# Hydraulic Characteristics of Continuous Submerged Jet Impinging on a Wall by Using Numerical Simulation and PIV Experiment

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## Abstract

**:**

## 1. Introduction

_{2}injectors with different nozzle distances using CFD simulations. Ni et al. [23] used CFD numerical calculation methods as a means of assessing the performance of dredging jet pumps, and used Design of Experiment (DOE) methods to effectively analysis the sensitivity of design parameters to dredging performance and the interaction between parameters. Zhang et al. [24] used a version of the RNG k-ε turbulence model and R600a working fluid to investigate the effect of nozzle position on entrainment ratio, pressure lift ratio and the performance of the ejector. Fu et al. [25] established an axisymmetric two-dimensional mathematical model for transonic compressible flow inside a steam ejector to investigate the flow characteristics inside steam ejectors, aiming at optimizing primary steam nozzle outlet section distance and mixing chamber throat diameter simultaneously. Lucas et al. [26] used the computational fluid dynamics simulation software Fluent to establish a two-phase flow CO

_{2}ejector heterogeneous model using a numerical simulation of the internal flow field of a CO

_{2}injector based on a non-homogeneous phase model. The phase change, pressure and velocity changes of the internal flow field of the ejector were analyzed. Ma et al. [27] established a rock breaking model based on the SPH method by using the constitutive relation of abrasive material, the corresponding material parameters and the state equations to analyze the particular feature of rock deformation and breakage in oil and gas resources exploitation. He et al. [28] established a coupling model for simulating the rock mass destruction process under the impact of a high-pressure water jet based on the arbitrary Lagrangian-Eulerian finite element method (ALE-FEM). Guo et al. [29] used CFD methods to numerically simulate the flow field inside the multi-port jet wind field model, in order to analyze the flow characteristics of the wind field under the action of a multi-port jet fan. The effects of different nozzle numbers and arrangements on the performance of the flow field were compared and analyzed, and the vortex dynamics theory was introduced to analyze the development of the distribution of the vortex structure in the flow field and its effect on fluid mixing. Most of the above scholars used ANSYS Fluent as a means of numerical simulation. However, Flow-3D as a unique CFD solution technology [30,31,32], often used in areas such as marine engineering and hydraulic engineering, has rarely been used for research in the field of impinging jets. At the same time, the velocity decay characteristics of the impinging jet and its coupling characteristics with the ambient fluid are still a hot topic of research in related fields, and no systematic conclusions and explanations have been formed for the above-mentioned hot topics. At present, there are many scholars who do not stick to one tool alone, but use both experiment, numerical simulation and theoretical analysis to investigate the object of study [11,33,34,35,36,37,38].

## 2. Governing Equations and Turbulence Model

_{t}is the eddy current viscosity coefficient, μ is the dynamic viscosity coefficient, the empirical constants c

_{ε1}and c

_{ε2}are 1.42 and 1.68, respectively, c

_{3}= 0.012, η

_{0}= 4.38, c

_{μ}= 0.085, the Prandtl numbers α

_{k}and α

_{ε}correspond to the turbulent kinetic energy k, and dissipation rate ε, which are both 0.7194.

## 3. Calculation Regions and Boundary Conditions

_{w}is 180 mm, and the overflow baffle keeps the water level constant. In order to match the numerical calculation with the experiments described below, a solid wall with a length L

_{s}of 5000 mm is set.

_{b}(V

_{b}is the average velocity of the section at the outlet of the jet pipe). The lower boundary is set as the wall boundary, and the fluid cannot cross this boundary. The left and right boundaries are set as the pressure boundary, the pressure on the specified boundary is 0 Pa and the water level is 180 mm.

## 4. Experimental Verification

^{3}; μ is the dynamic viscosity of the fluid, Pa·s. Therefore, the Reynolds number in the circular pipe can be found as 23,400.

_{max}(V

_{max}is the maximum velocity at the exit of the jet pipe) curve with the empirical formula V/V

_{max}= (1–2r/D)

^{1/n}for the fully developed turbulent velocity distribution of the circular pipe and the results obtained from the model experiment. When n is 6 or 7, the empirical formula is considered to be approximately consistent with the fully developed turbulent velocity distribution of the circular pipe. As can be seen from the figure, the results of the submerged impinging jet applying the RNG k-ε turbulence model are in good agreement with the empirical formulation and the results of the model experiments, indicating not only that the flow at the jet outlet is a fully developed circular pipe turbulence, but also that the RNG k-ε turbulence model is suitable for the numerical simulation of the current model. Figure 4b shows the same corroboration of the above point, from which it can be seen that the numerical results of the axial flow angle of the jet are in good agreement with the experimental results, especially during the development of the jet. However, it is undeniable that there are still some differences between the two in the impact region. It can be noted that the fluid diffusion in the impact region in the experiments is small and the flow angle does not change much, while the results of numerical simulations show that the flow angle of the fluid in the jet decreases rapidly during the impingement process, but the trend of change remains consistent with the experimental results. Considering the accidental or systematic errors between numerical simulation and experiment, the experimental and numerical simulation results can be considered to be well matched.

## 5. Results and Discussion

#### 5.1. Flow Field Analysis of Submerged Jet under Different Impinging Distances

_{b}at different impinging distances for a Reynolds number Re of 23,400. As can be seen from the figure, regardless of the value of the impinging distance, at r/D = 0 (at the axis of the jet pipe), V/V

_{b}first remains constant and then starts to fall, where this distance at which the velocity remains constant is called the length of potential core region H

_{p}/D (H

_{p}is the absolute length of the potential core region). The length of the potential core region H

_{p}/D determined by the “95% guideline” [43]. That is, the axial length from the hole to the point where u

_{m}decays to 0.95 u

_{j}(outlet velocity of the jet pipe). The figure shows that the length of the potential core region H

_{p}/D and the distance from the end of the potential core region to the impinging plate (H/D-H

_{p}/D) both increase with the increase of the impinging distance H/D. This is consistent with the conclusions drawn in the literature [44]. In the initial segment (IS), when r/D = ±0.5, V/V

_{b}first decreases and then increases. The reason is that r/D = ± 0.5 of the radial position of the jet leaving the jet pipe first by the environmental fluid hindrance effect leads to a decrease in its velocity, and then, due to the jet boundary layer and the environmental fluid mixing with each other to promote the environmental fluid by the jet mainstream structure volume absorption and movement with the jet, an increase in the flow rate of the ambient jet results.

_{p}/D and (H/D-H

_{p}/D) with H/D. As can be seen from the figure, H

_{p}/D is almost 0 when H/D is 1. This is caused by the limited space during the development of the jet, and the shear layer starts to decay rapidly before it touches the central potential nucleus region. When 2 ≤ H/D ≤ 5, both H

_{p}/D and (H/D-H

_{p}/D) increased linearly with the increase of H/D. When H/D exceeds 5, as H/D increases, the increment of H

_{p}/D begins to decrease. The reason for this is that when the impinging distance increases to a certain value (H/D ≥ 5), the longer the impinging distance is, the longer the jet will be affected by the resistance produced by the ambient static fluid while it is developing. The jet can no longer continue to develop at the same speed.

_{p}/H with H/D. H

_{p}/H dimensionless sizes the corresponding length and can better describe the relative length of the potential core region. As can be found from the figure, when 1 ≤ H/D ≤ 2, H

_{p}/H increases rapidly. However, when H/D is gradually increased in the range of 4 to 7, the magnitude of the increase in H

_{p}/H keeps decreasing, compared with H/D = 6, when H/D = 7, H

_{p}/H only increased by about 0.44%. At H/D = 8, H

_{p}/H starts to decrease compared to the former, it shows that under the condition of Re = 23,400, the relative length of the potential core region reaches the maximum value when H/D = 7. As a whole, when 5 ≤ H/D ≤ 8, H

_{p}/H remains basically unchanged. The value can be taken as the average of H

_{p}/H at 4 impinging distances, and is 0.673. Its maximum error with H

_{p}/H at 4 impinging distances is only 3.94%.

_{y}/V

_{x}), which can reflect the flow direction at any point in the jet. As can be seen from the figure, when r/D = 0, the φ at the end of the potential core zone is always around 90°. When the jet leaves the potential core zone, the size of φ on the axis (r/D = 0) starts to decrease gradually as the axial distance increases, the axial velocity on the axis starts to be smaller, and the circumferential velocity starts to increase, so the length of the potential core zone can also be defined as the length of the jet axis where φ remains around 90°. When r/D = ±0.5, with the increase of H/D, the range of φ at the end of the potential core region gradually approaches 90°. It shows that with the increase of H/D, the component of the water flow velocity at the end of the potential core region in the axial direction is gradually increasing.

_{b}, the horizontal velocity component |V

_{x}|/V

_{b}and the vertical velocity component |V

_{y}|/V

_{b}along the jet axis for different impinging distances. The position of the two sub-attenuation regions (potential core region and transition region), and the impinging region of the free jet region, can be clearly distinguished from the figure.

_{b}and |V

_{y}|/V

_{b}of the outlet section of the jet pipe (l/D = 0) are obviously smaller than the corresponding velocity of the section at other impinging distances. However, |V

_{x}|/V

_{b}is larger than the corresponding velocity of other impinging distances. The reason for this is the small impinging distance. The jet is strongly influenced by the wall, and the wall jet region forms earlier and influences the development of the jet in the free jet region.

_{b}remains basically unchanged. In the transition region, the magnitude of V/V

_{b}decreases with the increase of l/D. In the impinging region, |V

_{x}|/V

_{b}increases significantly with the increase of l/D. On the contrary, |V

_{y}|/V

_{b}decreases rapidly, and its decreasing trend is basically the same as that of V/V

_{b}. This shows that in the impinging region, although the axial velocity of the jet gradually decreases and the flow direction starts to change, the velocity of the jet in the axial direction (r/D = 0) is still greatly affected by its axial velocity, and is relatively less affected by the axial velocity.

_{b}with l/D in the impinging region is basically consistent with the linear function. Comparing the slope of the linear function under each impinging distance in Table 1, it can be found that while H/D is increasing, |k| is decreasing. It shows that within a certain range, the increase of H/D can reduce the decay rate of V/V

_{b}in the impinging region. After the impinging distance increases to 3, (k

_{i+1}−k

_{i}) gradually increases with the increase of the impinging distance. That is, the rate at which |k| decreases is gradually increasing. As shown in Figure 10, the |k| value shows an arc-like decrease. It was fitted using polynomials, and the second order polynomials fit the |k| values better. Its equation is as follows:

_{b}curve and the abscissa axis of V/V

_{b}= 0 at each impinging distance is the same as the impinging distance H/D. Then:

_{b}of the jet axis in the impinging region and l/D can be obtained when H/D ≥ 3:

#### 5.2. Analysis of Submerged Jet Flow Field under Different Reynolds Numbers

_{b}under different Reynolds numbers Re (H/D = 3). As can be found from the figure, for all Re, the relationship between V/V

_{b}and l/D remains the same at a certain radial position. The length of the potential core region H

_{p}/D is around 1.05. It is suggested that the value of Re has essentially no effect on the axial velocity distribution of the jet and the length of the potential core region.

_{b}and l/D at different radial positions both approximately intersect at the same point (l/D = 2.4, V/V

_{b}= 0.65) in the impinging region. Additionally, there is a significant change in speed before and after this point. At the beginning, the jet flows from the initial section through the transition section, and the velocity first remains constant and then slowly decreases, at which time the velocity on the jet axis is greater than the flow velocity on both axes. When the jet reaches the position of l/D =2.4, the velocity on the jet axis begins to drop sharply, and the magnitude of the velocity starts to be lower than the velocity on both axes.

_{b}, horizontal velocity component |V

_{x}|/V

_{b}and vertical velocity component |V

_{y}|/V

_{b}along the jet axis at different Reynolds numbers. As can be found from the figure, when the Reynolds number is 11,700, the velocity along the jet axis in the initial section and transition section is slightly larger than the velocity in this range at other Reynolds numbers. The variation in Reynolds number in the impinging region has less effect on the magnitude of the velocity along the axis of the jet. The possible reason is that the low-velocity jet, when the Reynolds number is small, enters the static water body from the jet pipe. The resistance is less than the resistance of a high velocity flow with a higher Reynolds number. As a result, it has been able to maintain its original pace of development relatively well.

## 6. Conclusions

- (1)
- Flow-3D can properly simulate the flow of continuous submerged jets impacting a wall. Meanwhile, the prediction results based on the RNG k-ε turbulence model are in good agreement with the empirical equations and the results of the model experiments, indicating that the RNG k-ε turbulence model is suitable for the numerical simulation of the submerged impinging jet.
- (2)
- The length of the potential core region H
_{p}/D increases with the increase of the impinging distance H/D. Among them, when 2 ≤ H/D ≤ 5, the length of the potential core region H_{p}/D increases linearly with the increase of the impinging distance H/D; when H/D > 5, the length of the potential core region H_{p}/D still increases with the increase of the impinging distance H/D, but the increase decreases. - (3)
- The length of the potential core region can be defined as the length at which the flow angle in the axial direction of the jet is kept near 90°. As the impinging distance H/D increases, the component in the direction of the flow axis at the end of the potential core region gradually increases.
- (4)
- The velocity V/V
_{b}of the jet axis in the impact zone decreases linearly with the development of the jet. At the same time, the action range of the vortex caused by the wall flow of the jet after impacting the flat plate is gradually reduced inward with the increase of the impinging distance H/D. - (5)
- The flow structure and flow characteristics of continuous submerged impinging jets are relatively independent of the Reynolds number Re; i.e., the length of the potential core region and the relationship between the axial velocity of the jet within the impinging zone and the impinging distance are also applicable to the conditions at other Reynolds numbers.

^{TM}system). At the same time, the hydraulic characteristics and scouring effects of impinging jets under the influence of multiple factors will be considered, such as multiphase media (gas phase, solid phase), nozzles of different shapes and sizes, differentiated jet inlet conditions (e.g., pulsing) and ambient fluids with different initial velocity conditions.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**(

**a**) Schematic diagram of the experimental setup; (

**b**) PIV images of vertical impinging jets with velocity fields.

**Figure 4.**(

**a**) Velocity distribution verification at the outlet of the jet pipe; (

**b**) Distribution of flow angle in the mid-axis of the jet [39].

**Figure 5.**Along-range distribution of the dimensionless axial velocity of the jet at different impact distances.

**Figure 12.**Along-range distribution of the dimensionless axial velocity of the jet at different Reynolds numbers.

**Table 1.**Slope and intercept of the linear variation of V/V

_{b}as a function of l/D in the impinging region at different impinging distances and their standard errors.

Impinging Distance H/D | Slope k | (k_{i+1}−k_{i}) × 10^{−2} | Standard Error K | Intercept b | Standard Error b |
---|---|---|---|---|---|

1 | −1.02709 | 0.03184 | 1.11785 | 0.01861 | |

11.10900 | |||||

2 | −0.91600 | 0.02053 | 1.89851 | 0.03141 | |

1.33400 | |||||

3 | −0.90266 | 0.02614 | 2.76997 | 0.06783 | |

0.78100 | |||||

4 | −0.89485 | 0.02336 | 3.65376 | 0.08207 | |

1.67900 | |||||

5 | −0.87806 | 0.02715 | 4.47909 | 0.12244 | |

2.86400 | |||||

6 | −0.84942 | 0.02813 | 5.19117 | 0.15494 | |

4.88700 | |||||

7 | −0.80055 | 0.02342 | 5.68175 | 0.1524 | |

7.09600 | |||||

8 | −0.72959 | 0.01698 | 5.89248 | 0.12745 |

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**MDPI and ACS Style**

Mi, H.; Wang, C.; Jia, X.; Hu, B.; Wang, H.; Wang, H.; Zhu, Y.
Hydraulic Characteristics of Continuous Submerged Jet Impinging on a Wall by Using Numerical Simulation and PIV Experiment. *Sustainability* **2023**, *15*, 5159.
https://doi.org/10.3390/su15065159

**AMA Style**

Mi H, Wang C, Jia X, Hu B, Wang H, Wang H, Zhu Y.
Hydraulic Characteristics of Continuous Submerged Jet Impinging on a Wall by Using Numerical Simulation and PIV Experiment. *Sustainability*. 2023; 15(6):5159.
https://doi.org/10.3390/su15065159

**Chicago/Turabian Style**

Mi, Hongbo, Chuan Wang, Xuanwen Jia, Bo Hu, Hongliang Wang, Hui Wang, and Yong Zhu.
2023. "Hydraulic Characteristics of Continuous Submerged Jet Impinging on a Wall by Using Numerical Simulation and PIV Experiment" *Sustainability* 15, no. 6: 5159.
https://doi.org/10.3390/su15065159