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2011,23(6):806-813 DOI: 10.1016/S1001-6058(10)60179-5

NUMERICAL STUDY OF HYDRODYNAMICS OF MULTIPLE TANDEM JETS IN CROSS FLOW*

XIAO Yang, TANG Hong-wu State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Nanjing 210098, China College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing 210098, China, E-mail: sediment_lab@hhu.edu.cn LIANG Dong-fang Department of Engineering, University of Cambridge, Cambridge, UK ZHANG Jiu-ding College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing 210098, China

(Received June 28, 2011, Revised October 14, 2011)

Abstract: The hydrodynamics of a single jet and four tandem jets in a cross flow are simulated by using the Computational Fluid Dynamics (CFD) software Fluent. The realizable k ε− model is used to close the Reynolds-Averaged equations. The flow chara- cteristics of the jets, including the jet trajectory, the velocity field and the turbulent kinetic energy are obtained with various jet-to- cross flow velocity ratios R in the range of 2.38-17.88. It is shown that a single jet penetrates slightly deeper than the first jet in a jet group at the same R , although the difference decreases with the decrease of R . It is also found that the way in which the velo- city decays along the centerline of the jet is similar for both a single jet and the first jet in a group, and the speed of the decay increa- ses with the decrease of R . The downstream jets in a group are found to behave differently due to the sheltering effect of the first jet in the group. Compared with the first jet, the downstream jets penetrate deeper into the cross flow, and the velocity decays moreslowly. The circulation zone between the two upstream jets in the front is stronger than those formed between the downstream jets.The Turbulent Kinetic Energy (TKE) sees a distinct double-peak across the cross-sections close to each nozzle, with low values inthe jet core and high values in the shear layers. The double-peak gradually vanishes, as the shear layers of the jet merge further away from the nozzle, where the TKE assumes peaks at the jet centerline.

Key words: multiple tandem jets, jet in cross flow, realizable k ε− model, flow dynamics

Introduction

Jets in cross flow configurations exist in many natural environments and industrial applications, such as pollutant dispersion, fuel injection and dilution holes in combustors. These flow fields have an impor- tant bearing on the control of the mixing process, the industrial design and the environmental risk assess- ment.

In this field, a single jet is much studied.

* Project supported by the National Natural Science Foun- dation of China (Grant Nos. 50879020, 51179055 and 51125034), the Special Fund of State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering (Grant No. 2010585512) and the Fundamental Research Funds for the Central Universities (Grant No. 2009B07614). Biography: XIAO Yang (1974-), Male, Ph. D., Associate Professor

Margason[1] and Lee and Chu[2] provided an extensive review in this respect. The velocity evolution in the jet center-plane[3], the vortex development[4], the jet tra- jectories[5], the scalar-mixing and transport proper- ties[6], the two-phase flow structure and the particle dispersion[7] were investigated, along with other asso- ciated phenomena.

Multiple jets in cross flow are widely used in dis- charging municipal wastewater due to their high mixing efficiency. Compared with a single jet in cross flow, multiple jets were not much studied. The earliest experimental studies on the twin jets in cross flow were performed by Ziegler and Wooler[8]. They obtai- ned data for trajectories of tandem air jets in cross flow. Their main conclusion is that the rear jet is less deflected than the front jet due to the sheltering effect. Issac and Schetz[9] proposed a momentum integral method to analyze the interaction between jets. Their results show that the trajectory of the rear jet is signi-

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ficantly modified by the upstream jet even for a sepa- ration as large as 7.5D , where D is the nozzle dia- meter. They carried out direct velocity measurements to show that the cross flow velocity decreases in the region between two jets[10]. Kolar et al.[11] used the standard crossed hot-wire anemometry to measure the mean-flow velocity and the turbulence statistics of the twin jets in cross flow with tandem and side-by-side arrangements. Yu et al.[12] performed a series of expe- riments on the interaction of tandem jet groups in cross flow using Laser Induced Fluorescence (LIF) and Particle Image Velocimetry (PIV). In their series 1 experiment, the jet interaction was studied for two, three, and four jets with jet spacing = 5s D . In their series 2 experiments, the effect of jet spacing =S2 , 3 , 5 ,10 ,15D D D D D was investigated with a twin jet. Their experimental results reveal a region of a reduced effective cross flow upstream of each down- stream jet due to the shear entrainment and the shelte- ring effect of the first jet. The downstream jets are found to be less deflected. The trajectories of all jets downstream of the leading jet are similar. The redu- ction in the effective cross flow velocity for the down- stream jets becomes larger for smaller jet spacing, but is found to be less dependent on the number of down- stream jets. Xiao et al.[13] developed a 3-D numerical model using Fluent, and validated the accuracy of this numerical model against the experimental data of Yu et al.[12]. More recently, Gutmark et al.[14] compared the dynamics of a single jet and tandem twin jets in cross flow at the same jet-to-crossflow velocity ratio of 3 using 2-D PIV. The penetration of the front jet was found to be comparable to that of a single jet in the near field. The rear jet was shielded by the front jet, which allowed the rear jet to penetrate deeper into the cross flow.

Fig.1 Flow regime definition of multiple tandem jets in cross flow

Most of existing studies focused on the twin jets. Other configurations of multiple jets, such as four jets group, have not been extensively studied. In this arti- cle, we apply the validated numerical model to con- duct a series of computations on the flow fields of a single jet and four tandem jets at a series of jet-to- cross flow velocity ratios.

Fig.2 Computational grid in jet center plane

Fig.3 A comparison of experimental and computed flow fields of four tandem jets in cross flow[13]

1. Computational setup The numerical model of this study is similar to

that in Xiao et al.[13]. Steady, three-dimensional and incompressible turbulent flows are considered. The realizable k ε− model is selected for the turbulent closure. The finite volume method is used to solve the governing equations. The quadratic upwind interpola- tion (Quick) approximation is selected for the spatial discretization. The SIMPLEC algorithm is employed for the velocity-pressure correction and the iterative

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Table 1 Computational parameters Run No. Jet velocity

0V (m/s) Cross flow velocity

aU (m/s) Jet to cross flow

ratio RMomentum length scale

ml (m)

1J01 and 4J01 7.15×10–3 3.0×10–3 2.38 2.1×10–4

1J02 and 4J02 7.15×10–3 1.53×10–3 4.67 4.1×10–4

1J03 and 4J03 7.15×10–3 8.1×10–4 8.83 7.8×10–4

1J04 and 4J04 7.15×10–3 4.0×10–4 17.88 1.58×10–3

solution of the discretized equations. The computation is stopped when the normalized residuals are smaller than 5×10–4 or the variation of the residual in the con- tinuity, velocity and turbulent transport equations cea- ses to change over time.

As shown in Fig.1, the four tandem jets are dis- charged horizontally into a perpendicular cross flow from the right side, with the jet diameter D being 0.01 m and the jet center-to-center spacing s being 5D . The numbering of the jets starts from the front upstream jet. The origin of the coordinate system is in the center of the first nozzle. ( , , )U V W are the velo- cities in the longitudinal, transversal and vertical ( , , )x y z directions, respectively. The single jet is formed by switching off the boundary conditions for the rear jets.

The computational domain is inside a rectangle box of 50 50 40D D D× × and four tandem pipes, of 10D in length. The center of the first jet is 10Ddownstream of the cross flow inlet plane (Figs.1 and 2). A total of 1 496 180 cells are used, including the fluid domain of 182×90×88 orthogonal grids and the four pipe domains each of 13 685 hexahedron grids. The minimum grid sizes are = 0.00125 mxΔ , =yΔ0.001 m , and = 0.00125 mzΔ , respectively.

The boundary conditions of the fluid domain are as follows. The upstream inflow boundary: = aU U ,where aU is the cross flow velocity, = = = 0V W C ,the turbulent intensity = 0.01I , the hydraulic dia- meter = 0.4 mHD , the turbulence length scale =l

0.07 HD , 2= 1.5( )ak U I , 3/4 3/2= /C k lμε , =Cμ

0.09 . The outflow boundary: / = / = /v x W x C∂ ∂ ∂ ∂ ∂= 0x∂ . Other boundaries: the free-slip boundary con-

ditions and the standard wall function are adopted. The boundary conditions of four pipes include: the interface condition is applied at the pipe outlet jun- ction. The pipe inlet: = 0U , 0=V V , where 0V is the jet velocity, = 0W , 0=C C , = 0.1I , =HD0.01 m , 0=k k , and 0=ε ε . The computational grid along the jet center plane is shown in Fig.2. An exa- mple of the computed flow fields, compared with the experimental results measured by PIV, is shown in

Fig.3. The jet velocity is not measured as shown in Fig.3(a), due to the big velocity difference between jets and cross flow.

In order to compare the results of a single jet and a four jet group, computations are performed for a single jet and four jets with the same initial jet velo- city and ambient velocity as shown in Table 1, where R is the jet-to-crossflow ratio, 1/2

0= /m al M U is the

momentum length scale and 2 20 0= / 4M D Vπ is the

jet momentum flux.

2. Results and discussions 2.1 Jet trajectory

The jet trajectory can be defined normally as the velocity trajectory or the concentration trajectory, which are the path through the locations of the maxi- mum time-averaged velocities and concentrations, res- pectively, in jet cross sections. The velocity trajectory is considered in this article.

Fig.4 The comparison of the velocity trajectories of the first jet and a single jet, with solid line representing the first jet and dashed line a single jet

Figure 4 shows the velocity trajectories of the first of the multiple tandem jets in comparison with a single jet with various R. It is found that, for the same R, the trajectories of a single jet are slightly higher than those of the first jet in the group. Gutmark et al.[14] found the similar phenomenon when investiga- ting tandem twin jets in cross flow. The main reason is that the fluid downstream of the first jet is entrained by both the first and the second jet, which contributes to the drop of the pressure downstream of the first jet.

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The larger pressure gradient causes the first jet to bend more than a single jet in a cross flow. The difference becomes smaller for smaller R , which suggests that the influence from the downstream jet decreases in a strong cross flow.

Fig.5 Velocity trajectories of four jets in various cases

Figure 5 compares the velocity trajectories of all jets in all cases listed in Table 1. The trajectories of the rear jets look similar, and are less deflected than the first jet, which means that there exists a slow cross

flow velocity region in front of the rear jets due to the sheltering of the front jet. When / 17y D > as shown in Fig.5(b), / 23y D > as shown in Fig.5(c) and /y

27D > as shown in Fig.5(d), the edges of the first and second jets start to merge, to induce the second jet to bend more than the third and fourth jets. When the cross flow is strong (such as that shown in Fig.5(a)), all rear jets are similar as the result of the rapid deve- lopment downstream.

Fig.6 Centerline velocity decay along with jet velocity tra- jectories

2.2 Mean velocity decay on the jet trajectory When leaving nozzles, the water entrains ambient

fluid to grow the jet volumes. This leads to a velocity and concentration decay gradually along the jet tra- jectories. Figure 6 shows the decay of the normalized centerline velocity of the first jet with various R ,where 0V is the mean jet velocity at the nozzle exit. In order to compare with the case of a single jet, the decay of a single jet is also shown in the same figure. The decay of the normalized centerline velocity is accelerated when R decreases. A stronger intera- ction between the jet and the cross flow can be obse- rved when R is smaller. The velocity decay of the first jet in a group and that of a single jet are almost identical. That means that the cross flow is the main factor to influence the decay of the normalized center- line velocity.

Fig.7 Centerline velocity decay along with jet velocity tra- jectories for different jets (Run 4J02)

Figure 7 compares the centerline velocity decay

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along jet velocity trajectories of all individual jets in Case 2. The velocity decays of the rear jets are similar, and are slower than those of the first jet. For instance, the velocity of rear jets would decay to approximately 46% of the discharging velocity at / = 15y D , where- as the first jet would already decay to 38% of the dis- charging velocity. This can be explained by the wea- kened strength of the ambient cross flow for the rear jets.

Fig.8 Normalized longitudinal velocity contour with jet cen- terline trajectories in the jet center plane ( = 0)z

2.3 Velocity field2.3.1 Longitudinal velocity U

The contour of the normalized longitudinal velo- city / aU U in the symmetric plane ( = 0)z is shown in Figs.8(a) (1J02) and 8b (4J02), where the velocity trajectories are overlaid. The contours close to the first jet in the group have a similar pattern as those close to a single jet. The velocity decreases gra- dually in front of the jet, and a reverse flow region is formed in the wake. Between the two adjacent jet nozzles, the flow region can be divided into two parts. One is the wake flow region near the former jet. Another is the upstream region near the latter jet. The reverse flow region between the first and second jets is apparently the largest. The cross flow velocities in front of the rear jets are similar and smaller than those in front of the first jet, which is why the first jet bends more. For instance, the normalized longitudinal velo- city in front of the first jet is about 0.8, whereas the velocities in front of the rear jets are only about 0.4.

Fig.9 Normalized longitudinal velocity contour in the cross section x-z plane and longitudinal velocity variation in jet center-plane at / =my l 0.5, 1.5, and 2.5

The contour of the normalized longitudinal velo- city / aU U in the -x z plane is shown in Fig.9, where are also marked the jet locations. The variation of the normalized longitudinal velocity along the cen-

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terline is shown above each contour plot. In the study of a single jet in cross flow, the momentum length scale ml is used to divide the jet region into the mo- mentum dominated near field ( / 1my l ) and the momentum dominated far field ( / 1my l ). Then, three vertical planes, with / my l equal to 0.5, 1.5 and 2.5, are selected to define the near field, the inter- mediate field and the far field, respectively. The near field shows the individual behavior of the four jets. The intermediate plane shows the interaction of the four jets. The far field shows the merged behavior of the four jets.

In the near field (Fig.9(a)), there are a velocity drop in front of the all jets caused by the blocking of the jet cores and a velocity peak within the jet regions caused by the entrainment of the jet. The accelerating area sees a symmetry in the shoulders of the all jets. The peak velocity due to the first jet in this area is larger than that due to the rear jets due to the shelte- ring of the first jet. Between nearby jets, the contours show a similar pattern. The only difference is that the velocity behind the first jet drops more than the other jets. In the intermediate region (Fig.9(b)), the edges of the first and second jets start to merge and they start to behave like a single unified jet. The third and fourth jets still remain separate. In the far field (Fig.9(c)), the first jet totally vanishes. The rear jets are merged together. However, the rear jets could still be vaguely observed, which indicates that the rear jets evolve slower than the first jet and the length scale lm needs to be increased for the rear jets[13].2.3.2 Transversal velocity V

The contour of the normalized transversal velo- city 0/V V in the symmetric plane ( = 0)z is shown in Fig.10 (Runs 1J02 and 4J02). When the jets enter into the cross flow, the jets start to bend. One can see two transversal velocity peaks: one is in the jet volume and the other is in the wake of the jet caused by the Counter Vortex Pair (CVP) for the single jet and the first jet in a group. Between the rear jets, one does not see a transversal velocity peak. The locations of the velocity peaks in the downstream of the second and third jets are in the merged region of the jets. Quantitative comparisons of transversal velocity pro- files among jets at various downstream locations are shown in Fig.11. At / =x D 0, 5, 10, 15, corre- sponding to the four jet locations, the jet velocities drop quickly after the jets are discharged into the cross flow. The velocity variations of the rear jets are simi- lar and more gradual than the first jet. For the first jet, with / =x D 1, 2.5, 4, two velocity maxima are obse- rved in the velocity profiles, where the upper one is

due to the deflected jet and the lower one is induced by the CVP. As the deflected jet entrains more ambient fluid, the upper peak also decreases accordi- ngly. The velocity profiles downstream of the rear jets generally follow the one peak trend. The interaction of jets prevents the CVP to be formed. These profiles exhibit similar features, although the location and the amplitude of the velocity peaks vary slightly.

Fig.10 Normalized transversal velocity contour in the jet center plane ( = 0)z

2.4 Turbulent Kinetic Energy (TKE)The TKE is defined as

2 2 2+ +=2

u v wk′ ′ ′

(1)

where u′ , v′ and w′ are the turbulent velocities and the overbar denotes the Reynolds average. The contours of the normalized TKE 2( / )ak U distribu- tion within the jet centerplane ( = 0)z for cases 1J02 and 4J02 are shown in Fig.12. The TKE of a single jet and that of the first of the four tandem jets have very similar distributions. The TKE distributions of three rear jets are very similar to each other. A quantitative comparison of the normalized TKE at different con- stant y planes is shown in Fig.13. At a very short dis- tance from the nozzle ( = 0.5 my l ), two TKE peaks are observed at the edge of each jet volume, which con-

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Fig.11 Normalized transversal velocity profiles of all jets along various downstream locations at 4J02. Solid line is the first jet, dashed is the second jet, triangle is the third jet and cross is the fourth jet

Fig.12 Contour lines of turbulent kinetic energy in jet center plane (dashed, 1J02 and solid, 4J02)

nect the low TKE regions of the jet core and the ambient flow region. The upstream peak is always higher than the downstream peak, which indicates the higher shear intensity on the upstream side of each jet. The upstream peak in the first jet takes the largest value among all jets due to the presence of the largest effective cross flow in front of the first jet. The peak value is about 3.2 for the first jet and nearly 2.5 for the rear jets. The double peaks gradually vanish and a single peak TKE distribution appears for each jet at higher y levels, corresponding to the merging of the shear layers. When = 1.0 my l , a single peak is formed with a similar value for all jets. After that, the boun- daries of the first and second jets start to overlap. The first jet finally loses its own identity and the peak TKE value of 0.91 is found in the second jet volume, while the last two jets downstream have similar TKE

values of 0.75. The turbulence intensity is seen to decrease consistently further away from the nozzles.

Fig.13 Normalized TKE profiles at different y locations along xdirection (4J02)

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3. Conclusions 3-D turbulent flows of a single jet and four

tandem jets in cross flow configurations are studied, based on a validated numerical model. The flow characteristics of the jets, including the jet trajectory, the velocity decay, the velocity evolution and the tur- bulent kinetic energy, are analyzed in detail. The main conclusions can be summarized as follows:

(1) The trajectories of a single jet are slightly higher than those of the first jet in a jet group at the same R. The differences decrease with the decrease of R . The trajectories of the rear jets are less deflected than that of the first jet, and with a similar shape.

(2) The velocity decay along the first of multiple jets is similar to that along a single jet. The decay of the normalized centerline velocity is faster at smaller R. The velocity decays in the rear jets are relatively slow, and also with a similar pattern.

(3) The flow fields of a single jet and the first jet in a jet group show a similar development in the near field. The region between two adjacent jets can be divided into two parts. One is the wake flow region downstream of the former jet. The other is upstream of the latter jet. The area of the reverse flow region between the first and second jets is larger than the others.

(4) The TKE of a single jet and that of the first of the four tandem jets show similar distributions. The three rear jets have similar TKE distributions. The jet core and the ambient flow have a low turbulence level, and the shear layers between these two regions have a high turbulence level. The jet core disappears some distance away from the nozzle, with a high turbulence intensity along the jet trajectory. The largest TKE value always occurs where the incoming flow starts to impinge on the jet.

AcknowledgementsThe authors thank Prof. Lee Joseph H. W. and

two anonymous reviewers for suggestions on impro- ving the article.

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