# Flux Vector Splitting Method of Weakly Compressible Water Navier-Stokes Equation and Its Application

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Control Equation and Flux Vector Splitting Method

#### 2.1. Weakly Compressible Water N-S Equation

_{ij}is defined by ${\tau}_{ij}=\mu (\frac{\partial {u}_{i}}{\partial {x}_{j}}+\frac{\partial {u}_{j}}{\partial {x}_{i}})-\frac{2}{3}{\delta}_{ij}\mu (\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z})$, where μ is the dynamic viscosity coefficient of fluid and δ is the Kronecker delta.

_{x}and Vis

_{y}are the viscous terms in the x and y direction, respectively. During numerical calculation, dimensionless equations are more often used. We select the reference length L

_{ref}, reference velocity u

_{ref}, reference density ρ

_{ref}, reference pressure p

_{ref}= ρ

_{ref}(u

_{ref})

^{2}, and reference time t

_{ref}= L

_{ref}/u

_{ref}. The above control equations can be written in the dimensionless form of flux, as follows:

**U**is the conservative flux and

**F**and

**G**are the flow flux vectors in the x and y direction, respectively. The detailed expressions are given in Equation (4).

_{∞}are determined by the experimental data. For example, Shyue [12] sets n = 5.5, p

_{∞}= 492 MP. Beig [14] wrote it as p = (n − 1)ρaT − B

_{0}, where the fitting parameters are n = 2.35, a = 1816 J/(kg·K), B

_{0}= 1000 MPa, and T is the water temperature. As Beig [14] pointed out, “The above parameters n, e, c, and B

_{0}are all obtained by fitting experimental data based on the water pressure range of the studied problem”. Therefore, the parameters A and B in the linear model (5) may vary depending on the problem. According to definition of dimensionless variable, the model (5) can be expressed in a dimensionless form. In this article, we choose ρ

_{ref}= 1000 kg/m

^{3}and p

_{ref}= 1 MPa. It is easily confirmed that the dimensionless and dimensional forms are same. Namely, the dimensionless equation of state is also written as p = Aρ − B. Under the standard condition, the water pressure is 0.1 MPa when water density is 1000 kg/m

^{3}. Therefore, there is a relationship A = B + 0.1. The dimensionless linear model can be written as:

_{ref}= $\sqrt{{p}_{ref}/{\rho}_{ref}}$ = 31.62 m/s. The dimensionless wave velocity is given by a′ = $\sqrt{dp/d\rho}$ = $\sqrt{B+0.1}$ = 47.15. Hence, the dimensional wave speed is a = a′ × u

_{ref}= 1491 m/s in water. Clearly, this wave speed derived from above model is consistent with the existing experimental value. The FVS method introduced in Section 2.2 is based on the linear model (7).

#### 2.2. Flux Vector Splitting Method

**A**and

**B**can be obtained by the flux derivative, as follows.

**A**and

**B**can be derived by $\left|\mathit{A}-\lambda \mathit{I}\right|=0$. For matrix

**A**, its three eigenvalues are λ

_{1}= u, λ

_{2}= u − a, and λ

_{3}= u + a. For matrix

**B**, its three eigenvalues are η

_{1}= v, η

_{2}= v − a, and η

_{3}= v + a. Then, we focus on the eigenvector of matrix

**A**. Its right eigenvector is defined by $\left(\mathit{A}-\lambda \mathit{I}\right)R=0$.

_{1}= u, the eigenvector is $\overrightarrow{{\mathit{r}}_{1}}={\left(\begin{array}{ccc}0& 0& 1\end{array}\right)}^{\mathrm{T}}$. For λ

_{2}= u − a, the eigenvector is $\overrightarrow{{\mathit{r}}_{2}}={\left(\begin{array}{ccc}1& u-a& v\end{array}\right)}^{\mathrm{T}}$. For λ

_{3}= u + a, the eigenvector is $\overrightarrow{{\mathit{r}}_{3}}={\left(\begin{array}{ccc}1& u+a& v\end{array}\right)}^{\mathrm{T}}$. Then, the right eigenvector

**R**is given by:

**AR**= λ

**R**. Then, the left eigenvector

**L**is derived by

**RL**=

**I**, where

**I**is the unit matrix. The detailed expression is as follows.

**LA**= λ

**L**. Then, the formulae of $R\Lambda $ and $R\Lambda \mathit{L}$ are given by:

_{1}= u, λ

_{2}= u − a, and λ

_{3}= u + a for matrix

**A**, and they are η

_{1}= v, η

_{2}= v − a, and η

_{3}= v + a for matrix

**B**. At this point, we have derived the FVS formula as given in the above Equation (15).

## 3. Results and Discussion

#### 3.1. Validation of Water Hammer

#### 3.2. Validation of Pulse Hydraulic Fracturing

#### 3.3. Discussion

## 4. Conclusions

**A**are λ

_{1}= u, λ

_{2}= u − a, and λ

_{3}= u + a. The eigenvalues of matrix

**B**are η

_{1}= v, η

_{2}= v − a, and η

_{3}= v + a. The detailed derivation processes are given in Section 2. Based on Formula (16), many kinds of high-order algorithms can be used to simulate the weakly compressible water flow. In this article, we take the fifth-order WENO algorithm as an example to show that the proposed FVS formula is reliable and valuable for the high-precision simulation of weakly compressible water flow.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

a | Wave speed, m/s | u | Streamwise velocity, m/s |

A | Slope in linear model, J/kg | v | Normal velocity, m/s |

B | Intercept in linear model, Pa | x | Streamwise coordinate, m |

d | Pipe diameter, m | y | Normal coordinate, m |

e | Internal energy per unit mass, J/kg | U | Conservative flux |

H | Width of the wedge, m | F | Flow flux in x direction |

L | Length of the pipe, m | G | Flow flux in y direction |

p | Water pressure, Pa | γ | Specific heat ratio |

R | Gas constant | λ | Eigenvalues of matrix A |

t | Time, s | η | Eigenvalues of matrix B |

T | Water temperature, K | ρ | Fluid density, kg/m^{3} |

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**Figure 1.**Water equation of state described by a linear model (7) which is consistent with the Tait model. Here the y-axis shows dimensional pressure which is computed by combining Equation (7) and the reference pressure p

_{ref}= 1 MPa. The experimental data are from reference [15].

**Figure 3.**Comparison of water pressure for the water hammer case, (

**a**) variation regularity of water head at the right endpoint of pipe, (

**b**) variation regularity of water head at the midpoint of pipe. Here exp is the experimental data from [23].

**Figure 4.**Comparison of water pressure distribution, (

**a**) simulation results from the MAC method, (

**b**) simulation results from the FVS-WENO method.

**Figure 5.**Comparison of water pressure curves along the symmetry axis marked by the dashed line in Figure 4.

**Figure 6.**Simulation results of water shock wave reflection in wedge-shaped structure based on the present FVS-WENO method. The red represents the high-pressure and the blue represents the low-pressure.

**Figure 7.**Experimental photos of water shock waves’ reflection in wedge-shaped structure from reference [28].

Parameter | Value |
---|---|

Pipe length/(m) | 36 |

Flow velocity/(m/s) | 0.1 |

Initial pressure/(water head, m) | 32 |

Water temperature/(°C) | 15.4 |

Wave speed/(m/s) | 1319 |

Closure time of valve/(s) | 0.009 |

Grid number | 1000 |

Parameter | Value |
---|---|

Wedge length/(m) | 0.1 |

Wedge width/(m) | 0.03527 |

Wedge angle/(°) | 20 |

Initial pressure/(MPa) | 1 |

Pulse pressure/(MPa) | 2 |

Wave speed/(m/s) | 1491 |

Grid number Nx | 1600 |

Grid number Ny | 564 |

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**MDPI and ACS Style**

Li, H.; Huang, B.
Flux Vector Splitting Method of Weakly Compressible Water Navier-Stokes Equation and Its Application. *Water* **2023**, *15*, 3699.
https://doi.org/10.3390/w15203699

**AMA Style**

Li H, Huang B.
Flux Vector Splitting Method of Weakly Compressible Water Navier-Stokes Equation and Its Application. *Water*. 2023; 15(20):3699.
https://doi.org/10.3390/w15203699

**Chicago/Turabian Style**

Li, Heng, and Bingxiang Huang.
2023. "Flux Vector Splitting Method of Weakly Compressible Water Navier-Stokes Equation and Its Application" *Water* 15, no. 20: 3699.
https://doi.org/10.3390/w15203699