# Numerical Model of Supersaturated Total Dissolved Gas Dissipation in a Channel with Vegetation

^{1}

^{2}

^{*}

## Abstract

**:**

_{in}of TDG in vegetation-affected flows was studied, and the quantitative relationships between the inner dissipation coefficient k

_{in}and the average flow velocity, average water depth, average water radius, Reynolds number, and vegetation density were characterized. Based on the simulation results, the distribution characteristics of the supersaturated TDG in water around vegetation and in the vertical, lateral, and longitudinal directions of the flume under different flow and vegetation densities were analyzed. A supersaturated TDG transportation and dissipation model for vegetation-affected flow is proposed and can be used to predict the impact of TDG in a floodplain.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Case

^{−1}and a measurement error of equal to or less than 1.5%. The TDG saturation level in the water was detected with a TGP (Total Dissolved Gas Pressure) detector composed of Point Four TGP portable trackers made by Pentair (Minnesota, USA), with a TGP measuring range of 0–200% and an accuracy of 2%.

_{V}(the percentage of the ratio between the projected area of vegetation in the Z direction and the area of the flume’s bottom) and five groups of flow, resulting in a total of 25 experimental cases as shown in Table 1. Figure 2 shows the arrangement of the Plexiglas columns.

#### 2.2. Model Assumption

#### 2.3. Model Equations

#### 2.3.1. Continuity Equation

^{−1}) represent the flow velocities in the x, y, and z directions, respectively; $\rho $ (kg·m

^{−3}) represents the density of the mixture of water and air; ${\rho}_{g}$ (kg·m

^{−3}) represents the density of air; ${\rho}_{w}$ (kg·m

^{−3}) represents the density of water; $\gamma $ represents the volume fraction of water.

#### 2.3.2. Momentum Equation

^{2}) represents the molecular viscosity coefficient; ${\mu}_{w}$ (Ns/m

^{2}) represents the molecular viscosity coefficient of water; ${\mu}_{g}$ (Ns/m

^{2}) represents the molecular viscosity coefficient of air; ${f}_{x}$, ${f}_{y}$ and ${f}_{z}$ are body force per unit mass including gravity and surface tension effects at the interface in the x, y, and z directions, respectively.

#### 2.3.3. k-ε Equations

^{2}/s

^{2}) represents the turbulent energy; $\epsilon $ (m

^{2}/s

^{3}) represents the rate of the energy dissipation; ${\mu}_{t}$ (Ns/m

^{2}) represents the turbulent viscosity; ${\sigma}_{k}$ represents the Prandtl number of the turbulent energy (here, ${\sigma}_{k}$ = 1.0); $\sigma \epsilon $ represents the Prandtl number of the rate of the energy dissipation (here, $\sigma \epsilon $ = 1.39); ${C}_{\epsilon 1}$ and ${C}_{\epsilon 2}$ are empirical constants (here, ${C}_{\epsilon 1}$ = 1.44 and ${C}_{\epsilon 2}$ = 1.92); ${G}_{k}$ is the production term of the turbulent kinetic energy caused by the average velocity gradient; ${C}_{\mu}$ is a constant (here, ${C}_{\mu}$ = 0.09).

#### 2.3.4. Transportation Equation of TDG

^{−3}min

^{−1}) represents the mass transfer coefficient, and ${G}_{eq}$ (%) is the percent saturation of TDG under standard atmospheric pressure. ${S}_{G}$, representing source term, is set in Section 2.6.

#### 2.4. Compute Region and Meshes

#### 2.5. Boundary Conditions

#### 2.5.1. Boundary Conditions of Flow Field

#### 2.5.2. Boundary Conditions of the TDG Concentration Field

^{−1}) and ${k}_{a}$ is the adsorption rate of the wall (m

^{−2}min

^{−1}) (here ${k}_{a}$ = 0.0046 m

^{−2}min

^{−1}).

#### 2.6. Model Parameters

#### 2.6.1. Parameterizing the TDG Source Term ${S}_{G}$

^{−1}) and $k$ is the turbulent kinetic energy (m

^{2}s

^{−2}).

^{−1}).

#### 2.6.2. The Formula of Inner Dissipation Coefficient

_{V}, as shown in Figure 4.

_{e}were also selected as important parameters to fit formula of the inner dissipation coefficient in water. The Reynolds number can be used to represent the effect of turbulence on the inner dissipation coefficient, and the resulting expression can be written as follows:

^{m}represents the correction factor due to the influence of the wall and $\varphi $, l, m, n, and $\phi $ are dimensionless constants that can be fitted from the experimental results of Huang (2017) [41].

^{−1}, Q = 3.5 L·s

^{−1}, Q = 5.5 L·s

^{−1}, Q = 7.5 L·s

^{−1}), the values of $\varphi $, l, m, n, and $\phi $ were found to be 3.0 × 10

^{−6}, 0.29, 2.3, 0.24 and −0.7, respectively.

^{−1}(shown in Table 1), and the calculated results are listed in Table 3. The calculated results are very close to the calibrated results (shown in Figure 5), indicating that Equation (24) can be used to calculate the inner dissipation coefficient of supersaturated TDG in vegetation affected flows.

#### 2.7. Verification of the Model

#### 2.7.1. Verification Case

_{V}= 0.6, Q = 9.5 L·s

^{−1}) and Case 15 (d

_{V}= 0.2, Q = 9.5 L·s

^{−1}) are selected as the verification cases for the numerical simulation. Case 15 and Case 25, which have the same lateral arrangement of Plexiglas columns but different longitudinal arrangements, are also selected as a verification of the influence of wake flow around the columns.

#### 2.7.2. Verification Results of the Flow Field

^{−1}, demonstrating that the above mentioned model can perfectly reflect the flow field distribution of the vegetation-affected flow.

#### 2.7.3. Verification Results of the TDG Concentration Field

## 3. Results and Discussions

#### 3.1. The Distribution of Supersaturated TDG around a Column

_{V}= 0.6, Q = 9.5 L·s

^{−1}) is shown in Figure 11 (the distance from the bottom of the flume is 0.09 m). The area downstream of the columns exhibited an obviously low saturation. The reason may be that the existence of columns blocks the flow, resulting in an area of low-speed wakes and vortices behind the columns. In this region, the decreased flow rate results in a longer retention time, allowing the supersaturated TDG to more fully dissipate.

#### 3.2. Vertical Distribution of Supersaturated TDG

_{V}= 0.6, Q = 9.5 L·s

^{−1}) in Figure 12. Figure 12 shows that the TDG saturation in the surface water is slightly lower than that in the subsurface water. The TDG saturation in the subsurface water decreases as the water depth increases. This feature is the result of the uneven distribution of vertical velocity and mass transfer in the free water surface. In vegetation-affected flows, the vertical distribution of flow velocity exhibits a pattern in which the deeper the water depth is, the lower the flow velocity is. Affected by this distribution, the upper water with a higher TDG saturation reaches the downstream section before the lower water. Furthermore, the lower flow velocity causes longer retention time of the lower water, and the supersaturated TDG in this part of the water is allowed to more fully dissipate.

#### 3.3. Planar Distribution of Supersaturated TDG

^{−1}for graphical analysis. In order to compare the distribution discipline of supersaturated TDG under the condition without vegetation and with vegetation, we chose the case without vegetation (Case 5, Figure 13a) and the case with maximum vegetation density (Case 25, Figure 13b) for comparative analysis. As can be seen from Figure 13a,b, the distributions of supersaturated TDG are quite different between cases without vegetation and with vegetation. For Case 5 without vegetation, the equal TDG saturation lines are concave backwards curves, while for Case 25 with maximum vegetation, the equal TDG saturation lines are convex forward curves. In addition, for comparing the distribution discipline of TDG saturation under different vegetation densities, we conducted a comparative analysis of case with low vegetation density (Case 15, Figure 13c) and Case 25. It can be seen that the curvature of the equal TDG saturation lines at the inflection point is larger when the vegetation density is larger.

#### 3.3.1. The Longitudinal Distribution of Supersaturated TDG

^{−1}and indicates that the existence of vegetation can significantly promote the dissipation of supersaturated TDG. With the same flow, the higher the density of vegetation is, the greater the obstruction of vegetation to water is, so that the retention time of supersaturated TDG in water increases and the supersaturated TDG is fully dissipated along the path. Due to the influence of the vertical column, the TDG saturation near the vertical column presents a zigzag distribution feature. In addition, because the initial saturation of the inlet section in cases with higher vegetation density is larger, the curve of TDG saturation in cases with higher vegetation intersects the curve of TDG saturation in cases with lower vegetation along the longitudinal direction. Figure 14b shows the longitudinal scattergram of supersaturated TDG in cases with different flows and the same vegetation density of 0.3. As seen from the Figure 14b, lower flow values are associated with greater dissipation of the supersaturated TDG throughout the whole process. The reason may be that the average water retention time is extended by the decrease in the flow, causing the supersaturated TDG in the water more fully dissipate. At the same time, when the flow rate is smaller, the retention effect of the column is better, and the serrated distribution characteristics of the supersaturated TDG are more obvious.

#### 3.3.2. The Lateral Distribution of Supersaturated TDG

^{−1}. In cases without vegetation, the middle TDG saturation is slightly higher than that of both sides. In cases with vegetation, the lateral distribution of TDG saturation is the opposite, characterized by low saturation in the middle and high saturation on both sides; the higher the vegetation density is, the more obvious this distribution characteristic is. The reason may be that vegetation acts as a barrier to water flow, causing the lateral flow velocity outside the flume’s wall boundary layer to present a phenomenon of low in the middle and high on both sides. As a result, when upstream water with higher TDG saturation is transported downstream, the water along the edge flows downstream before the central water, making the TDG saturation on both sides higher than that in the center. Figure 15b presents a lateral scattergram of supersaturated TDG in cases with different flows and the same vegetation density of 0.3. When the vegetation density is constant while the flow varies, the distribution characteristics of supersaturated TDG present the same trend, namely the TDG saturation in the middle is slightly lower than that on both sides. It seems that the effect of flow on the lateral distribution of supersaturated TDG is not as great as that of vegetation.

## 4. Conclusions and Prospect

- (1)
- A three-dimensional two-phase flow dynamics model was established to study the complex characteristics of three-dimensional flow under the effects of vegetation, and the model was verified by measurements of flow velocity in vegetation-affected flows in the experiments. The verification results indicated that the numerical simulation results of each section were basically consistent with the measured flow velocity distribution, and the calculation error of flow velocity was within 0.019 m·s
^{−1}. - (2)
- Dividing the dissipation process of supersaturated TDG in vegetation-affected flows into the liquid-gas free surface transfer and the inner dissipation, a three-dimensional supersaturated TDG transportation and dissipation model considering the influence of vegetation was established. In this model, the inner dissipation coefficient was introduced to characterize the inner dissipation of supersaturated TDG. A formula based on the average velocity, the average hydrodynamic radius, Reynolds number and vegetation density was developed to predict the inner dissipation coefficient of supersaturated TDG. The prediction model was verified with two individual cases, and this verification demonstrated that the simulation results of the dissipation process of supersaturated TDG in the flume were very close to the measured values. The transportation and dissipation model of supersaturated TDG established in this paper can be used to predict the transportation and dissipation process of supersaturated TDG dissipation in vegetation-affected flows.
- (3)
- The simulation results show that the water-blocking effect caused by a column formed an obvious area of low TDG saturation behind the column. In the vertical direction, the TDG saturation in the surface water was slightly lower than that in the subsurface water, and TDG saturation in the subsurface water decreased as the water depth increased. At the same time, affected by the water-blocking effect of the vegetation group, TDG saturation presented a distribution characterized by high values in the middle and low values on both sides in the lateral direction. In the longitudinal direction, the TDG saturation decreased gradually with downstream extent but showed serrated distribution characteristics in the region behind the column.
- (4)
- Because the existing measurement and numerical simulation methods have difficulty reflecting the influence of real vegetation on the flow field and the supersaturated TDG transportation and dissipation process, a Plexiglas vertical column with a square cross section was selected to simulate rigid hydrophilic plants. The effects of vegetation’s flexible characteristics and the presence of leaves on the transportation and dissipation processes of supersaturated TDG in flowing water remain to be studied. Limited by laboratory conditions, only small-scale mechanism experiments have been carried out. The parameters used in the three-dimensional TDG transportation and dissipation model were also calibrated by the experimental results. The applicability of this model for simulating the transportation and dissipation process of supersaturated TDG under vegetation-affected flows in large-scale flows remains to be further studied.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Wydoski, R.S.; Wick, E.J. Flooding and aquatic ecosystems. In Inland Flood Hazards: Human, Riparian, and Aquatic Communities; Wohl, E.E., Ed.; Cambridge University Press: New York, NY, USA, 2011; pp. 238–268. [Google Scholar]
- Reolid, M.; Rodríguez-Tovar, F.J.; Nagy, J. Ecological replacement of Valanginian agglutinated foraminifera during a maximum flooding event in the Boreal realm (Spitsbergen). Cretac. Res.
**2012**, 33, 196–204. [Google Scholar] [CrossRef] - Heydel, F.; Engels, J.G.; Feigs, J.T.; Vásquez, E.; Rudolph, B.; Rohwer, J.; Jensen, K. Adaptation to tidal flooding and rapid genetic divergence between a narrow endemic grass species and its widespread congener lead to an early stage of ecological speciation. Perspect. Plant Ecol.
**2017**, 27, 57–67. [Google Scholar] [CrossRef] - Bendix, J.; Cowell, C.M. Fire, floods and woody debris: Interactions between biotic and geomorphic processes. Geomorphology
**2010**, 116, 297–304. [Google Scholar] [CrossRef] - Chaleeraktrakoon, C.; Chinsomboon, Y. Dynamic rule curves for flood control of a multipurpose dam. J. Hydro-Environ. Res.
**2015**, 9, 133–144. [Google Scholar] [CrossRef] - Talukdar, S.; Pal, S. Impact of dam on inundation regime of flood plain wetland of punarbhaba river basin of barind tract of Indo-Bangladesh. Int. Soil Water Conserv. Res.
**2017**, 5, 109–121. [Google Scholar] [CrossRef] - Tao, J.P.; Yang, Z.; Cai, Y.P.; Wang, X.; Chang, J.B. Spatiotemporal response of pelagic fish aggregations in their spawning grounds of middle Yangtze to the flood process optimized by the Three Gorges Reservoir operation. Ecol. Eng.
**2017**, 103, 86–94. [Google Scholar] [CrossRef] - Quirino, B.A.; Carniatto, N.; Guglielmetti, R.; Fugi, R. Changes in diet and niche breadth of a small fish species in response to the flood pulse in a Neotropical floodplain lake. Limnologica
**2017**, 62, 126–131. [Google Scholar] [CrossRef] - Lemke, M.J.; Hagy, H.M.; Dungey, K.; Casper, A.F.; Lemke, A.M.; VanMiddlesworth, T.D.; Kent, A. Echoes of a flood pulse: Short-term effects of record flooding of the Illinois River on floodplain lakes under ecological restoration. Hydrobiologia
**2017**, 804, 151–175. [Google Scholar] [CrossRef] - Bolland, J.D.; Nunn, A.D.; Lucas, M.C.; Cowx, I.G. The habitat use of young-of-the-year fishes during and after floods of varying timing and magnitude in a constrained lowland river. Ecol. Eng.
**2015**, 75, 434–440. [Google Scholar] [CrossRef] [Green Version] - Li, R.; Li, J.; Li, K.F.; Deng, Y.; Feng, J.J. Prediction for supersaturated total dissolved gas in high-dam hydropower projects. Sci. China Ser. E
**2009**, 52, 3661–3667. [Google Scholar] [CrossRef] - Feng, J.J.; Li, R.; Ma, Q.; Wang, L.L. Experimental and field study on dissipation coefficient of supersaturated total dissolved gas. J. Cent. South Univ.
**2014**, 21, 1995–2003. [Google Scholar] [CrossRef] - Kamal, R.; Zhu, D.Z.; Mcarthur, M.; Leake, A. Field study on the dissipation of supersaturated total dissolved gases in a cascade reservoir system. In Proceedings of the World Environmental and Water Resources Congress 2016, West Palm Beach, FL, USA, 22–26 May 2016; pp. 452–460. [Google Scholar]
- Weitkamp, D.E.; Sullivan, R.D.; Swant, D.; DosSantos, J. Gas Bubble Disease in Resident Fish of the Lower Clark Fork River. Trans. Am. Fish. Soc.
**2003**, 132, 865–876. [Google Scholar] [CrossRef] - Wang, Y.M.; Liang, R.F.; Tuo, Y.C.; Li, K.F.; Hodegs, B.R. Tolerance and Avoidance Behavior towards Gas Supersaturation in Rock Carp Procypris rabaudi with a History of Previous Exposure. N. Am. J. Aquac.
**2015**, 77, 478–484. [Google Scholar] [CrossRef] - Li, R.; Hodegs, B.R.; Feng, J.J.; Yong, X.D. Comparison of supersaturated total dissolved gas dissipation with dissolved oxygen dissipation and reaeration. J. Environ. Eng.
**2013**, 139, 385–390. [Google Scholar] [CrossRef] - Ou, Y.M.; Li, R.; Hodges, B.R.; Feng, J.J.; Pu, C.X. Impact of temperature on the dissipation process of supersaturated total dissolved gas in flowing water. Fresenius Environ. Bull.
**2016**, 25, 1927–1934. [Google Scholar] - Huang, J.P.; Li, R.; Feng, J.J.; Xu, W.L.; Wang, L.L. Relationship investigation between the dissipation process of supersaturated total dissolved gas and wind effect. Ecol. Eng.
**2016**, 95, 430–437. [Google Scholar] [CrossRef] - Ou, Y.M.; Li, R.; Tuo, Y.C.; Niu, J.L.; Feng, J.J.; Pu, C.X. The promotion effect of aeration on the dissipation of supersaturated total dissolved gas. Ecol. Eng.
**2016**, 95, 245–251. [Google Scholar] [CrossRef] - Shen, X.; Li, R.; Huang, J.P.; Feng, J.J.; Hodges, B.R.; Li, K.F.; Xu, W.L. Shelter construction for fish at the confluence of a river to avoid the effects of total dissolved gas supersaturation. Ecol. Eng.
**2016**, 97, 642–648. [Google Scholar] [CrossRef] - Witt, A.; Stewart, K.; Hadjerioua, B. Predicting Total Dissolved Gas Travel Time in Hydropower Reservoirs. J. Environ. Eng.
**2017**, 143, 06017011. [Google Scholar] [CrossRef] - Yuan, Y.Q.; Feng, J.J.; Li, R.; Huang, Y.H.; Huang, J.P.; Wang, Z.H. Modelling the promotion effect of vegetation on the dissipation of supersaturated total dissolved gas. Ecol. Model.
**2018**, 386, 89–97. [Google Scholar] [CrossRef] - Pickett, J.; Rueda, H.; Herold, M. Total Maximum Daily Load for Total Dissolved Gas in the Mid-Columbia River and Lake Roosevelt; U.S. Environmental Protection Agency and Washington State Department of Ecology: Washington, DC, USA, 2004; pp. 1–79.
- Perkins, W.A.; Richmond, M.C. MASS2, Modular Aquatic Simulation System in Two Dimensions: Theory and Numerical Methods; Pacific Northwest National Laboratory Report; Pacific Northwest National Laboratory: Washington, DC, USA, 2004; pp. 1–73.
- Johnson, E.L.; Clabough, T.S.; Peery, C.A.; Bennett, D.H.; Bjornn, T.C.; Caudill, C.C.; Richmond, M.C. Estimating adult Chinook salmon exposure to dissolved gas supersaturation downstream of hydroelectric dams using telemetry and hydrodynamic models. River Res. Appl.
**2010**, 23, 963–978. [Google Scholar] [CrossRef] - Ma, Q.; Li, R.; Zhang, Q.; Hodges, B.R.; Feng, J.J.; Yang, H.X. Two-Phase Flow Simulation of Supersaturated Total Dissolved Gas in the Plunge Pool of a High Dam. Environ. Prog. Sustain.
**2016**, 35, 1139–1148. [Google Scholar] [CrossRef] - Witt, A.; Magee, T.; Stewart, K.; Hadjerioua, B. Development and implementation of an optimization model for hydropower and total dissolved gas in the Mid-Columbia River System. J. Water Res. Plan.
**2017**, 143, 04017063. [Google Scholar] [CrossRef] - Feng, J.J.; Li, R.; Liang, R.F.; Shen, X. Eco-environmentally friendly operational regulation: An effective strategy to diminish the TDG supersaturation of reservoirs. Hydrol. Earth Syst. Sci.
**2014**, 10, 14355–14390. [Google Scholar] [CrossRef] - Aberle, J.; Järvelä, J. Hydrodynamics of Vegetated Channels, In Rivers-Physical, Fluvial and Environmental Processess; Springer International Publishing Switzerland: Bern, Switzerland, 2015; pp. 255–277. [Google Scholar]
- Nepf, H.M. Flow and Transport in Regions with Aquatic Vegetation. Annu. Rev. Fluid Mech.
**2012**, 44, 123–142. [Google Scholar] [CrossRef] [Green Version] - Afzalimehr, H.; Najafabadi, E.F.; Gallichand, J. Effects of accelerating and decelerating flows in a channel with vegetated banks and gravel bed. Int. J. Sediment Res.
**2012**, 27, 188–200. [Google Scholar] [CrossRef] - Serio, F.D.; Meftah, M.B.; Mossa, M.; Termini, D. Experimental investigation on dispersion mechanisms in rigid and flexible vegetated beds. Adv. Water Resour.
**2017**, 120, 98–113. [Google Scholar] [CrossRef] - Hamidifar, H.; Omid, M.H.; Keshavarzi, A. Kinetic energy and momentum correction coefficients in straight compound channels with vegetated floodplain. J. Hydrol.
**2016**, 537, 10–17. [Google Scholar] [CrossRef] - Dijk, W.M.V.; Teske, R.; Lageweg, W.I.V.D.; Kleinhans, M.G. Effects of vegetation distribution on experimental river channel dynamics. Water Resour. Res.
**2013**, 49, 7558–7574. [Google Scholar] [CrossRef] [Green Version] - Yang, J.Q.; Kerger, F.; Neph, H.M. Estimation of the bed shear stress in vegetated and bare channels with smooth beds. Water Resour. Res.
**2015**, 51, 3647–3663. [Google Scholar] [CrossRef] [Green Version] - Richet, J.B.; Ouvry, J.F.; Saunier, M. The role of vegetative barriers such as fascines and dense shrub hedges in catchment management to reduce runoff and erosion effects: Experimental evidence of efficiency, and conditions of use. Ecol. Eng.
**2016**, 103, 455–469. [Google Scholar] [CrossRef] - Liu, C.; Nepf, H. Sediment deposition within and around a finite patch of model vegetation over a range of channel velocity. Water Resour. Res.
**2016**, 52, 600–612. [Google Scholar] [CrossRef] [Green Version] - Hu, Z.H.; Lei, J.R.; Liu, C.; Nepf, H. Wake structure and sediment deposition behind models of submerged vegetation with and without flexible leaves. Adv. Water Resour.
**2018**, 118, 28–38. [Google Scholar] [CrossRef] - Lu, J.; Dai, H.C. Three dimensional numerical modeling of flows and scalar transport in a vegetated channel. J. Hydro-Environ. Res.
**2017**, 16, 27–33. [Google Scholar] [CrossRef] - Lu, J.; Dai, H.C. Numerical modeling on pollution transport in flexible vegetation. Appl. Math. Model.
**2018**, 64, 93–105. [Google Scholar] [CrossRef] - Huang, Y.H. Impact of Vegetation on the Dissipation Process of Supersaturated Total Dissolved Gas. Master’s Thesis, Sichuan University, Sichuan, China, 2017. (In Chinese). [Google Scholar]
- Huai, W.X.; Hu, Y.; Zeng, Y.H.; Han, J. Velocity distribution for open channel flows with suspended vegetation. Adv. Water Resour.
**2012**, 49, 56–61. [Google Scholar] [CrossRef] - Yang, Z.H.; Bai, F.P.; Huai, W.X.; An, R.D.; Wang, H.Y. Modelling open-channel flow with rigid vegetation based on two-dimensional shallow water equations using the lattice Boltzmann method. Ecol. Eng.
**2017**, 106, 75–81. [Google Scholar] [CrossRef] - Beudin, A.; Kalra, T.S.; Ganju, N.K.; Wamer, J.C. Development of a coupled wave-flow-vegetation interaction model. Comput. Geosci.
**2016**, 100, 76–86. [Google Scholar] [CrossRef] - Politano, M.S.; Carrica, P.M.; Turan, C.; Weber, L. A multidimensional two-phase flow model for the total dissolved gas downstream of spillways. J. Hydraul. Res.
**2007**, 45, 165–177. [Google Scholar] [CrossRef] - Politano, M.S.; Carrica, P.; Weber, L. A multiphase model for the hydrodynamics and total dissolved gas in tailraces. Int. J. Multiph. Flow
**2009**, 35, 1036–1050. [Google Scholar] [CrossRef] - Fu, X.L.; Li, D.; Zhang, X.F. Simulations of the three-dimensional total dissolved gas saturation downstream of spillways under unsteady conditions. J. Hydrodyn.
**2010**, 22, 598–604. [Google Scholar] [CrossRef] - Geldert, D.A.; Gulliver, J.S.; Wilhelms, S.C. Modeling dissolved gas supersaturation below spillway plunge pools. J. Hydraul. Eng.
**1998**, 124, 513–521. [Google Scholar] [CrossRef]

**Figure 6.**Distribution of the simulated results and experimental results of the surface velocity vector.

**Figure 7.**Distribution of the simulated vertical average velocity and experimental average velocity of the section behind the vertical columns. Section 15-2 is the lateral section behind the sixth row of vertical columns in Case 15; Section 25-2 is the lateral section behind the seventh row of vertical columns in Case 25.

**Figure 8.**Distribution of the simulated vertical average velocity and experimental vertical average velocity of the lateral section downstream of the vertical columns. Section 15-3 is the lateral section 5 cm downstream of the sixth row of vertical columns in Case 15. Section 25-3 is the lateral section 5 cm downstream of the seventh row of vertical columns in Case 25.

**Figure 9.**Distribution of the simulated vertical average velocity and the experimental vertical average velocity of the longitudinal section between the front row and back row of vertical columns. (

**a**) Section 15-4 is the longitudinal section behind the second vertical column of sixth row (Y = 10.7 cm) in Case 15; (

**b**) Section 15-5 is the longitudinal section before the second vertical column of the seventh raw (Y = 14.3 cm) in Case 15; (

**c**) Section 25-4 is the longitudinal section behind the second vertical column of seventh row (Y = 7.1 cm) in Case 25; (

**d**) Section 25-5 is the longitudinal section before the second vertical column of seventh row (Y = 10.7 cm) in Case 25.

Case No. | Flow (L·s^{−1}) | The Vegetation Density | Lateral Space L_{y} (10^{−2} m) | Longitudinal Space L_{x} (10^{−2} m) | The Flow Depth h (10^{−2} m) | TDG Saturation of Entry Section (%) |
---|---|---|---|---|---|---|

1 | 1.5 | 0 | / | / | 2.2 | 144.5 |

2 | 3.5 | 0 | / | / | 3.8 | 144.4 |

3 | 5.5 | 0 | / | / | 4.7 | 145.6 |

4 | 7.5 | 0 | / | / | 5.6 | 147.1 |

5 | 9.5 | 0 | / | / | 6.6 | 148.1 |

6 | 1.5 | 0.1 | 12.5 | 60 | 2.5 | 144.2 |

7 | 3.5 | 0.1 | 12.5 | 60 | 4.4 | 144.6 |

8 | 5.5 | 0.1 | 12.5 | 60 | 5.3 | 144.5 |

9 | 7.5 | 0.1 | 12.5 | 60 | 6.4 | 146.6 |

10 | 9.5 | 0.1 | 12.5 | 60 | 7.5 | 147.9 |

11 | 1.5 | 0.2 | 7.1 | 60 | 2.8 | 144.9 |

12 | 3.5 | 0.2 | 7.1 | 60 | 4.6 | 147.4 |

13 | 5.5 | 0.2 | 7.1 | 60 | 5.8 | 148.5 |

14 | 7.5 | 0.2 | 7.1 | 60 | 7.2 | 148.6 |

15 | 9.5 | 0.2 | 7.1 | 60 | 8.6 | 149.0 |

16 | 1.5 | 0.3 | 12.5 | 20 | 3.1 | 145.3 |

17 | 3.5 | 0.3 | 12.5 | 20 | 4.7 | 145.9 |

18 | 5.5 | 0.3 | 12.5 | 20 | 6.6 | 147.5 |

19 | 7.5 | 0.3 | 12.5 | 20 | 7.8 | 147.8 |

20 | 9.5 | 0.3 | 12.5 | 20 | 9.0 | 148.6 |

21 | 1.5 | 0.6 | 7.1 | 20 | 3.2 | 145.7 |

22 | 3.5 | 0.6 | 7.1 | 20 | 5.9 | 147.5 |

23 | 5.5 | 0.6 | 7.1 | 20 | 7.6 | 148.5 |

24 | 7.5 | 0.6 | 7.1 | 20 | 8.4 | 149.3 |

25 | 9.5 | 0.6 | 7.1 | 20 | 9.6 | 149.7 |

Case No. | Flow Rate (L·s^{−1}) | Vegetation Density | TDG Saturation Upstream (%) | TDG Saturation Downstream (%) | k_{in} (s^{−1}) |
---|---|---|---|---|---|

1 | 1.5 | 0 | 144.5 | 137.2 | 4.5 × 10^{−5} |

2 | 3.5 | 0 | 144.4 | 138.9 | 5.5 × 10^{−5} |

3 | 5.5 | 0 | 145.6 | 140.2 | 6.0 × 10^{−5} |

4 | 7.5 | 0 | 147.1 | 142.0 | 6.5 × 10^{−5} |

5 | 9.5 | 0 | 148.1 | 143.9 | 7.8 × 10^{−5} |

6 | 1.5 | 0.1 | 144.2 | 135.2 | 4.3 × 10^{−5} |

7 | 3.5 | 0.1 | 144.6 | 137.8 | 5.2 × 10^{−5} |

8 | 5.5 | 0.1 | 144.5 | 138.3 | 5.8 × 10^{−5} |

9 | 7.5 | 0.1 | 146.6 | 141.1 | 6.3 × 10^{−5} |

10 | 9.5 | 0.1 | 147.9 | 143.5 | 7.5 × 10^{−5} |

11 | 1.5 | 0.2 | 144.9 | 133.5 | 4.0 × 10^{−5} |

12 | 3.5 | 0.2 | 147.4 | 139.2 | 5.0 × 10^{−5} |

13 | 5.5 | 0.2 | 148.5 | 141.5 | 5.5 × 10^{−5} |

14 | 7.5 | 0.2 | 148.6 | 142.6 | 6.0 × 10^{−5} |

15 | 9.5 | 0.2 | 149.0 | 144.1 | 7.2 × 10^{−5} |

16 | 1.5 | 0.3 | 145.3 | 131.3 | 3.7 × 10^{−5} |

17 | 3.5 | 0.3 | 145.9 | 135.9 | 4.7 × 10^{−5} |

18 | 5.5 | 0.3 | 147.5 | 139.0 | 5.2 × 10^{−5} |

19 | 7.5 | 0.3 | 147.8 | 140.8 | 5.8 × 10^{−5} |

20 | 9.5 | 0.3 | 148.6 | 143.0 | 7.0 × 10^{−5} |

21 | 1.5 | 0.6 | 145.7 | 128.2 | 3.0 × 10^{−5} |

22 | 3.5 | 0.6 | 147.5 | 134.2 | 4.1 × 10^{−5} |

23 | 5.5 | 0.6 | 148.5 | 138.0 | 4.5 × 10^{−5} |

24 | 7.5 | 0.6 | 149.3 | 140.2 | 5.0 × 10^{−5} |

25 | 9.5 | 0.6 | 149.7 | 142.7 | 6.3 × 10^{−5} |

**Table 3.**Comparison of the inner dissipation coefficient of supersaturated total dissolved gasbetween calculated results and experimental data.

Case No. | Flow (L·s^{−1}) | Vegetation Density | k_{in} Calculated (s^{−1}) | k_{in} Calibrated (s^{−1}) |
---|---|---|---|---|

5 | 9.5 | 0 | 8.1 × 10^{−5} | 8.0 × 10^{−5} |

10 | 9.5 | 0.1 | 7.7 × 10^{−5} | 7.5 × 10^{−5} |

15 | 95 | 0.2 | 7.3 × 10^{−5} | 7.2 × 10^{−5} |

20 | 9.5 | 0.3 | 7.0 × 10^{−5} | 7.0 × 10^{−5} |

25 | 9.5 | 0.6 | 6.1 × 10^{−5} | 6.2 × 10^{−5} |

Case No. | Flow (m^{3}·s^{−1}) | The Water Depth of Inlet Section (10^{−2} m) | The TDG Saturation of Inlet Section (%) |
---|---|---|---|

15 | 0.0095 | 7.4 | 149.0 |

25 | 0.0095 | 8.4 | 149.7 |

**Table 5.**Error analysis results. RSD: relative average deviation; STD: standard deviation; RMSE: root-mean-square error.

Section No. | Mean Error (m/s) | RSD (%) | STD | RMSE (m/s) |
---|---|---|---|---|

Section 15-1 | 0.018 | 7.2 | 0.02 | 0.023 |

Section 15-2 | 0.019 | 6.5 | 0.02 | 0.009 |

Section 15-3 | 0.015 | 6.8 | 0.02 | 0.004 |

Section 15-4 | 0.008 | 3.3 | 0.01 | 0.006 |

Section 15-5 | 0.013 | 6.4 | 0.02 | 0.008 |

Section 25-1 | 0.017 | 8.6 | 0.02 | 0.019 |

Section 25-2 | 0.010 | 4.7 | 0.01 | 0.007 |

Section 25-3 | 0.012 | 6.8 | 0.01 | 0.004 |

Section 25-4 | 0.011 | 5.5 | 0.01 | 0.008 |

Section 25-5 | 0.011 | 6.4 | 0.01 | 0.002 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yuan, Y.; Huang, Y.; Feng, J.; Li, R.; An, R.; Huang, J.
Numerical Model of Supersaturated Total Dissolved Gas Dissipation in a Channel with Vegetation. *Water* **2018**, *10*, 1769.
https://doi.org/10.3390/w10121769

**AMA Style**

Yuan Y, Huang Y, Feng J, Li R, An R, Huang J.
Numerical Model of Supersaturated Total Dissolved Gas Dissipation in a Channel with Vegetation. *Water*. 2018; 10(12):1769.
https://doi.org/10.3390/w10121769

**Chicago/Turabian Style**

Yuan, Youquan, Yinghan Huang, Jingjie Feng, Ran Li, Ruidong An, and Juping Huang.
2018. "Numerical Model of Supersaturated Total Dissolved Gas Dissipation in a Channel with Vegetation" *Water* 10, no. 12: 1769.
https://doi.org/10.3390/w10121769