# Numerical Simulation Analysis of the Formation and Morphological Evolution of Asymmetric Crescentic Dunes

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Research Methods and Parameter Setting

#### 2.1. Model of Dunes

#### 2.2. Simulation of Shear Stress on Dunes’ Surface by Wind

#### 2.3. Relation between Starting Velocity of Wind and Topographic Angle

#### 2.4. Simulation Methods of Sand Flux

#### 2.5. Evolution of Dune’s Morphology

_{75}is the median size of the sand particles (3 mm), and ${\tau}_{c}$ is the dimensionless critical shear stress (0.05).

#### 2.6. Simulation of Wind Flow Field

^{−5}·m

^{2}·s

^{−1}. $v$ represents the viscous coefficient of the air. u represents the calculated mean velocity of wind. ${\mathrm{V}}_{\mathrm{r}}$ is the ratio and the relative velocity between the sand particle and the wind flow field. To obtain the accurate and stable results, a constant uniform wind field was used in this study.

#### 2.7. Interaction Force between Sand Particles and Wind Flow

#### 2.8. Validation of the Accuracy of the Simulation

## 3. Results

#### 3.1. Simulation of the Effects of Bi-Directional Winds on Morphological Changes in Sand Dunes

#### 3.2. Simulation of the Effects of Sand Particles’ Size on Morphological Changes in Crescentic Dunes

#### 3.3. Simulation of the Effects of Topography on Morphological Changes in Crescentic Dunes

#### 3.4. Simulation of the Effects of Epiphytic Vegetation on Morphological Changes in Crescentic Dunes

#### 3.5. Simulation of the Effects of Dune Collisions on Morphological Changes in Crescentic Dunes

^{3}and 18.27 m

^{3}and the volume ratio was 0.23, while the volume of the initial two dunes in Figure 7b was 2.98 m

^{3}and 14.38 m

^{3}and the volume ratio was 0.20. The evolved volume of the two dunes in Figure 7a was 18.9 m

^{3}and 3.58 m

^{3}, respectively, while that in Figure 7b was 16.68 m

^{3}and 0.68 m

^{3}, respectively. Two dunes with a similar volume ratio evolved new dunes with a large volume difference after collision, indicating that the sand flux in the downwind direction was the reason for the evolution of the dunes. The volume ratio and collision position of the two dunes determined the future evolution of the dunes. The reflux zone of the leeward slope also greatly affected the sand flux from the front dune, thus redistributing the spatial distribution of the sand particles moving downwind. It is generally believed that the larger the volume of sand dunes, the slower the velocity of movement. When two dunes collide, if the crests overlap, they converge to form a larger dune with a lower migration velocity; if the crests do not overlap, and the local sustained sand flux is interrupted, a new small dune forms. The small dune, therefore, accelerates the separation of the two dunes because of the faster movement.

## 4. Discussion

#### 4.1. Influence of Bi-Directional Winds on the Asymmetry of Sand Dunes

#### 4.2. Influence of Different Size of Sand Particles on the Asymmetry of Sand Dunes

#### 4.3. Influence of Topography on the Asymmetry of Sand Dunes

#### 4.4. Influence of Epiphytic Vegetation on the Asymmetry of Sand Dunes

#### 4.5. Influence of Collision of Dunes on the Asymmetry of Sand Dunes

#### 4.6. Erosion and Mechanism of Asymmetric Crescentic Sand Dunes on the Environment

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Song, X.; Wang, T.; Xue, X.; Yan, C.; Li, S. Monitoring and analysis of aeolian desertification dynamics from 1975 to 2010 in the Heihe River Basin, northwestern China. Environ. Earth Sci.
**2015**, 74, 3123–3133. [Google Scholar] [CrossRef] - Silvestro, S.; Fenton, L.K.; Vaz, D.A.; Bridges, N.T.; Ori, G.G. Ripple migration and dune activity on Mars: Evidence for dynamic wind processes. Geophys. Res. Lett.
**2010**, 37, 95–108. [Google Scholar] [CrossRef] - Livingstone, I.; Wiggs, G.F.; Weaver, C.M. Geomorphology of desert sand dunes: A review of recent progress. Earth-Sci. Rev.
**2007**, 80, 239–257. [Google Scholar] [CrossRef] - Hersen, P.; Douady, S. Collision of barchan dunes as a mechanism of size regulation. Geophys. Res. Lett.
**2005**, 32, 97–116. [Google Scholar] [CrossRef] - White, B.R. Soil transport by winds on Mars. J. Geophys. Res. Solid Earth
**1979**, 84, 4643–4651. [Google Scholar] [CrossRef] - Albertson, J.D.; Parlange, M. Surface length scales and shear stress: Implications for land-atmosphere interaction over complex terrain. Water Resour. Res.
**1999**, 35, 2121–2132. [Google Scholar] [CrossRef] [Green Version] - Ma, X.J.; Yan, Y.; Song, X.; Liu, J.; Zhang, P.A. Traykovsk. Morphology and maintenance of steep dunes near dune asymmetry transitional areas on the shallow shelf (Beibu Gulf, northwest South China Sea). Mar. Geol.
**2019**, 412, 37–52. [Google Scholar] [CrossRef] - Bourke, M.C. Barchan dune asymmetry: Observations from Mars and Earth. Icarus
**2010**, 205, 183–197. [Google Scholar] [CrossRef] - Tsoar, H.; Parteli, E.J.R. Bidirectional winds, barchan dune asymmetry and formation of seif dunes from barchans: A discussion. Environ. Earth Sci.
**2016**, 75, 1237. [Google Scholar] [CrossRef] - Parteli, E.J.; Durán, O.; Bourke, M.C.; Tsoar, H.; Pöschel, T.; Herrmann, H. Origins of barchan dune asymmetry: Insights from numerical simulations. Aeolian Res.
**2014**, 12, 121–133. [Google Scholar] [CrossRef] [Green Version] - Bossard, V.; Lerma, A.N. Geomorphologic characteristics and evolution of managed dunes on the South West Coast of France. Geomorphology
**2020**, 367, 107312. [Google Scholar] [CrossRef] - Qian, G.; Yang, Z.; Dong, Z.; Luo, W.; Zhang, Z.; Lu, J. Long-term measurements of aeolian transport directional variations over a zibar surface in the northern Kumtagh Sand Sea. Geomorphology
**2020**, 371, 107452. [Google Scholar] [CrossRef] - Chen, G.; Wang, W.; Sun, C.; Li, J. 3D numerical simulation of wind flow behind a new porous fence. Powder Technol.
**2012**, 230, 118–126. [Google Scholar] [CrossRef] - Luna, M.C.D.M.; Parteli, E.J.; Herrmann, H.J. Model for a dune field with an exposed water table. Geomorphology
**2012**, 159–160, 169–177. [Google Scholar] [CrossRef] [Green Version] - Klose, M.; Shao, Y. Large-eddy simulation of turbulent dust emission. Aeolian Res.
**2013**, 8, 49–58. [Google Scholar] [CrossRef] - Hleibieh, J.; Wegener, D.; Herle, I. Numerical simulation of a tunnel surrounded by sand under earthquake using a hypoplastic model. Acta Geotech.
**2014**, 9, 631–640. [Google Scholar] [CrossRef] - Huang, H.; Bo, T.; Zheng, X. Numerical modeling of wind-blown sand on Mars. Eur. Phys. J. E
**2014**, 37, 80. [Google Scholar] [CrossRef] - Lecrivain, G.; Sevan, D.-M.; Thomas, B.; Hampel, U. Numerical simulation of multilayer deposition in an obstructed channel flow. Adv. Powder Technol.
**2014**, 25, 310–320. [Google Scholar] [CrossRef] - Chen, X.; Chen, L.; He, L. Modelling Three-dimensional Interfacial Flow with Sand Dunes. Procedia Eng.
**2015**, 126, 382–387. [Google Scholar] [CrossRef] [Green Version] - Sauermann, G.; Kroy, K.; Herrmann, H.J. A Continuum Saltation Model for Sand Dunes. Phys. Rev. E
**2001**, 64, 31305. [Google Scholar] [CrossRef] [Green Version] - Durán, O.; Herrmann, H. Modelling of saturated sand flux. J. Stat. Mech. Theory Exp.
**2006**, 7, P07011. [Google Scholar] [CrossRef] - Durán, O.; Parteli, E.J.; Herrmann, H.J. A continuous model for sand dunes: Review, new developments and application to barchan dunes and barchan dune fields. Earth Surf. Process. Landf.
**2010**, 35, 1591–1600. [Google Scholar] [CrossRef] - Schwämmle, V.; Herrmann, H.J. A model of Barchan dunes including lateral shear stress. Eur. Phys. J. E
**2005**, 16, 57–65. [Google Scholar] [CrossRef] - Coleman, S.E.; Nikora, V.I. Exner equation: A continuum approximation of a discrete granular system. Water Resour. Res.
**2009**, 45, 706–715. [Google Scholar] [CrossRef] - Kubatko, E.J.; Westerink, J.J. Exact Discontinuous Solutions of Exner’s Bed Evolution Model: Simple Theory for Sediment Bores. J. Hydraul. Eng.
**2007**, 133, 305–311. [Google Scholar] [CrossRef] [Green Version] - Giudice, A.L.; Nuca, R.; Preziosi, L.; Coste, N. Wind-blown particulate transport: A review of computational fluid dynamics models. Math. Eng.
**2019**, 1, 508–547. [Google Scholar] [CrossRef] - Abadie, S.; Morichon, D.; Grilli, S.; Glockner, S. Numerical simulation of waves generated by landslides using a multiple-fluid Navier–Stokes model. Coast. Eng.
**2010**, 57, 779–794. [Google Scholar] [CrossRef] - Mohotti, D.; Wijesooriya, K.; Dias-da-Costa, D. Comparison of Reynolds Averaging Navier-Stokes (RANS) turbulent models in predicting wind pressure on tall buildings. J. Build. Eng.
**2018**, 21, 1–17. [Google Scholar] [CrossRef] [Green Version] - Zhang, Y.; Rinoshika, A. Multi-scale vortical structure analysis on large eddy simulation of dune wake flow. J. Vis.
**2015**, 18, 95–109. [Google Scholar] [CrossRef] - Omidyeganeh, M.; Piomelli, U.; Christensen, K.T.; Best, J.L. Large eddy simulation of interacting barchan dunes in a steady, unidirectional flow. J. Geophys. Res. Earth Surf.
**2013**, 118, 2089–2104. [Google Scholar] [CrossRef] [Green Version] - Lu, J.; Wang, L.L.; Zhu, H.; Dai, H.C. Large eddy simulation of water flow over series of dunes. Water Sci. Eng.
**2011**, 4, 421–430. [Google Scholar] - Stoesser, T.; Braun, C.; Garcia-Villalba, M.; Rodi, W. Turbulence Structures in Flow over Two-Dimensional Dunes. J. Hydraul. Eng.
**2008**, 134, 44–55. [Google Scholar] [CrossRef] - Durán, O.; Schwämmle, V.; Herrmann, H. Breeding and solitary wave behavior of dunes. Phys. Rev. E Stat. Nonlinear Soft Matter Phys.
**2005**, 72, 021308. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bagnold, R.A. The Physics of Blown Sand and Desert Dunes 1941; Springer: Dordrecht, The Netherlands, 1981. [Google Scholar]
- Tsoar, H. Internal Structure and Surface Geometry of Longitudinal (Seif) Dunes. J. Sediment. Res.
**1982**, 52, 823–831. [Google Scholar] [CrossRef] - Rubin, D.M.; Tsoar, H.; Blumberg, D.G. A second look at western Sinai seif dunes and their lateral migration. Geomorphology
**2008**, 93, 335–342. [Google Scholar] [CrossRef] - Howard, A.D.; Morton, J.B.; Gad-El-Hak, M.; Pierce, D.B. Sand transport model of barchan dune equilibrium. Sedimentology
**1978**, 25, 307–338. [Google Scholar] [CrossRef] - Ewing, R.C.; Kocurek, G.A. Aeolian dune interactions and dune-field pattern formation: White Sands Dune Field, New Mexico. Sedimentology
**2010**, 57, 1199–1219. [Google Scholar] [CrossRef] - Wang, C.; Tang, Z.; Bristow, N.; Blois, G.; Christensen, K.; Anderson, W. Numerical and experimental study of flow over stages of an offset merger dune interaction. Comput. Fluids
**2017**, 158, 72–83. [Google Scholar] [CrossRef] - Kok, J.F.; Parteli, E.J.; Michaels, T.I.; Karam, D.B. The physics of wind-blown sand and dust. Rep. Prog. Phys. Phys. Soc.
**2012**, 75, 106901. [Google Scholar] [CrossRef] [Green Version] - Smith, A.B.; Jackson, D.W.; Cooper, J.A.G. Three-dimensional airflow and sediment transport patterns over barchan dunes. Geomorphology
**2016**, 278, 28–42. [Google Scholar] [CrossRef] - Pont, S.C.D. Dune morphodynamics. C. R. Phys.
**2015**, 16, 118–138. [Google Scholar] [CrossRef] - Niiya, H.; Awazu, A.; Nishimori, H. Stability of transverse dunes against perturbations: A theoretical study using dune skeleton model. Aeolian Res.
**2013**, 9, 63–68. [Google Scholar] [CrossRef] [Green Version] - Hanocha, G.; Yizhaqb, H.; Ashkenazyb, Y. Modeling the bistability of barchan and parabolic dunes. Aeolian Res.
**2018**, 35, 9–18. [Google Scholar] [CrossRef] - Prigozhin, L. Nonlinear dynamics of Aeolian sand ripples. Phys. Rev. E
**1999**, 60, 729–733. [Google Scholar] [CrossRef] [Green Version] - Jiang, H.; Huang, N.; Zhu, Y. Analysis of Wind-blown Sand Movement over Transverse Dunes. Sci. Rep.
**2014**, 4, 7114. [Google Scholar] [CrossRef] [Green Version] - Andreotti, B. A two-species model of aeolian sand transport. J. Fluid Mech.
**2005**, 510, 47–70. [Google Scholar] [CrossRef] [Green Version] - Raffaele, L.; Bruno, L.; Pellerey, F.; Preziosi, L. Windblown sand saltation: A statistical approach to fluid threshold shear velocity. Aeolian Res.
**2016**, 23, 79–91. [Google Scholar] [CrossRef] [Green Version] - Burkow, M.; Griebel, M. Numerical simulation of the temporal evolution of a three dimensional barchanoid dune and the corresponding sediment dynamics. Comput. Fluids
**2018**, 166, 275–285. [Google Scholar] [CrossRef] - Zhang, Y.; Deng, S.; Wang, X. RANS and DDES simulations of a horizontal-axis wind turbine under stalled flow condition using OpenFOAM. Energy
**2019**, 167, 1155–1163. [Google Scholar] [CrossRef] - Wu, Z.; Zhang, H.; Liu, S. Dynamic process simulation of a glacier on Qilian Mountain based on a thermo-mechanically coupled model. Sci. Total Environ.
**2021**, 781, 147027. [Google Scholar] - Zhang, H.; Wu, Z.; Hu, J.; Zhang, Z.; Xiao, B.; Ma, J. Numerical simulation of wind field and sand flux in crescentic sand dunes. Sci. Rep.
**2021**, 11, 4973. [Google Scholar] [CrossRef] [PubMed] - Anderson, R.S.; Haff, P.K. Wind modifacation and bed response during saltation in air. Acta Mech.
**1991**, 1, 21–51. [Google Scholar] - Parteli, E.J.R.; Duran, O.; Tsoar, H.; Schwammle, V.; Herrmann, H.J. Dune formation under bimodal winds. Proc. Natl. Acad. Sci. USA
**2009**, 106, 22085–22089. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bourke, M.C.; Goudie, A.S. Varieties of barchan form in the Namib Desert and on Mars. Aeolian Res.
**2009**, 1, 45–54. [Google Scholar] [CrossRef] [Green Version] - Folk, R.L. Longitudinal dunes of the northwestern edge of the Simpson Desert. Sedimentology
**2010**, 16, 5–54. [Google Scholar] [CrossRef] - Li, Y.; Yi, G. Numerical simulation of aeolian dusty sand transport in a marginal desert region at the early entrainment stage. Geomorphology
**2008**, 100, 335–344. [Google Scholar] [CrossRef] - Cheng, H.; Zou, X.; Zhang, C. Probability distribution functions for the initial liftoff velocities of saltating sand grains in air. J. Geophys. Res. Atmos.
**2006**, 111, 22205. [Google Scholar] [CrossRef] [Green Version] - Lancaster, N. Grain-Size Characteristics of Linear Dunes in the Southwestern Kalahari. J. Sediment. Res.
**1986**, 57, 573–574. [Google Scholar] [CrossRef] - Lancaster, N.; Nickling, W.G.; Neuman, C.M.; Wyatt, V.E. Sediment flux and airflow on the stoss slope of a barchan dune. Geomorphology
**1996**, 17, 55–62. [Google Scholar] [CrossRef] - Tsoar, H.; Levin, N.; Porat, N.; Maia, L.P.; Herrmann, H.J.; Tatumi, S.H.; Claudino-Sales, V. The effect of climate change on the mobility and stability of coastal sand dunes in Ceará State (NE Brazil). Quat. Res.
**2009**, 71, 217–226. [Google Scholar] [CrossRef] - Hesp, P.A.; Smyth, T.A. Nebkha flow dynamics and shadow dune formation. Geomorphology
**2017**, 282, 27–38. [Google Scholar] [CrossRef] [Green Version] - Lu, P.; Narteau, C.; Dong, Z.; Rozier, O.; Du Pont, S.C. Unravelling raked linear dunes to explain the coexistence of bedforms in complex dunefields. Nat. Commun.
**2017**, 8, 14239. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Parteli, E.; Kroy, K.; Tsoar, H.; Andrade, J.; Pöschel, T. Morphodynamic modeling of aeolian dunes: Review and future plans. Eur. Phys. J. Spéc. Top.
**2014**, 223, 2269–2283. [Google Scholar] [CrossRef] - Lancaster, N. The formation of seif dunes from barchans—Supporting evidence for Bagnold’s model from the Namib Desert. Zeitsch Geomorphol.
**1980**, 24, 160–167. [Google Scholar] [CrossRef] - Yang, Z.; Qian, G.; Han, Z.; Dong, Z.; Luo, W.; Zhang, Z.; Lu, J.; Liang, A.; Tian, M. Variation in grain-size characteristics as a function of wind direction and height in the Sanlongsha dune field of the northern Kumtagh Desert, China. Aeolian Res.
**2019**, 40, 53–64. [Google Scholar] [CrossRef] - Badr, T.; Harion, J. Numerical modelling of flow over stockpiles: Implications on dust emissions. Atmos. Environ.
**2005**, 39, 5576–5584. [Google Scholar] [CrossRef] - Li, J.; Dong, Z.; Zhang, Z.; Qian, G.; Luo, W.; Lu, J. Grain-size characteristics of linear dunes on the northern margin of Qarhan Salt Lake, northwestern China. J. Arid Land.
**2016**, 8, 375–388. [Google Scholar] [CrossRef] [Green Version] - Wakes, S.J.; Maegli, T.; Dickinson, K.J.; Hilton, M.J. Numerical modelling of wind flow over a complex topography. Environ. Model. Softw.
**2010**, 25, 237–247. [Google Scholar] [CrossRef] - Nicklingb, I.J.W.; William, G. Dynamics of secondary airflow and sediment transport over and in the lee of transverse dunes. Prog. Phys. Geogr.
**2002**, 26, 47–75. [Google Scholar] - Fernández-Montblanc, T.; Duo, E.; Ciavola, P. Dune reconstruction and revegetation as a potential measure to decrease coastal erosion and flooding under extreme storm conditions. Ocean Coast. Manag.
**2020**, 188, 105075. [Google Scholar] [CrossRef] - Charbonneau, B.R.; Dohner, S.M.; Wnek, J.P.; Barber, D.; Zarnetske, P.; Casper, B.B. Vegetation effects on coastal foredune initiation: Wind tunnel experiments and field validation for three dune-building plants. Geomorphology
**2021**, 378, 107594. [Google Scholar] [CrossRef] - García-Romero, L.; Delgado-Fernandez, I.; Hesp, P.A.; Hernández-Calvento, L.; Brito, A. Airflow dynamics, vegetation and aeolian erosive processes in a shadow zone leeward of a resort in an arid transgressive dune system. Aeolian Res.
**2019**, 38, 48–59. [Google Scholar] [CrossRef] - Wang, Y.; Chu, L.; Daryanto, S.; Lü, L.; Ala, M.; Wang, L. Sand dune stabilization changes the vegetation characteristics and soil seed bank and their correlations with environmental factors. Sci. Total Environ.
**2018**, 648, 500–507. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bo, T.-L.; Zheng, X.-J. Collision behaviors of barchans in aeolian dune fields. Environ. Earth Sci.
**2013**, 70, 2963–2970. [Google Scholar] [CrossRef]

**Figure 1.**Comparison of the simulated and the actual measured velocity of wind in the field at different locations on the main axis of the sand dune (the round blue dots were the measured data, and the square red blocks were the simulation results).

**Figure 2.**Changes in morphology of the dunes caused by bi-directional winds with different angles at r = 60% (the lateral form was state of changes over time). (

**a**) At an angle of 30°; (

**b**) at an angle of 60°; (

**c**) at an angle of 90°; (

**d**) at an angle of 120°; (

**e**) at an angle of 150°.

**Figure 3.**Typical two-dimensional migration trajectory of the sand particles with different sizes under the wind (red: 0.1 mm; green: 0.3 mm; blue: 0.6 mm).

**Figure 4.**Later evolution of the dune with different sizes of sand particles under bi-directional winds with an angle of 120° and r = 0.6 (with the same initial morphology as in Figure 2): (

**a**) with the particle size of 0.50 mm; (

**b**) with the particle size of 0.15 mm.

**Figure 5.**Morphological changes of crescentic dunes under different topographies. (

**a**) The angle between the left horn and the horizontal ground was θ; (

**b**) the angle between the right horn and the horizontal ground was θ; (

**c**) the angle between the horns and the horizontal ground was θ; (

**d**) the angle between the bottom of the windward toe and the horizontal ground was θ.

**Figure 6.**Evolution on morphology of crescentic dune under epiphytic vegetation barrier and unidirectional wind action (translucent green areas represented areas covered by vegetation). (

**a**) The initial morphology of the dune with vegetation distributed on the two horns; (

**b**) the two leeward toes and the windward toe disappeared; (

**c**) the original windward slope moved to the position of the original leeward slope; (

**d**) the dune became an inverted “U” shape; (

**e**) the dune was gradually stretched and lengthened; (

**f**) the dune formed a shallow “V” shape.

**Figure 7.**Morphological changes of crescentic dunes during collision migration. (

**a**) The two dunes have a volume ratio of 0.23 and a closer distance between the axes; (

**b**) the two dunes have a volume ratio of 0.2 and a longer distance between the axes.

**Figure 8.**Flow field around the dune. (

**a**

**1**) Dune with morphology 1 at the direction of wind of 0°; (

**a**

**2**) dune with morphology 1, the bi-directional winds’ angle of 30°; (

**b**

**1**) dune with morphology 2 at the direction of wind of 0°; (

**b**

**2**) dune with morphology 2, the bi-directional winds’ angle of 60°; (

**c1**) dune with morphology 3 at the direction of wind of 0°; (

**c2**) dune with morphology 3, the bi-directional winds’ angle of 90°; (

**d1**) dune with morphology 4 at the direction of wind of 0°; (

**d2**) dune with morphology 4, the bi-directional winds’ angle of 120°; (

**e1**) dune with morphology 5 at the direction of wind of 0°; (

**e2**) dune with morphology 5, the bi-directional winds’ angle of 150°. The morphology of the dune was the third subfigure of the evolution process corresponding to each angle in Figure 2. The numbers 1 and 2 behind the serial number represent different wind directions.

**Figure 9.**Evolution process of the break-out of extended horn. (

**a**) There was enough sand supply in the extended horn to produce displacement; (

**b**) the transportable sand supply became less abundant as the sand particles gradually dispersed; (

**c**) the extended horn broke and a new small dune formed in the tail; (

**d**) more small dunes emerged after the extended horn.

**Figure 10.**Flow field around the sand dunes (

**a**) with the same sizes of sand particles on the two horns (0.25 mm); (

**b**) with the different sizes of sand particles on the two horns (0.25 mm on the left horn and 0.20 mm on the right horn).

**Figure 11.**Differences in morphology of several types of crescentic dunes (the initial dune with the thickest right horn or thickest left horn or thickest middle part) affected by different slopes in the same orientation (with the angles between the right horn and the horizontal ground) under the action of unidirectional wind: (

**a**) at an angle of 5°; (

**b**) at an angle of 10°; (

**c**) at an angle of 20°.

Symbol | Parameter | Value | Unit |
---|---|---|---|

d | the size of sand particles | 250 | μm |

${\rho}_{fluid}$ | the density of the air | 1.225 | kg/m^{3} |

$\nu $ | the viscosity of the air | 1.75 × 10^{−5} | kg/ms |

${\rho}_{sand}$ | the density of sand particles | 2650 | kg/m^{3} |

${u}_{thredhold}$ | the threshold for velocity of shear | 0.2 | m/s |

${\theta}_{dyn}$ | the collapse angle | 33 | degree |

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**

Zhang, H.; Li, C.; Zhang, J.; Wu, Z.; Zhang, Z.; Hu, J.; Cao, L.; Song, L.; Ma, J.; Xiao, B.
Numerical Simulation Analysis of the Formation and Morphological Evolution of Asymmetric Crescentic Dunes. *Sustainability* **2022**, *14*, 8966.
https://doi.org/10.3390/su14148966

**AMA Style**

Zhang H, Li C, Zhang J, Wu Z, Zhang Z, Hu J, Cao L, Song L, Ma J, Xiao B.
Numerical Simulation Analysis of the Formation and Morphological Evolution of Asymmetric Crescentic Dunes. *Sustainability*. 2022; 14(14):8966.
https://doi.org/10.3390/su14148966

**Chicago/Turabian Style**

Zhang, Huiwen, Changlong Li, Jianhui Zhang, Zhen Wu, Zhiping Zhang, Jing Hu, Lei Cao, Longlong Song, Jianping Ma, and Bin Xiao.
2022. "Numerical Simulation Analysis of the Formation and Morphological Evolution of Asymmetric Crescentic Dunes" *Sustainability* 14, no. 14: 8966.
https://doi.org/10.3390/su14148966