# Numerical Simulations of 2D Hydraulic Jumps by a Parallel SPH Model

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Method

#### 2.1. Equations

**u**is the velocity, p represents the pressure,

**r**represents the position of a generic material point,

**g**= (0, 0, −9.81) m/s

^{2}is the gravitational acceleration, $\Gamma $ represents the viscous stress tensor, ${\rho}_{0}$ = 1000 kg/m

^{3}represents the reference density, and c

_{0}refers to the reference speed of sound. ${c}_{0}$ usually takes 10U

_{max}, where U

_{max}is the maximum wave speed for a simulation problem. The Mach number is 0.1 in the following simulations. The compression effect is O(M

^{2}). Therefore, the change of fluid density is not more than 1%.

^{−3}N$\xb7$s/m

^{2}for water, where υ refers to the kinematic viscosity of water; V = mρ is the volume of the particle, where m represents the particle mass; and ${W}_{b}({r}_{a})$ refers to the smoothing kernel function at b-th particle caused by a-th particle.

#### 2.2. Boundary Treatments

#### 2.3. Parallel Strategy

## 3. Numerical Test Cases

#### 3.1. Performance Analysis on Environment Variables

_{s}is compared in Figure 3 for a hydraulic jump test case to test the efficiency of the parallel SPH code. In Figure 3, the coordinate x is the block size of the scheduling strategy. The coordinate y is the parallel speed-up R

_{s}. The color of the line represents the scheduling strategy, where a red line is the result of a serial program. P

_{1}–P

_{4}represent discrete particle numbers in the calculation domain. The parallel code runs on an Intel(R) Core (TM) i5 CPU with 2 cores, 4 threads, and a main frequency of 3.2 GHz. The total particle numbers are the sum of the boundary particles and the fluid particles at the initial time. These did not change much when the simulation became stable. The parallel speed-up was calculated as:

_{p}represents the execution time of the parallel code while t

_{s}represents the execution time of the serial code.

#### 3.2. Open Channel Flow

_{0}was 0.1 m and 0.2 m, respectively. To obtain laminar flow, the dynamic viscosity μ was set to 1.0 $\times $ 10

^{−1}N$\xb7$s/m

^{2}for case 1 and 6.0 $\times $ 10

^{−1}N$\xb7$s/m

^{2}for case 2, respectively. The Reynolds numbers $Re=\rho {u}_{\mathrm{max}}h/\mu $ of the two test cases were 200 and 100. The channel bed was the non-slip boundary. A uniform velocity that was calculated using the formula in [9] was given to the inlet particles. The outlet boundary was free outflow. To analyze the adequacy of the particles, Figure 4 compares the velocity profile between the numerical results with different particle spacing and analytical solutions. The numerical results with a particle spacing h/Δx = 20 show a good agreement with the analytical solutions. Therefore, the particle spacing should ensure that the particle numbers along the water depth are not less than 20.

_{2}errors between the analytical and numerical velocity are given in Table 1. The L

_{2}errors with h/Δx = 20, which is less than or equal to 0.05, are quite small. This suggests that the present SPH model can accurately simulate the uniform laminar flow

#### 3.3. Hydraulic Jumps

_{1}was always 0.01 m. The outlet conjugate water depths obtained from the analytical formula were 0.012 m and 0.022 m for case 1 and case 2, respectively. The length of the numerical horizontal flume was L = 40h

_{1}. The inlet boundary conditions were specific uniform velocity U

_{1}and water depth h

_{1}. The outlet boundary conditions were specific uniform velocity U

_{2}. For the solid wall boundary, the slip boundary condition was adopted. The initial water level and velocity in the computational domain were specific uniform velocity U

_{2}and water depth h

_{1}. The initial pressure and density were calculated based on the hydrostatic pressure hypothesis. For ideal fluid, the viscosity was ignored. Therefore, the model was extended to simulate the inviscid flow by replacing the dynamic viscosity with an artificial viscosity coefficient. This artificial viscosity was mainly adopted to keep stability of calculation. Here, a formula $\mu ={\rho}_{0}\alpha h{c}_{0}/8$ was used. Following the study of Federico et al. [9], α = 0.02 was taken. It was found that the pressure fields were noisy when the Fr

_{1}was large. This noisiness of pressure fields was not found in Federico et al. [9], because only velocity fields of hydraulic jumps were provided, while pressure fields were not considered. Therefore, a Shepard filter was introduced into the model. To reduce loss of time, the Shepard filter was calculated every 30-time step, which proved to be sufficient [19]. The equation is ${\rho}_{a}^{new}={\displaystyle {\sum}_{b=1}^{N}{\rho}_{b}{W}_{ab}^{new}{m}_{b}/{\rho}_{b}=}{\displaystyle {\sum}_{b=1}^{N}{m}_{b}{W}_{ab}^{new}}$ where ${W}_{ab}^{new}={W}_{ab}/{\displaystyle {\sum}_{b=1}^{N}{W}_{ab}({m}_{b}/{\rho}_{b})}$. The space between particles was 0.005 m. Total time of 16 s was simulated.

_{1}.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Test Cases | Case 1 | Case 2 |
---|---|---|

h/Δx = 10 | 0.05 | 0.06 |

h/Δx = 20 | 0.03 | 0.05 |

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**MDPI and ACS Style**

Lin, J.; Mao, H.; Ding, W.; Jia, B.; Pan, X.; Jin, S. Numerical Simulations of 2D Hydraulic Jumps by a Parallel SPH Model. *Water* **2021**, *13*, 2536.
https://doi.org/10.3390/w13182536

**AMA Style**

Lin J, Mao H, Ding W, Jia B, Pan X, Jin S. Numerical Simulations of 2D Hydraulic Jumps by a Parallel SPH Model. *Water*. 2021; 13(18):2536.
https://doi.org/10.3390/w13182536

**Chicago/Turabian Style**

Lin, Jinbo, Hongfei Mao, Weiye Ding, Baozhu Jia, Xinxiang Pan, and Sheng Jin. 2021. "Numerical Simulations of 2D Hydraulic Jumps by a Parallel SPH Model" *Water* 13, no. 18: 2536.
https://doi.org/10.3390/w13182536