# SPH Simulation of Interior and Exterior Flow Field Characteristics of Porous Media

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Smooth Particle Hydrodynamics (SPH) Methodology

_{b}and ρb represent particle mass and density, W

_{ab}is a kernel function, r is the distance between the reference particle and the adjacent (m), h is the smoothing length (m). To ensure a linear and angular momentum conservation, the asymmetric pressure gradient terms can be obtained as follows:

_{a}W

_{ab}represents the kernel gradient taken relative to the position of the particle a and P represents the pressure; similarly, the divergence of the vector at a given particle a can be estimated by u:

## 3. SPH Model

#### 3.1. Equations for Flow Field Porous Media

_{w}is the external flow rate of the pore.

_{p}

_{1}is the conveying speed (m/s); n

_{w}is the porosity (/); K

_{p}is the permeability (/); C

_{f}is the nonlinear resistance coefficient (/).

#### 3.2. Numerical Model Solving Process

_{t}and r

_{t}represents the particle velocity and position at time t, respectively. The pressure term is based on the classical pressure Poisson equation that can be expressed as follows [23]:

_{0}represents the initial constant particle density; ρ

_{*}represents the central particle density after the prediction step, and P

_{t+1}is the pressure of the particles at the t+1 time step. In the calibration of the second step, the pressure gradient term is combined with the momentum equation to ensure incompressibility. Pressure can be used to correct particle velocity as follows:

_{t}

_{+1}is represents the particle velocity and r

_{t}

_{+1}represents position at the moment of t + 1.

#### 3.3. Boundary Conditions

#### 3.3.1. Free Surface Boundary

#### 3.3.2. Fluid-Structure Coupling Boundary

#### 3.3.3. Impermeable/Fixed Solid Wall Boundary

#### 3.3.4. Periodic Inflow and Outflow Boundaries Accompanied with a Damping Zone

^{3}/s, and the particle size D as 0.1 m. In order to further test the performance of our method, three time stages t = 18.0 s, t = 20.0 s, and t = 22.0 s were considered to analyze its inlet and outlet flow rate, respectively. x = 0.0 represents the inlet flow rate distribution curve; x = 60.0 represents the outlet flow rate distribution curve. Analyzing the tests shown in Figure 2, it is possible to notice that the flow distribution of the particles is very similar between inlet (x = 0.0) and outlet (x = 60.0). Furthermore, the velocity ranges from 0 m/s to 1.5 m/s for each time step displayed proving the periodic boundary conditions of the inlet and outlet.

## 4. Model Verification

^{−6}(Pa·s). Finally, the flow field characteristics were simulated at the porosity of 0.349 and 0.475.

## 5. Model Application

^{−6}(Pa·s), structure coordinates are 30.0, 0.0. The schematic diagram of this configuration is displayed in Figure 9.

#### 5.1. Flow Field Velocity Distribution Diagram of Different Volumes of Porous Media

#### 5.2. Analysis of Flow Field Inside and Outside the Porous Media

#### 5.3. Longitudinal and Vertical Flow Field Distribution under Different Porosity

#### 5.4. Convergence Verification of Pore Logistics Field Model

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Espinoza-Andaluz, M.; Velasco-Galarza, V.; Romero-Vera, A. On hydraulic tortuosity variations due to morphological considerations in 2D porous media by using the Lattice Boltzmann method. Math. Comput. Simul.
**2020**, 169, 74–87. [Google Scholar] [CrossRef] - Akbari, H. Modified moving particle method for modeling wave interaction with multi layered porous structures. Coast. Eng.
**2014**, 89, 1–19. [Google Scholar] [CrossRef] - Houreh, N.B.; Shokouhmand, H.; Afshari, E. Effect of inserting obstacles in flow field on a membrane humidifier performance for PEMFC application: A CFD model. Int. J. Hydrog. Energy
**2019**, 44, 30420–30439. [Google Scholar] [CrossRef] - Mei, T.-L.; Zhang, T.; Candries, M.; Lataire, E.; Zou, Z.-J. Comparative study on ship motions in waves based on two time domain boundary element methods. Eng. Anal. Bound. Elem.
**2020**, 111, 9–21. [Google Scholar] [CrossRef] - Basser, H.; Rudman, M.; Daly, E. SPH modelling of multi-fluid lock-exchange over and within porous media. Adv. Water Resour.
**2017**, 108, 15–28. [Google Scholar] [CrossRef] - Aganetti, R.; Lamorlette, A.; Thorpe, G.R. The relationship between external and internal flow in a porous body using the penalisation method. Int. J. Heat Fluid Flow
**2017**, 66, 185–196. [Google Scholar] [CrossRef] - Chan, H.C.; Leu, J.M.; Lai, C.J. Velocity and turbulence field around permeable structure:Comparisons between laboratory and numerical experiments. J. Hydraul. Res.
**2007**, 45, 216–226. [Google Scholar] [CrossRef] - Gui, Q.; Dong, P.; Shao, S.; Chen, Y. Incompressible SPH simulation of wave interaction with porous structure. Ocean Eng.
**2015**, 110, 126–139. [Google Scholar] [CrossRef] - Khayyer, A.; Gotoh, H.; Shimizu, Y.; Gotoh, K.; Falahaty, H.; Shao, S. Development of a projection-based SPH method for numerical wave flume with porous media of variable porosity. Coast. Eng.
**2018**, 140, 1–22. [Google Scholar] [CrossRef] - Vanneste, D.; Troch, P. 2D numerical simulation of large-scale physical model tests of wave interaction with a rubble-mound breakwater. Coast. Eng.
**2015**, 103, 22–41. [Google Scholar] [CrossRef] - Shao, S. Incompressible SPH flow model for wave interactions with porous media. Coast. Eng.
**2010**, 57, 304–316. [Google Scholar] [CrossRef] - Gnanasekaran, B.; Liu, G.-R.; Fu, Y.; Wang, G.; Niu, W.; Lin, T. A Smoothed Particle Hydrodynamics (SPH) procedure for simulating cold spray process—A study using particles. Surf. Coat. Technol.
**2019**, 377, 124812. [Google Scholar] [CrossRef] - Shadloo, M.S.; Oger, G.; Le Touze, D. Smoothed particle hydrodynamics method for fluid flows, towards industrial appliactions: Motivations, current state and challenges. Comput. Fluids
**2016**, 136, 11–34. [Google Scholar] [CrossRef] - Niu, X.; Zhao, J.; Wang, B. Application of smooth particle hydrodynamics (SPH) method in gravity casting shrinkage activity prediction. Comput. Part. Mech.
**2019**, 6, 803–810. [Google Scholar] [CrossRef] - Yin, J.P.; Shi, Z.X.; Chen, J.; Chang, B.H.; Yi, J.Y. Smooth particle hydrodynamics-based characteristics of a shaped jet from different materials. Strength Mater.
**2019**, 51, 85–94. [Google Scholar] [CrossRef] - Avesani, D.; Dumbser, M.; Chiogna, G.; Bellin, A. An alternative smooth particle hydrodynamics formulation to simulate chemotaxis in porous media. J. Math. Biol.
**2017**, 74, 1037–1058. [Google Scholar] [CrossRef][Green Version] - Eghtesad, A.; Knezevic, M. A new approach to fluid–structure interaction within graphics hardware accelerated smooth particle hydrodynamics considering heterogeneous particle size distribution. Comput. Part. Mech.
**2018**, 5, 387–409. [Google Scholar] [CrossRef] - Yang, H.X.; Li, R.; Lin, P.Z.; Wan, H.; Feng, J. Two-phase smooth particle hydrodynamics modeling of air-water interface in aerated flows. Sci. China Technol. Sci.
**2017**, 60, 479–490. [Google Scholar] [CrossRef] - Shao, S. Incompressible smoothed particle hydrodynamics simulation of multifluid flows. Int. J. Numer. Methods Fluids
**2011**, 69, 11. [Google Scholar] [CrossRef] - Khayyer, A.; Gotoh, H.; Shao, S.D. Corrected incompressible SPH method for accurate water-surface tracking in breaking waves. Coast. Eng.
**2008**, 55, 236–250. [Google Scholar] [CrossRef][Green Version] - Shao, S.; Gotoh, H. Turbulence particle models for tracking free surfaces. J. Hydraul. Res.
**2010**, 43, 276–289. [Google Scholar] [CrossRef][Green Version] - Wang, S.; Shu, A.; Rubinato, M.; Wang, M.; Qin, J. Numerical Simulation of Non-Homogeneous Viscous Debris-Flows based on the Smoothed Particle Hydrodynamics (SPH) Method. Water
**2019**, 11, 2314. [Google Scholar] [CrossRef][Green Version] - Shao, S.; Lo, E.Y.M. Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface. Adv. Water Resour.
**2003**, 26, 787–800. [Google Scholar] [CrossRef] - Monaghan, J.J. Smoothed Particle Hydrodynamics. Annu. Rev. Astron. Astrophys.
**1992**, 30, 543–574. [Google Scholar] [CrossRef] - Huang, C.-J.; Chang, H.-H.; Hwung, H.-H. Structural permeability effects on the interaction of a solitary wave and a submerged breakwater. Coast. Eng.
**2003**, 49, 1–24. [Google Scholar] [CrossRef] - Sun, P.N.; Colagrossi, A.; Marrone, S.; Antuono, M.; Zhang, A.M. Multi-resolution Delta-plus-SPH with tensile instability control: Towards high Reynolds number flows. Comput. Phys. Commun.
**2018**, 224, 63–80. [Google Scholar] [CrossRef] - Sun, P.N.; Colagrossi, A.; Le Touze, D.; Zhang, A.M. Extension of the δ-plus-SPH model for simulating vortex-induced-vibration problems. J. Fluids. Struct.
**2019**, 90, 19–42. [Google Scholar] [CrossRef] - Rubinato, M.; Martins, R.; Kesserwani, G.; Leandro, J.; Djordjevic, S.; Shucksmith, J. Experimental investigation of the influence of manhole grates on drainage flows in urban flooding conditions. In Proceedings of the 14th IWA/IAHR International Conference on Urban Drainage, Prague, Czech Republic, 10–15 September 2017. [Google Scholar]
- Rubinato, M.; Martins, R.; Kesserwani, G.; Leandro, J.; Djordjevic, S.; Shucksmith, J. Experimental calibration and validation of sewer/surface flow exchange equations in steady and unsteady flow conditions. J. Hydrol.
**2017**, 552, 421–432. [Google Scholar] [CrossRef] - Rubinato, M.; Lee, S.; Martins, R.; Shucksmith, J. Surface to sewer flow exchange through circular inlets during urban flood conditions. J. Hydroinform.
**2018**, 20, 564–576. [Google Scholar] [CrossRef][Green Version] - Martins, R.; Rubinato, M.; Kesserwani, G.; Leandro, J.; Djordjević, S.; Shucksmith, J.D. On the Characteristics of Velocities Fields in the Vicinity of Manhole Inlet Grates During Flood Events. Water Resour. Res.
**2018**, 54, 6408–6422. [Google Scholar] [CrossRef] - Nichols, A.; Rubinato, M. Remote sensing of environmental processes via low-cost 3D free-surface mapping. In Proceedings of the 4th IHAR Europe Congress, Liege, Belgium, 27–29 July 2016. [Google Scholar]
- Lopes, P.; Shucksmith, J.; Leandro, J.; de Fernandes Carvalho, R.; Rubinato, M. Velocities profiles and air-entrainment characterization in a scaled circular manhole. In Proceedings of the 13th ICUD, Sawarak, Malaysia, 7–12 September 2014. [Google Scholar]
- Rojas Arques, S.; Rubinato, M.; Nichols, A.; Shucksmith, J.D. Cost effective measuring technique to simultaneously quantify 2D velocity fields and depth-averaged solute concentrations in shallow water flows. Flow Meas. Instrum.
**2018**, 64, 213–223. [Google Scholar] [CrossRef] - Beg, M.N.A.; Carvalho, R.F.; Tait, S.; Brevis, W.; Rubinato, M.; Schellart, A.; Leandro, J. A comparative study of manhole hydraulics using stereoscopic PIV and different RANS models. Water Sci. Tech.
**2018**, 2017, 87–98. [Google Scholar] [CrossRef] - Cavelan, A.; Boussafir, M.; Rozenbaum, O.; Laggoun-Défarge, F. Organic petrography and pore structure characterization of low-mature and gas-mature marine organic-rich mudstones: Insights into porosity controls in gas shale systems. Mar. Pet. Geol.
**2019**, 103, 331–350. [Google Scholar] [CrossRef][Green Version] - Zheng, X.; Ma, Q.; Shao, S.; Khayyer, A. Modelling of Violent Water Wave Propagation and Impact by Incompressible SPH with First-Order Consistent Kernel Interpolation Scheme. Water
**2017**, 9, 400. [Google Scholar] [CrossRef] - Lo, E.Y.; Shao, S. Simulation of near-shore solitary wave mechanics by an incompressible SPH method. Appl. Ocean. Res.
**2002**, 24, 275–286. [Google Scholar]

**Figure 2.**Comparison of flow velocities at inlet and outlet boundary at different times at T = 18.0 s, T = 20.0 s and T = 22.0 s.

**Figure 3.**Comparison of flow velocities at inlet and outlet boundary at different times at t = 60.0 s, t = 70.0 s, t = 80.0 s and t = 90 s.

**Figure 5.**(

**a**) represents the velocity field using porosity = 0.349; (

**b**) represents the velocity field using porosity = 0.475. U: Longitudinal velocity (m/s).

**Figure 6.**Comparison of longitudinal and vertical velocity for porosity 0.475. Legend: Red open circle: numerical simulation of longitudinal velocity of smooth particle hydrodynamics (SPH); Green dash dot: numerical simulation of longitudinal velocity presented by Chan et al. [7]; Blue solid line: experimental longitudinal velocity values from Chan et al. [7]; Red solid circle: numerical simulation of vertical velocity of SPH; Green solid triangle: numerical simulation of vertical velocity of Chan et al. [7]; Blue solid square: experimental vertical velocity values from Chan et al. [7]. H: water depth (m); U: longitudinal flow rate (m/s), V: vertical flow rate (m/s), U0: Inlet boundary velocity (m/s). X is the distance between the particle and the left boundary of the pore structure, H is the height of the pore structure, and X/H is a dimensionless treatment of the X axis. The pore structure is placed at the origin of this system.

**Figure 7.**Comparison of longitudinal and vertical velocity for porosity 0.349 Legend: Red open circle: numerical simulation of longitudinal velocity of SPH; Green dash dot: numerical simulation of longitudinal velocity presented by Chan et al. [7]; Blue solid line: experimental longitudinal velocity values from Chan et al. [7]; Red solid circle: numerical simulation of vertical velocity of SPH; Green solid triangle: numerical simulation of vertical velocity of Chan et al. [7]; Blue solid square: experimental vertical velocity values from Chan et al. [7]. H: water depth (m); U: longitudinal flow rate (m/s), V: vertical flow rate (m/s), U0: Inlet boundary velocity (m/s). X is the distance between the particle and the left boundary of the pore structure, H is the height of the pore structure, and X/H is a dimensionless treatment of the X axis. The pore structure is placed at the origin of this system.

**Figure 8.**Solid purple triangle D = 0.0104; Dashed line when D = 0.01. X/H = −1.4 indicates the oncoming flow field of the pore structure; X/H = 1.67 indicates that the water flows through the pore structure field; X/H = 4.47 indicates the back vortex field of the pore; Y (m) indicates that the particles of the water flow from the horizontal plane vertical distance; U (m/s) represents the longitudinal water flow velocity; V (m/s) represents the vertical water flow velocity.

**Figure 10.**Flow field velocity distribution of different volumes of porous media at a porosity of 0.5. U: Longitudinal velocity (m/s). (

**a**) Pore structure: 0.6 × 0.6 (m

^{2}); (

**b**) Pore structure: 0.9 × 0.9 (m

^{2}); (

**c**) Pore structure: 1.2 × 1.2 (m

^{2}); (

**d**) Pore structure: 1.5 × 1.5 (m

^{2})

**Figure 11.**Characteristics of longitudinal and vertical flow fields when the vertical section of the pore is 0.6 m × 0.6 m; solid line: x = 25.0; x = 30.1; x = 31.0; dashed line: x = 26.0; x = 30.2; x = 32.0; dotted line: x = 27.0; x = 30.3; x = 33.0. Long dashed line: x = 28.0; x = 30.4; x = 34.0; double dotted line: x = 29.0; x = 30.5; x = 35.0; dotted line: x = 36.0.

**Figure 12.**Characteristics of longitudinal and vertical flow fields when the vertical section of the pore is 0.9 m × 0.9 m; solid line: x = 25.0; x = 30.1; x = 31.0; dashed line: x = 26.0; x = 30.2; x = 32.0; dotted line: x = 27.0; x = 30.3; x = 33.0. Long dashed line: x = 28.0; x = 30.4; x = 34.0; double dotted line: x = 29.0; x = 30.5; x = 35.0; dotted line: x = 36.0.

**Figure 13.**Characteristics of longitudinal and vertical flow fields when the vertical section of the pore is 1.2 m × 1.2 m; solid line: x = 25.0; x = 30.1; x = 31.0; dashed line: x = 26.0; x = 30.2; x = 32.0; dotted line: x = 27.0; x = 30.3; x = 33.0. Long dashed line: x = 28.0; x = 30.4; x = 34.0; double dotted line: x = 29.0; x = 30.5; x = 35.0; dotted line: x = 36.0.

**Figure 14.**Characteristics of longitudinal and vertical flow fields when the longitudinal section of the pore is 1.5 m × 1.5 m; solid line: x = 25.0; x = 30.1; x = 31.0; dashed line: x = 26.0; x = 30.2; x = 32.0; dotted line: x = 27.0; x = 30.3; x = 33.0. Long dashed line: x = 28.0; x = 30.4; x = 34.0; double dotted line: x = 29.0; x = 30.5; x = 35.0; dotted line: x = 36.0.

**Figure 15.**Longitudinal velocity distribution of the flow field in the upstream area of the pore structure when the porosity is 0.5, 0.25 and 0.125 and x = 25.0–29.0 m. (

**a**) n

_{w}= 0.5; (

**b**) n

_{w}= 0.25; (

**c**) n

_{w}= 0.125.

**Figure 16.**Longitudinal velocity distribution of the flow field inside and directly above the pore structure when the porosity is 0.5, 0.25 and 0.125 and x = 30.1–30.5.0 m. (

**a**) n

_{w}= 0.5; (

**b**) n

_{w}= 0.25; (

**c**) n

_{w}= 0.125.

**Figure 17.**Longitudinal velocity distribution in the back vortex flow field when the porosity is 0.5, 0.25, and 0.125 and x = 31.0–36.0 m. (

**a**) n

_{w}= 0.5; (

**b**) n

_{w}= 0.25; (

**c**) n

_{w}= 0.125.

**Figure 18.**Vertical velocity distribution of the flow field in the upstream area of the pore structure when the porosity is 0.5, 0.25 and 0.125 and x = 25.0–29.0 m. (

**a**) n

_{w}= 0.5; (

**b**) n

_{w}= 0.25; (

**c**) n

_{w}= 0.125.

**Figure 19.**Vertical velocity distribution of the flow field inside and directly above the pore structure when the porosity is 0.5, 0.25 and 0.125 and x = 30.1–30.5 m. (

**a**) n

_{w}= 0.5; (

**b**) n

_{w}= 0.25; (

**c**) n

_{w}= 0.125.

**Figure 20.**Vertical velocity distribution in the back vortex flow field when the porosity is 0.5, 0.25 and 0.12 and x = 31.0–36.0 m. (

**a**) n

_{w}= 0.5; (

**b**) n

_{w}= 0.25; (

**c**) n

_{w}= 0.125.

**Figure 21.**Longitudinal velocity distribution of the inflow field with different particle spacing (

**a**), longitudinal velocity distribution of the sea current flowing through the porous media with different particle spacing (

**b**), and longitudinal velocity distribution of the rear vortex field with different particle spacing (

**c**).

**Figure 22.**Vertical velocity distribution of the inflow field with different particle spacing (

**a**), vertical velocity distribution of the sea current flowing through the porous media with different particle spacing (

**b**), and vertical velocity distribution of the rear vortex field with different particle spacing (

**c**).

**Figure 23.**Back vortex field of the pore structure when the porosity is 0.349. U: Longitudinal velocity (m/s).

**Figure 24.**Back vortex field of the pore structure when the porosity is 0.475. U: Longitudinal velocity (m/s).

Particle Size D (m) | Total Number of Particles | Total Physical Simulation Time (s) | CPU Time | CPU Cores |
---|---|---|---|---|

0.0104 | 13,914 | 100 | 27 h 05 min | 4 |

0.01 | 15,000 | 100 | 27 h 14 min | 4 |

0.096 | 16,225 | 100 | 8 h 28 min | 20 |

0.1 | 15,000 | 100 | 8 h 23 min | 20 |

0.104 | 13,914 | 100 | 8 h 18 min | 20 |

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**MDPI and ACS Style**

Wu, S.; Rubinato, M.; Gui, Q. SPH Simulation of Interior and Exterior Flow Field Characteristics of Porous Media. *Water* **2020**, *12*, 918.
https://doi.org/10.3390/w12030918

**AMA Style**

Wu S, Rubinato M, Gui Q. SPH Simulation of Interior and Exterior Flow Field Characteristics of Porous Media. *Water*. 2020; 12(3):918.
https://doi.org/10.3390/w12030918

**Chicago/Turabian Style**

Wu, Shijie, Matteo Rubinato, and Qinqin Gui. 2020. "SPH Simulation of Interior and Exterior Flow Field Characteristics of Porous Media" *Water* 12, no. 3: 918.
https://doi.org/10.3390/w12030918