# Theoretical Model and Solution of Dynamic Evolution in Initial Stage of Lacustrine Delta

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Theoretical Mode of Plane Jet Boundary Layer in the Initial Stage of Delta

#### 2.1. Theoretical Model

_{0}, and the velocity of entrainment and the mixing part between the jet core and the static liquid is less than u

_{0}, which is called the jet boundary layer. The jet zone with a longitudinal velocity not equal to zero is a combination of the upper and lower boundary layers bounded by the center line, as shown in Figure 1.

_{x}, u

_{y}are the velocities in x and y directions respectively; h is water depth; ${\rho}_{m},{\rho}_{s},\Delta \rho ,{\rho}_{0}$ are sediment-laden flow density, sediment density, density difference between sediment and clear water, and mixed density of bed sediment and saturated pore water respectively; y

_{0}is the height of the bed sediment bottom; h

_{0}is the height of the deposit; S is the average sediment concentration of the cross-section; ε

_{x}and ε

_{y}are turbulent diffusion coefficients in x and y directions, respectively; f

_{x}, f

_{y}are unit mass forces in x and y directions, respectively; $\upsilon $ is the dynamic viscosity coefficient; ${\tau}_{sx},{\tau}_{sy}$ are wind stresses in x and y directions, respectively; n is the roughness coefficient.

#### 2.2. Similarity Solution Method of the Model

#### 2.3. Calculation of Erosion and Deposition of Bed Surface

_{m}is the maximum sediment concentration of muddy water to maintain fluid characteristics and can be calculated as follows [31]:

_{l}is in millimeters; ${q}_{blx}={q}_{bl}\frac{{u}_{x}}{\sqrt{{u}_{x}^{2}+{u}_{y}^{2}}}$, ${q}_{bly}={q}_{bl}\frac{{u}_{y}}{\sqrt{{u}_{x}^{2}+{u}_{y}^{2}}}$, and ${q}_{bl}$ is the bedload transport rate of group l, which can be calculated according to the method of Levy [31]:

_{l}= 0.25–23 mm, h/d

_{l}= 5–500, U/u

_{lc}= 1–3.5.

## 3. Validation of the Model

^{3}/s was selected. The selection of grain size mainly considered that sediment particles were easy to flocculate if they were too fine, whereas it should not be too large combined with the selected discharge value; thus, a kind of natural sediment having a median grain size d = 0.3 mm was selected in our experiment.

_{s}= 2650 kg/m

^{3}and grain size d = 0.3 mm was used at the entrance and the bottom of the flume) and can prevent the erosion from reaching the bottom.

_{s}/t and z

_{s}in dimensionless form. The following nondimensional variables are introduced:

_{s}of sections x = 0.4 m and x = 0.5 m, as shown in Figure 4d–e.

_{s}/t on the x = 0.5 m section lies at y = −0.05 m and y = 0.2 m, followed by y = 0.05 m. Combined with the analysis of experimental data, the calculation result in this paper is more reasonable.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Joshi, P.B. Hydromechanics of tidal jets. J. Waterw. Port Coast. Ocean. Div.
**1982**, 108, 239–253. [Google Scholar] [CrossRef] - Ortega-Sánchez, M.; Losada, M.A.; Baquerizo, A. A global model of a tidal jet including the effects of friction and bottom slope. J. Hydraul. Res.
**2008**, 46, 80–86. [Google Scholar] [CrossRef] - Chen, X.; Zheng, B.M.; Hu, C.H. Stochastic analysis of sediment movement in the Three Gorges Reservoir. J. Sediment Res.
**2013**, 12, 6–11. [Google Scholar] - Falcini, F.; Jerolmack, D.J. A potential vorticity theory for the formation of elongate channels in river deltas and lakes. J. Geophys. Res. Earth Surf.
**2010**, 115, F04038. [Google Scholar] [CrossRef] - Nardin, W.; Mariotti, G.; Edmonds, D.A.; Guercio, R.; Fagherazzi, S. Growth of river mouth bars in sheltered bays in the presence of frontal waves. J. Geophys. Res. Earth Surf.
**2013**, 118, 872–886. [Google Scholar] [CrossRef] - Leonardi, N.; Canestrelli, A.; Sun, T.; Fagherazzi, S. Effect of tides on mouth bar morphology and hydrodynamics. J. Geophys. Res. Ocean.
**2013**, 118, 4169–4183. [Google Scholar] [CrossRef] - Abramovich, G.N. The Theory of Turbulent Jets; The MIT Press: Cambridge, UK, 2003. [Google Scholar] [CrossRef]
- Rajaratnam, N. Turbulent Jets; Elsevier: Amsterdam, The Netherlands, 1976. [Google Scholar]
- Schlichting, H.; Gersten, K. Boundary-Layer Theory; Springer: Berlin/Heidelberg, Germany, 2017. [Google Scholar]
- Rajaratnam, N.; Stalker, M.J. Circular wall jets in coflowing streams. J. Hydraul. Div.
**1982**, 108, 187–198. [Google Scholar] [CrossRef] - Özsoy, E.; Ünlüata, Ü. Ebb-tidal flow characteristics near inlets. Estuar. Coast. Shelf Sci.
**1982**, 14, 251–263. [Google Scholar] [CrossRef] - Taylor, R.B.; Dean, R.G. Exchange Characteristics of Tidal Inlets. Coast. Eng. Proc.
**1974**, 14, 132. [Google Scholar] [CrossRef] - Wang, F.C. The dynamics of a river-bay-delta system. J. Geophys. Res. Ocean.
**1984**, 89, 8054–8060. [Google Scholar] [CrossRef] - Muto, T.; Steel, R.J. Retreat of the front in a prograding delta. Geology
**1992**, 20, 967. [Google Scholar] [CrossRef] - Parker, G.; Muto, T. 1D numerical model of delta response to rising sea level. In Proceedings of the Third IAHR Symposium, River, Coastal and Estuarine Morphodynamics, Barcelona, Spain, 1–5 September 2003; pp. 1–10. [Google Scholar]
- Edmonds, D.A.; Shaw, J.B.; Mohrig, D. Topset-dominated deltas: A new model for river delta stratigraphy. Geology
**2011**, 39, 1175–1178. [Google Scholar] [CrossRef] - Price, W.E., Jr. Simulation of alluvial fan deposition by a random walk model. Water Resour. Res.
**1974**, 10, 263–274. [Google Scholar] [CrossRef] - Han, Q.W. Delta deposition in reservoir(i). J. Lake Sci.
**1995**, 2, 107–118. [Google Scholar] - Han, Q.W. Delta deposition in reservoir(ii). J. Lake Sci.
**1995**, 3, 213–225. [Google Scholar] - Jiménez-Robles, A.M.; Ortega-Sánchez, M.; Losada, M.A. Effects of basin bottom slope on jet hydrodynamics and river mouth bar formation. J. Geophys. Res. Earth Surf.
**2016**, 121, 1110–1133. [Google Scholar] [CrossRef] - Xie, J.H. The Simulation of River; Water Conservancy and Electric Power Press: Beijing, China, 1990; pp. 8–20. [Google Scholar]
- Barenblatt, G.I. Scaling, Self-similarity, and Intermediate Asymptotics: References; Cambridge University Press: Cambridge, UK, 1996. [Google Scholar]
- Forthmann, E. Turbulent Jet Expansion; Archive of Applied Mechanics: Washington, DC, USA, 1936. [Google Scholar]
- Albertson, M.L.; Dai, Y.; Jensen, R.A.; Rouse, H. Diffusion of Submerged Jets. Trans. Am. Soc. Civ. Eng.
**1950**, 115, 639–664. [Google Scholar] [CrossRef] - Bradbury, J.L. An Investigation into the Structure of a Turbulent Plane Jet. Ph.D. Thesis, Queen Mary, University of London, London, UK, 1963. [Google Scholar]
- Swenson, J.B.; Voller, V.R.; Paola, C.; Parker, G.; Marr, J.G. Fluviodeltaic Sedimentation: A Generalized Stefan Problem. Eur. J. Appl. Maths.
**2000**, 11, 433–452. [Google Scholar] [CrossRef] - State Key Laboratory of Hydraulics and Mountain River Development and Protection of Sichuan University. Hydraulics; Higher Education Press: Beijing, China, 2016; pp. 210–217. [Google Scholar]
- Yu, C.Z. Introduction to Environmental Hydrodynamics; Tsinghua University Press: Beijing, China, 1992; pp. 202–206. [Google Scholar]
- Shirazi, M.A.; Davis, L.R. Workbook of Thermal Plume Prediction 2; National Environmental Research Center: Nagpur, India, 1974. [Google Scholar]
- Stolzenbach, K.D.; Harleman, D.R.F. An Analytical and Experimental Investigation of Surface Discharges of Heated Water; Cambridge University Press: Cambridge, UK, 1971. [Google Scholar]
- Qian, N.; Wan, Z.H. Mechanics of Sediment Movement; Science Press: Beijing, China, 1991; p. 231. [Google Scholar]
- Hoyal, D.C.J.D.; Sheets, B.A. Morphodynamic evolution of experimental cohesive deltas. J. Geophys. Res. Earth Surf.
**2009**, 114, F02009. [Google Scholar] [CrossRef] - Clarke, L.; Quine, T.A.; Nicholas, A. An experimental investigation of autogenic behaviour during alluvial fan evolution. Geomorphology
**2010**, 115, 278–285. [Google Scholar] [CrossRef] - Zhang, X.; Wang, S.; Wu, X.; Xu, S.; Li, Z. The development of a laterally confined laboratory fan delta under sediment supply reduction. Geomorphology
**2016**, 257, 120–133. [Google Scholar] [CrossRef]

**Figure 4.**Calculated and measured values of erosion and deposition of each section. (

**a**). Our solution Zc* and measured value Zm* of section of x = 0.1 m. (

**b**). Our solution Zc* and measured value Zm* of section of x = 0.2 m. (

**c**). Our solution Zc* and measured value Zm* of section of x = 0.3 m. (

**d**). Our solution Zc* and measured value Zm* of section of x = 0.4 m. (

**e**). Our solution Zc* and measured value Zm* of section of x = 0.5 m.

Quantity | Value |
---|---|

discharge Q (cm^{3}/s) | 100 |

depth h (cm) | 2.0 |

slope J (/) | 1% |

particle size d (mm) | 0.3 |

roughness n (/) | 0.02 |

volumetric sediment supplied rate S_{v} (/) | 0.2% |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Xin, W.; Yang, Y.; Xie, W.; Ji, Z.; Bai, Y.
Theoretical Model and Solution of Dynamic Evolution in Initial Stage of Lacustrine Delta. *Water* **2022**, *14*, 2522.
https://doi.org/10.3390/w14162522

**AMA Style**

Xin W, Yang Y, Xie W, Ji Z, Bai Y.
Theoretical Model and Solution of Dynamic Evolution in Initial Stage of Lacustrine Delta. *Water*. 2022; 14(16):2522.
https://doi.org/10.3390/w14162522

**Chicago/Turabian Style**

Xin, Weiyan, Yanhua Yang, Wude Xie, Ziqing Ji, and Yuchuan Bai.
2022. "Theoretical Model and Solution of Dynamic Evolution in Initial Stage of Lacustrine Delta" *Water* 14, no. 16: 2522.
https://doi.org/10.3390/w14162522