Mixing of a Rosette Jet Group in a Crossflow

In many coastal cities, partially treated wastewater is often discharged into the receiving water through submarine outfall diffusers to minimize environmental impact. In modern outfall designs, the wastewater is typically discharged through risers fitted circumferentially with 2–8 horizontal nozzles (Fig. 1). The mixing of this rosette jet group discharge with the ambient flow involves the complicated interaction of coflowing, oblique-flowing, cross-flowing, and counterflowing jets. Because of the complexity of the discharge configuration, ocean outfall design has typically resorted to comprehensive physical model experiments. In addition, the “line plume approximation” is often adopted; the near field dilution and wastefield characteristics are assumed to depend on the discharge volume and buoyancy flux per unit diffuser length without consideration of jet momentum or riser spacing effects.

The problem of merging buoyant jets from an outfall was first considered by Liseth (1976) for an alternating jet diffuser in stagnant fluid, for a two-dimensional section of a diffuser. In the absence of an ambient current to supply the entrainment demanded by the jets, a “starved plume” condition can result; because of pressure interaction, the horizontal jets discharging in opposite directions on each side of the diffuser bend back toward each other, forming a vertically rising line plume. Isaacson et al. (1983) performed a physical model study for the San Francisco outfall in which rosette jet groups were tested under stratified and unstratified stagnant and flowing ambient, for inter-riser spacing up to 0.5 $H$ (water depth). They found that the dilution did not depend strongly on whether the discharge ports were evenly spaced along the diffuser or were clustered together as risers (up to eight per riser), provided that the total port area per unit diffuser length remained unchanged. The minimum dilution was also found to increase with decreasing port size (or increasing number of nozzles), and ultimately approached that of a line source with the same discharge per unit length. Roberts et al. (1989a, b, c) performed an experimental investigation for multiple $T$-shaped risers in a linearly stratified crossflow, and found that dilution was only weakly dependent on the source momentum flux and port spacing for typical diffuser arrangements. Roberts and Snyder (1993a, b) performed physical model experiments of the rosette diffuser fitted with 8–12 horizontal nozzles for the Boston outfall. Counter to previous studies, it was found that a 12-nozzle riser resulted in lower dilutions than an 8-nozzle riser for both stagnant and flowing ambient. More recently, Daviero and Roberts (2006) and Tian et al. (2004a, b, 2006) carried out a more detailed investigation on the effect of port spacing on the initial dilution of $T$-shaped (2-jet) risers. In a related study, a theoretical approach to treat jet merging in stagnant water based on the momentum superposition principle was reported (Yannopoulos and Noutsopoulos 2006a, b). Seo and Yeo (2002) reported a study on the initial dilution of rosette jet groups located in shallow water. Based on their experimental data and dimensional analysis, relations for predicting the dilution of rosette jet groups were established under specific ambient conditions. Other than experimental studies, an attempt to use computational fluid dynamics (CFD) modeling to simulate an ocean outfall with a number of rosette risers was reported (Law et al. 2002); the simulation required a run time of four days.

Although the “line plume approximation” adopted in several major outfall studies is applicable in deep water and sufficiently far from the outfall, it is typically not applicable to the site conditions of many Asian cities where the coastal water depth is in the order of 5–20 m (Table 1). There appear to be scant experimental studies of rosette jet group in shallow crossflow, and dimensional analyses based on the limited data are not generally applicable. On the other hand, CFD modeling of outfall discharges is primarily limited to the study of detailed research issues (e.g., Kuang and Lee 2006; Kuang et al. 2006); the advantages of CFD as a design and environmental assessment tool are not apparent, considering the vast range of length scales (ranging from the jet diameter in the order of 0.1 m to a distance of 5–100 m) of the flow phenomenon, the large computational costs and time, and issues of numerical convergence and accuracy. Robust models are needed for predicting the initial dilution of buoyant wastewater effluent from rosette diffusers under any flowing ambient condition.

A significant issue arising from the previous studies was to determine the number of nozzles that should be used on a riser to optimize the performance of an outfall. The anomaly observed by Roberts and Snyder (1993b) demonstrated that it is not always true that higher dilutions result from a larger number of nozzles (per riser) or a larger number of risers. It is not clear, however, whether this is attributable to the dynamic interaction of jets, as observed by Liseth (1976), or the kinematic interaction of jets attributable primarily to the overlapping of plumes. If the latter is true, it is of interest to see if the initial dilution of the jet group can be predicted by treating each jet independently and computing the jet merging in a three-dimensional manner.

This paper presents a comprehensive experimental study on the mixing of a rosette buoyant and nonbuoyant jet group. Detailed cross-sectional measurements are made of the scalar concentration field downstream of the bent-over jets. The trajectories of multiple and individual jets discharging at various angles are measured. The dynamic and kinematic interaction of the rosette jets are investigated through the measured data. A general model for predicting the jet group dilution of a rosette jet group in a crossflow is presented. The theoretical predictions are compared with experimental data of an unstratified crossflow and data of a stratified crossflow from previous studies. The implication of the findings on outfall design is also discussed.

Consider a jet group clustered around the circumference of an outfall riser, discharging into a horizontal ambient current $U_a$ (Fig. 1). The individual jets in a rosette jet group discharge at different angles $\theta$ relative to the ambient current; e.g., as jets in coflow, crossflow, and counterflow. Each individual jet mixes with the ambient current by shear entrainment close to the source and vortex entrainment further away, and is bent-over by the current. If the jet momentum is not directed in the ($x-z$) plane, where $z=\mathrm{\text{vertical direction}}$, the jet will have a three-dimensional trajectory. The mixing of the jet group with the crossflow hence traces out a rosette-shaped pattern, as shown in Fig. 1(a). The merging of the multiple jets is complex (Isaacson et al. 1983; Seo and Yeo 2002), and depends on the jet momentum and buoyancy flux, ambient current velocity, and the riser and port spacing.

Downstream of the riser, the bent-over jets merge and the plume cross sections overlap (Fig. 2). For practical ocean outfall designs, the port diameter ($D$) is typically an order of magnitude smaller than the riser diameter $D_R$; for example, $D/D_R\approx 0.1$ and 0.05 for the Hong Kong Harbour Area Treatment Scheme (Table 1) and the Boston Outfall, respectively (e.g., Roberts and Snyder 1993a). Provided an adequate ambient flow is available for jet entrainment, it is reasonable to assume (to be confirmed by experiments) that the dynamic interaction of the widely separated jets can be neglected as a first approximation; the jets can be treated as independent of each other. By mass conservation, the mass flux of merged multiple jets can be written aswhere $A_m$, $C_m$, $u_m$ = cross-sectional area, cross-sectional average concentration, and velocity of the merged jets, respectively; $A_i$, $C_i$, $u_i$ = cross-sectional area, cross-sectional average concentration, and velocity of an individual jet, respectively; and $N$ = number of individual jets [Fig. 2(b)]. In the bent-over phase of a jet in crossflow, experiments have shown that the jet horizontal excess momentum is dissipated within a short distance owing to the mixing with ambient crossflow (e.g., Lee and Chu 2003), and the horizontal jet velocity is approximately the same as the crossflow velocity, with $u_i\approx u_m\approx U_a$. At a downstream $x$-section, the cross-sectional average concentration of the merged jets is then given as

The cross-sectional area of the merged jets $A_m$ is smaller than the sum of individual jet areas owing to the overlapping area $A_o$ [Fig. 2(b)], with $A_m=\sum _{i=1}^N{A}_i-A_o$. The average concentration of the merged jets from a rosette diffuser can then be obtained from Eq. (2), where the concentration and cross section of individual jets can be predicted from a near field jet model, or obtained directly from experiments. The average dilution of the jet group can then be defined as $S=C_o/C_m$, where $C_o=\mathrm{\text{source concentration}}$.

Experiments of rosette jet groups discharging from a riser are performed for a representative range of ambient and discharge velocities in a $12\times 0.4\times 0.5\mathrm{m}$ deep recirculating flume (Fig. 3). The discharge flow is fed from a constant head tank and measured by a calibrated Tokyo Keiso rotameter; the ambient velocity is measured by an acoustic Doppler velocimeter (ADV) placed upstream of the rosette diffuser at the same height as the port. Buoyancy of the source fluid is obtained by addition of ethanol to water; the temperature of the source fluid and the ambient fluid are measured by an electronic thermometer, and the density is measured using a Kyoto Electronics density meter. Rhodamine 6G is used as the tracer dye, with a source concentration $\sim 0.3\mathrm{mg}/\mathrm{L}$. The concentration field and trajectory of the jet group is measured using the laser-induced fluorescence (LIF) technique. A 5 W argon-ion laser source and a high speed rotating mirror is used to produce the laser sheet for cross-sectional concentration field measurement. In situ calibration is carried out before each set of experiments to determine the response of camera to the dye concentration for ethanol-water mixtures of 0, 20, 40, and 60%. Source fluid of known dye concentration is added and mixed in steps to a water-filled Perspex cell placed at the position of the laser sheet. For each step, the LIF image of the resulting homogeneous solution is captured and the corresponding fluorescent light intensity (gray level) is determined after subtraction of background image. The light intensity varies approximately linearly with the dye concentration. A second order polynomial determined experimentally is used to eliminate any nonlinearity. The error associated with the jet flow and ambient velocity measurement can be estimated to be 1–2%; the maximum deviation of ambient velocity from the flume sectional average velocity is approximately 5%; the error associated with the LIF technique attributable to laser attenuation is estimated to be approximately 2% (Lai 2009). The maximum error of the concentration measurement is estimated to be within 5–10%.

Six- and eight-nozzle risers are considered; the jets are uniformly clustered around the circumference of the riser and discharge horizontally into the ambient flow; for symmetry, only half of the nozzles are used. The jet nozzle diameter varies in the range of $D=0.25-0.49\mathrm{cm}$; the riser diameter is $D_R=3.6\mathrm{cm}$. Two series of experiments are carried out: (1) for nonbuoyant and buoyant jet trajectory (Series 1a and 1b, respectively); and (2) for scalar tracer concentration field (Series 2) measurements. For experiments with nonbuoyant momentum jets, the laser sheet is located horizontally at the center plane of jets, and an overhead charged-coupled device (CCD) camera is used to capture the top-view of the LIF image. 400 LIF images ($576\times 768\mathrm{pixels}$) are taken with interval of 0.1 s and the time-averaged data of these images are used (after background subtraction) for analysis. For the buoyant jet group, food dye is used to trace the multiple buoyant jet trajectories, because the planar LIF technique is not suitable for determining the three-dimensional buoyant jet trajectories. In these experiments, suitable lighting is provided to give the best visualization effect. Both the top and side views of the jet are simultaneously captured using CCD cameras. 200 images are taken with interval of 0.1 s and time-averaged for later analysis.

The experimental parameters for the nonbuoyant and buoyant rosette jet group trajectory experiments are summarized in Table 2 and 3, respectively. For the buoyant jet-groups, the jet densimetric Froude number, $\mathsf{F}=U_o/\sqrt{g(\Delta \rho /{\rho }_o)D}\approx 17-54$ (where $U_o=\mathrm{\text{jet velocity}}$; $\Delta \rho =\mathrm{\text{density difference of jet and ambient}}$; ${\rho }_o=\mathrm{\text{jet density at source}}$; and $g=\mathrm{\text{gravitational acceleration}}$); the jet Reynolds number $\mathsf{R}=U_{o}D/V\approx 1,700–5,000$ (where $v$ is the kinematic viscosity of water); the ratio of jet to ambient velocity, $K=U_o/U_a\approx 4.8-10$. The water depth is kept at $H=0.3\mathrm{m}$.

In Series 2 experiments, a vertical laser sheet for LIF measurement is produced perpendicular to the ambient flow direction at 20–80$D$ from the riser. A CCD camera fitted with a Nikkor 35 mm lens and Hoya orange filter is enclosed in a water tight Perspex casing and placed under water to capture the cross-sectional LIF image (Fig. 3). The time-averaged concentration field is obtained from 400 images over 40 s, which is considered to be long enough to produce stable and repeatable time-averaged images. In most of the Series 1 and 2 experiments, observations are made on (1) the multiple jet group; and (2) each of the individual jets discharging in isolation. Using the three-jet group of Run TJ3A1K8 (simulating the symmetrical half of a six-jet group) as an example (Table 2), experiments are conducted as follows: single jet discharge at 30°, single jet discharge at 90°, single jet discharge at 150°; then three jets discharging together (TJ3A1K8). In this way, the effect of jet merging on the jet trajectory and the initial dilution can be studied. For the cross-sectional concentration field measurements, supplementary experiments are also carried out in a $12\times 5\mathrm{m}$ $\mathrm{wide}\times 1\mathrm{m}$ deep shallow water basin (Run no. C1J6-8, C2J6-12 in Table 4). Six-, eight-, and 12-jet nozzles are used (i.e., no symmetry plane is used and the full rosette jet group is tested). The jet nozzle diameter is $D=0.25-0.75\mathrm{cm}$. The water depth was kept at $H=0.3\mathrm{m}$. The cross-sectional concentration field of the jet group is measured at 20$D$–80$D$. The width of the basin enabled the whole concentration field to be captured without the effect of lateral boundaries, especially for a weak crossflow. The experimental parameters for the cross-sectional scalar field measurements are summarized in Table 4.

The experimental observations of the jet trajectories and cross-sectional concentration field are presented. First, the observed degree of interaction between adjacent jets is presented and compared to the trajectory of individual single jets for representative jet-groups. The observed jet merging process is described; the interpretation of the experimental data is greatly facilitated by using the JETLAG/VISJET model.

JETLAG is a generalized Lagrangian integral model in which the downstream development of a jet or plume element from the jet exit is computed in the Lagrangian frame with appropriate mixing and entrainment hypothesis (Lee and Cheung 1990; Lee and Chu 2003). Top-hat average velocity and concentration profiles are assumed for the plume element. The model has been validated against laboratory and field results on both trajectory and centerline concentration data for a wide range of discharge and ambient flow conditions. In particular, the model is well-suited for predicting rosette buoyant jet discharges in a crossflow, which are typically characterized by three-dimensional jet trajectories. VISJET is the latest version of JETLAG which includes an updated entrainment hypothesis, an interactive computer interface, and virtual reality visualization (Lee et al. 2000; Lee and Chu 2003). One significant feature of VISJET is the capability of visualizing the jet merging process along any cut plane; in addition, the degree of jet merging (plume overlap) on a given downstream cross section can be quantified. Further details on the model can be found at http://www.aoe-water.hku.hk/visjet.

The jet group dilution can be predicted by the VISJET model, which allows effective visualization of jet merging and the “plume elements” in any cutting plane. Furthermore, the visual merged jet area [$A_m$ in Eq. (2)] is also computed numerically by using three-dimensional computer graphics techniques (Lee et al. 2000; http://www.aoe-water.hku.hk/visjet). Fig. 8(c) shows the predicted visual boundaries of the plume elements and how they intersect at different vertical cross sections downstream.

For a bent-over jet and plume in crossflow, both experiments and turbulence model computations have revealed the existence of added mass (Chu 1996; Lee and Chu 2003). For integral models that neglect added mass, the predicted jet half-width ($B$) is larger than the visual half-width (“top-hat” profile) by a factor of $\sqrt{1+k}$, where $k=\mathrm{\text{added mass coefficient}}=1$ for advected line puffs and thermals (Chu and Lee 1996). Because the added mass flux effect is not considered in the JETLAG model, the visual half-width can be determined as $B_v=B/\sqrt{1+k}=B/\sqrt{2}$ (Chu 1996). This physical interpretation is supported by extensive experiments.

For a bent-over jet group in crossflow, the flux-average dilution $S$ is given bywhere $Q$ = local volume flux of the jet group; $Q_o$ = source discharge flow; and $A_{mf}$ = merged jet area; $A_{mf}=2A_m$ by simple scaling.

The VISJET predicted flux-average dilution [Eq. (3)] is compared to the observed multiple jet dilution [based on the average concentration within the $0.25C_{\max }$ boundary and Eq. (2)] in Fig. 10. The predicted dilutions agree well with the experimental data for both buoyant and nonbuoyant jet groups.

It has been demonstrated that for typical riser geometries, the dynamic interaction among clustered jets on an outfall riser is negligible in a current. The initial dilution of merging jet groups can be reasonably estimated from a purely kinematic point of view by using a robust integral jet model with advanced visualization capability. It is certainly of interest to approach optimal outfall design from a more theoretical basis than is hitherto possible. Two important issues are addressed: (1) the optimal number of nozzles to be used on a given riser geometry; (2) the optimal number of risers for a given discharge and diffuser. Both issues can be systematically examined by using VISJET.

A comprehensive experimental and mathematical investigation has been performed on the interaction of rosette jet groups from ocean outfalls in an unstratified crossflow. The multiple jet trajectory and scalar concentration field of rosette jet groups are measured for representative riser configurations and jet to ambient current ratios. For typical outfall design conditions, the experiments show that dynamic interaction among adjacent jets is insignificant. Consequently, the jet trajectory can be predicted for each individual jet as if it is independent of the others. On the basis of this observation, the average dilution of a rosette jet group can be predicted by accounting for the jet merging. Predictions of the jet trajectory and jet group dilution by the VISJET model are in good agreement with experimental data. Furthermore, the predicted dilutions are also in agreement with various rosette jet group experiments for unstratified, uniformly stratified, and nonlinearly stratified ambient conditions. The present contribution is twofold. First, the study of merging plumes is possible only with a general Lagrangian jet model that predicts the mixing of jets discharges at different angles into a stratified crossflow. Second, the merging process and the initial dilution can be quantified with the aid of advanced computer graphics and visualization capability. The present findings demonstrate that in the presence of an ambient current, the initial dilution of merging jets can be predicted by accounting for plume overlap in a purely kinematic manner. The proposed model is capable of predicting the changes in dilution with the number of nozzles or the number of risers, and helps to resolve the apparent anomalies of previous experimental observations. The proposed method can be used as a preliminary design tool to study the performance of an ocean outfall fitted with rosette risers. For stagnant or close to stagnant ambient conditions, the dynamic interaction has been shown theoretically and experimentally to be significant; a semianalytical model has been developed to account for the interaction (Lai and Lee 2008; Lai 2009).

This work is supported by the Hong Kong Research Grants Council (Project No. UNSPECIFIEDHKU 7518/03), and in part by a grant from the University Grants Committee (Project No. UNSPECIFIEDAoE/P-04/04). The assistance of Chris Lai in the experimental investigation is gratefully acknowledged.