (2). (

**a**) A sample case where the hydraulic jump is contained within the expansion zone. The distance from the center of the constriction to downstream boundary condition is ${L}_{d}=250$ m, the length of the segment with maximum contraction is ${L}_{b}=10$ m, the length of the contracting/expanding zones is ${L}_{e}=50$ m, and the channel widths ${b}_{1}$ and ${b}_{2}$ are 40 m and 15 m, respectively, according to the schematic in Figure 3a. The discharge is $Q=500$ m${}^{3}$/s, and Manning’s coefficient is $n=0.0389$. (

**b**) A sample case where the hydraulic jump is expelled outside of the expansion zone. In this case ${L}_{d}=100$ m and ${L}_{e}=20$ m.

**Figure 5.**Model validation: supercritical flow in a channel contraction. (

**a**) Schematic description of the flume geometry, according to the experiments of Ippen and Dawson [16], recently revisited by [44]. We show the steady-state water depth for a flow discharge $Q=0.041$ m${}^{3}$/s, inlet water depth $h=0.03$ m, and Manning friction coefficient $n=0.013$. (

**b**) Comparison between experimental observations of the water surface profile along the channel centerline [44], and model predictions.

**Figure 6.**Model validation: supercritical flow in a channel contraction: impact of bottom friction. We repeat the analysis of Figure 5 using several values of the Manning coefficient, and show the maps of water depth (2D simulations) and 1D profiles for $n=0.011$ (panel

**a**), $n=0.01$ (panel

**b**), $n=0.0075$ (panel

**c**) and $n=0.005$ (panel

**d**).

**Figure 7.**Model validation: transcritical flow in a Parshall flume. (

**a**) Schematic description of the flume geometry, according to the experiments of [43], and steady-state water depth for a flow discharge $Q=0.0145$ m${}^{3}$/s. (

**b**) Comparison between experimental observations of water surface profile [43] and model predictions.

**Figure 8.**Impact of grid refinement on the comparison between the 2D shallow water equations and experimental observations of flow past a Parshall flume. We show the maps of water depth (panels

**a**–

**f**) and a comparison between water depth profiles along the axis of the flume (panel

**g**) for several grid sizes, $\Delta x$.

**Figure 9.**Model validation: transcritical flow in a Khafagi flume. Schematic description of the flume geometry, according to the experiments of [24], and steady-state water depth for a flow discharge $Q=22$ L/s.

**Figure 10.**Model validation: transcritical flow in a Khafagi flume. Comparison between experimental observations of water surface profile [24] and model predictions, for discharges of $Q=22$ L/s (

**a**) and $Q=10$ L/s (

**b**). We consider a channel that is short enough for the hydraulic jump to be repelled out of the simulated domain.

**Figure 11.**Transition to subcritical flow in Khafagi’s experiments [24]. We illustrate the comparison between model predictions and experimental results using the discharge $Q=10$ L/s. Hydraulic jumps are described as shocks by the models, and as smooth water surface variations in the experimental data, due to the complex internal structure of the roller in the jump region. We use solid lines for profiles along the channel axis in 2D simulations, broken lines for the 1D theory, and circles for the experimental measurements. Line colors correspond to different imposed tailwater levels.

**Figure 12.**Predicting the transition to subcritical flow in Khafagi’s flume experiments: comparison between models and experiments for four discharges $Q=10,14,17.5$ and 22 L/s (panels

**a**–

**d**, respectively). We use solid lines for profiles along the channel axis in 2D simulations, broken lines for the 1D theory, and circles for the experimental measurements. Line colors correspond to different imposed tailwater levels.

**Figure 13.**Predicting the jump location downstream of a channel contraction: problem set-up. We consider a long channel with slope ${S}_{0}=0.002$, length of 2000 m and width ${b}_{1}=40$. A symmetric, contraction reduces the width to a value ${b}_{2}$ over a distance ${L}_{e}=20$ m. We solve steady-state flow for several values of the discharge Q, minimum width ${b}_{2}$ and imposed downstream water depth, ${h}_{d}$, and compare the predicted location of the jump, ${x}_{j}$, using the 1D theory and the 2D shallow-water Equations (along the channel axis).

**Figure 14.**Comparison between 1D (solid lines) and 2D (circles) predictions of the jump location downstream of a channel contraction: summary of results. We plot the jump position, ${x}_{j}$, as a function of imposed downstream water depth, ${h}_{d}$, for three flow discharges, $Q=500,350$ and 200 m${}^{3}$/s (panels

**a**,

**b**and

**c**, respectively), and several width ratios, ${b}_{1}/{b}_{2}=2,4,5$. The jump is repelled out of the expansion zone when ${x}_{j}>{L}_{e}$, and the flow becomes fully subcritical when ${x}_{j}=0$.

**Figure 15.**Onset of transcritical flow in the 1D and 2D models. By reducing the distance to the outlet, ${L}_{d}$, we control the available energy at the downstream end of the contraction. For sufficiently small available energy, the flow is forced to undergo a regime change at the contraction. The basic model parameters are the same as those in Figure 3, Figure 16 and Figure 17: ${b}_{1}=40$ m, ${b}_{2}=26.5$ m, ${L}_{e}=20$ m, $Q=500$ m${}^{3}$/s, and $n=0.04$. The distance to the outlet is ${L}_{d}=500$ m (panel

**a**), ${L}_{d}=250$ m (panel

**b**), and ${L}_{d}=125$ m (panel

**c**). The 1D theory predicts an earlier onset of transcritical conditions (panel

**d**). In fact, it predicts that even the hydraulically-long case, ${L}_{d}=4000$ m is transcritical, while the 2D simulations predict near-critical conditions up to ${L}_{d}=250$ m.

**Figure 16.**Impact of grid refinement on 2D shallow-water simulations of transcritical flow. We show maps of water depth computed using different grid sizes in the contraction region, $\Delta x=5,1.5,1,0.15$ m (panels

**a**–

**d**respectively). The length of the expansion zone is ${L}_{e}=100$ m and the distance to the outlet is ${L}_{d}=250$ m. The coarse grid solution is quasi-1D, showing an excellent match with the water profile computed using the classical 1D theory (panel

**e**). As the grid is refined, and the simulations capture the complex 2D flow features, the 1D and 2D predictions deviate significantly. In particular, the sharp hydraulic jump of the 1D theory is replaced by an intricate sequence of oblique standing waves (panel

**d**).

**Figure 17.**Impact of grid refinement on 2D shallow-water simulations of transcritical flow. We show maps of water depth computed using different grid sizes in the contraction region, $\Delta x=5,1.5,1,0.15$ m (panels

**a**–

**d**respectively). The length of the expansion zone is ${L}_{e}=50$ m and the distance to the outlet is ${L}_{d}=125$ m. The coarse grid solution agrees with the water profile computed using the classical 1D theory (panel

**e**). As the grid is refined, and the simulations capture the complex 2D flow features, the 1D and 2D predictions deviate significantly. In particular, the sharp hydraulic jump of the 1D theory is replaced by an intricate sequence of oblique standing waves (panel

**d**).

**Figure 18.**Role of the contraction geometry on the emergence of spatial flow patterns. We show maps of water depth (panels

**a**–

**d**) and Froude number (panels

**e**–

**h**), for the same flow conditions of Figure 3, Figure 16 and Figure 17, with ${L}_{d}=125$ m, and several values of the length of the expansion zone, ${L}_{e}=10,20,50,$ and 100 m. For a smooth transition, ${L}_{e}=100$ m, the 1D theory captures the gradually varied water surface profile remarkably well. Two-dimensional effects dominate the flow for the sharpest contraction (${L}_{e}=10$ m, panels

**a**and

**e**): while the location of the hydraulic jump seems to be correctly predicted by the 1D theory, the overall structure of the water surface downstream from the expansion zone. The abruptness of the expansion and contraction segments affects the Froude number in the expansion zone and downstream from it, illustrating the strong flow-focusing effect of an abrupt change in channel width.

**Figure 19.**Flow patterns at abrupt contractions and large width ratios, ${b}_{1}/{b}_{2}$. We show maps of water depth (panels

**a**–

**d**) and Froude number (panels

**e**–

**h**) for different width ratios, leading to increasing effective Froude numbers in the supercritical flow region. We increase the Froude number downstream of the contraction by keeping the basic flow conditions of Figure 3, Figure 16 and Figure 17, in particular ${b}_{1}=40$ m, and considering an abrupt contraction, ${L}_{e}=1$ m, while reducing the width of the narrowest segment, ${b}_{2}$. The width ratios are ${b}_{1}/{b}_{2}=$ 4 (panels

**a**and

**e**), 2.67 (panels

**b**and

**f**), 2 (panels

**c**and

**g**), and 1.6 (panels

**d**and

**h**). We compare the 1D and 2D predictions of the water surface profile (panel

**i**). We use dots for profiles along the channel axis in 2D simulations, and solid lines for the 1D theory. Line colors correspond to the different width ratios.

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Cueto-Felgueroso, L.; Santillán, D.; García-Palacios, J.H.; Garrote, L.
Comparison between 2D Shallow-Water Simulations and Energy-Momentum Computations for Transcritical Flow Past Channel Contractions. *Water* **2019**, *11*, 1476.
https://doi.org/10.3390/w11071476

**AMA Style**

Cueto-Felgueroso L, Santillán D, García-Palacios JH, Garrote L.
Comparison between 2D Shallow-Water Simulations and Energy-Momentum Computations for Transcritical Flow Past Channel Contractions. *Water*. 2019; 11(7):1476.
https://doi.org/10.3390/w11071476

**Chicago/Turabian Style**

Cueto-Felgueroso, Luis, David Santillán, Jaime H. García-Palacios, and Luis Garrote.
2019. "Comparison between 2D Shallow-Water Simulations and Energy-Momentum Computations for Transcritical Flow Past Channel Contractions" *Water* 11, no. 7: 1476.
https://doi.org/10.3390/w11071476