# Plunging Circular Jets: Experimental Characterization of Dynamic Pressures near the Stagnation Zone

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Arrangement and Test Program

#### 2.2. About Scale Effects

## 3. Impact Pressures at the Stagnation Zone—Results and Discussion

- The mean dynamic pressure coefficient:$${C}_{p}=\frac{{P}_{mean}-Y}{\varphi \xb7{V}_{j}^{2}/2g}$$
- The fluctuating dynamic pressure coefficient:$${C}_{p}^{\prime}=\frac{\sigma}{\varphi \xb7{V}_{j}^{2}/2g}$$

#### 3.1. Radial Study of Mean Dynamic Pressures—$0.00\le r/D\le 0.69$

- (a)
**Core persistence (core length ${\mathit{Y}}_{\mathit{c}}$) for $\mathit{r}/\mathit{D}$ = 0.00**: ${Y}_{c}$ is defined as the depth required to develop the core jet and the starting depth to observe a decrease in $Cp$. For the lowest velocities with $F{r}_{pool}\le 1.92$, the core length ${Y}_{c}$ did not develop deeper than $8.3D$. For the highest velocities with $F{r}_{pool}\ge 1.92$ the core length ${Y}_{c}$ did not develop deeper than $9.7D$. The pool depth of $12.5D$ imposed an important decrease in $Cp$ for all tests, except the one with the lowest velocity ($F{r}_{pool}=0.98$). Generally, the core length ${Y}_{c}$ was deeper when comparing the ${Y}_{c}$ obtained by other authors, as represented in Figure 4. Table 3 summarizes the different core lengths ${Y}_{c}$ observed following each jet velocity. By using Table 3, it is possible to define shallow and deep pools. The fundamental question is whether the jet core (with a centreline velocity equal or close to the jet issuance velocity) prevails in depth and impacts the plunge pool bottom. In terms of pressure signature, one can define any pool with stagnation pressures equal or close to $\varphi \xb7{V}_{j}^{2}/2g$ as shallow pools. The Limited Depth Diffusion Model (LDDM) compiled and presented in Manso et al. [26] provides an adequate framework.A shallow pool depth Y is thus considered for a given velocity when the core jet impacts the pool bottom ($Y<{Y}_{c}$). A deep pool depth Y is considered for a given velocity when the core jet is already diffused and, thus, a developed jet is impacting the pool bottom ($Y>{Y}_{c})$. A schematic representation is available on the transverse section of Figure 2.- (b)
**Maximum mean dynamic pressure for $\mathit{r}/\mathit{D}$ = 0.00**: Increase up to the maximum $Cp$ value until $Y/D$ between 5–10, except for $F{r}_{pool}=0.98$ (V = 5.0 m/s). This maximum of $Cp$ with respect to $Y/D$ is “shifting to the right” when increasing velocity, as can be more easily seen with blue dots on Figure 5.- (c)
**Comparison of mean dynamic pressure with the literature for $\mathit{r}/\mathit{D}$ = 0.00**: From point (a) and (b), the results show a certain discrepancy when compared with the previous authors’ results at stagnation. Although this discrepancy could be explained with some different flow conditions, facility geometry, type of measurements, choice of how to compute the kinetic energy correction factor $\varphi $, the main idea regarding this difference is related to the core persistence ${Y}_{c}$, and thus to the definition of shallow and deep pools. Most previous researchers found a core jet impact for pool depth ratios up to 4.0–7.5 $Y/D$. In the present study, the authors measured high values of stagnation pressures, typical of the impact of the jet core for various jet velocity values. The most relevant difference with the previous authors is the persistence of the jet core impact up to 9.7 $Y/D$, for velocities V of 14.7 and 18.0 m/s (cf. Table 3). This long persistence of jet centreline velocities, with little velocity decay in depth, leading to high stagnation pressures at impact, reflects the stable and compact character of the jets generated at LNEC’s facility and the lower disturbance provided by pool flow features as compared to other similar facilities that were explored in the past [2,3,4,5,12,13,14,15,17].

- (a)
**The influence of velocity inside the jet diameter footprint ($\mathit{r}/\mathit{D}<\mathbf{0}.\mathbf{5}$)**: Increasing velocity shows a “shifting to the right” for the maximum mean dynamic pressure at distance $r/D$ = 0.00–0.25–0.35. The study of the stagnation zone ($r/D$ = 0.00–0.35–0.69–1.04) was made in Duarte et al. [27], concentrating on the influence of air entrainment and pressure inside fissures. The influence of these velocities on the results was already observed in this research but was not explored in terms of the $Cp$ coefficient with different velocities and pool depths along the flat bottom, which makes it difficult to compare the results.- (b)
**Pressure evolution inside the jet diameter footprint ($\mathit{r}/\mathit{D}<\mathbf{0}.\mathbf{5}$)**: $\frac{C{p}_{r/D}}{C{p}_{0}}$, the pressure ratio between $C{p}_{r/D}$, at a radial distance $r/D$, and $C{p}_{0}$, at the jet centreline $r/D=0.00$, respectively, for each pool depth considered Y, shows an important decrease in mean dynamic pressure, moving away from the jet centreline, even inside the jet diameter footprint ($r/D<0.5$). Table 4 summarizes results of this ratio, representing the remaining pressure at a distance of r. For example, at $r/D=0.35$, the average of the ratios $\frac{C{p}_{r/D=0.35}}{C{p}_{0}}$ is 72% for shallow pools and 79% for deep pools.- (c)
**Pressure outside the jet diameter footprint ($\mathit{r}/\mathit{D}=\mathbf{0}.\mathbf{69}$)**: Differing from previous observations made inside the jet diameter footprint, no increase in $Cp$ values was found while increasing $Y/D$ for $r/D=0.69$. This is nearly constant with the pool depth, with a smooth decrease. In addition, at $r/D=0.69$, the average of the ratios $\frac{C{p}_{r/D=0.69}}{C{p}_{0}}$ drops to 41% for shallow pools and 44% for deep pools.

#### 3.2. Radial Study of Fluctuating Dynamic Pressures — $0.00\le r/D\le 0.69$

- (a)
**Influence of acquisition frequency on measured pressure fluctuations**: It should be said that higher values of fluctuating pressures $C{p}^{\prime}$ were measured using a frequency acquisition of 2400 Hz compared to values measured using 600 Hz. This was especially observed for higher velocities (V = 12.0–14.7–18.0 m/s) and shallow pool depths ($Y/D$ < 6–8), where higher values of $C{p}^{\prime}$ were noticed for 2400 Hz of about 10–30%. Comparing power spectra from 600 Hz and the ones with 2400 Hz showed a similar loss of information concerning turbulence with high frequencies and a low energy content. The sensor outside the jet diameter footprint was more subjected to differences in $C{p}^{\prime}$. This observation could bring to the unsafe side of engineering practice if too low acquisition frequency is used. This is why only the results of fluctuations recorded at 2400 Hz are presented.- (b)
**Comparison of fluctuations with the literature for $\mathit{r}/\mathit{D}$ = 0.00**: A good envelope was found for $C{p}^{\prime}$ data from the plot of Bollaert [2]. The closest sensor to the jet centreline in Bollaert’s work was at $r/D=0.35$ but, following the results in Figure 8, it is still relevant to plot values of sensors at different radial distance $r/D$ on the same graph. Indeed, distance from the jet axis has a far lesser influence on pressure fluctuations $C{p}^{\prime}$ compared to the influence on $Cp$ when looking at sensors of up to $r/D\le 0.5$, i.e., those still inside the jet diameter’s footprint. Regarding other author contributions, (Bollaert & Schleiss [19]; Castillo et al. [17]), one can observe the wide range of trends observed when best fitting their data to the evaluation of the fluctuations. A peak in fluctuations was obtained in nearly all previous studies around a pool depth of $Y/D$ = 4–6. However, as can be seen in Figure 8, this is not what was observed in the present study.- (c)
**Maximum value of fluctuations with respect to $\mathit{Y}/\mathit{D}$**: In Ervine et al. 1997 [15], the maximum is around $C{p}^{\prime}=0.2$ for $Y/D=6$. Globally, a concentration of values is detected around this point, but the same maximum peak for $Y/D=6$ cannot be clearly observed. In the dataset at our disposal, maximum values of $Cp$ appear between $Y/D$ = 6 and 11. As the velocity V increases ($F{r}_{pool}$), the maximum value of $C{p}^{\prime}$ is “shifting to the right”, meaning that increasing the pool depth ratio $Y/D$ can have a counter-productive effect on the pressure fluctuations acting on the pool bottom. A similar phenomenon was already observed for $Cp$, and is more visible for $C{p}^{\prime}$ in the Figure 8 with fewer data plotted in one graph. More tests should be performed to obtain a more relevant statistical study, over a wider range of velocities, to observe if this “shift to the right” is confirmed and progresses with velocities higher than $V=18$ m/s.

#### 3.3. Power of the Jet in the Frequency Domain—$0.00\le r/D\le 0.69$

**Pool depth**: shallow pool ($Y/D=4.2$) and deep pool ($Y/D=12.5$).**Radial distance from jet axis**: inside the jet diameter footprint ($r/D<0.50$) and outside ($r/D>0.50$).**Velocity of the jet**: influence of velocity V on distribution of spectral content (from V = 5.0 m/s to 18.0 m/s).

- (a)
- Differences in variance distribution inside ($r/D<0.50$) and outside the jet diameter footprint ($r/D>0.50$). Inside, the three considered sensors ($r/D$ = 0.00–0.25–0.35) follow the exact same trends, as opposed to the sensor outside at $r/D=0.69$ (comparing colored spectra and grey/black spectra on the Figure).
- (b)
- Differences in variance distribution for shallow and deep pool depths; thus, the impact of a predominantly core jet against a more developed one. As the pool depth increases, a reduction in the high frequencies’ participation in the spectrum is observed. At a deep pool depth ($Y/D=12.5$), most energy is dissipated on low frequencies with a high amplitude, represented by large eddies of turbulent flow recirculating due to the deflection and limited size of the facility;
- (c)
- Similarly to the trend observed in $Cp$, $C{p}^{\prime}$, the “shift to the right” with increasing velocity is noticed with the small concave shape formed in the spectra. This breakpoint of the decay is the starting frequency of the dissipation zone, which leads to the cascade of energy, distributing the energy from large-scale eddies to small ones with higher frequencies. Thus, by increasing the velocity of the jet, more energy is shifted to high-frequency phenomena. This transition point, influenced by velocity, moves from approximately 15 Hz (V = 5.0 m/s) to 100 Hz (V = 18.0 m/s) for the shallow pool and from 3 Hz (V = 5.0 m/s) to 25 Hz (V = 18.0 m/s) for the deep pool. The influence of velocity on the spectral distribution was also observed in Manso et al. [28], studying laterally confined plunge pools.
- (d)
- Following previous researches and confirmed by EPFL researchers [19,21,27,28], there is a predominance of linear slope decay of ${f}^{-1}$ for shallow pools, even at high frequencies, and a sudden decay of ${f}^{-5/3}$ or ${f}^{-7/3}$ for deep pools. In Figure 9, the latter observation on the predominance of slopes for deep pools can be validated, but no linear decay with slope ${f}^{-1}$ is perceived at high frequencies, meaning that the minimum depth $Y/D$ = 4.2 available likely does not represent a shallow pool depth with core impact conditions. Moreover, EPFL experiments were recorded at 1000 Hz, 2.4 times less than on the graphs presented here, and it was shown that, in some cases, information can be lost if a sufficiently high acquisition frequency is not used. This is why, to the authors’ knowledge, no such slope of ${f}^{-20/3}$ was already observed on the highest range of frequencies ($f>500$ Hz).

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## List of Symbols

r | radial horizontal distance from jet axis |

D | diameter of the jet at issuance |

B | thickness of the jet at impingement (rectangular case) |

Y | pool depth |

${W}_{pool}$ | pool width |

H | jet travel distance in the air |

${L}_{b}$ | jet break-up length in the air |

${Y}_{c}$ | jet core persistence (core development length) |

V | velocity at issuance |

${V}_{j}$ | velocity at impact with water mattress |

$\varphi $ | correction factor for non-uniform distribution at nozzle exit: $\varphi \xb7{V}_{j}$ |

g | gravitational acceleration; $g=9.81$ m/s${}^{2}$ |

$\sigma $ | standard deviation of a data sample |

${u}^{\prime}$ | root-mean-square (RMS) value of the axial component of turbulent velocity |

$Tu$ | turbulence intensity; $Tu={u}^{\prime}/V$ |

$Re$ | Reynolds number; $Re=(V\xb7D)/\nu $ |

$Fr$ | Froude number; $Fr=V/\sqrt{g\xb7D}$ |

$F{r}_{pool}$ | Froude pool number; $F{r}_{pool}=V/\sqrt{g\xb7{W}_{pool}}$ |

$\nu $ | kinematic viscosity $\nu =1.15\xb7{10}^{-6}$ m${}^{2}$/s at 15 °C |

Q | discharge |

$Cp$ | mean dynamic pressure coefficient |

$C{p}_{r/D}$ | mean dynamic pressure coefficient at a distance $r/D$ (can be written as $C{p}_{r}$) |

$C{p}_{0}$ | mean dynamic pressure coefficient at a distance $r/D=0$ (stagnation point) |

$C{p}^{\prime}$ | fluctuating dynamic pressure coefficient |

${P}_{mean}$ | mean pressure value of a data sample |

${S}_{xx}$ | spectral power content [Unit${}^{2}$/Hz] |

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**Figure 1.**Photographs of the experimental facility at LNEC, Lisbon: (

**a**) General view, (

**b**) View from a flapgate.

**Figure 4.**Mean dynamic pressure coefficient $Cp$ [-] as a function of the pool depth ratio $Y/D$—at a radial distance $r/D=0.00$—Jet break-up length ratio $H/{L}_{b}\le 0.68$.

**Figure 5.**Mean dynamic pressure coefficient $Cp$ [-] as a function of the pool depth ratio $Y/D$—at a radial distance $r/D$ = 0.00–0.25–0.35–0.69—Jet break-up length ratio $H/{L}_{b}\le 0.68$.

**Figure 6.**Variation of $\frac{C{p}_{r/D}}{C{p}_{0}}$ with $r/Y$ out from jet centreline—difference made between shallow and deep pools.

**Figure 7.**Fluctuating dynamic coefficient $C{p}^{\prime}$ [-] as a function of the pool depth ratio $Y/D$—at a radial distance $r/D=0.00$—Jet break-up length ratio $H/{L}_{b}\le 0.68$.

**Figure 8.**Fluctuating dynamic coefficient $C{p}^{\prime}$ [-] as a function of the pool depth ratio $Y/D$—at a radial distance $r/D$ = 0.00–0.25–0.35–0.69—Jet break-up length ratio $H/{L}_{b}\le 0.68$.

**Figure 9.**Evaluation of influence of $Y/D$ (shallow/deep pool), radial distance $r/D$ and velocity V on non-dimensional spectral content ${P}_{xx}/{\sigma}^{2}$ [Hz${}^{-1}$]. (

**a**) V = 5.0 m/s; (

**b**) V = 7.4 m/s; (

**c**) V = 9.8 m/s; (

**d**) V = 12.0 m/s; (

**e**) V = 14.7 m/s; (

**f**) V = 18.0 m/s.

Q [l/s] | V [m/s] $\left({\mathit{Fr}}_{\mathit{pool}}\right)$ | Y [m] $(\mathit{Y}/\mathit{D})$ |
---|---|---|

20 | 5.0 (0.98) | 0.3 (4.2) |

30 | 7.4 (1.45) | 0.4 (5.6) |

40 | 9.8 (1.92) | 0.5 (6.9) |

49 | 12.0 (2.35) | 0.6 (8.3) |

60 | 14.7 (2.88) | 0.7 (9.7) |

73 | 18.0 (3.53) | 0.8 (11.1) |

0.9 (12.5) |

**Table 2.**Jet test conditions in terms of velocity at issuance V, at impact ${V}_{j}$, kinetic energy correction factor $\varphi $, Reynolds $Re$, Froude $Fr$, Froude Pool $F{r}_{pool}$ numbers and jet break-up length ratio $H/{L}_{b}$.

V [m/s] | ${\mathit{V}}_{\mathit{j}}$ [m/s] | $\mathit{\varphi}$ [-] | $\mathit{Re}$ * ${10}^{5}$ [-] | $\mathit{Fr}$ [-] | ${\mathit{Fr}}_{\mathit{pool}}$ [-] | $\mathit{H}/{\mathit{L}}_{\mathit{b}}$ [-] |
---|---|---|---|---|---|---|

5.0 | 5.2–6.2 | 1.000 | 3.1 | 5.9 | 0.98 | 0.04–0.68 |

7.4 | 7.5–8.3 | 1.083 | 4.7 | 8.8 | 1.45 | 0.04–0.59 |

9.8 | 9.9–10.5 | 1.075 | 6.1 | 11.7 | 1.92 | 0.03–0.54 |

12.0 | 12.1–12.6 | 1.058 | 7.5 | 14.3 | 2.35 | 0.03–0.50 |

14.7 | 14.8–15.2 | 1.019 | 9.2 | 17.5 | 2.88 | 0.03–0.46 |

18.0 | 18.1–18-4 | 1.096 | 11.3 | 21.4 | 3.53 | 0.03–0.43 |

V [m/s] $\left({\mathit{Fr}}_{\mathit{pool}}\right)$ | ${\mathit{Y}}_{\mathit{c}}$ [m] |
---|---|

5.0 (0.98) | <4.2 D |

7.4 (1.45) | 5.6 D |

9.8 (1.92) | 8.3 D |

12.0 (2.35) | 8.3 D |

14.7 (2.88) | 9.7 D |

18.0 (3.53) | 9.7 D |

**Table 4.**Summarizing results of the radial mean dynamic pressure ratio $\frac{C{p}_{r/D}}{C{p}_{0}}$ [%].

$\mathit{r}/\mathit{D}$ | Min | Max | Average for Shallow Pools | Average for Deep Pools |
---|---|---|---|---|

$0.25$ | 49% | 96% | 82% | 76% |

$0.35$ | 62% | 93% | 72% | 79% |

$0.69$ | 28% | 67% | 41% | 44% |

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**MDPI and ACS Style**

Jamet, G.; Muralha, A.; Melo, J.F.; Manso, P.A.; De Cesare, G.
Plunging Circular Jets: Experimental Characterization of Dynamic Pressures near the Stagnation Zone. *Water* **2022**, *14*, 173.
https://doi.org/10.3390/w14020173

**AMA Style**

Jamet G, Muralha A, Melo JF, Manso PA, De Cesare G.
Plunging Circular Jets: Experimental Characterization of Dynamic Pressures near the Stagnation Zone. *Water*. 2022; 14(2):173.
https://doi.org/10.3390/w14020173

**Chicago/Turabian Style**

Jamet, Grégoire, António Muralha, José F. Melo, Pedro A. Manso, and Giovanni De Cesare.
2022. "Plunging Circular Jets: Experimental Characterization of Dynamic Pressures near the Stagnation Zone" *Water* 14, no. 2: 173.
https://doi.org/10.3390/w14020173