# 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

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**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