# A WCSPH Particle Shifting Strategy for Simulating Violent Free Surface Flows

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

## 3. Numerical Method

#### 3.1. The Discrete System of Equations

#### 3.2. Enhanced Particle Shifting Technique

#### 3.2.1. Free Surface Detection Algorithm

#### 3.2.2. Modified Particle Shifting Equation

## 4. Validation

#### 4.1. Oscillating Droplet Uunder a Central Conservative Force Field

#### 4.2. Evolution of an Initially-Square Fluid Patch

#### 4.3. Dam Break Problem

#### 4.4. Sloshing in a Rectangular Tank

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Free surface particle detection concept. $\theta $ is the angle of vision and $2\Delta {x}_{0}$ is the vision distance. Where $\Delta {x}_{0}$ denotes the initial particle spacing.

**Figure 2.**Principals of the free surface particles detection, Step 1 presents the free surface particles detection conditions and Step 3 presents the definition of inner particles near to the free surface particles.

**Figure 3.**The free surface particles detected for the dam-break example. The free surface particle of the fluid are colored with black color (${D}_{FS}$), whereas for the particle colored with yellow and green colors represent respectively, the inner particle near (${D}_{IFS}$) and far (${D}_{IN}$) from the free surface region. The first row (

**a**) shows the results via the method proposed by Marrone et al. [45], while the rows (

**b**,

**c**) represent the results using Jandaghian and Shakibaeinia [47] and the present approaches, respectively.

**Figure 4.**Different parameters used to define $\varphi $ for particle i as ${\mathit{r}}_{\mathit{i}}\in {D}_{IFS}$.

**Figure 5.**Oscillating droplet under a central conservative force field. R is the raduis of the intial circular drop, a and b are the semi-major and semi-minor axis of the evolving ellipse, respectively.

**Figure 6.**Oscillating droplet under a central conservative force field: Pressure and the dimensionless parameter $\varphi $ evolution in time. The right half of the droplet is colored by the pressure field, and the left half is colored by the $\varphi $ paremeter field.

**Figure 7.**Oscillating droplet under a central conservative force field: The particle are colored with pressure and dimensionless $\varphi $- parameter on the left and the right half of the fluid droplet, respectively at the time $t=2.44$ s. (

**a**) under the resolution of $dx=\frac{R}{100}$ in the snapshot (

**a**) and $dx=\frac{R}{50}$ in (

**b**). The snapshot (

**c**) shows a zoom on the $\varphi $ field obtained through the selected red rectangle of the snapshot (

**b**).

**Figure 8.**Oscillating droplet under a central conservative force field: horizontal semi major axis evolution in time $a\left(t\right)$. The black solid line presents the theoretical results obtained by solving the system (18) [60]. The red dotted line represents the results obtained by the present particle shifting formulation for the fluid droplet of resolution $dx=\frac{R}{50}$, whereas for the magenta dash-dot line is for $dx=\frac{R}{100}$. The blue dashed line represents the results obtained by the particle shifting technology (IPST) proposed by Wang et al. [37]. The bottom part (

**b**) of the figure represents the zoom of selected red rectangle in (

**a**).

**Figure 9.**Oscillating droplet under a central conservative force field: Vertical semi minor axis evolution in time $b\left(t\right)$. The black solid line presents the theoretical results obtained by solving the system (18) [60]. The red dotted line represents the results obtained by the present particle shifting formulation for the fluid droplet of resolution $dx=\frac{R}{50}$, whereas for the magenta dash-dot line is for $dx=\frac{R}{100}$. The blue dashed line represents the results obtained by the particle shifting technology (IPST) proposed by Wang et al. [37]. The bottom part (

**b**) of the figure represents the zoom of selected red rectangle in (

**a**).

**Figure 10.**Oscillating droplet under a central conservative force field: Kinetic energy evolution in time ${E}_{k}\left(t\right)$. The black solid line presents the theoretical results obtained by solving the system (18) [60]. The red dotted line represents the results obtained by the present particle shifting formulation for the fluid droplet of resolution $dx=\frac{R}{50}$, whereas for the magenta dash-dot line is for $dx=\frac{R}{100}$. The blue dashed line represents the results obtained by the particle shifting technology (IPST) proposed by Wang et al. [37]. The bottom part (

**b**) of the figure represents the zoom of selected red rectangle in (

**a**).

**Figure 11.**Oscillating droplet under a central conservative force field: Potential Energy evolution in time ${E}_{p}\left(t\right)$. The black solid line presents the theoretical results obtained by solving the system (18) [60]. The red dotted line represents the results obtained by the present particle shifting formulation for the fluid droplet of resolution $dx=\frac{R}{50}$, whereas for the magenta dash-dot line is for $dx=\frac{R}{100}$. The blue dashed line represents the results obtained by the particle shifting technology (IPST) proposed by Wang et al. [37]. The bottom part (

**b**) of the figure represents the zoom of selected red rectangle in (

**a**).

**Figure 12.**Oscillating droplet under a central conservative force field: Total energy evolution in time ${E}_{T}\left(t\right)$. The black solid line presents the theoretical results obtained by solving the system (18) [60]. The red dotted line represents the results obtained by the present particle shifting formulation for the fluid droplet of resolution $dx=\frac{R}{50}$, whereas for the magenta dash-dot line is for $dx=\frac{R}{100}$. The blue dashed line represents the results obtained by the particle shifting technology (IPST) proposed by Wang et al. [37].

**Figure 13.**Oscillating droplet under a central conservative force field: Convergence studies with present SPH and $\delta $-SPH [23] schemes. The dotted line with small circles denotes the convergence curve of the $\delta $-SPH scheme and the dotted line with small squares denotes the convergence curve of the present SPH model. The dashed and continues lines represent $1\mathrm{st}$ and $2\mathrm{nd}$ order of convergence slops, respectively.

**Figure 15.**Evolution of an initially-square fluid patch: Pressure field obtained at dimentionless times $t\omega =\{0.6,1.2,2.04\}$. The black bold points present the free surface evaluated by the Lagrangian Finite Difference Method [61].

**Figure 16.**Evolution of an initially-square fluid patch: Pressure field obtained and evolving shape of the square path at dimensionless times $t\omega =\{6,8,9.5\}$.

**Figure 17.**Evolution of an initially-square fluid patch: The fluid particles distribution at the dimensionless time $t\omega =2.2$. The left column presents the results obtained by using our particle shifting approach (here a poor detection free surface is presented), whereas for the right column shows the results obtained using by using IPST [37]. The three last rows of the figure present the zooms on the three selected regions.

**Figure 18.**Evolution of an initially-square fluid patch: The fluid particles distribution at the dimensionless time $t\omega =2.2$ in the case of poor detection free surface. The particles are colored with dimensionless $\varphi $-parameter field. The second row presents the zooms on the three selected regions.

**Figure 19.**Evolution of an initially-square fluid patch: Time evolution of the pressure at the center of the fluid path. The red dotted line represents the results obtained by the present particle shifting formulation for the particle resolution of $dx=\frac{L}{100}$, whereas for the magenta dash-dot line, it corresponds to the particle resolution of $dx=\frac{L}{200}$. The black dotted line with diamond marker denotes the pressure results obtained by BEM-MEL solver [61]. The second row of the figure presents a zoom region.

**Figure 20.**Evolution of an initially-square fluid patch: Dissipation of the kinetic energy. The red dotted line presents the results obtained by the present particle shifting formulation for the particle resolution of $dx=\frac{L}{100}$, The blue dashed line represents the results obtained by using IPST [37] for the particle resolution of $dx=\frac{L}{100}$. whereas for the magenta dash-dot line presents the results obtained by the present particle shifting formulation for the particle resolution of $dx=\frac{L}{200}$.

**Figure 22.**Dam break flow: Snapshots of the computational results at dimensionless times $t\sqrt{g/H}=\{1.61,5.45,6.14,10.24,20.02\}$. The left column presents the pressure field, the middle column denotes the $\varphi $-parameter field and the right column presents the restriction field (no shifting is applied on the yellow region).

**Figure 23.**Dam break flow: Snapshots of pressure field at dimensionless times $t\sqrt{g/H}=\{6.00,14.80,20.03\}$, The left column depicts the results obtained by the present method. The right column presents the results obtained with IPST [37]. The solid magenta line shows the position of final free surface at equilibrium, should the total volume conserve.

**Figure 24.**Dam break flow: time evolution of the water front, the green solid line presents the analytical solution from shallow water theory [76], The black triangles denotes the experimental data given by Buchner [69]. The blue dash-dot line represents the results obtained by using IPST [37], whereas for the magenta dashed line presents the results obtained by the present particle shifting formulation.

**Figure 25.**Dam break flow: History of the pressure signals recorded at P. The black triangles denotes the experimental data given by Buchner [69]. The blue dash-dot line represents the results obtained by using IPST [37], whereas for the magenta dashed line presents the results obtained by the present particle shifting formulation.

**Figure 26.**Dam break flow: Time evolution of mechanical energy. The blue dash-dot line represents the results obtained by using IPST [37], whereas for the magenta dashed line presents the results obtained by the present particle shifting formulation. The black reference curve is taken from Marrone et al. [23].

**Figure 28.**Sloshing in a rectangular tank: Snapshots of the computational results at times $t=\{1.74,2.32,3.00,3.99,6.90,7.60\}$ s. The left column presents the pressure filed, the middle column denotes the $\varphi $-parameter filed and the right column presents the restriction field (no shifting is applied on the yellow region).

**Figure 29.**Sloshing in a rectangular tank: Snapshots of pressure field at times $t=\{6.80,8.00,8.50\}$ s, The first row depicts the results obtained by the present method. The second row presents the results obtained with IPST [37].

**Figure 30.**Sloshing in a rectangular tank: time history of the impact pressure at the three installed probes ${P}_{1}$, ${P}_{2}$ and ${P}_{3}$. figures (

**a**–

**c**) present the comparison between the present numerical and experimental pressure signals obtained at the probes ${P}_{1}$, ${P}_{2}$ and ${P}_{3}$, respectively. The black triangles denotes the experimental data given by Rafiee et al. [80]. The blue dashed line represents the results obtained by using IPST [37], whereas for the magenta dash-dot line presents the results obtained by the present particle shifting formulation.

**Figure 31.**Sloshing in a rectangular tank: Snapshots of pressure field at times $t=10.25$ s and $t=11.70$ s. The first row depicts the results obtained by the present method. The second row presents the results obtained with IPST [37].

**Table 1.**Dam break flow: comparison table of the computational times using our present particle shifting formulation and IPST [37] for simulating 5 s of the dam break physical time under 12,800 particles.

Particle Shifting Model | Computational Time in s |
---|---|

Present particle shifting formulation | 6813 |

IPST [37] | 6608 |

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**MDPI and ACS Style**

Krimi, A.; Jandaghian, M.; Shakibaeinia, A.
A WCSPH Particle Shifting Strategy for Simulating Violent Free Surface Flows. *Water* **2020**, *12*, 3189.
https://doi.org/10.3390/w12113189

**AMA Style**

Krimi A, Jandaghian M, Shakibaeinia A.
A WCSPH Particle Shifting Strategy for Simulating Violent Free Surface Flows. *Water*. 2020; 12(11):3189.
https://doi.org/10.3390/w12113189

**Chicago/Turabian Style**

Krimi, Abdelkader, Mojtaba Jandaghian, and Ahmad Shakibaeinia.
2020. "A WCSPH Particle Shifting Strategy for Simulating Violent Free Surface Flows" *Water* 12, no. 11: 3189.
https://doi.org/10.3390/w12113189