Numerical investigation of velocity ratio effect in combined wall and offset jet flows *

Nidhal Hnaien, Salwa Marzouk , Habib Ben Aissia, Jacques Jay

1. Unité de Métrologie et des systèmes énergétiques, Ecole National d'ingénieurs de Monastir, Tunisie

2. Centre de Thermique de Lyon (CETHIL–UMR CNRS 5008), INSA de Lyon, France

Abstract: In the present work, we propose to numerically study a combined turbulent wall and offset jet flow (this combination will be denoted WOJ). Several turbulence models were tested in this study: the standard -ωk, SST -ωk, standard -kε, RNG, and realizable -kε model. A parametric study was performed to determine the effect of offset ratio H and the velocity ratio r on the longitudinal and transverse positions of the merge point (MP), the combined point (CP), the upper vortex center (UVC) and the lower vortex center (LVC). Correlations that predict the position of these characteristic points of the WOJ flow as a function of H and r have been provided. Results show that any increase in the velocity ratio causes a displacement of the MP, CP, UVC and LVC further upstream along the longitudinal direction. Concerning the transverse positions, the increase of velocity ratio results in a deviation of the merge point (MP) and the lower vortex center (LVC) toward the strong jet (LWJ) whereas the transverse position of combined point(CP) and the upper vortex center (UVC) is almost independent of the velocity ratio.

Key words: Finite volume, combined point, merge point, offset jet, velocity ratio

Introduction

The multi-jet flows play an important role in various industrial applications such as boiler, burners,fuel injection systems, vertical takeoff and landing of air planes, ejectors, heating and air conditioning systems ...etc. The combined wall and offset jet flow(wall offset jet (WOJ)) (Fig. 1) is frequently encountered in many industrial processes especially in the heat exchanger and the wastewater evacuation process which makes interesting the study of the WOJ flow dynamic characteristics. The first reported work on this kind of flow was that of Wang and Tan[1], who have experimentally studied the WOJ flow characteristics using particle image velocimetry (PIV). They considered water as the working fluid. The Reynolds number and the offset ratio were respectively 10 000 and 2. The different flow characteristics such as the average velocity, the shear stress and the jet halfwidth at different shear layers are also measured.Results show that the zone in the vicinity of nozzleplate is characterized by a periodic large-scale Karman-like vortex shedding. This phenomenon results in a periodic interaction between the wall jet(LWJ) and the offset jet (UOJ). Vishnuvardhanarao and Das[2]numerically studied the heat transfer mechanism in the WOJ flow using the standard -kε turbulence model. The flow is assumed steady and incompressible. The Reynolds numbers considered were taken between 10 000 and 40 000 and the Prandtl number was =0 7Pr .. These authors considered two different thermal boundary conditions:constant wall heat flux and constant wall temperature.Vishnuvardhanarao and Das[2]have found that far downstream of the nozzle plate, the local Nusselt number Nu increases with increasing the mass flow ejected by elevating either the wall jet (LWJ) or the offset jet (UOJ) initial velocity. They also showed that the local Nusselt number and consequently the heat transfer exchanged between the wall and the whole flow is higher when the wall is under constant heat flux boundary condition. Vishnuvardhanarao and Das[2]also noticed that for an UOJ initial velocity of 0.25 m/s and that of LWJ equal to 1 m/s which corresponds to a velocity ratio of 4, the average Nusselt number is maximum and it is higher by 5%compared with the reference value.

The numerical study of the turbulent WOJ flow is also achieved by Kumar and Das[3]using the high Reynolds number standard -kε turbulence model.The Reynolds number, the jet spacing and the turbulence intensity adopted were respectively 20 000,9 and 5%. The Kumar and Das[3]numerical code was validated on the experimental results of Pelfrey and Liburdy[4]and Vishnuvardhanarao and Das[5]for a single offset jet (SOJ) flow. The Kumar and Das[3]numerical results were compared with various available data in the literature. A discussion was also performed on the pressure and the turbulent kinetic energy fields. The maximum velocity decay and the half width evolution are also presented. Li et al.[6]have numerically studied the WOJ flow using the large eddy simulation (LES) method. Governing equations describing this flow are discretized using the finite volume method with an orthogonal mesh of 6.17×106nodes. The longitudinal velocity and turbulence intensity evolution for different longitudinal positions are presented and their distribution is in good agreement with experimental results of Wang and Tan[1]. Li et al.[6]showed that the interaction between the LWJ and UOJ is greater in the zone near the nozzle plate (the converging zone) and that these two jets begin the merging process at the merge point.In the second study, Li et al.[7]have numerically investigated the interaction between the LWJ and the UOJ in a WOJ flow for different offset ratio H and velocity ratio r using several turbulence models.The comparison between different turbulence models shows that the realizable -kε model is the most effective for the prediction of this type of flow. Li et al.[7]have noticed that for a fixed offset ratio, the flow pattern change from a single offset jet to that of a single wall jet flow when increasing the velocity ratio.They have also showed that when increasing the velocity ratio while maintaining constant the UOJ equal to 1m/s, the WOJ flow maximum velocity decay rate increase.

Mondal and Das[8]numerically studied using two-dimensional unsteady RANS equations the nozzle spacing effect on the zone in the vicinity of the nozzle plate (the converging zone) of a WOJ flow. The numerical simulation was performed for different nozzle spacing and nozzle width. These authors noted that when the non-dimensional nozzle spacing was taken between 0.7 and 2.1, the converging region is characterized by the presence of large scale Karmanlike vortices. However, outside this range, two steady vortices with opposite direction are formed in the converging zone. The experimental setup of Pelfrey and Liburdy[4]on a SOJ flow with a Reynolds number of 15 000 and an offset ratio of 7 was recently numerically simulated by Kumar[9]. The nozzle width was d =12. 5 mm and the nozzle outlet velocity was 17.4 m/s. After validating the geometric configuration of Pelfrey and Liburdy[4], Kumar[9]added in parallel to the offset jet a second wall jet to form a WOJ flow.The purpose of his study was to investigate the effect of the wall jet (LWJ) addition on the whole WOJ flow structure. The numerical analysis performed by Kumar[9]compares the SOJ flow characteristic with those of WOJ flow for an offset ratio H ranging from 3 to 15 at an interval of 2. Correlations that predict the longitudinal and transverse positions of the merge point, combined points and the recirculation vortex centers are also provided with respect to a single parameter which is the offset ratio H.

Through all the work that we have just mentioned, it appears that the majority of studies on the WOJ flow are concentrated mainly in the determination of the different zones of the flow field and on the offset ratio H effect on the MP, CP,UVC and LVC positions seen their important roles in the flow development, lunching and interruption of the merging process. In most WOJ flow studies,correlations that predict the positions of these various characteristic points are limited to a single parameter which is the offset ratio H. However, to our knowledge, there is no significant attempt to study the velocity ratio effect on the positions of these characteristic points for heights offset ratios H. This study aims to fill these gaps. It is important to note that this is the first time that correlations that predict the longitudinal and transverse positions of the merge point, the combined points and the vortices centers are provided as a function of two parameters together which are the offset ratio H and the velocity ratio r. Thus, the main objective of the present study is to investigate the simultaneous effect of the velocity ratio and the offset ratio on the WOJ flow characteristics. This study allowed us to provide correlations that predict the longitudinal and transverse positions of the merge point, the combined point and the vortex centers. Indeed, these correlations may be useful in the quick estimation of these different characteristics points that manage this type of flow in various industrial applications.

1. Mathematical formulation

1.1 . Geometric configuration

The geometric configuration of Kumar[9]is used to study the influence of the velocity ratio in the WOJ flow (Fig. 1).

Both nozzles are identical, rectangular, each ofaligned in the direction of the transverse flow (along y axis). The offset ratio H is set equal to 9, the turbulence intensity and the Reynolds number at each nozzle outlet were respectively I = 5% and Re = 15 000 which relate to an initial velocity value u0=17.4 m/s. The computed region dimensions are chosen so as not to affect the flow spreading. Several configurations were tested for finally adopt the following dimensions 100d and 50d respectively along the longitudinal and the transverse directions.Figure 2 shows the domain dimensions as well as the location of the Cartesian coordinates system adopted.

Fig. 1 Schematic diagram of combined wall and offset jet flow(WOJ)

Fig. 2 Grid size and boundary conditions

1.2 Hypothesis

The system of equations governing the flow is written in the Cartesian coordinate system whose origin σ is located on the horizontal wall axis (Fig.1). The following hypothesis will be also adopted:

(1) The flow is two-dimensional and isothermal.

(2) The work fluid is air assumed incompressible and the thermo-physical properties are constant.

(3) The flow is supposed in a steady state.

(4) The jet is emitted in the longitudinal direction.

(5) The flow is assumed to be turbulent and fully developed.

1.3 Governing equations

The Reynolds-average Navier-Stocks (RANS)equations can be written in Cartesian tensor form as follows:

Additional terms now appear which represent the effect of turbulence. These Reynolds stresses must be modeled in order to close Eq. (2). One common method employs the Boussinesq approximation to relate the Reynolds stress to the mean velocity gradient as follows

The turbulent kinetic energy k and its specific dissipation rate “ω” noted w are obtained from the following equations:

For k-ω model with high Reynolds number,α = 1. The dissipation of the turbulent kinetic energyand that of its specific dissipation rateare given as follows:

1.4 Grid distribution and boundary conditions

A non uniform grid is adopted along the longitudinal and the transverse directions. Indeed, a fine grid is used near the nozzle plate and a little looser further. As shown in Fig. 2, an uniform grid is used for the [A], [B] and [C] segments with a spacing s = 0.05. The [D] segment has a non uniform grid with spacing of 0.5 and an expansion ratio e =1.0450 which result in a nodes number of 280 on these segments ([A], [B], [C] and [D]). Note that the grid spacing s is dimensionless based on the nozzle width d and the expansion ratio e is a non-dimensional coefficient. For [E] segment, the grid is non uniform with s = 0.50 and e = 1.0185 which give the node number of 200. It is noted that all opposite segments have the same grid size. Consequently we obtained a total number of quadratic cells over the entire area as follows

This particular choice of nodes number will be discussed later in the grid size sensitivity section. To complete the problem, besides the equations mentioned above, it is necessary to take into account the boundary conditions which are summarized in Table 1.

Table 1 Boundary and emission conditions

1.5 Numerical method

In this work, the transport equation associated with the boundary and emission conditions are solved numerically by the finite volume method and by using the computational fluid dynamics software “Fluent”.The computed region is divided into finite number of sub-regions called “control volume”. The resolution method is to integrate on each control volume the transport equations such as the momentum conservation, mass conservation, the turbulence kinetic energy k and the specific dissipation rate of the turbulence kinetic energy w. These equations are discretized using the second order upwind. The coupling velocitypressure is based on the SIMPLEC algorithm. The disposal method of Gauss-Seidel associated with a relaxation technique is used to solve the resulting tri-diagonal matrix. A non uniform grid is adopted in the longitudinal and transverse directions, a fine grid is used near the nozzles and a little looser further. The convergence of the global solution is obtained when the normalized residuals fall below 10-4. We have verified that the increase in this accuracy had practically no influence on the results. The normalized residual is expressed as followsis the variable such as u, v, k, q or ω and,are the coefficients of the discretized equations andis the source term.

2. Results and discussion

2.1 Velocity field validation and grid sensitivity study

Figure 3(a) shows the prediction of the longitudinal mean velocity transverse distribution on the plane X =7 using the standard k-omega turbulence model for H = 9 and R e = 15 000.

Fig. 3 Transverse distributions of the longitudinal velocities at X=7

Fig. 4 Longitudinal velocity U, static pressure P and turbulent kinetic energy K contours for =9H and =Re 15000

As shown in Fig. 3(a), the numerical results are given using three different grid sizes such as(180/125), (280/200) and (330/333). These grids contain respectively 22 500, 56 000 and 109 800 quadratic cells. It seems clear that the velocity profile predicted by the grid (280/200) and (330/333) are almost identical while a remarkable difference was found between the prediction by the grids (180/125)and (280/200). It seems important to note that the grid(180/125) has a maximum discrepancy of 7.3%compared with the results of Kumar[9]while the grid(280/200) and (330/333) have almost the same maximum discrepancies approximately of 1.3%compared with Kumar[9]results. Thus, the grid(280/200) is sufficient to obtain a numerical solution independent of the grid size and validated by the numerical results of Kumar[9]. We presented in Fig.3(b), the transverse distribution of the longitudinal velocity on the plane =7X with (280/200) grid size using five different turbulence models: standard

k -ω , SST k -ω , standard k -ε , RNG k -ε and realizable k -ε . The longitudinal velocity profile predicted by the various turbulence models are in agreement with the numerical results of Kumar[9],except the k -ε realizable turbulence models witch over-predicts the velocity profile for 2.5 ≤Y ≤ 5 .Whereas for Y > 5 , the standard k -ω model provides the best agreement with the Kumar[9] results.Thus, the standard k -ω turbulence model with(280/200) grid size will be adopted in this numerical study.

2.2 Flow characteristics

We present in Fig. 4 the longitudinal velocity U contours (Fig. 4(a)), the static pressure P contours(Fig. 4(b)) and the turbulent kinetic energy K contours (Fig. 4(c)) in the three WOJ flow zones namely the convergence zone forthe merging zone for (and the combined zonefor an offset ratio =H 9, a Reynolds number R e = 15000 and a velocity ratio =1r .

The convergence zone is characterized by a negative longitudinal velocity U which highlights a recirculation flow (Fig. 4(a)) and by a negative static pressure P defining a depression zone (Fig. 4(b)).This last zone is characterized by an increase in the turbulent kinetic energy due to the strong interaction between the LWJ and the UOJ that attract and mix at the merge point defined at X =10.48. At this merge point, the static pressure P (Fig. 4(b)) and the turbulent kinetic energy K (Fig. 4(c)) have high values while the longitudinal velocity is zero (Fig.4(a)). Downstream of the merge point is located the merging zone. In this zone occurs the two jets (LWJ and UOJ) merging process. This region is characterized by a gradual increase of the longitudinal velocity (Fig. 4(a)), by a decrease in the static pressure(Fig. 4((b)) and by a decrease in the turbulent kinetic energy (Fig. 4(c)). Beyond the merging zone, we found the combined zone defined for X≥ 1 8.32, in this zone, the two jets combine and their resulting flow behaves like a single wall jet. In this latter zone(combined zone), the longitudinal velocity, the static pressure and the kinetic turbulent energy decrease continuously according to the longitudinal distances(Fig. 4).

2.3 Offset ratio H effect (=1)r

To highlight the offset ratio H effect on the MP, CP, UVC and LVC positions, we present in Fig.5 the longitudinal velocity contours U =0, the transverse velocity contours V =0 and the dU / d Y =0 contours for different offset ratio H and for a velocity ratio =1r .

Fig. 5 U = 0, V =0 and d U / d Y =0 contours for r = 1,Re = 15000 and for different offset ratios

Fig. 6 Static pressure (1000)P contours for =1r and for two different offset ratios

The =0U and =0V contours intersection defines three characteristic points of the WOJ flow in the recirculation zone namely the merge point (MP),the upper vortex center (UVC) and the lower vortex center (LVC)[9]while the combined point is identified0 contour extremity (Fig. 5). It is clear from this figure that the MP, CP, UVC and LVC longitudinal positions increase when increasing the offset ratio H. In addition, these characteristic points move transversely away from the wall (higher Y values) when the offset ratio H increases.

This effect of the offset ratio H on the various characteristic points longitudinal positions can be explained from Fig. 6. This figure in which are plotted the static pressure contours for two different offset ratio H = 5 (Fig. 6(a)) and H = 15 (Fig. 6(b))shows that increasing H results in a decrease in the recirculation zone depression compared with the atmospheric pressure (Pminis -110×10-3and -64×10-3respectively for H = 5 and H =15). This results in lower deviation of the UOJ toward the LWJ and consequently towards the horizontal wall for H = 15 compared to H =5. Therefore, the various characteristic points (MP, CP, UVC and LVC) are less attracted toward the nozzle plate (defined at X =0)and less deflected toward the horizontal wall (defined at Y =0).

Figure 7 shows the variation of the merge point(Fig. 7(a)) and combined point (Fig. 7(b)) longitudinal position as a function of the offset ratio H for a velocity ratio =1r . The experimental and numerical results for the WOJ flow[1,3,9]and for two parallel jets flow[10-14]are used for comparison. As already noticed in Fig. 5, the Fig. 7 also shows that increasing H results in an increase inpandaccording to linear functions defined for =1r and for 5≤≤H 15 by the following equations:

Equation (17) shown in Fig. 7(a) and which predictmpX is consistent with the results of Wang and Tan[1], Kumar and Das[3], Kumar[9], Nasr and Lai[10], Durve et al.[12], Hnaien et al.[13]and Wang et al.[14]while certain difference is noticed when comparing with the study of Anderson and Spall[11]forpredicted by Eq. (18) (Fig. 7(b)),our numerical results are in good agreement with those of Anderson and Spall[11], Kumar and Das[3],Kumar[9], Durve et al.[12], Hnaien et al.[13]while they differ from other studies[10,1,14]. The gap observed between the numerical simulation and experimental results are probably due to two reasons: (1) In addition to the experimental techniques variety for each study,the initial and emissions conditions adopted are not completely identical. (2) The geometric configurations of the WOJ flow and that of two parallel jets flow free parallel jet (FPJ) are not identical. Indeed, in the WOJ configuration, the LWJ is a wall jet that develops along a horizontal wall located at =0y ,while in the FPJ configuration, the wall jet becomes a free jet (absence of the horizontal wall).

The evolution of the merge and combined points transverse positions with respect to the offset ratio H is shown in Fig. 8 for a velocity ratio =1r . It is noted thatandincrease linearly when increasing H following respectively these equations:

Fig. 8 Transverse positions of merge and combined points with respect to H for =1r and for FPJ and WOJ flows

Figure 9 shows the U = 0, V =0 andcontours for an offset ratio H = 9 and for different velocity ratios.

The analysis of these figures shows that when increasing the velocity ratio, the MP, CP, UVC and LVC approach the nozzle plate defined at =0X while deviating toward the strong jet (LWJ) and therefore toward the wall defined at =0Y . This remark is in good agreement with our previous work[13], with that of Elbanna and Sabbaght[15], Durve et al.[12]in the case of two parallel jets flow (FPJ).These authors noted that when increasing the velocity ratio, the lower jet (with lower outlet velocity) is attracted toward the strong jet (with higher outlet velocity). The flow pattern presented by the stream functions is also showed in Fig. 9 for the same offset ratio =9H and for a velocity ratio =1r . These figures clearly show the two counter rotated vortices

2.4 Velocity ratio effect

Fig. 9 U = 0, V =0 and dU / dY =0 contours and streamline pattern for H = 9 and for different velocity ratios

(UVC and LVC) for the different considered velocity ratios.

The deviation phenomenon of the flow characteristic points (MP, CP, UVC and LVC) can be explained from the static pressure contours for =9H and for two different velocity ratios (see Fig. 10). The pressure reduction rate produced by the LWJ or the UOJ depends on the amount of the entrained fluid which in turn depends on the outlet velocity. For high velocity ratios, the jet with lower outlet velocity (UOJ)produces low pressure reduction rate compared with that produced by the jet with high outlet velocity(LWJ).

Fig. 10 Static pressure (1000 )P contours for =9H and for two different velocity ratios

Therefore, as shown in Fig. 10(b), for a high velocity ratio, the pressure at the UOJ peripheryis greater than that at the LWJ peripherywhereas when r=1 (Fig. 10(a)), the two jets have almost the same pressures at their peripheries (69- ×Therefore, when the velocity ratio increases, the lower jet (UOJ) with higher pressure at its periphery is deflected toward the strong jet (LWJ) with lower pressure at its periphery.

Figure 9 also shows that increasing the velocity ratio causes a reduction of the merging zone length(this length is defined as the difference between the merge and combined points longitudinal positions).This is obvious since by increasing r, the UOJ outlet velocity decreases while keeping constant that of the LWJ, therefore the UOJ mass flow decreases and rapidly mixed with that of the LWJ. It also seems important to note that the UVC and LVC longitudinal positions decrease and approach the same location when elevating the velocity ratio. Indeed, for =2r ,the UVC and the LVC have almost the same longitudinal position

2.4.1 Merge and combined points

Figure 11 shows the merge point (Fig. 11(a)) and the combined point (Fig. 11(b)) longitudinal positions as a function of the offset ratio H for different velocity ratio r.

Fig. 11 Longitudinal positions of the merge and combined points

This figure shows that increasing H leads to an increase inandfollowing polynomial functions for different velocity ratios. For fixed value of H,anddecrease for higher value of velocity ratio (Fig. 11). Indeed, for >1r , the MP and CP move further upstream along the longitudinal direction indicating an acceleration in the merging process launch. This figure shows also that the slope ofandevolution as a function of H decreases as the velocity ratio increases. This means that the H effect on the MP and CP longitudinal positions is less noticed for high velocity ratios. This can be explained by the fact that the transverse velocity amplitude in >1r case is greater than that in the =1r case. On the other hand, when r is increased from 1 to 2,decreases by 23%, 32.1%and 41.5% respectively for =5H , =7H and(Fig. 11(a)), likewisedecreases by 49.8%, 50.5% and 59.1% (Fig. 11(b)). We can then conclude that the velocity ratio effect on the MP and CP longitudinal positions is more accentuated for higher offset ratios H. This can be explained by the fact that the MP and CP upstream displacement in the longitudinal direction when increasing the velocity ratio are damped by the shearing effect of the horizontal wall. Or from Fig. 5, we note that increasing the offset ratio H result that the MP and CP transversally move away from the horizontal wall which weakened the wall shearing effect on these points. Indeed, when the offset ratio H increases from 5 to 15, the MP and CP transversally deviate from the wall respectively by 41.6% and 64.6%. Thus for elevated values of H and r, the merge and combined points displacement becomes less damped by the horizontal wall shearing effect.

Fig. 12 Scatter plots relative to longitudinal positions of the merge and combined points

Knowing the merge and combined points positions is essential to have a complete idea about the lunching and interruption of the mixing process in the WOJ flow. These positions are strongly affected by the offset ratio H and the velocity ratio r.Therefore, it is desirable to determine an estimating method of the merge and combined points positions based on these important parameters (Hand r).

In this regard, we proposed empirical correlations that predict the longitudinal positions of these points.Fig. 12 present the scatter plots relating tocurves as a function of Ln()Hfor an offset ratio H between 5 and 15 and a velocity ratio r between 1 and 2, the scatter plots may be represented by the following linear functions:

Equations (21), (22) allow us to deduce correlations that attach the MP and CP longitudinal positions to H and r. These correlations are as follows:

Figures 14(a), 14(b) respectively show the merge and combined point’s transverse evolution for 515H≤≤ and 12r≤≤. These figures reveal thatandincrease when increasing H for the various considered values of velocity ratios. It is clear from Fig. 14(a) that thecurve slope as a function of H slightly decreases when r increases, which means that the offset ratio H effect on theevolution is slightly less observed for high velocity ratios. For fix value of the offset ratio H, the increase in the velocity ratio results in adecrease,therefore, the merge point is laterally deflected towards the strong jet (LWJ) and subsequently towards the horizontal wall. When the velocity ratio increases from 1 to 2, the merge point undergoes a lateral deviation of 8.6% and 12.2% respectively for H =5 and H = 15. Therefore the effect of the velocity ratio inmpY is sparsely influenced by the offset ratio H. For an offset ratio =9H , when the velocity ratio r varied from 1 to 2, the merge point is deviated laterally by 13.1%. On the other hand, in the numerical study of Durve et al.[12]on two parallel jets flow (FPJ) and respecting the same Cartesian coordinates origin adopted in this present study, the merge point is deviated by 43% for the same velocity ratio r interval (from 1 to 2). Then we can conclude that the velocity ratio effect is more pronounced in FPJ flow compared to WOJ flow. This is obvious since the horizontal wall in the WOJ flow partially damp the lateral deviation of the merge point toward the strong jet (LWJ).

Fig. 13 longitudinal positions of the merge and combined points compared to correlations (23) and (24)

For 1.5r≤ , the combined point transverse positionincreases linearly as a function of H with an increase rate which is independent of the velocity ratio r (Fig. 14(b)). For fixed offset ratio H, when the velocity ratio increases, the combined point transverse position slightly decreases. Indeed, for r≤ 1 .5, the Ycpcurves for different r may be considerate similar and represented by a single linear equation

For >1.5r , the combined point transverse positiondecreases as r increases for <11H ,which mean that the merge point slightly deviate toward the strong jet (LWJ). While for 11H= , the transverse position of the CP is the same for >1.5r ,while for >11H ,increases when increasing the velocity ratio. Thus, the combined point transversely deviates this time toward the lower jet (UOJ).

Fig. 14 Transverse positions of the merge and combined points

From Eq. (26), we can deduce the following correlation which attaches the merge point transverse position to H and r

Fig. 15 Scatter plots relative to transverse position of the merge point and comparison with correlation (27)

2.4.2 Vortices centers

In WOJ flow, two vortices with opposite directions are observed in the recirculation zone: an upper vortex (UVC) in the side of the UOJ and another lower (LVC) in the side of the LWJ, these vortices(Fig. 9) are responsible for the UOJ deviation toward the LWJ and consequently toward the horizontal wall.

In Fig. 16, we plot the curves which represent the UVC (Fig. 16(a)) and LVC (Fig. 16(b)) longitudinal positions as a function of H for different r.

Fig. 16 Longitudinal positions of the upper and lower vortex centers

The figure analysis shows that increasing the offset ratio H result in longitudinal displacement of the UVC and LVC further downstream. In addition,the H effect on UVC and LVC longitudinal positions is less observed for high velocity ratios. This results a reduction in the(Fig. 16(a)) and(Fig. 16(b)) increase rate for higher value of velocity ratio r. For =5H , we note that the UVC and LVC longitudinal positions are almost equal and independent of the velocity ratio. This observation is in good agreement with Kumar[9]who found thatandare almost the same for low offset ratio H. For H >5, the increase in velocity ratio causes longitudinal displacement of the UVC and LVC further upstream (anddecrease). The same figure shows that an increase in the velocity ratio r from 1 to 2 lead to a longitudinal displacement of the UVC further upstream by 40.6% and 49.9%respectively for H=9 and H=15. On the other hand,the LVC positions shift further upstream by 20.1%and 33.3% in the same interval of r. These percentages allows us to conclude that the velocity ratio effect onandeis more pronounced for high offset ratio H and thatis more influenced by the velocity ratio variation compared to. This can be explained by the fact that the UVC and LVC movement is damped by two type of resistance; one is applied by the horizontal wall on the lower vortex and the other applied by the surrounding fluid on the upper vortex. The resistance exerted by the wall on the lower vortex is greater than that exerted by the surrounding fluid on the upper vortex. Therefore, the UVC is more influenced by the velocity ratio compared to the LVC.

Figures 17(a), 17(b) show respectively the UVC and LVC transverse positions with respect to H for different r.

Fig. 17 Transverse positions of the upper and lower vortex centers

It is clear from Fig. 17(a) for all velocity ratios considered that the UVC and LVC transverse positions increase linearly when increasing the offset ratio H. The same figure shows that the H effect on the UVC transverse location is independent of r,this is clear from the constant value of theuvcY curve slope regardless of the velocity ratio. On the other hand, for fixed offset ratio H, the UVC transverse position slightly decreases when increasing r.Thereafter, we can considerate that theuvcY curves as a function of H are the same for all velocity ratios considered and can be described by the flowing equation

The Fig. 17(b) analysis shows an increase in the transverse position of the LVC when H increases for all velocity ratios. The decrease in thecurve slope for high value of r indicates a weaker H effect on the LVC transverse displacement for higher values of r. For fixed offset ratio H, increasing the velocity ratio results in a reduction of the LVC transverse position which reflect a LVC deviation toward the strong jet (LWJ) and subsequently toward the horizontal wall.

3. Conclusions

The present work has numerically investigated turbulent flow in which a wall and an offset jet flow(WOJ) combined. This study is based on the finite volume method using the computational fluid dynamics software “Fluent”. The objective of this study is to examine the effect of the velocity ratio r and the offset ratio H on the dynamic characteristics of the WOJ flow. The discussion focuses mainly on the -ωk model's validity to predict the characteristics points of WOJ flow such as the merge point(MP), the combined point (CP), the upper vortex center (UVC) and the lower vortex center (LVC). The present work can be used for a quick estimate of these points' position that enormously affect the WOJ flow in different industrial applications. The results derived from this study may be summarized as follows:

(1) For a fixed velocity ratio, increasing the offset ratio H results in a longitudinal displacement of the MP, CP, UVC and LVC further downstream(,,andincrease). The H effect on the MP, CP, UVC and LVC longitudinal position is less observed for high velocity ratios.Correlation that predict the MP and CP longitudinal positions are respectively given by Eqs. (23), (24).

(2) For a fixed velocity ratio, when increasing H, the MP, CP, UVC and LVC move further away from the wall in the transverse direction (andincrease). Correlation that predicts the MP transverse position is given by Eq. (27).

(3) The H effect on the MP and LVC transverse positions is less observed when increasing the velocity ratio while the increase in the velocity ratio r has virtually no effect on theandevolution respectively for 12r≤≤ and 1 5r.≤which are given as a function of the offset ratio respectively by Eqs. (25), (28). For a velocity ratio beyond =1 5r ., the H effect onevolution increase for an offset ratio value above H = 11.

(4) For a fixed offset ratio H, increasing the velocity ratio results in a longitudinal displacement of the MP, CP, UVC and LVC further upstream in the longitudinal direction (,,anddecrease. This decrease is more observed for higher offset ratio H.

(5) For a fixed offset ratio H, increasing the velocity ratio produces a transverse deviation of the MP and LVC toward the strong jet (LWJ) and subsequently toward the horizontal wall. This deviation is more intense for higher offset ratio H.On the other hand, the UVC transverse position is almost independent of the velocity ratio (slightly decreases as r increases). For the CP, its transverse position is also independent of the velocity ratio only when 1 5r.≤ (a slight decrease is noted as r increases). Beyond =1 5r . and for an offset ratio more than =11H , increasing the velocity ratio causes a deviation of the CP this time towards the lower jet (UOJ).

(6) The velocity ratio effect on the MP transverse position is more observed in the FPJ flow compared to the WOJ flow since the horizontal wall shearing effect in the WOJ flow partially damp the transverse displacement of the MP when changing the velocity ratio.

(7) The UVC longitudinal displacement is more affected by the velocity ratio variation compared with that of LVC because of the higher resistance applied by the horizontal wall on the lower vortex compared with that applied by the surrounding fluid on the upper vortex.

Acknowledgement

The helpful comments and suggestions of the reviewers are gratefully acknowledged by the authors.