Numerical Simulation of an Offset Jet in Bounded Pool with Deflection Wall

The k-ε turbulentmodel andVOFmethodswere used to simulate the three-dimensional turbulence jet. Numerical simulationswere carried out for three different kinds of jets in a bounded pool with the deflection wall with angles of 0, 3, 6, and 9. The numerical simulation agrees well with the experimental data. The studies show that the length of the potential core zone increases with the increase of the deflection angle. The velocity distribution is consistent with the Gaussian distribution and almost not affected by the deflection angle in potential core zone. The decay rates of flow velocity in the transition zone are 1.195, 1.281, 1.439, and 1.532 corresponding to the unilateral deflection angles, 0, 3, 6, and 9, respectively.The decay rates of velocity in the transition zone are 1.928 and 2.835 corresponding to the bilateral deflection angles 3 and 6. It is also found that the spread of velocity is stronger in the vertical direction as the deflection angles become smaller. The spread rates of velocity with unilateral deflection wall are higher than those with bilateral deflection walls in the horizontal plane in the pool.


Introduction
Jet is an important motion type of fluid and it can be divided into the laminar jet and the turbulent jet according to its turbulent intensity.In hydraulic engineering, the jets almost belong to the turbulent jets [1].In more detail, As shown in Figure 1, the jet can be divided into three regions: (I) core region or potential flow zone: the max velocity in this zone still maintains the velocity at the jet exit; (II) transition zone: the velocity in this zone decays rapidly; (III) fully development zone.When it moves into the limited pool, the flow is known as confined jets, such as hydraulics jump for energy dissipation.The flow is called wall jet when the main flow is closed to the boundary.The flow is called offset jet when the jet distances away from the boundary.Rajaratnam [2] has discussed these categories in detail.The hydraulic jump with sudden enlargement and drop was commonly used to be an energy dissipator in hydraulic engineering and it is actually a kind of submerged offset jet.According to the pressure in bottom, floor can be divided into three regions: A recirculation region, where jet is not affected by the bottom floor, B impingement region, where the jet attaches the bottom floor due to the pressure, and C wall jet region, where the flow pattern is similar to the wall jet.
Submerged offset jets occur in many engineering applications, so submerged offset jet has received a lot of attention from researchers.However, there have been fewer investigations of jets with small offset ratios (/ < 5) [3][4][5].Nasr and Lai [5] determined the mean velocities and turbulence characteristics of a turbulent plane offset jet with a small offset ratio of 2.125 and a Reynolds number of 11,000 using LDA.Yang and Yeh [6] have numerically studied the field of an offset jet with Reynolds number of 15,000 for offset ratios of 3, 7, and 11.In their work, it was revealed that when the flow is fully developed near the exit and when the walls are far away from the region of concern, zero normal gradient boundary condition could be used at the entrainment and exit boundary.Agelin-Chaab and Tachie [7] reported that, in the early region of the flow development, the Reynolds number and the offset ratio have significant effects on the decay of the maximum mean velocity, but the decay and spread rates were found to be nearly independent of the offset ratio at lager downstream distances.Assoudi et al. [8] had investigated the mean velocity and turbulence characteristics of the turbulent offset jet using PIV technique at 3 velocity ratios and 2 offset ratios; their results indicated that increasing the offset ratio gives a better distribution of the jet within the flow field, giving rise to a better dynamic mixture.Nyantekyi-Kwakye et al. [9] studied experimentally the jet with different height ratios (0, 2 and 4); their work revealed that the large scale structures within the inner layer increase with wall normal distance from the pool bottom.Further, Rathore and Das [10] reported a numerical investigation for the turbulent offset jet and compared with results proposed by Launder and Sharma [11] and Yang and Shih [12].They observed a similar profile in the wall jet region due to the resemblance of an offset jet with that of a wall jet flow in the wall jet region.Durand et al. [13] used ADV measuring the velocity of offset jet at the Reynold numbers of 34,000, 53,000, and 86,000; their results indicated the wall normal location of maximum mean velocity and jet spread to be independent of Reynold number.Kishore and Dey [14] provided an experimental investigation of hydraulic characteristics of submerged offset jets by ADV; the streamwise velocity profiles were measured at different sections.The work of Assoudi et al. [15] studied the mean flow behavior of a three-dimensional turbulent offset jet issuing into quiescent ambient; they reported high offset height and it is shown that reattachment point depends strongly on the jet form and the offset height.
In addition, most of the previous studies limit the jet pools to a rectangle.However, sometimes a deflection stilling basin can easily be used to different upstream and downstream conditions; it is more economical than costly transitional structure.The gradually expanding hydraulic jump had been studied by Omid et al. [16].Besides, Khalif and Moccorquodale [17] carried out a mathematical model of strip integral to investigate radiative hydraulic jump.
With the rapid development of computer power and numerical methods, it is more feasible to use numerical simulation to study the complex turbulent jets.Yue-Tzu Yang (1994), Gu [18], Karim and Ali [19], and Mohammad (2011) show that the k- turbulence model is an effective method to study the jet.In this paper, the standard k- turbulence model and VOF multiphase flow model were used to study the submerged offset jet in no deflection pool, unilateral deflection (deflection angle: 3 ∘ , 6 ∘ , and 9 ∘ ), and bilateral deflection (deflection angle: 3 ∘ , 6 ∘ ), so the understanding of the flow pattern of the submerged offset jet in a pool with unilateral and bilateral deflection wall can be deepened.

Numerical Model
2.1.Calculation Model.ANSYS 15.0 ICEM was used to carry out the numerical models for six types of the jet pools, as shown in Table 1.The computational area contained a circular jet exit, a pool, an end sill, and a tailwater section.The diameter of jet exit is  = 5 m, the offset height is  = 4 m (offset ratio is / = 0.8), the width of the start section of the pool is  = 8 m (expansion ratio / = 0.625), the length of the pool is  = 134.5 m, the height of the pool is  = 40 m, and the height of the end sill is He = 17 m.All of cases were meshed using structural grids.

Governing Equations.
The standard  −  model was considered to be useful when applied to offset jet flow, Mondal et al. [20], Kumar [21], and Mondal et al. [22].In addition, Assoudi (2015) believed that the standard  −  model is more appropriate for the prediction of the turbulent offset jet with small offset ratio (the offset ratio is 0.8 in this paper).So the standard  −  turbulence model presented by Launder and Spalding (1972) was used.The equations of the turbulent kinetic energy, , and its dissipation rate, , are as follows: (1)  equation: (2)  equation: Here, When  is average density,   is the average velocity component in the th direction;  is the dynamic viscosity.  = 0.09,   = 1.0,   = 1.3,  1 = 1.44, and  2 = 1.92 are empirical constants;  is time, and  is pressure.
The volume of fluid (VOF) was used to track the air-water interface;  and  in (1) and ( 2) are given by the following equation: where   is the volume fraction of water,   and   are the density of water and air, respectively, and   and   are the viscosities of water and air, respectively.The governing equations were discretized based on the finite volume method using the SIMPLE algorithm for the pressure-velocity coupling.The gradient was calculated by least squares cell-based method.The pressure was discretized by PRESTO! and volume fraction was discretized by Geo-Reconstruct.The second-order upwind scheme was used for the momentum.The first-order upwind was selected for discretizing the turbulent kinetic energy and the dissipation rate.(c) Wall boundary: it is assumed that there is a no-slip velocity boundary condition; near-wall regions of the velocity are analyzed via the wall function.

Boundary Conditions
(d) Free surface: at the free surface, the inlet pressure boundary is chosen; the pressure value is standard atmospheric pressure.

Grid Testing.
In order to verify the accuracy of the numerical model, the grid convergence index (GCI) (Celik et al., 2008) was selected to test the uncertainty in the numerical results with different grid sizes.The procedure is summarized below.
Step 1. Define a representative grid size ℎ.
where Δ  is the volume and  is the total number of grids used for computations.
Step 4. Calculate the extrapolated values from Step 5. Calculate the approximate relative error  21  , extrapolated relative error  21  ext , and fine-grid convergence index CGI 21  fine : In this case, the streamwise mean velocity () at different positions on the -axis was selected as the variable .The sampling point is selected on the jet exit axis, and the / range is 0 to 10 due to the high velocity flow in this range which is the focus of this study.Table 2 shows examples of the calculation procedure for the three selected grids.According to Table 1, the numerical uncertainty in the fine-grid solution for the velocity at / = 7.6 was 2.71% which corresponded to ±0.50 m/s.GCI index values are plotted in the form of error bars, as shown in Figure 2. Based on the grid convergence index, the maximum discretization uncertainty was 7.9% which corresponded to ±1.73 m/s. the discretization uncertainty value ranged from 0.01% to 7.9%, with an average of 1.47%.Based on these results and considering the computational accuracy and time, the number of grids was reasonably set to 379,432.

Model Validation.
In order to verify the numerical model, the physical model experiments were performed in the State Key Laboratory of Hydraulic and Mountain River Engineering, Sichuan University.The physical model consisted of an upper water tank, a circular tube, a circular jet exit, a pool, an end sill, a tailwater section, a measuring weir, and a reservoir.The physical model scale is 1/35.The size of a jet pool with unilateral 6 ∘ deflection angle was identical to that in the numerical model; other parameters are shown in Table 3.The time average pressure of the bottom floor and the height of water surface had been measured; the layout of the pressure measuring point is shown in Figure 3.It is evident from Figure 4(a) that the water surface height from computation agrees with the corresponding experimental results.Figures 4(b), 4(c), and 4(d) show the pressure values on the bottom of the pool.It can be seen that the numerical simulations were in good agreement with the experimental results; the maximum errors in the pressure were 8.01%.The numerical results thus produced acceptable error values.

Numerical Results and Analysis
3.1.Flow Pattern.When the jet enters the quiescent body of water, there is a velocity shear between the entering and the ambient water.The velocity shear is unavoidably disturbed during flow, loss of stability, and develops flow oscillations in the shear layer.These oscillations will roll up to form vortexes with increase in size and strength with the axial distance.The vortexes will influence the entrainment of the ambient water and the mixing of the ambient water and the jet water.(1) For the jet pool with no deflection, the volume of ambient water is relatively small due to the expansion ratio / = 0.625; the spread of the flow section is greatly restricted by the walls on both sides.From Figure 5(a), there is very small vertical axis vortex on both sides of the front part of the jet, then disappearing fast as the flow section spread to the side walls.(2) For the jet pool with unilateral deflection wall, the ambient water on both sides of the jet is asymmetrical; the restriction of the jet flow section on the side of the deflection wall is less compared to the side of the nondeflection wall; there are recirculation zone and the vertical axis vortex due to the jet flow interaction with the ambient water on the side of deflection wall.It is evident from Figures 5(b), 5(c), and 5(d) that the recirculation zone and vertical axis vortex are larger as the deflection angle increase.
(3) For the jet pool with bilateral deflection, the process of mixing between flow jet and ambient water less affected by the side walls, the oscillations have enough conditions to produce vortexes, if the deflection angle is large enough.As shown in Figures 5(e) and 5(f) the recirculation zone and vertical axis vortex are symmetrically dispersed on both sides of the jet; the size of the vortexes increases as the deflection angle increases.

The Velocity Attenuation of Mainstream.
Due to the lateral transmission of momentum, the entrained fluid gets momentum and flows forward with the original jet, and the original fluid momentum decreases and the velocities are lost to form a certain velocity gradient.The results of the mixing: the jet flow of section continues to expand, while the velocity is decreasing.In the mainstream direction, the velocity decay can be divided into three zones, including (I) potential core zone-the velocity in this zone still maintains the velocity at the jet exit; (II) transition zone-the velocity in this zone decays rapidly; (III) fully development zone-the velocity in this zone is small.Figure 6 shows the attenuation of maximum mean velocity Um for all types of jet pool.It can be found that, whether the jet pool is the unilateral deflection or bilateral deflection, the length of the potential core increases as the deflection angle increases.For the same deflection angle, the length of the potential core in the bilateral deflection jet pool is longer than that in unilateral deflection pool.The results show that if the deflection level of the jet pool is higher, flow is less affected by wall restrictions in the potential core zone, indicating that the turbulence intensity of the flow in potential core zone also is smaller, and the energy dissipation effect is worse.
Clark [23] reported that the decaying law of jet mean velocity in the transition zone is  ∝ (/) − , where     is called decay rate.Figure 7 shows that, within transition zone, the deflection angle of the pool has significant effect on the decay rate.The decay rate, , varies between 1.281∼1.532and 1.195∼2.835for unilateral deflection pool and bilateral deflection pool, respectively.The decay rate, , is lowest at the no deflection pool and increases with increasing deflection level.
As the flow oscillations will roll up to form vortexes, the kinetic energy from the jet flow is transferred to the turbulent vortexes.As a result, the pool consists of vortexes of different size.This energy is then handed down to smaller and smaller scales through an inviscid process called vortex stretching [24].At the smallest scales, the vortexes kinetic energy is dissipated by viscous action.For the transition zone, if the deflection level of the jet pool is higher, this indicates that the deflection level of the pool is higher; the effect of mixing on the velocity decay is more pronounced.

Velocity Distribution on Vertical Plane.
A jet spreads in the vertical directions as it entrains the surrounding fluid.
The vertical spread was expanded in both the inner and outer shear layer.It was expected that the pool floor would limit the spread in the inner layer.The velocity distribution in vertical direction at the end of potential core zone with different deflection angles can be seen in Figure 8.It can be found that the velocity distribution in the vertical direction is basically the same at the end of the potential core zone; it conforms with the Gaussian distribution.The results show that the velocity distribution characteristics in the potential core zone have little difference.Vertical half-width ( 1/2 ) distributions along the streamwise direction were shown in Figure 9 and it can be clearly seen that when / ≤ 8, the deflection angle of the jet pool (no deflection, unilateral deflection, and bilateral deflection) has little effect on  1/2 .Besides, in the potential core zone, the diffusion law of the velocity in the vertical direction is not affected by the deflection angle of the pool.In addition, Figure 9 also shows that when / > 8, the jet flow  1/2 (vertical half width) is higher when the pool is with no deflection or bilateral deflection.
Figure 10 shows the distribution of velocity on the bottom floor (Us) of the pools in streamwise direction.It can be seen that Us grows rapidly in / = 5∼13.The max velocity and velocity gradient are shown in Table 4, for the cases of unilateral deflection; when the unilateral deflection angle is 3 ∘ , 6 ∘ , and 9 ∘ , the maximum Us variation is very small, which is 0.29, 0.26, and 0.26, respectively, and these are similar to the value of pool with no deflection wall.For the cases of bilateral deflection, it is shown that as the deflection angle increases the maximum Us decreases.In terms of gradients of velocity on bottom floor ( * ), for the jet pools with unilateral deflection wall, the maximum  * decreases from 0.12 to 0.04 as the unilateral angle increased from 3 ∘ to 9 ∘ . * are 0.07 and 0.03, when the jet pools are with the bilateral deflection angle of 3 ∘ and 9 ∘ , respectively. * obtained in this paper is within the range of 0.03 ≤  * ≤ 0.16.
The variation rules of maximum Us and the maximum  * reflect the jet flow spread in the vertical direction.The above results indicate that the deflection angle has significant effect on the decay rate.This indicates the deflection angle and the jet spread in the inner layer have strongly contact; the larger the deflection level, the higher the lateral spread rate of jet flow; the corresponding vertical spread rate is weakened.velocity distribution in the horizontal direction for different cases is substantially the same at the end of the potential core zone and conforms to the Gaussian distribution.This indicates that the deflection wall has no significant effect on the jet spread in horizontal direction when the jet is still in the potential core zone.
Figure 11(b) shows the distribution of velocity on both side walls (U) of the pools in streamwise direction (unilateral deflection); Table 5 shows the maximum U and the corresponding location for all cases.The studies show that the jet pool is in the form of unilateral deflection; the maximum velocity U at the right side wall during the change of the deflection angle from 3 ∘ to 9 ∘ is always relatively higher than nondeflection wall.It is shown that the one side wall deflection of the pool has a little effect on the lateral spread rate of anther side.Figure 11(c) shows the distribution of velocity on both side walls (U) of the pools in streamwise direction (bilateral deflection); it is obvious that the jet flows attach the wall which need more distance from jet exit as the deflection angle increases; at the same time, the maximum U decreases as the deflection angle increases.Figure 11(d) compared the velocity in the deflection wall

Figure 1 :
Figure 1: The sketch of submerged offset jet inside a unilateral deflection pool.Note.Divided by velocity decay rate in streamwise direction: (I) core region or potential flow zone, (II) transition zone, and (III) full development zone.Divided by pressure in bottom floor: A recirculation region, B impingement region, and C wall jet region.
(a) Inlet boundary: the inlet is treated as an inlet velocity boundary, where the velocity is set to 30.66 m/s, so the mean velocity in the jet exit is 37.48 m/s.(b)Outlet boundary: the outlet is treated as an outlet pressure boundary.

Figure 2 :Figure 3 :
Figure 2: Fine-grid solution with discretization error bars computed using the GCI index.

Figure 4 :
Figure 4: Comparison of numerical and experimental values.

Figure 5 :
Figure 5: Instantaneous streamlines on different deflection patterns and deflection angle of the jet pool.

Figure 6 :Figure 7 :
Figure 6: Distribution of maximum mean velocity for all types of jet pool.Note.The f.t.means fitting curve.

Figure 11 :
Figure 11: (a) Velocity U distribution in horizontal direction at the end of the potential core.(b) Distribution of velocity on the both side wall (Us) of the pools in streamwise direction (unilateral deflection).(c) Distribution of velocity on both side walls (Us) of the pools in streamwise direction (bilateral deflection).(d) Comparing the velocity in the deflection wall between unilateral deflection pool and bilateral deflection pools.Note.The R means the right side wall, and the L means the left side wall.

Table 1 :
The jet pool types parameters.

Table 2 :
Sample calculations of discretization error using the GCI method.

Table 3 :
The parameters of the experiment.

Table 4 :
The max velocity and the max velocity gradient on the bottom floor of pools.

Table 5 :
The maximum U and the corresponding location for all test cases.