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Near-field dilution of a turbulent jet discharged into coastal waters: Effect of regular waves
Ocean Engineering 140 (2017) 29–42
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Near-field dilution of a turbulent jet discharged into coastal waters: Effect of regular waves
MARK
⁎
Zhenshan Xua, Yongping Chena,b, , Yana Wangc, Changkuan Zhanga a b c
College of Harbor, Coastal and Offshore Engineering, Hohai University, Nanjing 210098, China State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, China School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China
A R T I C L E I N F O
A BS T RAC T
Keywords: Dilution Near-field Jet Wave-following-current Regular wave
The effect of regular waves on three-dimensional scalar structures of a vertical round jet in the wave-followingcurrent environment is investigated. The wave effect is represented by two dimensionless parameters, i.e. the wave-to-current velocity ratio Rwc and Strouhal number St. The jet concentration distribution, including 13 cases in the wave-following-current environment and 1 case in the current-only environment, is obtained using the large eddy simulation method and validated by experimental data. The results show that the characteristics of the distinctive ‘effluent clouds’ phenomenon, namely the effluent cloud size and the distance between adjacent effluent clouds, are strongly dependent on Rwc and St. As a result, different structures of the timeaveraged concentration distribution are found in the transverse planes, and they are classified into three types: one-peak type, two-peak type, and three-peak type. The area of the concentration contour of C =0.25Cm, where Cm is the time-averaged cross-sectional maximum concentration, is defined as the jet visual area that represents the jet dilution characteristic. It highlights the existence of an optimal wave-to-current velocity ratio (approximately 0.6 here) for the highest dilution of a jet in the wave-following-current environment, which provides useful guidelines for selecting the sites of wastewater or brine discharge projects.
1. Introduction There is an urgent global requirement for the ocean disposal of urban treated wastewater and brine from desalination plants via submarine outfalls, as shown in Fig. 1 (Roberts and Snyder, 1993; Yang, 1995; Voutchkov, 2011). However, such plants are point sources of pollution, posing a potential threat to local human and wildlife communities (Roberts et al., 2010; Mendonça et al., 2013; Stark et al., 2016). A vital step in disposal management is selecting the site of submarine outfalls such that the wastewater or brine is quickly and effectively diluted by the ocean currents. Therefore, it is important to understand the effect of coastal waters on the mixing and dispersion of such discharge. From the viewpoint of environmental hydraulics, the movement of discharged wastewater or brine usually forms a turbulent jet, identified as a near-field process or a far-field process according to different time and distance scales. The time and distance scales in a near-field process are of the order of minutes and tens or hundreds of metres, whereas those in a far-field process are of the order of hours and kilometres, respectively (Roberts, 1999). It is important to study near-field mixing and dispersion, which primarily affects jet dilution in the far field and
⁎
partially dominates the entire process (Zhao et al., 2011; Chan et al., 2013). In the near field, there are two types of factors that affect the movement and dilution of the wastewater- or brine-formed jet, as shown in Fig. 2. One involves parameters associated with the jet itself, namely the initial velocity w0, the jet density ρ0, the geometry/diameter of the jet orifice d (circular or slit), and the discharge angle θ0. The other involves several ambient fluid parameters, such as water depth D, ambient fluid density ρa, and ambient fluid velocity Ua. A certain variable P representing the jet characteristic is determined by the following equation:
P = f (w0 , ρ0 , d , θ0, D, ρa , Ua )
By comparing the values of ρ0 and ρa, the jet is classified as a buoyant (e.g. salinity- or temperature-induced buoyancy) or nonbuoyant jet. As for the angle θ0, a jet outfall could be set horizontally, vertically, or obliquely in a discharge project. This study focuses only on the near-field movement and dilution of a vertical non-buoyant jet discharged into coastal waters. It should be noted that understanding such a jet is a prerequisite for studying an oblique buoyant jet. In coastal areas, the tidal current is one of the leading dynamic factors that affect the jet movement. Its action could be considered as
Corresponding author at: College of Harbor, Coastal and Offshore Engineering, Hohai University, Nanjing 210098, China.
http://dx.doi.org/10.1016/j.oceaneng.2017.05.003 Received 3 November 2016; Received in revised form 10 March 2017; Accepted 5 May 2017 Available online 18 May 2017 0029-8018/ © 2017 Elsevier Ltd. All rights reserved.
(1)
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the ‘effluent clouds’ phenomenon. This indicates that the wave action has a significant influence on the jet movement in the wave-current coexisting environment. However, only one case of a jet in such an environment was considered in their study. Xia and Lam (2004) studied a downward jet in an oscillating crossflow and found that the unsteadiness in the oscillating crossflow leads to an increase in the time-averaged jet width in the symmetrical plane. At present, there remains a lack of knowledge regarding the effect of wave periods and wave heights on the jet structure (particularly the three-dimensional structure) in the wave-current coexisting environment. Actually, instead of focusing on the jet-wave-current interaction, many researchers (e.g. Chyan and Hwung, 1993; Koole and Swan, 1994; Hsiao et al., 2011) have studied different flow structures of a jet in the wave-only environment by considering the relative strength between jets and waves. Mori and Chang (2003), Ryu et al. (2005), Chang et al. (2009) defined the wave-to-jet momentum ratio RM and identified it as a key parameter for describing the wave-jet interaction. With increasing RM, the horizontal jet in waves (similar to the vertical jet in waves) can be classified into three patterns: symmetric motion, asymmetric motion, and discontinuous motion (Mori and Chang, 2003). When the wave action is weak, the distribution of the time-averaged axial velocity or concentration in the cross sections follows a Gaussian shape. When the wave action is sufficiently enough, such a distribution will be flattopped or bi-peaked, especially in the jet deflection region. This has been confirmed by Chyan and Hwung (1993), Koole and Swan (1994), Hsiao et al. (2011), Xu et al. (2014). Thus, the jet undergoes a faster decay of centreline velocity, with a larger spreading width and higher dilution rate at higher RM. These conclusions serve as valuable references for the studies of the wave effect on the jet structure in the wave-current coexisting environment. Following the work of Xu et al. (2016), this study aims to investigate the three-dimensional scalar structure of a vertical round jet in the wave-current coexisting environment, using the method of large eddy simulation (LES). Initially, the wave-current coexisting environment is set to the regular wave-following-current flow. This paper provides a detailed description of the wave effects on the jet initial dilution as well as some guidelines for selecting sites for wastewater or brine discharge projects.
Main Pipe
Sea water
Wastewater/Brine
Diffusers
Fig. 1. Ocean disposal of wastewater or brine (modified from the diagram of Voutchkov (2011)).
Fig. 2. Factors affecting jet motion in the near field.
the effect of a series of steady currents over a long period. Under the current effect, the vertical jet is deflected and divided into the near zone, curvilinear zone, and vortex zone (Subramanya and Porey, 1984). The deflection is primarily determined by the jet-to-current velocity ratio, Rjc. Jets in the current-only environment with Rjc in the range of 2–25 have been investigated extensively in previous studies (e.g. Hodgson and Rajaratnam, 1992; Kelso et al., 1996; Moawad and Rajaratnam, 1998; Muppidi and Mahesh, 2005; Megerian et al., 2007). As Rjc increases, the jet reaches a greater height and the jet shear layer instability is weakened (Alves et al., 2008). Kelso et al. (1996) experimentally showed that the folding and rolling up of the jet shear layer very close to the jet orifice contribute to the formation of the counter-rotating vortex pair (CVP). Consequently, the jet trajectories and dilution are closely related to Rjc. For example, Wong (1991) expressed the vertical centreline minimum dilution Sc of a jet in the current-only environment as
⎛ x ⎞2/3 Sc d = 1.271 ⎜ ⎟ ⎝ Lm ⎠ Lm
2. Methodology 2.1. Dimensional analysis of jet characteristic parameters The characteristic parameters affecting the jet motion are shown in Fig. 2 and expressed by Eq. (1). For a vertical round jet, dimensional analysis yields the following dimensionless parameters:
⎛ ρ d w ⎞ P = f ⎜Rej, a , , 0 ⎟ ρ0 D Ua ⎠ ⎝
(2)
where
Lm =
M01/2 = 0.886Rjc d u0
(4)
where d is the jet diameter and Rej is the jet Reynolds number, which is defined as
(3)
Rej=ρ0
and M0 is the jet initial momentum flux. Nevertheless, the jet flow structure remains virtually unchanged with increasing Rjc. Four largescale coherent structures can be identified from a vertical jet in the current-only environment: jet shear layer vortices, horseshoe vortices, wake vortices, and CVP. Among these, CVP is a salient feature of the jet, which leads to a kidney-shaped and bifurcated cross-sectional scalar field, with distinct double concentration maxima (Lee et al., 2002). In addition to the tidal current, wave motion is a common dynamic factor in coastal areas. It usually coexists with the tidal current. As discussed by Xu et al. (2016), the three-dimensional flow structure of a vertical round jet in the wave-current coexisting environment is significantly distinct from that in the current-only environment, e.g.
w0 d υ
(5)
where υ is the kinematic coefficient of viscosity. The Reynolds number is sufficiently large in this study, and the jet motion belongs to the turbulent flow. The jet is non-buoyant, i.e. ρa/ρ0=1, and the ratio of the jet diameter to the water depth is assumed to be extremely small, i.e. d/ D ≈0. From these assumptions, Eq. (4) is simplified into
⎛w ⎞ P = f ⎜ 0⎟ ⎝ Ua ⎠
(6)
It can be seen that the velocity ratio w0/Ua is a key factor that determines the movement and dilution of the non-buoyant vertical round jet. For a jet in the current-only environment, the velocity ratio 30
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w0/Ua is defined as the ratio of the jet initial velocity w0 to the current velocity u0 (Kelso et al., 1996), i.e.
w0 u0
Rjc =
Table 1 Numerical cases of the jet in the wave-following-current environment.
It is a universal parameter for describing the jet and current interaction. For a jet in the wave-only environment, Chang et al. (2009) found that the wave-to-jet momentum ratio is the most important parameter for characterizing the wave effect on the jet diffusion in the wave-only environment. The ratio was expressed by Mori and Chang (2003) as
RM1 =
gH2 dw02
(8)
and by Ryu et al. (2005) as
RM 2 =
2gk w H2 w02
(9)
where g is the acceleration due to gravity, H is the wave height, and k is the wave number. It should be noted that the wave force varies with the water depth and increases with the wave height or wave period. The two above-mentioned momentum ratios were defined for a horizontal jet and only considered the effect of wave height. Hence, they are not suitable for a vertical jet in the wave-only environment. Chyan and Hwung (1993) found that the deflection region of wave-jet interaction has characteristics similar to a crossflow jet and recommended the use of the wave characteristic velocity to study the wave-jet interaction. Considering that the wave force near the jet orifice has a critical influence on the jet deflection, we chose the maximum horizontal particle velocity at the jet orifice position as the wave characteristic velocity. Following the small-amplitude wave theory, this velocity is expressed as
u wa = Hω
Case
Jet-tocurrent velocity ratio Rjc
Wavetocurrent velocity ratio Rwc
Strouhal number St
Jet initial velocity w0 (m/ s)
Current velocity u0 (m/s)
Wave period T (s)
Wave height H (cm)
J0 J1 J2 J3 J4 J5 J6 J7 J8 J9 J10 J11 J12 J13
10 10 10 10 10 10 10 10 10 10 10 10 10 10
/ 0.2 0.6 1.0 0.2 0.6 1.0 1.4 0.2 0.4 0.6 0.8 1.0 1.4
/ 0.2 0.2 0.2 0.133 0.133 0.133 0.133 0.1 0.1 0.1 0.1 0.1 0.1
0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05
/ 1.0 1.0 1.0 1.5 1.5 1.5 1.5 2.0 2.0 2.0 2.0 2.0 2.0
/ 1.15 3.45 5.75 0.64 1.92 3.20 4.48 0.54 1.08 1.62 2.16 2.70 3.78
(7)
cosh(kh 0 ) 2 sinh(kD )
Table 2 Parameters in the experimental runs. Run
Jet initial velocity w0 (m/s)
Current velocity u0 (m/s)
Wave period T (s)
Wave Height H (cm)
E1 E2 E3
0.58 0.58 1.22
0.055 0.055 0.133
1.0 1.4 1.4
3.0 3.0 7.0
(10)
where ω is the wave angular frequency and h0 is the height of the jet orifice from the bottom. Then, the jet-to-wave velocity ratio Rjw can be defined as
Rjw =
w0 u wa
Fig. 3. Computational domain of numerical model.
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are primarily dependent on Rjc, the structure characteristic of CVP, which is the most significant feature of the crossflow jet, remains virtually unchanged when Rjc ranges from 2 to 25. For simplicity, only one jet-to-current velocity ratio is considered in this study, namely Rjc =10, which is a common value in previous studies. When an effluent jet having a diameter of 0.5 m is discharged into an ocean current at 0.8 m/s, waves with periods of 3–8 s induce unsteadiness at a Strouhal number in the range of 0.08–0.2 (Xia and Lam, 2004). The Strouhal number in this study is set to three values, namely 0.2, 0.133, and 0.1. The jet diameter is 0.01 m and the current velocity is 0.05 m/s. Then, the wave period will be 1.0 s, 1.5 s, and 2.0 s. The wave-to-current velocity ratio Rwc is set from 0.2 to 1.4, considering different deflection situations of the jet in one wave cycle.
For a jet in the wave-following-current environment, the instantaneous jet-to-current velocity ratio changes continuously. The flow is governed by three dimensionless parameters similarly to a jet in an oscillating crossflow, which has been studied as the sinusoidal movement of the jet in a steady crossflow by Lam and Xia (2001), Xia and Lam (2004). These are the jet-to-current velocity ratio Rjc, the wave-tocurrent velocity ratio Rwc (the unsteadiness parameter in the study of Lam and Xia (2001)), and the Strouhal number St. These parameters are expressed as
Rwc =
St =
u wa u0
d u0 T
(12) (13)
If the velocity ratio Rwc is smaller than 1, the jet flow will always be deflected in the current direction. If the velocity ratio Rwc is larger than 1, the jet deflection will occur in two directions in one wave period. As discussed above, the key parameter w0/Ua in Eq. (6) is expressed in different forms in different environments, i.e. Rjc for the current-only environment, Rwj for the wave-only environment, and Rjc, Rwc, and St for the wave-following-current environment. Based on the dimensional analysis described above, 1 case of the jet in the current-only environment and 13 cases of the jet in the wavefollowing-current environment with different wave, current, and jet parameters are summarised in Table 1. Although the jet characteristics
2.2. Numerical model 2.2.1. Governing equations Xu et al. (2016) developed a σ-coordinate three-dimensional LES model based on spatially filtered Navier-Stokes equations to investigate the flow structure of a vertical round jet in the wave-current coexisting environment. The equations are expressed as
∂ui ∂Xk =0 ∂Xk ∂xi 31
(14)
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Fig. 4. Comparison between numerical results and experimental data for Run E3.
residual scalar flux Qj, which includes the information of the smallscale components, is modelled as
∂ui ∂ui ∂Xk ∂u ∂Xk 1 ∂p ∂Xk + + uj i =− + gi ∂τ ∂Xk ∂t ∂Xk ∂xj ρ ∂Xk ∂xi + (ν + νt )
∂Xk ∂ ⎛ ∂ui ∂Xm ⎞ ⎜ ⎟ ∂xj ∂Xk ⎝ ∂Xm ∂xj ⎠
Qj = −αt
(15)
∂c ∂c ∂Xk ∂c ∂Xk ∂Xk ∂ ⎛ ⎛ ν ν ⎞ ∂c ∂Xm ⎞ ⎜⎜ ⎟ + + uj = + t⎟ ⎝ ∂τ ∂Xk ∂t ∂Xk ∂xj ∂xj ∂Xk ⎝ Pr Prt ⎠ ∂Xm ∂xj ⎠
(16)
(17)
where Δx1, Δx2, and Δx3 are the grid sizes in the coordinates of x1, x2, and x3, respectively. To obtain the scalar field of the jet in the wave-following-current environment, the tracer mass conservation equation should also be considered:
⎞ ∂c ∂c ∂Xk ∂c ∂Xk ∂Xk ∂ ⎛ ∂c ∂Xm ⎜α + + uj = − Qj ⎟ ∂τ ∂Xk ∂t ∂Xk ∂xj ∂xj ∂Xk ⎝ ∂Xm ∂xj ⎠
2.2.2. Boundary conditions and numerical methods The details of the boundary conditions and numerical methods related to the flow module have been provided by Xu et al. (2016). The boundary conditions and numerical methods for the scalar transport are as follows. The zero-gradient condition is specified at the surface and bottom boundaries, while the radiation condition, coupled with the hydrostatic pressure conditions, is specified at the lateral boundaries. At the outflow boundary, the radiation condition combined with a sponger layer is imposed for the concentration to reduce the reflection.
(18)
where c is the large-scale component of the concentration and α is the molecular diffusion coefficient, which can be calculated as
α=
ν Pr
(21)
To obtain reliable concentration results, the values of Pr and Prt in the above equation should be carefully determined. Typical values for Prt are found to be in the range 0.3–0.9 from previous studies (Chua and Antonia, 1990; Pham et al., 2006; Rodi et al., 2013). It should be noted that the eddy viscosity νt in the jet region is much greater than the kinematic viscosity ν. As a result, the concentration results of the model are dominated by the second part (νt / Prt). In other words, the concentration results are more sensitive to Prt as compared to Pr. Hence, only the value of Prt will be calibrated in Sub-section 2.2.3. Following the choice of Ghaisas et al. (2015), we set the Pr value to 0.7 in this study.
where Cs is the Smagorinsky constant. The value of Cs should be calibrated and chosen based on the type of the flow (Yang et al., 2015). Its calibration is shown in Sub-section 2.2.3. Sij is the resolved strain rate tensor, and Δ is a representative grid spacing defined as
Δ = (Δx1 Δx2 Δx3)1/3
(20)
where αt is the turbulent diffusion coefficient and Prt is the turbulent Prandtl number. The final form of the governing equation for scalar transport is given by
where ui (i=1, 2, 3) and p are the large-scale components of velocity and pressure, respectively; ρ is the water density; gi is the acceleration due to gravity; t, xi and τ, Xi are the temporal and spatial coordinates in the Cartesian and σ-coordinate systems, respectively; ν is the kinematic viscosity; and νt is the eddy viscosity, which can be obtained from the Smagorinsky model (Smagorinsky, 1963) as
νt = (Cs Δ)2 2Sij Sij
∂csl ν ∂c = t sl ∂xj Prt ∂xj
(19)
where Pr is the Prandtl number. This represents the relationship between the momentum diffusivity and the mass diffusivity. The 32
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Fig. 5. Comparison between numerical results and experimental data for Run E1 and E2.
The static water depth in the flume was maintained at 0.5 m throughout the experiments. The waves were generated by a piston-type paddle movement and dissipated by a wave absorber installed at the tail of the flume. The currents were generated via a flow control valve at one end and a V-notch weir at the other end. A round acrylic pipe having a diameter (d) of 1.0 cm was installed in the mid-section of the flume. The jet was discharged vertically through the pipe at the centreline of the flume, and the jet orifice was 10.0 cm above the bottom. The jet source was supplied by a constant head tank above the wave flume, using an adjustable valve to control the volume flow rate. Further, the
At the jet inlet boundary, the perturbations of concentration are specified by the azimuthal forcing method proposed by Zhou et al. (2001). The operator splitting method (Lin and Li, 2002), which splits the solution procedure of Eq. (21) into advection and diffusion steps, is employed so that different numerical schemes can be used for the solution of different physical processes.
2.2.3. Model validation Experiments designed to validate the numerical model were conducted using a 46.0-m-long, 0.5-m-wide, and 1.0-m-deep wave flume. 33
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Fig. 5. (continued)
with pure water. The jet samples were collected from the physical flume by using a peristaltic suction pumping system, which consisted of a group of capillary tubes of diameter 1 mm, peristaltic pumps, and several graduated cylinders. The spacing between neighbouring tubes was 1.0 cm. The conductivity of each sample was measured using a conductivity probe. Table 2 summarises the parameters of the three experimental runs. The velocities and concentrations at 8 vertical lines in the symmetrical plane were measured in each run. The parameters for the numerical model were identical to those
jet was composed of sodium chloride solution with an initial concentration of 3.2 mg/L. The density difference between the jet and pure water was 0.026‰ and the influence of buoyancy was ignored. A 16MHz side-looking micro-acoustic Doppler velocimeter (micro-ADV) was used to measure the jet flow velocity. It is not easy to measure the concentration directly. Fortunately, the low concentration sodium chloride solution is conductive, and there is a linear relationship between the concentration of the solution and its conductivity. In this study, the concentration was calibrated as c =0.515* λ/1000, where c is the concentration and λ is the conductivity increment on comparing 34
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Fig. 6. Instantaneous flow pattern of the jet in the vertical symmetrical plane.
The calibration of the Smagorinsky constant Cs and the turbulent Prandtl number Prt was conducted using the following procedure. The comparison between experimental data in Run E3 and the corresponding numerical results are shown in Fig. 4, including the time-averaged velocity Um and the time-averaged concentration c . The time-averaged velocity Um is expressed as
used in the physical experiments. The computational domain was the same as that of Xu et al. (2016), i.e. 4.50 m long, 0.50 m wide, and 0.50 m deep, as shown in Fig. 3. The numerical results obtained using a 1.50-m-long damping zone and 10.0-m-long damping zone were quite close to each other. The time consumed in the computation using a 10.0-m-long damping zone was over 2 times that consumed using a 1.50-m-long damping zone, even using non-uniform grids. Hence, a 1.50-m-long damping zone was used in this study. The plane where y/d =0 is called the symmetrical plane and the planes where the value of x/ d is constant are called transverse planes. The non-uniform grid system C (GSC) with 205×99×126 nodes used by Xu et al. (2016) was employed here as well.
Um = (u 2 + w 2 )0.5
(22)
where u and w are the horizontal and vertical components of the timeaveraged velocity. Three values of Cs (0.14, 0.17, and 0.20) and four values of Prt (0.4, 0.53, 0.65, and 0.8) were tested in this study. It was found that the numerical results are quite sensitive to Cs and Prt values, 35
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3. Results 3.1. Phase-averaged scalar structure in the symmetrical plane Fig. 6 shows the jet phase-averaged flow pattern in the symmetrical plane for four typical wave phases in 13 cases, as well as the flow pattern of the jet in the current-only environment. The ‘effluent clouds’ phenomenon is observed in all cases of the jet in the wave-followingcurrent environment. Compared with the jet in the current-only environment, the effluent clouds are the most distinct characteristics of the jet under the combined effects of wave and current. The jet body is distinguished as the ‘effluent clouds’ part and the jet bent-over part. The formation mechanism and dynamics of the effluent clouds have been clarified by Xu et al. (2016). The wave-induced oscillating flow leads to the formation of the effluent clouds, whose frequency is the same as the wave frequency. At the outset of effluent cloud development, the clouds are separated from one another. Nevertheless, it is difficult to determine the size of the effluent clouds in the symmetrical plane. Compared to the other contours, the contour C =0.02c0 shaped the effluent cloud outline or size (including the horizontal length Lh and the vertical length Lv) better. As shown in Fig. 7, the definition of the effluent cloud size focuses on the leftmost effluent cloud in the symmetrical plane at the down-zero crossing wave phase. If other values of the contour (e.g., C =0.03c0) are chosen, the horizontal length Lh will decrease while the vertical length Lv will increase as shown in Fig. 8. However, regardless of whether C =0.02c0 or C =0.03c0 is chosen, the features of the wave-induced changes of Lh and Lv remain the same. The effluent cloud size in cases J8–J13 is shown in Fig. 8. It is found that the effluent cloud size is primarily determined by the waveto-current velocity ratio Rwc. The jet flow will always be deflected in the current direction if Rwc < 1.0. The vertical length Lv increases with Rwc. The horizontal lengths Lh in cases J8, J9, and J10 (Rwc < 0.8) are nearly the same. However, Lh in case J11 (Rwc =0.8) increases because of the inclination of the effluent cloud. If the wave-to-current velocity ratio keeps increasing, namely Rwc ≥1.0, jet deflection will occur in two different directions in one wave cycle and the vertical length Lv of effluent clouds will stop increasing or might even decrease. However, the horizontal length Lh will keep increasing, because the jet discharged in a direction opposite to the current direction becomes a part of the effluent cloud and the corresponding discharge amount increases with Rwc. The centre-to-centre distance between adjacent effluent clouds De has a direct correlation with the Strouhal number, and it can be expressed as
Fig. 7. Definition of effluent cloud size.
Fig. 8. Effluent cloud size in cases J8–J13.
especially for the velocity and concentration distribution at a downstream location relatively far away from the jet orifice (i.e., x/d =10 or 13). The cases when Cs was 0.20 and Prt was 0.53 were the most optimal. The comparison between the numerical results (using Cs =0.20 and Prt =0.53) and the experimental data for Run E1 and E2 are shown in Fig. 5. The numerical results in each run are in good agreement with the experimental data. Hence, the Smagorinsky constant Cs was set to 0.2 and the turbulent Prandtl number Prt was set to 0.53 in this study. The numerical model could correctly reproduce the characteristics of the velocity and concentration distribution with the depth, such as the twin-peak distribution of the concentration. This proves that the LES model can be used to study the jet movement and dilution in different wave-following-current conditions.
De =
d = u0 T St
(23)
For example, the distance De equals 7.5 cm in cases J4–J7, while it is 10 cm in cases J8–J13. When moving downstream with the current, the effluent clouds interact with the surrounding water and expand in
Fig. 9. Instantaneous scalar field of the jet in the transverse plane (x/d =8) in the wave trough phase in cases J8–J13.
36
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Fig. 10. Instantaneous scalar field of the jet in the transverse plane (x/d =3) in the wave trough phase in cases J8–J13.
Fig. 11. Instantaneous scalar field of the jet in the transverse plane (x/d =10.5) in cases J1–J3 and that in the transverse plane (x/d =16) in cases J4–J7 in the wave trough phase.
not a counter-example of jet fluid transported to the effluent clouds, because this is only the centre section of an effluent cloud while the effluent cloud has a large horizontal length Lh. The effluent cloud in cases with Rwc larger than 1 could reach the position at the middle of the two adjacent effluent clouds, as shown in Fig. 10. Fig. 10 shows the phase-averaged scalar field of the jet in the transverse plane (x/d =3) in the wave trough phase in cases J8–J13. The transverse plane of x/d =3 is located at the middle of the two adjacent effluent clouds, and the effluent clouds have not merged with each other. All distributions exhibit double peaks, which is a characteristic similar to that of a jet in the current-only environment. However, when Rwc is greater than or equal to 1, the effluent clouds reach such a position, and this leads to another high concentration region above the double peaks. Fig. 11 shows the phase-averaged scalar field of the jet in the transverse plane (x/d =10.5) in the wave trough phase in cases J1–J3 and that in the transverse plane (x/d =16) in the wave trough phase in cases J4–J7. These two planes are located at the middle of the two adjacent effluent clouds, and the effluent clouds have merged with each other. The distributions show one peak or two peaks or three peaks in different cases. This will lead to the corresponding types of time-averaged concentration distributions in the transverse planes, which will be discussed in detail in the following section. It is found that the two parameters Rwc and St have a significant influence on the characteristics of the effluent clouds, including the size of the effluent clouds, the distance between adjacent effluent clouds, and the concentration distribution in the ‘effluent clouds’ part. They will further affect the time-averaged structure of the jet in the wavefollowing-current environment.
all directions, with an increase in both the horizontal length Lh and the vertical length Lv. When the horizontal length Lh of the effluent cloud is greater than the distance De, the effluent cloud will merge with the adjacent ones. The merging location of two effluent clouds appears to be highly dependent on the parameters St and Rwc. The merging location in case J1 or case J2 has a shorter horizontal distance from the jet orifice position compared with that in case J4 or case J5 and case J8 or case J10, which have the same value of Rwc but smaller value of St. Among the cases in which St =0.1, the merging between adjacent effluent clouds is first found in case J13 owing to its largest horizontal length Lh of the effluent cloud. Another interesting finding is the discontinuity of the effluent clouds in case J7 and case J13, which is quite similar to the discontinuous motion of the jet in the wave-only environment identified by Mori and Chang (2003). The ‘effluent clouds’ part of the jet in these cases is discharged when the flow is against the current direction. In the initial stage of formation, the effluent cloud itself is continuous. Then, the flow changes toward the current direction, and at the same time, the flow velocity increases to a large value in an extremely short time. The effluent clouds are barely able to maintain their continuity in such a strong dynamic environment.
3.2. Phase-averaged scalar structure in the transverse planes Fig. 9 shows the phase-averaged scalar field of the jet in the transverse plane (x/d =8) in the wave trough phase in cases J8–J13. The transverse plane of x/d =8 is located at the centre of the effluent cloud, and the effluent cloud has not merged with the adjacent ones. When the wave-to-current velocity ratio is small (Rwc =0.2), the ‘effluent clouds’ phenomenon is insignificant and a double-peak distribution related to the CVP structure can be observed in this transverse plane. When the wave-to-current velocity ratio increases (Rwc ≥0.6), another concentration peak, which is in the ‘effluent clouds’ part, is found in this transverse plane. This suggests that more jet fluid is transported to the effluent clouds as Rwc increases. However, if the wave-to-current velocity ratio is larger than 1, the concentration peak will no longer increase or even decrease. It should be noted that this is
3.3. Time-averaged scalar structure in the transverse planes Fig. 12 shows the time-averaged scalar field of the jet in different transverse planes (x/d =10–60) in all 14 cases. The data were obtained by averaging the values of the instantaneous concentration in 20 wave cycles. For the jet in the current-only environment, distinct double concentration maxima are observed in each transverse plane. Such a structure is certainly related to the counter-rotating vortex pair (CVP). 37
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Fig. 12. Time-averaged scalar field of the jet in different transverse planes (x/d =10–60).
When the Strouhal number is small, say St =0.1 or 0.133, the concentration distribution consists of three high-concentration regions in the transverse planes close to the jet orifice (x/d =10) in each case. Of the three high-concentration regions, one is due to the ‘effluent clouds’ phenomenon and the others are related to the CVP. As the jet moves downstream, the concentration distribution in the transverse planes will evolve in two different ways. It is either dominated by the high-concentration region due to the ‘effluent clouds’ or dominated by the high-concentration regions related to the CVP, with the other highconcentration region(s) becoming weaker and eventually disappearing.
For the jet in the wave-following-current environment, no uniform structure can be identified for all these cases, even in the different transverse planes of each case. The time-averaged structure has a strong relationship with the parameters Rwc and St. When the wave-to-current velocity ratio is small (Rwc =0.2), the concentration distribution is quite similar to that in the current-only environment, with double maxima in each transverse plane, regardless of whether St equals 0.2, 0.133, or 0.1. The characteristics of the concentration distribution when Rwc is greater than 0.2 will be discussed next. 38
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Fig. 13. Integrated concentration distribution in cases J2, J5, and J10.
3.4. Integrated time-averaged concentration distribution in the transverse planes Xia and Lam (2004) found that there is a continuous transport from the jet bent-over part to the ‘effluent clouds’ part. It is difficult to find relevant evidence from Fig. 12, which shows that all concentration peaks in the transverse planes decrease along the current direction. In this study, the concentration distribution in each transverse plane is integrated along the y direction, denoted as cl. Fig. 13 shows the integrated concentration distribution in cases J2, J5, and J10. Two concentration peaks exist in each transverse plane. As the jet moves downstream with the current, both peaks keep moving upward and the peak at the lower position becomes smaller while the other one becomes increasingly larger. This indicates a bottom-up transport of the jet fluid, which is consistent with the results of Xia and Lam (2004), even though they only analysed the data on vertical symmetrical lines. Fig. 14 shows the effects of Rwc on the integrated concentration distribution in the transverse plane of x/d =20. The increase in Rwc leads to a higher concentration peak in the ‘effluent clouds’ part and a lower peak in the jet bent-over part. Further, a larger value of Rwc implies a lower position of the concentration peak in the jet bent-over part. However, there is no clear tendency for the peak position in the effluent cloud region. This shows that the increase in Rwc contributes to the mass transport from the jet bent-over part to the ‘effluent clouds’ part.
Fig. 14. Effects of Rwc on the integrated concentration distribution in the transverse plane of x/d =20.
3.5. Jet visual area The jet-affected region is representative of the jet dilution characteristic. In general, it is bounded by the jet edge defined by the concentration contour of C =0.25Cm, where Cm is the time-averaged maximum scalar concentration in the transverse planes. As shown by Lee and Chu (2003), such a definition is in good agreement with the jet visual boundary. They defined the vertical and horizontal half-widths Rv and Rh of the crossflow jet based on the time-averaged scalar field in the transverse planes, as shown in Fig. 15. Instead of using these two width parameters, the area of the concentration contour of C =0.25Cm, which is called the jet visual area (A25%) in this study, is used to represent the jet dilution characteristic. Fig. 16 shows the effect of the Strouhal number St on the jet visual area A25% in the wave-following-current environment. The wave effect results in expansion of the jet visual area. Moreover, the jet visual area increases as the Strouhal number decreases. One reason is that a smaller value of St means a larger distance between adjacent effluent
Fig. 15. Definition diagram of the jet visual area.
If the high-concentration region due to the ‘effluent clouds’ dominates the evolution, two high-concentration regions will appear again. The new structure is quite similar to that related to the CVP. Finally, the concentration distribution far from the jet orifice (x/d =60 or more) has two high-concentration regions. When the Strouhal number is large, say St =0.2, it seems that the evolution process is preferentially dominated by the high-concentration region owing to the ‘effluent clouds’, and then, the concentration distribution gradually exhibits double maxima, as shown in case J2 and case J3. 39
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Fig. 16. Effect of the Strouhal number St on the jet visual area A25%.
Fig. 17. Effect of the wave-to-current velocity ratio Rwc on the jet visual area A25%.
differences in the jet visual area induced by the Strouhal number become more prominent. The effect of the wave-to-current velocity ratio Rwc on the jet visual area A25% in the wave-following-current environment is shown in Fig. 17, which indicates that Rwc has a significant influence on the jet
clouds. This implies that there is more surrounding water to dilute each effluent cloud and that the merging of adjacent effluent clouds occurs further downstream. When the wave-to-current velocity ratio is extremely small (Rwc =0.2), the increase in the jet visual area with the decreasing Strouhal number is insignificant. As Rwc increases, the
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insufficient. Thus, the jet visual area will keep decreasing as the waveto-current velocity ratio increases. It should be noted that a more accurate value of the optimal ratio requires further investigation for real wastewater and brine discharge projects. Nevertheless, this study suggests that the wave effect should be taken into consideration when designing a wastewater or brine discharge project. The wave and current dynamics at the discharge site should be in an appropriate range in order to eliminate adverse environmental effects. In summary, two approaches for enhancing the dilution of wastewater or brine discharged into the ambient environment, especially in tide- or current- dominated areas, were presented in this study. One is to sway the diffusers at a certain frequency and amplitude, and the other is to oscillate the initial velocity of the wastewater or brine in a sinusoidal manner. The swaying or oscillating period is of the order of seconds and the amplitude can be determined on the basis of the local current velocity.
Fig. 18. Relationship between the jet visual area and the wave-to-current velocity ratio in the transverse plane (x/d =40).
visual area. An unexpected finding is that an increase in Rwc might lead to a decrease in the jet visual area. This indicates that an excessively strong wave might be not conducive to jet dilution in the wavefollowing-current environment.
5. Conclusion In this study, a numerical model based on the large eddy simulation (LES) method was established to study the three-dimensional scalar structures of a turbulent jet in different wave-following-current conditions. The main conclusions are as follows.
4. Discussion For a jet in the current-only environment, the cross-sectional scalar field has double concentration maxima (Lee et al., 2002). By contrast, the concentration distribution with three high-concentration regions is the most typical one for a jet in the wave-following-current environment. All the other distributions, including one high-concentration type and two high-concentration types, are considered as either the upper part or the lower part of the most typical one. The parameters Rwc and St together determine the dominant part of the distribution in a certain transverse plane. In general, an increase in Rwc or a decrease in St leads to a higher concentration in the ‘effluent clouds’ part and a lower concentration in the jet bent-over part. An important finding with regard to the time-averaged concentration distribution is the transformation from one-peak type to two-peak type, e.g. in cases J2, J3, J7, and J13. One possible reason is discussed here. The ‘effluent clouds’ maintain part of the jet initial vertical momentum and keep moving upwards, as shown in Fig. 11 of Xu et al. (2016). When interacting with the surrounding water, the effluent clouds cannot cross over the interface completely. This will certainly result in lateral flow. Then, a structure similar to CVP will emerge. This is similar to the mechanism of the existing CVP structure in a crossflow jet, which was suggested by Broadwell and Breidenthal (1984). The results on the jet visual area imply that there is an optimal wave-to-current velocity ratio for the highest dilution of a jet in the wave-following-current environment. Fig. 18 shows the relationship between the jet visual area and the wave-to-current velocity ratio in the transverse plane (x/d =40). It indicates that the optimal ratio is around 0.6 for the cases in this study. For the cases with the same value of St, such as cases J8–J13, the distance between adjacent effluent clouds remains unchanged. When the wave-to-current velocity ratio increases from 0 to 0.6, the vertical length Lv of the effluent cloud increases significantly with a small change in the horizontal length Lh, while the concentration of the effluent clouds gradually increases simultaneously. The concentration of the effluent clouds can be diluted to a low level by the surrounding water. The jet visual area increases with the wave-tocurrent velocity ratio. If the wave-to-current velocity ratio is in the range of 0.8–1.0, it will be difficult to reduce the high concentration inside the effluent clouds to the same level. Hence, the jet visual area decreases as the wave-to-current velocity ratio increases up to 1.0. If the wave-to-current velocity ratio keeps increasing, the horizontal length Lh of the effluent cloud increases sharply even though the concentration level does not decrease significantly. Further, the amount of surrounding water to be mixed with the effluent clouds is
(1) The regular wave effect on the jet dilution can be ascribed to the wave-to-current velocity ratio Rwc and the Strouhal number St. (2) The characteristics of the most distinct ‘effluent clouds’ phenomenon has a strong relationship with the abovementioned parameters; the effluent cloud size is primarily determined by the waveto-current velocity ratio Rwc, and the distance between adjacent effluent clouds is inversely proportional to the Strouhal number St. (3) As the jet moves downstream, the evolution process of the timeaveraged concentration distribution is either dominated by the high-concentration region owing to the ‘effluent clouds’ or dominated by the high-concentration regions related to the CVP structure. An increase in Rwc or a decrease in St leads to a higher concentration in the ‘effluent clouds’ part and a lower concentration in the jet bent-over part. (4) There exists an optimal wave-to-current velocity ratio (approximately 0.6 in this study) for the highest dilution of a jet in the wave-following-current environment. This study should be a useful reference for the selection of sites in wastewater or brine discharge projects. Swaying the diffusers or oscillating the initial velocity could result in enhancement of wastewater or brine dilution in the surrounding water. Further study is recommended for obtaining quantitative results. Acknowledgement This work was partly supported by the Fundamental Research Funds for the Central Universities (Grant No. 2016B13214) and the National Natural Science Foundation of China (Grant Nos. 51379072, 51509080). References Alves, L.S.B., Kelly, R.E., Karagozian, A.R., 2008. Transverse-jet shear-layer instabilities. Part 2. linear analysis for large jet-to-crossflow velocity ratio. J. Fluid Mech. 602, 383–401. Chan, S.N., Thoe, W., Lee, J.H.W., 2013. Real-time forecasting of Hong Kong beach water quality by 3D deterministic model. Water Res. 47, 1631–1647. Chang, K.A., Ryu, Y., Mori, N., 2009. Parameterization of neutrally buoyant horizontal round jet in wave environment. J. Waterw. Port Coast. Ocean Eng. 135 (3), 100–107. Chua, L.P., Antonia, R.A., 1990. Turbulent Prandtl number in a circular jet. Int. J. Heat Mass Tran. 33 (2), 331–339. Chyan, J.M., Hwung, H.H., 1993. On the interaction of a turbulent jet with waves. J.
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Eng. 125 (6), 564–573. Roberts, P.J.W., Snyder, W.H., 1993. Hydraulic model study for Boston Outfall. I: riser Configuration. J. Hydraul. Eng. 119 (9), 970–987. Rodi, W., Constantinescu, G., Stoesser, T., 2013. Large-eddy Simulation in Hydraulics. CRC Press, 35–36. Ryu, Y., Chang, K.A., Mori, N., 2005. Dispersion of neutrally buoyant horizontal round jet in wave environment. J. Hydraul. Eng. 131 (12), 1088–1097. Smagorinsky, J., 1963. General circulation experiments with the primitive equations- I. the basic experiment. Mon. Weather Rev. 91, 99–164. Stark, J.S., Corbett, P.A., Dunshea, G., Johnstone, G., King, C., Mondon, J.A., Power, M.L., Samuel, A., Snape, I., Riddle, M., 2016. The environmental impact of sewage and wastewater outfalls in Antarctica: an example from Davis station, East Antarctica. Water Res. 105, 602–614. Subramanya, K., Porey, P.D., 1984. Trajectory of a turbulent cross jet. J. Hydraul. Res. 22 (5), 343–354. Voutchkov, N., 2011. Overview of seawater concentrate disposal alternatives. Desalination 273, 205–219. Wong, C.F., 1991. Advected line Thermals and Puffs (M. Phil. Thesis). Department of Civil and Structural Engineering, University of Hong Kong, Hong Kong. Xia, L.P., Lam, K.M., 2004. Unsteady effluent dispersion in a round jet interacting with an oscillating cross-flow. J. Hydraul. Eng. 130 (7), 667–677. Xu, Z.S., Chen, Y.P., Zhang, C.K., Li, C.W., Wang, Y.N., Hu, F., 2014. Comparative study of a vertical round jet in regular and random waves. Ocean Eng. 89, 200–210. Xu, Z.S., Chen, Y.P., Tao, J.F., Pan, Y., Sowa, D.M.A., Li, C.W., 2016. Three-dimensional flow structure of a non-buoyant jet in a wave-current coexisting environment. Ocean Eng. 116, 42–54. Yang, L., 1995. Review of Marine outfall systems in Taiwan. Water Sci. Technol. 32 (2), 257–264. Yang, Z.Y., 2015. Large-eddy simulation: past, present and the future. Chin. J. Aeronaut. 28 (1), 11–24. Zhao, L., Chen, Z., Lee, K., 2011. Modelling the dispersion of wastewater discharges from offshore outfalls: a review. Environ. Res. 19, 107–120. Zhou, X., Luo, K.H., Williams, J.J.R., 2001. Large-eddy simulation of a turbulent forced plume. Eur. J. Mech. B-Fluid. 20, 233–254.
Hydraul. Res. 31 (6), 791–810. Ghaisas, N.S., Shetty, D.A., Frankel, S.H., 2015. Large eddy simulation of turbulent horizontal buoyant jets. J. Turbul. 16 (8), 772–808. Hodgson, J.E., Rajaratham, N., 1992. An experimental study of jet dilution in crossflows. Can. J. Civil Eng. 19, 733–743. Hsiao, S.C., Hsu, T.W., Lin, J.F., et al., 2011. Mean and turbulence properties of a neutrally buoyant round jet in a wave environment. J. Waterw. Port Coast. Ocean Eng. 137 (3), 109–122. Kelso, R.M., Lim, T.T., Perry, A.E., 1996. An experimental study of round jets in crossflow. J. Fluid Mech. 306, 111–144. Koole, R., Swan, C., 1994. Measurements of a 2-D non-buoyant jet in a wave environment. Coast. Eng. 24, 151–169. Lam, K.M., Xia, L.P., 2001. Experimental simulation of a vertical round jet issuing into an unsteady cross-flow. J. Hydraul. Eng. 127 (5), 369–379. Lee, J.H.W., Kuang, C.P., Chen, G.Q., 2002. The structure of a turbulent jet in a crossflow-effect of jet-crossflow velocity. China Ocean Eng. 16 (1), 1–20. Lin, P.Z., Li, C.W., 2002. A σ-coordinate three-dimensional numerical model for surface wave propagation. Int. J. Numer. Meth. Fl. 38 (11), 1045–1068. Megerian, S., Davitian, J., Alves, L.S.B., Karagozian, A.R., 2007. Transverse-jet shearlayer instabilities. Part 1. Experimental studies. J. Fluid Mech. 593, 93–129. Mendonça, A., Losada, M., Reis, M.T., Neves, M.G., 2013. Risk assessment in submarine outfall projects: the case of Portugal. J. Environ. Manag. 116, 186–195. Moawad, A.K., Rajaratnam, N., 1998. Dilution of circular turbulent nonbuoyant surface jets in cross-flows. J. Hydraul. Eng. 124 (5), 474–483. Mori, N., Chang, K.A., 2003. Experimental study of a horizontal jet in a wavy environment. J. Eng. Mech. 129 (10), 1149–1155. Muppidi, S., Mahesh, K., 2005. Study of trajectories of jets in crossflow using direct numerical simulations. J. Fluid Mech. 530, 81–100. Pham, M.V., Plourde, F., Kim, S.D., Balachandar, S., 2006. Large-eddy simulation of a pure thermal plume under rotating conditions. Phys. Fluids 18, 015101. Roberts, D.A., Johnston, E.L., Knott, N.A., 2010. Impacts of desalination plant discharges on the marine environment: a critical review of published studies. Water Res. 44, 5117–5128. Roberts, P.J.W., 1999. Modeling Mamala Bay outfall plumes. I: near field. J. Hydraul.
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Near-field dilution of a turbulent jet discharged into coastal waters: Effect of regular waves
MARK
⁎
Zhenshan Xua, Yongping Chena,b, , Yana Wangc, Changkuan Zhanga a b c
College of Harbor, Coastal and Offshore Engineering, Hohai University, Nanjing 210098, China State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, China School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China
A R T I C L E I N F O
A BS T RAC T
Keywords: Dilution Near-field Jet Wave-following-current Regular wave
The effect of regular waves on three-dimensional scalar structures of a vertical round jet in the wave-followingcurrent environment is investigated. The wave effect is represented by two dimensionless parameters, i.e. the wave-to-current velocity ratio Rwc and Strouhal number St. The jet concentration distribution, including 13 cases in the wave-following-current environment and 1 case in the current-only environment, is obtained using the large eddy simulation method and validated by experimental data. The results show that the characteristics of the distinctive ‘effluent clouds’ phenomenon, namely the effluent cloud size and the distance between adjacent effluent clouds, are strongly dependent on Rwc and St. As a result, different structures of the timeaveraged concentration distribution are found in the transverse planes, and they are classified into three types: one-peak type, two-peak type, and three-peak type. The area of the concentration contour of C =0.25Cm, where Cm is the time-averaged cross-sectional maximum concentration, is defined as the jet visual area that represents the jet dilution characteristic. It highlights the existence of an optimal wave-to-current velocity ratio (approximately 0.6 here) for the highest dilution of a jet in the wave-following-current environment, which provides useful guidelines for selecting the sites of wastewater or brine discharge projects.
1. Introduction There is an urgent global requirement for the ocean disposal of urban treated wastewater and brine from desalination plants via submarine outfalls, as shown in Fig. 1 (Roberts and Snyder, 1993; Yang, 1995; Voutchkov, 2011). However, such plants are point sources of pollution, posing a potential threat to local human and wildlife communities (Roberts et al., 2010; Mendonça et al., 2013; Stark et al., 2016). A vital step in disposal management is selecting the site of submarine outfalls such that the wastewater or brine is quickly and effectively diluted by the ocean currents. Therefore, it is important to understand the effect of coastal waters on the mixing and dispersion of such discharge. From the viewpoint of environmental hydraulics, the movement of discharged wastewater or brine usually forms a turbulent jet, identified as a near-field process or a far-field process according to different time and distance scales. The time and distance scales in a near-field process are of the order of minutes and tens or hundreds of metres, whereas those in a far-field process are of the order of hours and kilometres, respectively (Roberts, 1999). It is important to study near-field mixing and dispersion, which primarily affects jet dilution in the far field and
⁎
partially dominates the entire process (Zhao et al., 2011; Chan et al., 2013). In the near field, there are two types of factors that affect the movement and dilution of the wastewater- or brine-formed jet, as shown in Fig. 2. One involves parameters associated with the jet itself, namely the initial velocity w0, the jet density ρ0, the geometry/diameter of the jet orifice d (circular or slit), and the discharge angle θ0. The other involves several ambient fluid parameters, such as water depth D, ambient fluid density ρa, and ambient fluid velocity Ua. A certain variable P representing the jet characteristic is determined by the following equation:
P = f (w0 , ρ0 , d , θ0, D, ρa , Ua )
By comparing the values of ρ0 and ρa, the jet is classified as a buoyant (e.g. salinity- or temperature-induced buoyancy) or nonbuoyant jet. As for the angle θ0, a jet outfall could be set horizontally, vertically, or obliquely in a discharge project. This study focuses only on the near-field movement and dilution of a vertical non-buoyant jet discharged into coastal waters. It should be noted that understanding such a jet is a prerequisite for studying an oblique buoyant jet. In coastal areas, the tidal current is one of the leading dynamic factors that affect the jet movement. Its action could be considered as
Corresponding author at: College of Harbor, Coastal and Offshore Engineering, Hohai University, Nanjing 210098, China.
http://dx.doi.org/10.1016/j.oceaneng.2017.05.003 Received 3 November 2016; Received in revised form 10 March 2017; Accepted 5 May 2017 Available online 18 May 2017 0029-8018/ © 2017 Elsevier Ltd. All rights reserved.
(1)
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the ‘effluent clouds’ phenomenon. This indicates that the wave action has a significant influence on the jet movement in the wave-current coexisting environment. However, only one case of a jet in such an environment was considered in their study. Xia and Lam (2004) studied a downward jet in an oscillating crossflow and found that the unsteadiness in the oscillating crossflow leads to an increase in the time-averaged jet width in the symmetrical plane. At present, there remains a lack of knowledge regarding the effect of wave periods and wave heights on the jet structure (particularly the three-dimensional structure) in the wave-current coexisting environment. Actually, instead of focusing on the jet-wave-current interaction, many researchers (e.g. Chyan and Hwung, 1993; Koole and Swan, 1994; Hsiao et al., 2011) have studied different flow structures of a jet in the wave-only environment by considering the relative strength between jets and waves. Mori and Chang (2003), Ryu et al. (2005), Chang et al. (2009) defined the wave-to-jet momentum ratio RM and identified it as a key parameter for describing the wave-jet interaction. With increasing RM, the horizontal jet in waves (similar to the vertical jet in waves) can be classified into three patterns: symmetric motion, asymmetric motion, and discontinuous motion (Mori and Chang, 2003). When the wave action is weak, the distribution of the time-averaged axial velocity or concentration in the cross sections follows a Gaussian shape. When the wave action is sufficiently enough, such a distribution will be flattopped or bi-peaked, especially in the jet deflection region. This has been confirmed by Chyan and Hwung (1993), Koole and Swan (1994), Hsiao et al. (2011), Xu et al. (2014). Thus, the jet undergoes a faster decay of centreline velocity, with a larger spreading width and higher dilution rate at higher RM. These conclusions serve as valuable references for the studies of the wave effect on the jet structure in the wave-current coexisting environment. Following the work of Xu et al. (2016), this study aims to investigate the three-dimensional scalar structure of a vertical round jet in the wave-current coexisting environment, using the method of large eddy simulation (LES). Initially, the wave-current coexisting environment is set to the regular wave-following-current flow. This paper provides a detailed description of the wave effects on the jet initial dilution as well as some guidelines for selecting sites for wastewater or brine discharge projects.
Main Pipe
Sea water
Wastewater/Brine
Diffusers
Fig. 1. Ocean disposal of wastewater or brine (modified from the diagram of Voutchkov (2011)).
Fig. 2. Factors affecting jet motion in the near field.
the effect of a series of steady currents over a long period. Under the current effect, the vertical jet is deflected and divided into the near zone, curvilinear zone, and vortex zone (Subramanya and Porey, 1984). The deflection is primarily determined by the jet-to-current velocity ratio, Rjc. Jets in the current-only environment with Rjc in the range of 2–25 have been investigated extensively in previous studies (e.g. Hodgson and Rajaratnam, 1992; Kelso et al., 1996; Moawad and Rajaratnam, 1998; Muppidi and Mahesh, 2005; Megerian et al., 2007). As Rjc increases, the jet reaches a greater height and the jet shear layer instability is weakened (Alves et al., 2008). Kelso et al. (1996) experimentally showed that the folding and rolling up of the jet shear layer very close to the jet orifice contribute to the formation of the counter-rotating vortex pair (CVP). Consequently, the jet trajectories and dilution are closely related to Rjc. For example, Wong (1991) expressed the vertical centreline minimum dilution Sc of a jet in the current-only environment as
⎛ x ⎞2/3 Sc d = 1.271 ⎜ ⎟ ⎝ Lm ⎠ Lm
2. Methodology 2.1. Dimensional analysis of jet characteristic parameters The characteristic parameters affecting the jet motion are shown in Fig. 2 and expressed by Eq. (1). For a vertical round jet, dimensional analysis yields the following dimensionless parameters:
⎛ ρ d w ⎞ P = f ⎜Rej, a , , 0 ⎟ ρ0 D Ua ⎠ ⎝
(2)
where
Lm =
M01/2 = 0.886Rjc d u0
(4)
where d is the jet diameter and Rej is the jet Reynolds number, which is defined as
(3)
Rej=ρ0
and M0 is the jet initial momentum flux. Nevertheless, the jet flow structure remains virtually unchanged with increasing Rjc. Four largescale coherent structures can be identified from a vertical jet in the current-only environment: jet shear layer vortices, horseshoe vortices, wake vortices, and CVP. Among these, CVP is a salient feature of the jet, which leads to a kidney-shaped and bifurcated cross-sectional scalar field, with distinct double concentration maxima (Lee et al., 2002). In addition to the tidal current, wave motion is a common dynamic factor in coastal areas. It usually coexists with the tidal current. As discussed by Xu et al. (2016), the three-dimensional flow structure of a vertical round jet in the wave-current coexisting environment is significantly distinct from that in the current-only environment, e.g.
w0 d υ
(5)
where υ is the kinematic coefficient of viscosity. The Reynolds number is sufficiently large in this study, and the jet motion belongs to the turbulent flow. The jet is non-buoyant, i.e. ρa/ρ0=1, and the ratio of the jet diameter to the water depth is assumed to be extremely small, i.e. d/ D ≈0. From these assumptions, Eq. (4) is simplified into
⎛w ⎞ P = f ⎜ 0⎟ ⎝ Ua ⎠
(6)
It can be seen that the velocity ratio w0/Ua is a key factor that determines the movement and dilution of the non-buoyant vertical round jet. For a jet in the current-only environment, the velocity ratio 30
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w0/Ua is defined as the ratio of the jet initial velocity w0 to the current velocity u0 (Kelso et al., 1996), i.e.
w0 u0
Rjc =
Table 1 Numerical cases of the jet in the wave-following-current environment.
It is a universal parameter for describing the jet and current interaction. For a jet in the wave-only environment, Chang et al. (2009) found that the wave-to-jet momentum ratio is the most important parameter for characterizing the wave effect on the jet diffusion in the wave-only environment. The ratio was expressed by Mori and Chang (2003) as
RM1 =
gH2 dw02
(8)
and by Ryu et al. (2005) as
RM 2 =
2gk w H2 w02
(9)
where g is the acceleration due to gravity, H is the wave height, and k is the wave number. It should be noted that the wave force varies with the water depth and increases with the wave height or wave period. The two above-mentioned momentum ratios were defined for a horizontal jet and only considered the effect of wave height. Hence, they are not suitable for a vertical jet in the wave-only environment. Chyan and Hwung (1993) found that the deflection region of wave-jet interaction has characteristics similar to a crossflow jet and recommended the use of the wave characteristic velocity to study the wave-jet interaction. Considering that the wave force near the jet orifice has a critical influence on the jet deflection, we chose the maximum horizontal particle velocity at the jet orifice position as the wave characteristic velocity. Following the small-amplitude wave theory, this velocity is expressed as
u wa = Hω
Case
Jet-tocurrent velocity ratio Rjc
Wavetocurrent velocity ratio Rwc
Strouhal number St
Jet initial velocity w0 (m/ s)
Current velocity u0 (m/s)
Wave period T (s)
Wave height H (cm)
J0 J1 J2 J3 J4 J5 J6 J7 J8 J9 J10 J11 J12 J13
10 10 10 10 10 10 10 10 10 10 10 10 10 10
/ 0.2 0.6 1.0 0.2 0.6 1.0 1.4 0.2 0.4 0.6 0.8 1.0 1.4
/ 0.2 0.2 0.2 0.133 0.133 0.133 0.133 0.1 0.1 0.1 0.1 0.1 0.1
0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05
/ 1.0 1.0 1.0 1.5 1.5 1.5 1.5 2.0 2.0 2.0 2.0 2.0 2.0
/ 1.15 3.45 5.75 0.64 1.92 3.20 4.48 0.54 1.08 1.62 2.16 2.70 3.78
(7)
cosh(kh 0 ) 2 sinh(kD )
Table 2 Parameters in the experimental runs. Run
Jet initial velocity w0 (m/s)
Current velocity u0 (m/s)
Wave period T (s)
Wave Height H (cm)
E1 E2 E3
0.58 0.58 1.22
0.055 0.055 0.133
1.0 1.4 1.4
3.0 3.0 7.0
(10)
where ω is the wave angular frequency and h0 is the height of the jet orifice from the bottom. Then, the jet-to-wave velocity ratio Rjw can be defined as
Rjw =
w0 u wa
Fig. 3. Computational domain of numerical model.
(11)
are primarily dependent on Rjc, the structure characteristic of CVP, which is the most significant feature of the crossflow jet, remains virtually unchanged when Rjc ranges from 2 to 25. For simplicity, only one jet-to-current velocity ratio is considered in this study, namely Rjc =10, which is a common value in previous studies. When an effluent jet having a diameter of 0.5 m is discharged into an ocean current at 0.8 m/s, waves with periods of 3–8 s induce unsteadiness at a Strouhal number in the range of 0.08–0.2 (Xia and Lam, 2004). The Strouhal number in this study is set to three values, namely 0.2, 0.133, and 0.1. The jet diameter is 0.01 m and the current velocity is 0.05 m/s. Then, the wave period will be 1.0 s, 1.5 s, and 2.0 s. The wave-to-current velocity ratio Rwc is set from 0.2 to 1.4, considering different deflection situations of the jet in one wave cycle.
For a jet in the wave-following-current environment, the instantaneous jet-to-current velocity ratio changes continuously. The flow is governed by three dimensionless parameters similarly to a jet in an oscillating crossflow, which has been studied as the sinusoidal movement of the jet in a steady crossflow by Lam and Xia (2001), Xia and Lam (2004). These are the jet-to-current velocity ratio Rjc, the wave-tocurrent velocity ratio Rwc (the unsteadiness parameter in the study of Lam and Xia (2001)), and the Strouhal number St. These parameters are expressed as
Rwc =
St =
u wa u0
d u0 T
(12) (13)
If the velocity ratio Rwc is smaller than 1, the jet flow will always be deflected in the current direction. If the velocity ratio Rwc is larger than 1, the jet deflection will occur in two directions in one wave period. As discussed above, the key parameter w0/Ua in Eq. (6) is expressed in different forms in different environments, i.e. Rjc for the current-only environment, Rwj for the wave-only environment, and Rjc, Rwc, and St for the wave-following-current environment. Based on the dimensional analysis described above, 1 case of the jet in the current-only environment and 13 cases of the jet in the wavefollowing-current environment with different wave, current, and jet parameters are summarised in Table 1. Although the jet characteristics
2.2. Numerical model 2.2.1. Governing equations Xu et al. (2016) developed a σ-coordinate three-dimensional LES model based on spatially filtered Navier-Stokes equations to investigate the flow structure of a vertical round jet in the wave-current coexisting environment. The equations are expressed as
∂ui ∂Xk =0 ∂Xk ∂xi 31
(14)
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Fig. 4. Comparison between numerical results and experimental data for Run E3.
residual scalar flux Qj, which includes the information of the smallscale components, is modelled as
∂ui ∂ui ∂Xk ∂u ∂Xk 1 ∂p ∂Xk + + uj i =− + gi ∂τ ∂Xk ∂t ∂Xk ∂xj ρ ∂Xk ∂xi + (ν + νt )
∂Xk ∂ ⎛ ∂ui ∂Xm ⎞ ⎜ ⎟ ∂xj ∂Xk ⎝ ∂Xm ∂xj ⎠
Qj = −αt
(15)
∂c ∂c ∂Xk ∂c ∂Xk ∂Xk ∂ ⎛ ⎛ ν ν ⎞ ∂c ∂Xm ⎞ ⎜⎜ ⎟ + + uj = + t⎟ ⎝ ∂τ ∂Xk ∂t ∂Xk ∂xj ∂xj ∂Xk ⎝ Pr Prt ⎠ ∂Xm ∂xj ⎠
(16)
(17)
where Δx1, Δx2, and Δx3 are the grid sizes in the coordinates of x1, x2, and x3, respectively. To obtain the scalar field of the jet in the wave-following-current environment, the tracer mass conservation equation should also be considered:
⎞ ∂c ∂c ∂Xk ∂c ∂Xk ∂Xk ∂ ⎛ ∂c ∂Xm ⎜α + + uj = − Qj ⎟ ∂τ ∂Xk ∂t ∂Xk ∂xj ∂xj ∂Xk ⎝ ∂Xm ∂xj ⎠
2.2.2. Boundary conditions and numerical methods The details of the boundary conditions and numerical methods related to the flow module have been provided by Xu et al. (2016). The boundary conditions and numerical methods for the scalar transport are as follows. The zero-gradient condition is specified at the surface and bottom boundaries, while the radiation condition, coupled with the hydrostatic pressure conditions, is specified at the lateral boundaries. At the outflow boundary, the radiation condition combined with a sponger layer is imposed for the concentration to reduce the reflection.
(18)
where c is the large-scale component of the concentration and α is the molecular diffusion coefficient, which can be calculated as
α=
ν Pr
(21)
To obtain reliable concentration results, the values of Pr and Prt in the above equation should be carefully determined. Typical values for Prt are found to be in the range 0.3–0.9 from previous studies (Chua and Antonia, 1990; Pham et al., 2006; Rodi et al., 2013). It should be noted that the eddy viscosity νt in the jet region is much greater than the kinematic viscosity ν. As a result, the concentration results of the model are dominated by the second part (νt / Prt). In other words, the concentration results are more sensitive to Prt as compared to Pr. Hence, only the value of Prt will be calibrated in Sub-section 2.2.3. Following the choice of Ghaisas et al. (2015), we set the Pr value to 0.7 in this study.
where Cs is the Smagorinsky constant. The value of Cs should be calibrated and chosen based on the type of the flow (Yang et al., 2015). Its calibration is shown in Sub-section 2.2.3. Sij is the resolved strain rate tensor, and Δ is a representative grid spacing defined as
Δ = (Δx1 Δx2 Δx3)1/3
(20)
where αt is the turbulent diffusion coefficient and Prt is the turbulent Prandtl number. The final form of the governing equation for scalar transport is given by
where ui (i=1, 2, 3) and p are the large-scale components of velocity and pressure, respectively; ρ is the water density; gi is the acceleration due to gravity; t, xi and τ, Xi are the temporal and spatial coordinates in the Cartesian and σ-coordinate systems, respectively; ν is the kinematic viscosity; and νt is the eddy viscosity, which can be obtained from the Smagorinsky model (Smagorinsky, 1963) as
νt = (Cs Δ)2 2Sij Sij
∂csl ν ∂c = t sl ∂xj Prt ∂xj
(19)
where Pr is the Prandtl number. This represents the relationship between the momentum diffusivity and the mass diffusivity. The 32
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Fig. 5. Comparison between numerical results and experimental data for Run E1 and E2.
The static water depth in the flume was maintained at 0.5 m throughout the experiments. The waves were generated by a piston-type paddle movement and dissipated by a wave absorber installed at the tail of the flume. The currents were generated via a flow control valve at one end and a V-notch weir at the other end. A round acrylic pipe having a diameter (d) of 1.0 cm was installed in the mid-section of the flume. The jet was discharged vertically through the pipe at the centreline of the flume, and the jet orifice was 10.0 cm above the bottom. The jet source was supplied by a constant head tank above the wave flume, using an adjustable valve to control the volume flow rate. Further, the
At the jet inlet boundary, the perturbations of concentration are specified by the azimuthal forcing method proposed by Zhou et al. (2001). The operator splitting method (Lin and Li, 2002), which splits the solution procedure of Eq. (21) into advection and diffusion steps, is employed so that different numerical schemes can be used for the solution of different physical processes.
2.2.3. Model validation Experiments designed to validate the numerical model were conducted using a 46.0-m-long, 0.5-m-wide, and 1.0-m-deep wave flume. 33
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Fig. 5. (continued)
with pure water. The jet samples were collected from the physical flume by using a peristaltic suction pumping system, which consisted of a group of capillary tubes of diameter 1 mm, peristaltic pumps, and several graduated cylinders. The spacing between neighbouring tubes was 1.0 cm. The conductivity of each sample was measured using a conductivity probe. Table 2 summarises the parameters of the three experimental runs. The velocities and concentrations at 8 vertical lines in the symmetrical plane were measured in each run. The parameters for the numerical model were identical to those
jet was composed of sodium chloride solution with an initial concentration of 3.2 mg/L. The density difference between the jet and pure water was 0.026‰ and the influence of buoyancy was ignored. A 16MHz side-looking micro-acoustic Doppler velocimeter (micro-ADV) was used to measure the jet flow velocity. It is not easy to measure the concentration directly. Fortunately, the low concentration sodium chloride solution is conductive, and there is a linear relationship between the concentration of the solution and its conductivity. In this study, the concentration was calibrated as c =0.515* λ/1000, where c is the concentration and λ is the conductivity increment on comparing 34
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Fig. 6. Instantaneous flow pattern of the jet in the vertical symmetrical plane.
The calibration of the Smagorinsky constant Cs and the turbulent Prandtl number Prt was conducted using the following procedure. The comparison between experimental data in Run E3 and the corresponding numerical results are shown in Fig. 4, including the time-averaged velocity Um and the time-averaged concentration c . The time-averaged velocity Um is expressed as
used in the physical experiments. The computational domain was the same as that of Xu et al. (2016), i.e. 4.50 m long, 0.50 m wide, and 0.50 m deep, as shown in Fig. 3. The numerical results obtained using a 1.50-m-long damping zone and 10.0-m-long damping zone were quite close to each other. The time consumed in the computation using a 10.0-m-long damping zone was over 2 times that consumed using a 1.50-m-long damping zone, even using non-uniform grids. Hence, a 1.50-m-long damping zone was used in this study. The plane where y/d =0 is called the symmetrical plane and the planes where the value of x/ d is constant are called transverse planes. The non-uniform grid system C (GSC) with 205×99×126 nodes used by Xu et al. (2016) was employed here as well.
Um = (u 2 + w 2 )0.5
(22)
where u and w are the horizontal and vertical components of the timeaveraged velocity. Three values of Cs (0.14, 0.17, and 0.20) and four values of Prt (0.4, 0.53, 0.65, and 0.8) were tested in this study. It was found that the numerical results are quite sensitive to Cs and Prt values, 35
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3. Results 3.1. Phase-averaged scalar structure in the symmetrical plane Fig. 6 shows the jet phase-averaged flow pattern in the symmetrical plane for four typical wave phases in 13 cases, as well as the flow pattern of the jet in the current-only environment. The ‘effluent clouds’ phenomenon is observed in all cases of the jet in the wave-followingcurrent environment. Compared with the jet in the current-only environment, the effluent clouds are the most distinct characteristics of the jet under the combined effects of wave and current. The jet body is distinguished as the ‘effluent clouds’ part and the jet bent-over part. The formation mechanism and dynamics of the effluent clouds have been clarified by Xu et al. (2016). The wave-induced oscillating flow leads to the formation of the effluent clouds, whose frequency is the same as the wave frequency. At the outset of effluent cloud development, the clouds are separated from one another. Nevertheless, it is difficult to determine the size of the effluent clouds in the symmetrical plane. Compared to the other contours, the contour C =0.02c0 shaped the effluent cloud outline or size (including the horizontal length Lh and the vertical length Lv) better. As shown in Fig. 7, the definition of the effluent cloud size focuses on the leftmost effluent cloud in the symmetrical plane at the down-zero crossing wave phase. If other values of the contour (e.g., C =0.03c0) are chosen, the horizontal length Lh will decrease while the vertical length Lv will increase as shown in Fig. 8. However, regardless of whether C =0.02c0 or C =0.03c0 is chosen, the features of the wave-induced changes of Lh and Lv remain the same. The effluent cloud size in cases J8–J13 is shown in Fig. 8. It is found that the effluent cloud size is primarily determined by the waveto-current velocity ratio Rwc. The jet flow will always be deflected in the current direction if Rwc < 1.0. The vertical length Lv increases with Rwc. The horizontal lengths Lh in cases J8, J9, and J10 (Rwc < 0.8) are nearly the same. However, Lh in case J11 (Rwc =0.8) increases because of the inclination of the effluent cloud. If the wave-to-current velocity ratio keeps increasing, namely Rwc ≥1.0, jet deflection will occur in two different directions in one wave cycle and the vertical length Lv of effluent clouds will stop increasing or might even decrease. However, the horizontal length Lh will keep increasing, because the jet discharged in a direction opposite to the current direction becomes a part of the effluent cloud and the corresponding discharge amount increases with Rwc. The centre-to-centre distance between adjacent effluent clouds De has a direct correlation with the Strouhal number, and it can be expressed as
Fig. 7. Definition of effluent cloud size.
Fig. 8. Effluent cloud size in cases J8–J13.
especially for the velocity and concentration distribution at a downstream location relatively far away from the jet orifice (i.e., x/d =10 or 13). The cases when Cs was 0.20 and Prt was 0.53 were the most optimal. The comparison between the numerical results (using Cs =0.20 and Prt =0.53) and the experimental data for Run E1 and E2 are shown in Fig. 5. The numerical results in each run are in good agreement with the experimental data. Hence, the Smagorinsky constant Cs was set to 0.2 and the turbulent Prandtl number Prt was set to 0.53 in this study. The numerical model could correctly reproduce the characteristics of the velocity and concentration distribution with the depth, such as the twin-peak distribution of the concentration. This proves that the LES model can be used to study the jet movement and dilution in different wave-following-current conditions.
De =
d = u0 T St
(23)
For example, the distance De equals 7.5 cm in cases J4–J7, while it is 10 cm in cases J8–J13. When moving downstream with the current, the effluent clouds interact with the surrounding water and expand in
Fig. 9. Instantaneous scalar field of the jet in the transverse plane (x/d =8) in the wave trough phase in cases J8–J13.
36
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Fig. 10. Instantaneous scalar field of the jet in the transverse plane (x/d =3) in the wave trough phase in cases J8–J13.
Fig. 11. Instantaneous scalar field of the jet in the transverse plane (x/d =10.5) in cases J1–J3 and that in the transverse plane (x/d =16) in cases J4–J7 in the wave trough phase.
not a counter-example of jet fluid transported to the effluent clouds, because this is only the centre section of an effluent cloud while the effluent cloud has a large horizontal length Lh. The effluent cloud in cases with Rwc larger than 1 could reach the position at the middle of the two adjacent effluent clouds, as shown in Fig. 10. Fig. 10 shows the phase-averaged scalar field of the jet in the transverse plane (x/d =3) in the wave trough phase in cases J8–J13. The transverse plane of x/d =3 is located at the middle of the two adjacent effluent clouds, and the effluent clouds have not merged with each other. All distributions exhibit double peaks, which is a characteristic similar to that of a jet in the current-only environment. However, when Rwc is greater than or equal to 1, the effluent clouds reach such a position, and this leads to another high concentration region above the double peaks. Fig. 11 shows the phase-averaged scalar field of the jet in the transverse plane (x/d =10.5) in the wave trough phase in cases J1–J3 and that in the transverse plane (x/d =16) in the wave trough phase in cases J4–J7. These two planes are located at the middle of the two adjacent effluent clouds, and the effluent clouds have merged with each other. The distributions show one peak or two peaks or three peaks in different cases. This will lead to the corresponding types of time-averaged concentration distributions in the transverse planes, which will be discussed in detail in the following section. It is found that the two parameters Rwc and St have a significant influence on the characteristics of the effluent clouds, including the size of the effluent clouds, the distance between adjacent effluent clouds, and the concentration distribution in the ‘effluent clouds’ part. They will further affect the time-averaged structure of the jet in the wavefollowing-current environment.
all directions, with an increase in both the horizontal length Lh and the vertical length Lv. When the horizontal length Lh of the effluent cloud is greater than the distance De, the effluent cloud will merge with the adjacent ones. The merging location of two effluent clouds appears to be highly dependent on the parameters St and Rwc. The merging location in case J1 or case J2 has a shorter horizontal distance from the jet orifice position compared with that in case J4 or case J5 and case J8 or case J10, which have the same value of Rwc but smaller value of St. Among the cases in which St =0.1, the merging between adjacent effluent clouds is first found in case J13 owing to its largest horizontal length Lh of the effluent cloud. Another interesting finding is the discontinuity of the effluent clouds in case J7 and case J13, which is quite similar to the discontinuous motion of the jet in the wave-only environment identified by Mori and Chang (2003). The ‘effluent clouds’ part of the jet in these cases is discharged when the flow is against the current direction. In the initial stage of formation, the effluent cloud itself is continuous. Then, the flow changes toward the current direction, and at the same time, the flow velocity increases to a large value in an extremely short time. The effluent clouds are barely able to maintain their continuity in such a strong dynamic environment.
3.2. Phase-averaged scalar structure in the transverse planes Fig. 9 shows the phase-averaged scalar field of the jet in the transverse plane (x/d =8) in the wave trough phase in cases J8–J13. The transverse plane of x/d =8 is located at the centre of the effluent cloud, and the effluent cloud has not merged with the adjacent ones. When the wave-to-current velocity ratio is small (Rwc =0.2), the ‘effluent clouds’ phenomenon is insignificant and a double-peak distribution related to the CVP structure can be observed in this transverse plane. When the wave-to-current velocity ratio increases (Rwc ≥0.6), another concentration peak, which is in the ‘effluent clouds’ part, is found in this transverse plane. This suggests that more jet fluid is transported to the effluent clouds as Rwc increases. However, if the wave-to-current velocity ratio is larger than 1, the concentration peak will no longer increase or even decrease. It should be noted that this is
3.3. Time-averaged scalar structure in the transverse planes Fig. 12 shows the time-averaged scalar field of the jet in different transverse planes (x/d =10–60) in all 14 cases. The data were obtained by averaging the values of the instantaneous concentration in 20 wave cycles. For the jet in the current-only environment, distinct double concentration maxima are observed in each transverse plane. Such a structure is certainly related to the counter-rotating vortex pair (CVP). 37
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Fig. 12. Time-averaged scalar field of the jet in different transverse planes (x/d =10–60).
When the Strouhal number is small, say St =0.1 or 0.133, the concentration distribution consists of three high-concentration regions in the transverse planes close to the jet orifice (x/d =10) in each case. Of the three high-concentration regions, one is due to the ‘effluent clouds’ phenomenon and the others are related to the CVP. As the jet moves downstream, the concentration distribution in the transverse planes will evolve in two different ways. It is either dominated by the high-concentration region due to the ‘effluent clouds’ or dominated by the high-concentration regions related to the CVP, with the other highconcentration region(s) becoming weaker and eventually disappearing.
For the jet in the wave-following-current environment, no uniform structure can be identified for all these cases, even in the different transverse planes of each case. The time-averaged structure has a strong relationship with the parameters Rwc and St. When the wave-to-current velocity ratio is small (Rwc =0.2), the concentration distribution is quite similar to that in the current-only environment, with double maxima in each transverse plane, regardless of whether St equals 0.2, 0.133, or 0.1. The characteristics of the concentration distribution when Rwc is greater than 0.2 will be discussed next. 38
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Fig. 13. Integrated concentration distribution in cases J2, J5, and J10.
3.4. Integrated time-averaged concentration distribution in the transverse planes Xia and Lam (2004) found that there is a continuous transport from the jet bent-over part to the ‘effluent clouds’ part. It is difficult to find relevant evidence from Fig. 12, which shows that all concentration peaks in the transverse planes decrease along the current direction. In this study, the concentration distribution in each transverse plane is integrated along the y direction, denoted as cl. Fig. 13 shows the integrated concentration distribution in cases J2, J5, and J10. Two concentration peaks exist in each transverse plane. As the jet moves downstream with the current, both peaks keep moving upward and the peak at the lower position becomes smaller while the other one becomes increasingly larger. This indicates a bottom-up transport of the jet fluid, which is consistent with the results of Xia and Lam (2004), even though they only analysed the data on vertical symmetrical lines. Fig. 14 shows the effects of Rwc on the integrated concentration distribution in the transverse plane of x/d =20. The increase in Rwc leads to a higher concentration peak in the ‘effluent clouds’ part and a lower peak in the jet bent-over part. Further, a larger value of Rwc implies a lower position of the concentration peak in the jet bent-over part. However, there is no clear tendency for the peak position in the effluent cloud region. This shows that the increase in Rwc contributes to the mass transport from the jet bent-over part to the ‘effluent clouds’ part.
Fig. 14. Effects of Rwc on the integrated concentration distribution in the transverse plane of x/d =20.
3.5. Jet visual area The jet-affected region is representative of the jet dilution characteristic. In general, it is bounded by the jet edge defined by the concentration contour of C =0.25Cm, where Cm is the time-averaged maximum scalar concentration in the transverse planes. As shown by Lee and Chu (2003), such a definition is in good agreement with the jet visual boundary. They defined the vertical and horizontal half-widths Rv and Rh of the crossflow jet based on the time-averaged scalar field in the transverse planes, as shown in Fig. 15. Instead of using these two width parameters, the area of the concentration contour of C =0.25Cm, which is called the jet visual area (A25%) in this study, is used to represent the jet dilution characteristic. Fig. 16 shows the effect of the Strouhal number St on the jet visual area A25% in the wave-following-current environment. The wave effect results in expansion of the jet visual area. Moreover, the jet visual area increases as the Strouhal number decreases. One reason is that a smaller value of St means a larger distance between adjacent effluent
Fig. 15. Definition diagram of the jet visual area.
If the high-concentration region due to the ‘effluent clouds’ dominates the evolution, two high-concentration regions will appear again. The new structure is quite similar to that related to the CVP. Finally, the concentration distribution far from the jet orifice (x/d =60 or more) has two high-concentration regions. When the Strouhal number is large, say St =0.2, it seems that the evolution process is preferentially dominated by the high-concentration region owing to the ‘effluent clouds’, and then, the concentration distribution gradually exhibits double maxima, as shown in case J2 and case J3. 39
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Fig. 16. Effect of the Strouhal number St on the jet visual area A25%.
Fig. 17. Effect of the wave-to-current velocity ratio Rwc on the jet visual area A25%.
differences in the jet visual area induced by the Strouhal number become more prominent. The effect of the wave-to-current velocity ratio Rwc on the jet visual area A25% in the wave-following-current environment is shown in Fig. 17, which indicates that Rwc has a significant influence on the jet
clouds. This implies that there is more surrounding water to dilute each effluent cloud and that the merging of adjacent effluent clouds occurs further downstream. When the wave-to-current velocity ratio is extremely small (Rwc =0.2), the increase in the jet visual area with the decreasing Strouhal number is insignificant. As Rwc increases, the
40
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insufficient. Thus, the jet visual area will keep decreasing as the waveto-current velocity ratio increases. It should be noted that a more accurate value of the optimal ratio requires further investigation for real wastewater and brine discharge projects. Nevertheless, this study suggests that the wave effect should be taken into consideration when designing a wastewater or brine discharge project. The wave and current dynamics at the discharge site should be in an appropriate range in order to eliminate adverse environmental effects. In summary, two approaches for enhancing the dilution of wastewater or brine discharged into the ambient environment, especially in tide- or current- dominated areas, were presented in this study. One is to sway the diffusers at a certain frequency and amplitude, and the other is to oscillate the initial velocity of the wastewater or brine in a sinusoidal manner. The swaying or oscillating period is of the order of seconds and the amplitude can be determined on the basis of the local current velocity.
Fig. 18. Relationship between the jet visual area and the wave-to-current velocity ratio in the transverse plane (x/d =40).
visual area. An unexpected finding is that an increase in Rwc might lead to a decrease in the jet visual area. This indicates that an excessively strong wave might be not conducive to jet dilution in the wavefollowing-current environment.
5. Conclusion In this study, a numerical model based on the large eddy simulation (LES) method was established to study the three-dimensional scalar structures of a turbulent jet in different wave-following-current conditions. The main conclusions are as follows.
4. Discussion For a jet in the current-only environment, the cross-sectional scalar field has double concentration maxima (Lee et al., 2002). By contrast, the concentration distribution with three high-concentration regions is the most typical one for a jet in the wave-following-current environment. All the other distributions, including one high-concentration type and two high-concentration types, are considered as either the upper part or the lower part of the most typical one. The parameters Rwc and St together determine the dominant part of the distribution in a certain transverse plane. In general, an increase in Rwc or a decrease in St leads to a higher concentration in the ‘effluent clouds’ part and a lower concentration in the jet bent-over part. An important finding with regard to the time-averaged concentration distribution is the transformation from one-peak type to two-peak type, e.g. in cases J2, J3, J7, and J13. One possible reason is discussed here. The ‘effluent clouds’ maintain part of the jet initial vertical momentum and keep moving upwards, as shown in Fig. 11 of Xu et al. (2016). When interacting with the surrounding water, the effluent clouds cannot cross over the interface completely. This will certainly result in lateral flow. Then, a structure similar to CVP will emerge. This is similar to the mechanism of the existing CVP structure in a crossflow jet, which was suggested by Broadwell and Breidenthal (1984). The results on the jet visual area imply that there is an optimal wave-to-current velocity ratio for the highest dilution of a jet in the wave-following-current environment. Fig. 18 shows the relationship between the jet visual area and the wave-to-current velocity ratio in the transverse plane (x/d =40). It indicates that the optimal ratio is around 0.6 for the cases in this study. For the cases with the same value of St, such as cases J8–J13, the distance between adjacent effluent clouds remains unchanged. When the wave-to-current velocity ratio increases from 0 to 0.6, the vertical length Lv of the effluent cloud increases significantly with a small change in the horizontal length Lh, while the concentration of the effluent clouds gradually increases simultaneously. The concentration of the effluent clouds can be diluted to a low level by the surrounding water. The jet visual area increases with the wave-tocurrent velocity ratio. If the wave-to-current velocity ratio is in the range of 0.8–1.0, it will be difficult to reduce the high concentration inside the effluent clouds to the same level. Hence, the jet visual area decreases as the wave-to-current velocity ratio increases up to 1.0. If the wave-to-current velocity ratio keeps increasing, the horizontal length Lh of the effluent cloud increases sharply even though the concentration level does not decrease significantly. Further, the amount of surrounding water to be mixed with the effluent clouds is
(1) The regular wave effect on the jet dilution can be ascribed to the wave-to-current velocity ratio Rwc and the Strouhal number St. (2) The characteristics of the most distinct ‘effluent clouds’ phenomenon has a strong relationship with the abovementioned parameters; the effluent cloud size is primarily determined by the waveto-current velocity ratio Rwc, and the distance between adjacent effluent clouds is inversely proportional to the Strouhal number St. (3) As the jet moves downstream, the evolution process of the timeaveraged concentration distribution is either dominated by the high-concentration region owing to the ‘effluent clouds’ or dominated by the high-concentration regions related to the CVP structure. An increase in Rwc or a decrease in St leads to a higher concentration in the ‘effluent clouds’ part and a lower concentration in the jet bent-over part. (4) There exists an optimal wave-to-current velocity ratio (approximately 0.6 in this study) for the highest dilution of a jet in the wave-following-current environment. This study should be a useful reference for the selection of sites in wastewater or brine discharge projects. Swaying the diffusers or oscillating the initial velocity could result in enhancement of wastewater or brine dilution in the surrounding water. Further study is recommended for obtaining quantitative results. Acknowledgement This work was partly supported by the Fundamental Research Funds for the Central Universities (Grant No. 2016B13214) and the National Natural Science Foundation of China (Grant Nos. 51379072, 51509080). References Alves, L.S.B., Kelly, R.E., Karagozian, A.R., 2008. Transverse-jet shear-layer instabilities. Part 2. linear analysis for large jet-to-crossflow velocity ratio. J. Fluid Mech. 602, 383–401. Chan, S.N., Thoe, W., Lee, J.H.W., 2013. Real-time forecasting of Hong Kong beach water quality by 3D deterministic model. Water Res. 47, 1631–1647. Chang, K.A., Ryu, Y., Mori, N., 2009. Parameterization of neutrally buoyant horizontal round jet in wave environment. J. Waterw. Port Coast. Ocean Eng. 135 (3), 100–107. Chua, L.P., Antonia, R.A., 1990. Turbulent Prandtl number in a circular jet. Int. J. Heat Mass Tran. 33 (2), 331–339. Chyan, J.M., Hwung, H.H., 1993. On the interaction of a turbulent jet with waves. J.
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