# Numerical Study of the Velocity Decay of Offset Jet in a Narrow and Deep Pool

^{*}

## Abstract

**:**

_{exit}) in the early region of flow development, and a fitted formula is presented to estimate the length of the potential core zone (L

_{PC}). Analysis of the flow field for OR = 1 showed that the decay of cross-sectional streamwise maximum mean velocity (U

_{m}) in the transition zone could be fitted by power law with the decay rate n decreased from 1.768 to 1.197 as the ER increased, while the decay of U

_{m}for OR = 2 or 3 was observed accurately estimated by linear fit. Analysis of the flow field of circular offset jet showed that U

_{m}for OR = 2 decayed fastest due to the fact that the main flow could be spread evenly in floor-normal direction. For circular jets, the offset ratio and expansion ratio do not affect the spread of streamwise velocity in the early region of flow development. It was also observed that the absence of sudden expansion of offset jet is analogous to that of a plane offset jet, and the flow pattern is different.

## 1. Introduction

_{x}, L

_{y}, L

_{z}, S, d, h

_{t}, U

_{j}, and U

_{m}represent the length, width, and height of jet pool, offset height, the diameter of jet exit, depth of tailwater, bulk velocity, and cross-sectional streamwise maximum mean velocity, respectively. Likewise, if the jet exit is rectangular, a

_{0}, b

_{0}represents the height and width of the rectangle, respectively. The flow field in jet pool can be divided into three zones by the decay of U

_{m}: (a) the potential core zone, where U

_{m}was equal to the jet exit velocity, U

_{j}; (b) transition zone, where the decay of U

_{m}was rapidly in this zone; and (c) fully development zone, where the decay of U

_{m}became very slow.

_{0}= 28,475–80,730, where OR = S/a

_{0}(for circular jet exit a

_{0}= d), SR = h

_{t}/S, ER = L

_{y}/b

_{0}, and R

_{0}= (U

_{0}a

_{0})/υ, υ is the coefficient of kinematic viscosity of water. They concluded important characteristic length of submerged offset jets, such as the length of the recirculation region and impingement region, are expressed as a function of the R

_{0}, OR, and SR. In another attempt, Bhuiyan [5] analyzed the characteristics of submerged turbulent plane offset jets (OR = 0.5–3.6) in channels with rough beds and shallow tailwater depths. The results indicated that for an offset height larger than the jet thickness, the peak velocity, the flow momentum decay faster in the downstream direction in an offset jet than in a turbulent plane wall jet. Durand et al. [6] studied the effect of Reynolds number on 3D offset jets both experimentally and numerically for R

_{0}= 34,000, 53,000 and 86,000. The results indicated the floor-normal location of maximum mean velocity and jet spread to be independent of Reynolds number. The investigated by Nyantekyi-Kwakye [7] on a 3D rectangular offset jet, performed at three offset ratios (OR = 0, 2, and 4) and revealed that large-scale structures dominate the inner layer of the wall jet region. For 3D circular offset jets (with OR ranging from 0–3.5), Agelin-chaab and Tachie [8] using a planar particle image velocimetry (PIV) system conducted the experiments to study the velocity. They observed that OR influenced both the decay of U

_{m}and growth of the shear layer within the developing region. Besides, there have been other experimental studies on 3D offset jets [9,10,11].

## 2. Numerical Simulation

#### 2.1. Mathematical Model

_{i}is component of velocity in the x

_{i}direction; $\epsilon $ is turbulent energy dissipation rate; ${\mu}_{t}$ is dynamic turbulent viscosity; ${\sigma}_{k}$, ${\sigma}_{\epsilon}$ is turbulent Prandtl number for k and $\epsilon $, respectively; ${G}_{k}$ is generation of turbulent kinetic energy due to mean velocity gradients; and ${\mu}_{t}$ and ${G}_{k}$ can be determined as

#### 2.2. Simulation Setup

^{®}, Canonsburg, PA, USA) was utilized to develop the numerical models for 13 types of offset jet. Hexahedral grids were used throughout the computational domain. The grid meshing is shown in Figure 2. A Cartesian coordinate system is used so that the origin is at the center of the intersection of the offset wall and floor. The computational domain consists of the pipe and jet pool. The jet is discharged from offset wall offset by a height S above the floor. The total pipe length is 1 m, and the type of pipe can be divided into the circular pipe and rectangular pipe. The length, L

_{x}, and height, L

_{z}, of the pool are 8 m and 0.8 m, respectively. For the circular pipe, the offset ratio (OR) varied from 1 to 3, the expansion ratio (ER) varied from 3 to 4.8 as the width of the jet pool increased from 0.3 m to 0.48 m. For the rectangular pipe, the aspect ratio (AR) of the exit varied from 0.33 to 11.46. Moreover, the area of exit remains constant for all shape of exits. The parameters of all cases are given in Table 1, and the jet exits and the offset wall are shown in Figure 3.

#### 2.3. Initial Conditions and Boundary Conditions

_{i}, P, k, and ε) are initialized at zero. The computed data are stored at every alternate time step for post-processing. All computations are conducted on an Inter core i7 3.60 GHz Windows machine.

- Inflow boundary: the inlet was treated as an inlet velocity boundary with the velocity was set as 5 m/s;
- Outflow boundary: pressure outlet boundary was selected at the outlet, the depth of tailwater was fixed at 0.55 m with the help of user-defined function (UDF);
- Free surface: pressure inlet was employed, and its value was the standard atmospheric pressure, the operating pressure and density were selected as 101,325 Pa and 1.225 kg/m
^{3}, respectively. - Wall boundary: for the parameters investigated in this study, the data near the wall was ineffective. No slip boundary condition is considered for velocity. To avoid the fine mesh required to resolve the viscous sub-layer near the boundary, so standard wall function method has been used.

#### 2.4. Numerical Discretizations

^{®}, Canonsburg, PA, USA) was utilized to perform the simulation. The governing equations are discretized using the implicit Finite Volume Method (FVM). The SIMPLE algorithm, using a relationship between velocity and pressure corrections to enforce mass conservation and to obtain the pressure field, was applied to couple the velocity and pressure. The least-squares cell-based method was used to calculate the gradient. PRESTO! was used to discretize the pressure and Geo-Reconstruct was used for the volume fraction. The second-order upwind scheme was used for the momentum and the first-order upwind for the turbulent kinetic energy and the dissipation rate with ANSYS Fluent’s default under relaxation values for all parameters. The time step was Δt = 0.0001~0.001 s and the iteration number was always less than 2.

#### 2.5. Grids Sensitivity and Model Validation

_{i}is the average grid size on the ith grid and h

_{1}> h

_{2}> h

_{3}.

## 3. Results and Analysis

#### 3.1. Velocity Attenuation

_{m}). Previous studies indicated that U

_{m}is constant within the potential core region, followed by a rapid decay with streamwise distance in transition zone [32,33]. The entrained ambient fluid gets momentum since the jet enters the pool, the section of jet flow continues to expand, and the velocity is decreasing as the results of the mixing of jet and ambient fluid. Figure 7a shows the distribution of U

_{m}normalized by the jet exit velocity, U

_{j}. Normalized values of U

_{m}were observed decayed sharply within the region x/d > L

_{PC}and the trend of decay of U

_{m}varies greatly in the transition zone for different circular offset jet. It can be seen from Figure 7b that the length of the potential core zone (L

_{PC}) varied from 5.0 to 5.5 with changes in ER, and a higher ER indicated a higher L

_{PC}.

_{PC}varies greatly with AR of the rectangular jet exit due to the circumference of exit (L

_{exit}) were changed. When L

_{exit}increased, the L

_{PC}decreased sharply, a fitting formula was used to estimate the relationship of L

_{exit}and L

_{PC}as indicated in Figure 7d, the empirical equation for L

_{PC}was employed as ${L}_{PC}/d=2100{\mathrm{e}}^{-2\left({L}_{exit}/d\right)}+0.5$.

_{m}over distance x (scaled by d) for Case R-AR11 is shown in Figure 7c. The most striking feature of Case R-AR11 is that there is no sudden expansion (ER = 1). Thus, the Case R-AR11 is analogous to that of a plane offset jet. The decay of U

_{m}in the recirculation region is quite sharp, the U

_{m}drops to a local minimum value due to an increase in pressure resulting from the jet impingement after the reattachment of the jet. The decay of U

_{m}is rather gradual in the wall jet region. A similar observation was reported by Gu [16] and Dey [4].

_{m}/U

_{j}decay profiles can be grouped based on their characteristics. The power law of the form ${U}_{m}/{U}_{j}=C{\left(x/d\right)}^{-n}$ is typically used to describe the velocity decay in the transition zone [6,32,34,35], where C and n are a proportionality symbol and decay rate, respectively. The decay rate indicated the extent of entrainment and boundary effects on the jet. Figure 8 shows the variation of U

_{m}/U

_{j}for all cases with the normalized longitudinal coordinate x/d within the transition zone. Rajaratman reported n = 0.5 for a plane free jet by considering simplified conservation laws and entrainment hypotheses [36]. The wall jet was observed to be accurately estimated by using the power law. However, the distribution of U

_{m}/U

_{j}for the offset jets are not accurately described by the power law, which should be replaced by a linear fit [32].

_{m}/U

_{j}

_{,}for the higher offset ratio (OR = 2 or 3) circular jet are not accurately described by the power law. Therefore, a linear fit was used to estimate the decay rates for higher offset ratio jet, the slope κ the linear equation was considered as the decay rate. Decay rate κ values of 0.110, 0.096 and 0.090 with R

^{2}above 0.94 were obtained for ER = 3, 4 and 4.8, respectively, when OR = 2. Decay rate values of 0.081, 0.048 and 0.049 with R

^{2}above 0.968 were obtained for ER = 3, 4 and 4.8, respectively, when OR = 3. Above analysis and as shown in Figure 8a–c indicated that when the offset ratio is moderate (OR = 2) produces a faster decay than that of higher or lower offset ratio (OR = 1 or 3). The reason is that two circulating vortices were developed below and above jet, respectively. The circulating vortices develop reverse flows against the inflow-jet direction. Negative momentum of the reverse flows reduced the inflow momentum slowing the jet rapidly down.

_{m}/U

_{j}for 1 ≤ AR ≤ 3 is expressed well by power law form with the decay rate n varies from 1.068 to 1.228 (Table 3), and the U

_{m}decay in transition zone does not show substantial differences (Figure 8d). Therefore, it appears reasonable to assume the effect of the aspect ratio of the rectangular jet on the mainstream attenuation is mainly to influence the length of the potential core zone.

_{PC}) and decay profiles in the transition zone could be used to roughly calculate the length of the energy dissipation region (L

_{ED}). For example, a square offset jet is issued to a pool. The side length of the square is 2.67 m (equivalent diameter is 3 m), the offset height is 3 m, the width of the pool is 12 m, and the height of tail water is 16.5 m. If the U/U

_{m}= 0.2 as the end of the length of the energy dissipation region, the distance U/U

_{m}decreases from 1 to 0.2 is called L

_{tr}. The L

_{PC}can be described by:

_{PC}= 6.60 m calculated by ${L}_{PC}/d=2100{\mathrm{e}}^{-2\left({L}_{exit}/d\right)}+0.5$. The L

_{tr}= 39.53 m calculated by $0.2=-0.096\left(x/d\right)+1.465$. As the result, the length of energy dissipation region is 43.13 m.

#### 3.2. Vertical Velocity Spread

_{m}and d, respectively. Here, z

_{c}is the distance from the center of exit to the floor, that is, z − z

_{c}= 0, where is the axis of the jet exits. Vertical profiles of U are shown in Figure 10a for ER is 4. The effect of offset ratio (OR) on the vertical velocity spread was not evident within the region x/d < 9. It can be noted that the velocity distribution in the floor-normal direction conforms with the Gaussian distribution within the region of 5 ≤ x/d ≤ 9. However, as the jet travels away from the jet exits (x/d > 9), the data of jets at different offset height are no longer overlapped. The location of the maximum of U/U

_{m}for OR = 2 is almost at the axis of the jet exits. The location of the maximum of U/U

_{m}is above and below the axis of the jet exits for OR = 3 and OR = 1, respectively. The results reveal offset height changes travel direction of the bulk of the jet.

_{m}in the symmetry plane for the offset jets, and the lower and upper lines represent the loci of ${z}_{0.5}^{-}$ (U = 0.5 U

_{m}, the inner separation line), and ${z}_{0.5}^{+}$ (U = 0.5 U

_{m}, the outer separation line), respectively. The jet between ${z}_{0.5}^{-}$ and ${z}_{0.5}^{+}$ was considered as the main flow in the present paper. The ${L}_{Z0.5}^{-}$ is the distance from the exit to where the ${z}_{0.5}^{-}$ reaches the floor, and ${L}_{Z0.5}^{+}$ is the distance from the exit to where the ${z}_{0.5}^{+}$ reaches the water surface. For circular exits, the main flow for OR = 1 bends to the floor at x/d = 12 (${L}_{Z0.5}^{-}$ < 12 and ${L}_{Z0.5}^{+}$ > 46), then the flow spreads from floor to water surface (Figure 11a). The main flow for OR = 2 spreads evenly in the floor-normal direction at x/d > 20, the values of ${L}_{Z0.5}^{-}$ and ${L}_{Z0.5}^{+}$ are roughly equivalent (Figure 11b). Furthermore, the main flow for OR = 3 bends to the water surface, with the ${L}_{Z0.5}^{+}$ < 20 and ${L}_{Z0.5}^{-}$ > 42, which means the flow in the pool is spreading from water surface to floor (Figure 11c). For rectangular exits (Figure 11d), the main flow for AR = 1 is identical to that of a circular jet. The main flow for AR = 2 spreads from floor to water surface. The main flow for AR = 3 bends sharply to water surface as the jet enters the pool, the reason is that jet is too thin to effectively maintain a stable flow pattern. As the expansion ratio increases to 11.48, the spread range of the main flow in floor-normal direction is small, and the high velocity region is close to the floor.

#### 3.3. Lateral Velocity Spread

_{m}are lower for the expansion ratios, ER, are lower within the region y/d > 0.5. Figure 10 indicates that a more confining enclosure enhances the diffusion of the velocity than a less confining condition. The velocity profiles in the x-y plane of the jet with rectangular exits are not obtained. The flow patterns are not stable as the expansion ratio was changed [37].

## 4. Conclusions

- The development of the jet within the potential core zone was observed to be dependent on the circumference of jet exits. It was noted that the length of the potential core zone decreases with the circumference of exit.
- The results showed that the development of the U
_{m}in the transition zone could be divided into two modes. That is, (a) the decay of U_{m}could be estimated from power law fit with a decay rate n of 1.089–1.451 as the offset ratio is lower (OR = 1). (b) A linear fit was used to estimate the decay rates for higher offset ratio jet (OR = 2 or 3). It was observed that U_{m}decay is the fastest as the offset ratio (OR) is 2. Further, the results indicate that a more confining enclosure produces a more rapid jet decay than a less confining condition. - The spread rate of the circular jet is not affected by the expansion ratios and offset ratio in the early region of jet decay. The main flow for OR = 2 was traveled straightly, and the main flow was toward the floor and the water surface as the OR = 1 and OR = 3, respectively.
- The absence of the sudden expansion coupled with the high aspect ratio of the exit contributed to form a unique flow pattern compared with other cases.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

x | Stream wise direction |

y | Lateral direction |

z | Floor direction |

L_{x} | Length of the jet pool in the streamwise direction (m) |

L_{y} | The width of the jet pool in the lateral direction (m) |

L_{z} | The height of the jet pool in the floor-normal direction (m) |

d | The diameter of circular jet exit (m) or equivalent diameter of rectangular exit (m) |

a_{0} | The height of the rectangular exit (m) |

b_{0} | Width of rectangular exit (m) |

h_{t} | The height of tailwater (m) |

S | Offset height of jet exit (m) |

U | Mean velocity in streamwise (m/s) |

U_{m} | Cross-Sectional streamwise maximum mean velocity (m/s) |

U_{j} | Velocity in jet exit (m/s) |

L_{PC} | Length of potential core |

L_{exit} | The circumference of jet exit |

z_{c} | Distance from center of exit to floor, ${z}_{c}=S+0.5d$ or ${z}_{c}=S+0.5{a}_{0}$ |

${z}_{0.5}^{+}$ | The loci of where U = 0.5 U_{m} above the jet |

${z}_{0.5}^{-}$ | The loci of where U = 0.5 U_{m} below the jet |

${L}_{Z0.5}^{-}$ | Distance from the exit to where the ${z}_{0.5}^{-}$ reaches floor |

${L}_{Z0.5}^{+}$ | Distance from the exit to where the ${z}_{0.5}^{+}$ reaches water surface |

n | Decay rate used in power law |

κ | Decay rate used in the linear fit |

R_{0} | Reynolds number |

OR | Offset ratio of jet exit (S/d or S/a_{0}) |

ER | The expansion ratio of jet exit (L_{y}/d or L_{y}/b_{0}) |

AR | Aspect ratio of jet exit (b_{0}/a_{0}) |

## References

- SoaresFrazão, S. Wastewater Hydraulics; Springer: Berlin/Heidelberg, Germany, 2010; pp. 842–843. [Google Scholar]
- Liu, M.; Rajaratnam, N.; Zhu David, Z. Mean Flow and Turbulence Structure in Vertical Slot Fishways. J. Hydraul. Eng.
**2006**, 132, 765–777. [Google Scholar] [CrossRef] - Chen, J.-G.; Zhang, J.-M.; Xu, W.-L.; Li, S.; He, X.-L. Particle image velocimetry measurements of vortex structures in stilling basin of multi-horizontal submerged jets. J. Hydrodyn.
**2013**, 25, 556–563. [Google Scholar] [CrossRef] - Dey, S.; Ravi Kishore, G.; Castro-Orgaz, O.; Ali, S.Z. Hydrodynamics of submerged turbulent plane offset jets. Phys. Fluids
**2017**, 29, 065112. [Google Scholar] [CrossRef] - Bhuiyan, F.; Habibzadeh, A.; Rajaratnam, N.; Zhu David, Z. Reattached Turbulent Submerged Offset Jets on Rough Beds with Shallow Tailwater. J. Hydraul. Eng.
**2011**, 137, 1636–1648. [Google Scholar] [CrossRef] - Durand, Z.M.J.; Clark, S.P.; Tachie, M.F.; Malenchak, J.; Muluye, G. Experimental Study of Reynolds Number Effects on Three-Dimensional Offset Jets. In Proceedings of the 12th International Conference on Nanochannels, Microchannels, and Minichannels, Chicago, IL, USA, 3–7 August 2014. [Google Scholar] [CrossRef]
- Nyantekyi-Kwakye, B.; Tachie, M.F.; Clark, S.P.; Malenchak, J.; Muluye, G.Y. Experimental study of the flow structures of 3D turbulent offset jets. J. Hydraul. Res.
**2015**, 53, 773–786. [Google Scholar] [CrossRef] - Agelin-Chaab, M.; Tachie, M.F. Characteristics and structure of turbulent 3D offset jets. Int. J. Heat Fluid Flow
**2011**, 32, 608–620. [Google Scholar] [CrossRef] - Li, J.-N.; Zhang, J.-M.; Peng, Y. Characterization of the mean velocity of a circular jet in a bounded basin. J. Zhejiang Univ. Sci. A
**2017**, 18, 807–818. [Google Scholar] [CrossRef] - Kumar Rathore, S.; Kumar Das, M. Effect of Freestream Motion on Heat Transfer Characteristics of Turbulent Offset Jet. J. Therm. Sci. Eng. Appl.
**2015**, 8, 011021. [Google Scholar] [CrossRef] - Camino, G.A.; Zhu, D.Z.; Rajaratnam, N. Jet diffusion inside a confined chamber. J. Hydraul. Res.
**2012**, 50, 121–128. [Google Scholar] [CrossRef] - Zhang, Z.; Guo, Y.; Zeng, J.; Zheng, J.; Wu, X. Numerical Simulation of Vertical Buoyant Wall Jet Discharged into a Linearly Stratified Environment. J. Hydraul. Eng.
**2018**, 144, 06018009. [Google Scholar] [CrossRef] [Green Version] - Assoudi, A.; Habli, S.; Mahjoub Saïd, N.; Bournot, H.; Le Palec, G. Three-dimensional study of turbulent flow characteristics of an offset plane jet with variable density. Heat Mass Transf.
**2016**, 52, 2327–2343. [Google Scholar] [CrossRef] - Mondal, T.; Guha, A.; Das, M.K. Computational study of periodically unsteady interaction between a wall jet and an offset jet for various velocity ratios. Comput. Fluids
**2015**, 123, 146–161. [Google Scholar] [CrossRef] - Vishnuvardhanarao, E.; Das, M.K. Computation of Mean Flow and Thermal Characteristics of Incompressible Turbulent Offset Jet Flows. Numer. Heat Transf. A
**2007**, 53, 843–869. [Google Scholar] [CrossRef] - Gu, R. Modeling Two-Dimensional Turbulent Offset Jets. J. Hydraul. Eng.
**1996**, 122, 617–624. [Google Scholar] [CrossRef] - Hnaien, N.; Marzouk, S.; Ben Aissia, H.; Jay, J. CFD investigation on the offset ratio effect on thermal characteristics of a combined wall and offset jets flow. Heat Mass Transf.
**2017**, 53, 2531–2549. [Google Scholar] [CrossRef] - Abraham, J.; Magi, V. Computations of Transient Jets: RNG k-e Model Versus Standard k-e Model; SAE: Warrendale, PA, USA, 1997. [Google Scholar]
- Nasr, A.; Lai, J.C.S. A turbulent plane offset jet with small offset ratio. Exp. Fluids
**1998**, 24, 47–57. [Google Scholar] [CrossRef] - Launder, B.E.; Sharma, B.I. Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc. Lett. Heat Mass Transf.
**1974**, 1, 131–137. [Google Scholar] [CrossRef] - Bai, Z.; Wang, Y.; Zhang, J. Pressure distributions of stepped spillways with different horizontal face angles. In Proceedings of the Institution of Civil Engineers-Water Management; Thomas Telford Ltd.: London, UK; pp. 1–12.
- Li, S.; Zhang, J. Numerical Investigation on the Hydraulic Properties of the Skimming Flow over Pooled Stepped Spillway. Water
**2018**, 10, 1478. [Google Scholar] [CrossRef] - Hirt, C.W.; Nichols, B.D. Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries. J. Comput. Phys.
**1981**, 39, 201–225. [Google Scholar] [CrossRef] - Peng, Y.; Mao, Y.F.; Wang, B.; Xie, B. Study on C–S and P–R EOS in pseudo-potential lattice Boltzmann model for two-phase flows. Int. J. Mod. Phys. C
**2017**, 28, 1750120. [Google Scholar] [CrossRef] - Peng, Y.; Wang, B.; Mao, Y. Study on Force Schemes in Pseudopotential Lattice Boltzmann Model for Two-Phase Flows. J. Math. Probl. Eng.
**2018**, 2018, 9. [Google Scholar] [CrossRef] - Peng, Y.; Zhang, J.; Meng, J. Second-order force scheme for lattice Boltzmann model of shallow water flows. J. Hydraul. Res.
**2017**, 55, 592–597. [Google Scholar] [CrossRef] - Peng, Y.; Zhang, J.M.; Zhou, J.G. Lattice Boltzmann Model Using Two Relaxation Times for Shallow-Water Equations. J. Hydraul. Eng.
**2016**, 142, 06015017. [Google Scholar] [CrossRef] - Peng, Y.; Zhou, J.G.; Burrows, R. Modeling Free-Surface Flow in Rectangular Shallow Basins by Using Lattice Boltzmann Method. J. Hydraul. Eng.
**2011**, 137, 1680–1685. [Google Scholar] [CrossRef] - Peng, Y.; Zhou, J.G.; Burrows, R. Modelling solute transport in shallow water with the lattice Boltzmann method. Comput. Fluids
**2011**, 50, 181–188. [Google Scholar] [CrossRef] - Van Doormaal, J.P.; Raithby, G.D. Enhancements of the Simple Method for Predicting Incompressible Fluid Flows. Numer. Heat Transf.
**1984**, 7, 147–163. [Google Scholar] [CrossRef] - Celik, I.; Ghia, U.; Roache, P.; Freitas, C.J. Procedure for estimation and reporting of uncertainty due to discretization in CFD applications. J. Fluids Eng.
**2008**, 130, 078001. [Google Scholar] [CrossRef] - Nyantekyi-Kwakye, B.; Clark, S.P.; Tachie, M.F.; Malenchak, J.; Muluye, G. Flow characteristics within the recirculation region of three-dimensional turbulent offset jet. J. Hydraul. Res.
**2015**, 53, 230–242. [Google Scholar] [CrossRef] - Li, X.; Wang, Y.; Zhang, J. Numerical Simulation of an Offset Jet in Bounded Pool with Deflection Wall. Math. Probl. Eng.
**2017**, 2017, 11. [Google Scholar] [CrossRef] - Nyantekyi-Kwakye, B.; Tachie, M.F.; Clark, S.P.; Malenchak, J.; Muluye, G.Y. Acoustic Doppler velocimeter measurements of a submerged three-dimensional offset jet flow over rough surfaces. J. Hydraul. Res.
**2017**, 55, 40–49. [Google Scholar] [CrossRef] - Padmanabham, G.; Lakshmana Gowda, B.H. Mean and Turbulence Characteristics of a Class of Three-Dimensional Wall Jets—Part 1: Mean Flow Characteristics. J. Fluids Eng.
**1991**, 113, 620–628. [Google Scholar] [CrossRef] - Rajaratnam, N. Turbulents Jets; Elsevier: New York, NY, USA, 1976. [Google Scholar]
- Ohtsu, I.; Yasuda, Y.; Ishikawa, M. Submerged Hydraulic Jumps below Abrupt Expansions. J. Hydraul. Eng.
**1999**, 125, 492–499. [Google Scholar] [CrossRef]

**Figure 1.**the sketch of the submerged offset jet in a chamber. Flow divided by a relationship with bottom floor: ① recirculation region, ② impingement region, and ③ wall jet region.

**Figure 4.**(

**a**)Velocity profiles on jet exit axis with three grid sizes; (

**b**) Results from grid 2, with discretization error bars computed using GCI.

**Figure 6.**Model validation: (

**a**) Heights of the water surface in streamwise direction; (

**b**) Velocity distribution in streamwise direction (central line of exit); (

**c**) Velocity distribution in lateral direction; (

**d**) Velocity distribution in floor-normal direction.

**Figure 7.**(

**a**) U

_{m}decay for circular offset jets; (

**b**) the relationship of L

_{PC}, ER, and OR for circular offset jets; (

**c**) U

_{m}decay for rectangular offset jets (AR ≤ 3); (

**d**) the relationship of L

_{PC}and L

_{exit}.

**Figure 8.**(

**a**) power law fit to U

_{m}for circular offset jets as OR = 1; (

**b**) linear fit to U

_{m}for circular offset jets as OR = 2; (

**c**) linear fit to U

_{m}for circular offset jets as OR = 3; (

**d**) power law or linear fit to U

_{m}for rectangular offset jets.

**Figure 9.**Velocity vectors of offset jet with different offset ratio: (

**a**) OR = 1 (C-O1-E4); (

**b**) OR = 2 (C-O2-E4); (

**c**) OR = 3 (C-O3-E4).

**Figure 10.**Streamwise mean velocity distribution in symmetry (x–z) plane: (

**a**) circular offset jets (ER = 4); (

**b**) rectangular offset jets.

**Figure 11.**Contour of U/U

_{m}in symmetry (x–z) plane: (

**a**) circular offset jets (ER = 3); (

**b**) circular offset jets (ER = 4); (

**c**) circular offset jets (ER = 4.8); (

**d**) rectangular offset jets.

Case Name | Jet Exit Shape | d (m) | a_{0} (m) | b_{0} (m) | S (m) | L_{y} (m) | Offset Ratio (OR) | Expansion Ratio (ER) | AR (b_{0}/a_{0}) |
---|---|---|---|---|---|---|---|---|---|

C-O1-E3 | circular | 0.1 | - | - | 0.1 | 0.3 | 1 | 3 | - |

C-O1-E4 | circular | 0.1 | - | - | 0.1 | 0.4 | 1 | 4 | - |

C-O1-E5 | circular | 0.1 | - | - | 0.1 | 0.48 | 1 | 4.8 | - |

C-O2-E3 | circular | 0.1 | - | - | 0.2 | 0.3 | 2 | 3 | - |

C-O2-E4 | circular | 0.1 | - | - | 0.2 | 0.4 | 2 | 4 | - |

C-O2-E5 | circular | 0.1 | - | - | 0.2 | 0.48 | 2 | 4.8 | - |

C-O3-E3 | circular | 0.1 | - | - | 0.3 | 0.3 | 3 | 3 | - |

C-O3-E4 | circular | 0.1 | - | - | 0.3 | 0.4 | 3 | 4 | - |

C-O3-E5 | circular | 0.1 | - | - | 0.3 | 0.48 | 3 | 4.8 | - |

R-AR1 | rectangular | - | 0.089 | 0.089 | 0.1 | 0.3 | 1.13 | 3.39 | 1 |

R-AR2 | rectangular | - | 0.125 | 0.063 | 0.1 | 0.3 | 1.60 | 2.39 | 2 |

R-AR3 | rectangular | - | 0.153 | 0.051 | 0.1 | 0.3 | 1.95 | 1.95 | 3 |

R-AR11 | rectangular | - | 0.1 | 0.026 | 0.1 | 0.3 | 3.82 | 1.00 | 11.46 |

Decay Rate | n (OR = 1) | κ (OR = 2) | κ (OR = 3) |
---|---|---|---|

Fit Equation | Power Law | Linear Equation | |

ER = 3 | 1.768 | 0.110 | 0.081 |

ER = 4 | 1.448 | 0.096 | 0.049 |

ER = 4.8 | 1.197 | 0.090 | 0.048 |

Decay Rate | n (OR = 1) |
---|---|

Fit Equation | Power Law |

AR = 1 | 1.227 |

AR = 2 | 1.229 |

AR = 3 | 1.068 |

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## Share and Cite

**MDPI and ACS Style**

Li, X.; Zhou, M.; Zhang, J.; Xu, W.
Numerical Study of the Velocity Decay of Offset Jet in a Narrow and Deep Pool. *Water* **2019**, *11*, 59.
https://doi.org/10.3390/w11010059

**AMA Style**

Li X, Zhou M, Zhang J, Xu W.
Numerical Study of the Velocity Decay of Offset Jet in a Narrow and Deep Pool. *Water*. 2019; 11(1):59.
https://doi.org/10.3390/w11010059

**Chicago/Turabian Style**

Li, Xin, Maolin Zhou, Jianmin Zhang, and Weilin Xu.
2019. "Numerical Study of the Velocity Decay of Offset Jet in a Narrow and Deep Pool" *Water* 11, no. 1: 59.
https://doi.org/10.3390/w11010059