1 Introduction

“Jets in crossflow” are considered as a common and frequently present configuration in several domains and applications. In accordance with the aim of the handled application, whether single, twin or multiple jets are introduced within the transverse flow. This configuration is typically found in heat transfer applications where the jets are emitted normally or obliquely to a wall that we want to maintain at a proper temperature during high temperature thermal processes. This function is found in combustor wall cooling and in the cooling of gas turbine blades where more than a jet is used to enhance the performance of each individual jet further downstream of its injection point. Other uses for jets in crossflow are evident in lifting jets which provide an aircraft with V/STOL (vertical or short take-off and landing) capabilities. In this example, the efflux from the lifting jets interferes with the mainstream around the aircraft and significantly alters its aerodynamic behavior. We also cross such a configuration in smoke stacks exhausting from industrial chimneys; here the aim is to track the dispersal of the emitted pollutants in order to optimize it by reducing its drawbacks on the atmosphere. Similarly, we find this type of jets in the dispersion of liquid effluents into streams and rivers, which is relevant to water pollution. All these examples may require, as mentioned, more than a jet in crossflow, in order to enhance the efficiency of the handled application. In the literature several works treated the single and multiple jets in crossflow configuration. Nevertheless, a recent interest is raised by the intermediate twin jet configuration even if the consideration of this special case is not new in itself since it goes back to the seventies when it was first studied by Ziegler and Wooler [1] by means of a physical elaborated model. Through that work, an explanation could be given to the deflection of the emitted jets in terms of pressure gradient and entrainment and the role of the presence of each jet on the other was dynamically specified. In most of the available works, different orientations of the configuration are evaluated (tandem, side by side or one in front of another) and compared or not either one to another or to other configurations with a different number of emitted jets (single and/or multiple jets).

Herein, we have to mention that too little work was exclusively devoted to the inline tandem jets in crossflow. Ohanian and Rahai [2] are, to our knowledge, the only authors having accomplished that. Their work consisted in a numerical modeling of two turbulent planer jets with variable velocity ratios (0.5, 1.0, and 2.0), space between the jets’ nozzles (d, 2d, and 3d) and imposed kinetic energies in order to evaluate the impact of these parameters on the mixing and diffusion processes that take place within the resulting flow field.

Other authors have dealt with the inline twin jets in crossflow’s configuration by comparing its main features with the ones relative to other arrangement cases like Isaac and Schetz [3]. In fact, the latters have theoretically analyzed differently arranged twin jets emitted under a different inclination angle: 60° and 90° for tandem jets and 90° for the side by side arrangement. This work elaborated by means of physical modeling and a momentum integral method could specify the influence of the presence of each jet on the other and the role of the jets’ spread on their behavior. That influence was also highlighted under the variation of a further parameter: the jets’ nozzles’ distance

Kolar et al. [4–7] considered experimentally the same orientations by means of the standard crossed hot-wire anemometry technique in order to characterize the vorticity distribution and the turbulent vorticity processes in a try to understand their mechanism and origin. In some of these same papers, an analogy was drawn between the twin differently arranged jets behavior and the one relative to the single jet configuration. That allowed for example to compare the size of the developed counter rotating vortex pair (CVP) in both cases or even to compare the whole elaborated vorticity field. DiMicco et al. [8] proceeded the same but in order to compare the jets’ deflection and penetration and the effect the variation of the jets’ spacing could bring to them which led to an analogy between the single and the double jet configurations in crossflow.

Further authors, however, did not only compare the twin jets’ results with available data relative to the single jet configuration; they proceeded rather to the establishment of the results relatively to both configurations. The paper of Makihata and Miyai [9] enters within this context since it considered by means of a finite difference method a twin and a single jet in crossflow at various velocity ratios (between 1.2 and 10.6), initially inclined with an angle of 45° and under a nozzles interval ratio of 0.8. The main purpose of the authors was to track the developing vortices during their transition from a circular to a kidney shaped structure. Toy et al. [10] examined experimentally twin jets normally emitted within a crossflow by means of the flow visualization and the video digital imaging technique in order to evaluate the extent of the mixing region under a velocity ratio of 8. A further experimental exploration [11] of the same configurations was carried out by means of a real-time, quantitative, video image analysis in order to track the overall jet growth rates, the widths of the interface regions, and the spectra of the interface fluctuations. It was also question of testing the influence of the velocity ratio (from 4 to 10), the nozzles’ spacing (from 1 to 5 diameters) and the nozzles’ orientation relatively to the crossflow on the different characterizing the vortical flow features and particularly the vorticity field in the different configurations. A similar confrontation was carried out by Khan [12] in order to evaluate the scalar mixing of the resulting flowfield under an injection ratio equivalent to 6. This paper particularly examined the mean concentration and the fluctuation intensities of the emitted jets under the different geometrical conditions. It also compared the jets’ trajectories, the mean concentration decay and the spreading rates in all cases. The paper of Ibrahim and Gutmark [13] and the master’s research of Ibrahim [14] are the most recent works concerning the comparison of the single and double jets in crossflow. Both of them consisted in particle image velocimetry (PIV) experiments carried out in order to evaluate the blowing ratio impact on the jets’ dispersion. 3.2, 4.8 and 8 were the values tested for the single jet and 3 the one adopted for the twin jets. Several parameters were examined in order to compare the twin configuration behaviors such as the jets’ trajectory and penetration, the deflection of the jets’ trajectory, the jets’ mass entrainment, the windward and leeward jets’ spread, the size, location and magnitude of the reverse flow region and the turbulent kinetic energy (TKE), etc.

Other papers were dedicated to the comparison of double and multiples jets in crossflow. The work of Ziegler and Wooler [15] belongs to this category of papers since it could examine the internal mixing taking place within the jet cores and the induced blockage effects in the twin configuration cases. Different parameters were varied such as the stratification applied to the emitted jet and then calculations were made on the jet centerlines and on the induced surface static pressures.

As to Isaac and Jakubowski [16], they considered all the possible configurations (single, double and multiple jets in crossflow) where the jets were emitted normally to the oncoming crossflow. The experimental data tracked in a wind tunnel by means of the hot-wire anemometry showed some flow similarities (velocity and turbulence parameters) between the single and twin jets in the region where x/d > 10. This similarity also exists between the single and the multiple jets cases and could be observed on the turbulence mechanisms, the profiles of the mean velocities, and the profiles of Reynolds stresses. Xiao [17] considered the same geometries in a confined crossflow with constant boundary conditions for the jets and the duct flow. In the double and triple jets system, the center-to-center distances between the jets were 0.68 and 0.20 m, respectively. The blockage effect, the velocity decay and the recirculation of the mainstream were evaluated in all configurations and for the different spacing tested.

We see then that too few works were exclusively devoted to the tandem double jets in crossflow, that the most examined parameters in this configuration (in the others too) are the jets’ velocity ratio and the jets’ spacing and that the most studied features are the velocity and vorticity fields. Further parameters may be explored like the height of the jets’ nozzles, the geometry of the jets’ cross-section, the initial jets’ streamwise inclination, etc., the latter is precisely the parameter we are going to evaluate in this paper due to its great impact on the resulting flowfield and on the global expansion and mixing of the jets within the environing flow. We will particularly stress on the impact of the initial inclination on the pollutants’ dispersion and on the mixing process within the oncoming crossflow due to the relevance of the issue and its direct consequences on the atmosphere and then on the environment. For the matter numerical simulations based on the finite volume method were conducted under the different inclination cases. The adopted geometric and dynamic conditions are deduced from the experimental replica on which we could follow the different dynamic features by means of the PIV. The experiments have also provided us with reference data to validate the simulated model. Two different turbulent closure models were tested: the classic standard k–ε model and the Reynolds stress model (RSM) second order model. Once the validation obtained and the turbulent model chosen relatively to its better fitting to experiments, we will introduce further conditions on the model : the temperature gradient and the fume emission; and determine the impact of the inclination factor on the expansion of the polluted plume and its mixing stages within the environing flow. It’s true that previous digital video visualizations [18] have already allowed the experimental examination of the mass entrainment evolution under different velocity ratios and inter jets’ nozzles distances, but not for different inclination angles.

2 Presentation of the problem

The interaction between a double jet and a cooler crossing flow is a source of a complex resulting flowfield. This complexity is expressed through the generation of an important vortical system closely dependant on all the different affecting parameters (nozzles’ spacing, injection, height, velocity rate, inclination angle, Reynolds number, nozzles arrangement, etc.). Four main vortical structures are, however, always present in the resulting flowfield; only their magnitude may vary under the variation of one of the above mentioned factors. They consist in the horseshoe vortices (HSV), the wake vortices (WV), the shear layer vortices (SLV) and the contrarotating vortex pair (CVP) (Fig. 1).

Fig. 1
figure 1

Progressive evolution of the twin jets during their emission through the oncoming crossflow and the associated diagram of vortex system

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Regardless of the different interfering parameters, a constant process takes place between the twin interacting flows; it consists in the proper evolution of each jet and then the joining of the corresponding plumes far downstream; as for the vortical structures, they may develop further deeper in the vertical or streamwise directions; it depends on the influent parameters; in our case, this parameter will consist in the injection inclination which may further off or enclose the jets’ propagation from the injection plate. In this paper, however, our aim is to evaluate the influence of this parameter but on other influent features such as the mixing process and the pollutants’ dispersion among the oncoming crossflow. This influence will be demonstrated both experimentally in a wind tunnel by means of the PIV technique and numerically by means of the finite volume method and a turbulent closure model to validate after confrontation with the measured data.

3 Experimental set-up

The experimentally considered configuration consists in two tandem jets incline with an oncoming crossflow. The twin jets have the same dimensions (d = 10 mm) and are spaced with a center-to-center distance D = 3d. The injection pipes are circular but the initial 60° inclination angle results in elliptical cross-sections with identical major (d/sinα) and minor (d) diameters. To situate the different flows and scale the different features, a Cartesian coordinate system is introduced and its origin is placed within the center of the rear jet (Fig. 2). The choice of this particular system is induced by the asymmetry of the resulting flow that develops in spite of the symmetry of the configuration [19]. Both of them are fed air from the same main air supply and are seeded with glycerin particles whose diameter is approximately equivalent to 1 μm (the seeding density ≈30 particles ml−1 of pure jet fluid). As to the main air flow, it is seeded with oil droplets of approximately 0.8 μm diameter, and is introduced into the tunnel at the ambient temperature (T ∞). The measurements are carried out in a wind tunnel whose test section allows the introduction of a velocity range of about 0–16 m/s. The experiments were performed in an open circuit discharging to the atmosphere outside the laboratory and with a parallel-sided, closed working Sect. 3 m in length and 0.2 m × 0.3 m in cross-section. For optical access, it has a side made of Pyrex; the interior of the wind tunnel was painted black to reduce the possible reflections. The turbulence intensity level of the crossflow was less than 0.2%.

Fig. 2
figure 2

Geometric configuration of the twin inclined jets among the whole domain

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The experimentally tracked values were depicted by means of the PIV technique. It has the advantage of providing instantaneous velocity fields over global domains. As its name suggests, the PIV technique records the position over time of small tracer particles introduced into the flow to extract the local fluid velocity. Thus, PIV represents a quantitative extension of the qualitative flow visualization techniques that have been practiced for several decades [20]. It is based upon a TSI PowerView system, including a 50 mJ dual YAG laser which produces two flat pulses, the duration of one ranging from 5 × 10−9 to 10−8 s, a Power View 4M high resolution cross-correlation camera (2k × 2k resolution, 12 bits), a synchronizer and “Insight” Windows-based software for acquisition, processing and post-processing (Fig. 3). This software allows the synchronization of pulsations according to the observed phenomena, and the adjustment of the time step between two images. This time step was 70 μs. In order to avoid errors, the velocity vectors were calibrated at 130 μm/pixel and limited to the representation of the velocity field in the regions where the luminance was strong enough. The final fields were averages performed with 100 successive acquisitions. For more extended details about the experimental conditions, accuracy, etc., see reference Mahjoub et al. [21] and about the principle of the technique see Prasad’s work [20].

Fig. 3
figure 3

Different required elements for the PIV technique measurements

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4 Computational set-up

The handled geometrical configuration is considered and reproduced numerically. Consideration is given to a steady, three-dimensional, incompressible and turbulent flow. The equations describing this flow are obtained in a Cartesian coordinates’ system whose origin is situated, as mentioned above, in the center of the first injection nozzle. That gives us the following equations to handle:

$$ \frac{{\partial \left({\bar{\rho}{\tilde{\it{u}}}_{\it{i}}} \right)}}{{\partial {\it{x}}_{i}}} = 0 $$
(1)
$$ \frac{{\partial \left({\bar{\rho}{\tilde{\it{u}}}_{\it{j}} {\tilde{\it{T}}}} \right)}}{{\partial {\it{x}}_{\it{j}}}} = \frac{\partial}{{\partial {\it{x}}_{\it{j}}}}\left[{\left({\frac{\mu}{\it Pr} + \frac{{\mu_{\it{t}}}}{{\sigma_{\it{t}}}}} \right)\frac{{\partial {\tilde{\it{T}}}}}{{\partial {\it{x}}_{\it{j}}}}} \right] $$
(2)
$$ \frac{{\partial \left({\bar{\rho}{\tilde{\it{u}}}_{\it{j}} {\tilde{\it{f}}}} \right)}}{{\partial {\it{x}}_{\it{j}}}} = \frac{\partial}{{\partial {\it{x}}_{\it{j}}}}\left[{\left({\frac{\mu}{\it{Sc}} + \frac{{\mu_{\it{t}}}}{{\sigma_{\it{t}}}}} \right)\frac{{\partial {\tilde{\it{f}}}}}{{\partial {\it{x}}_{\it{j}}}}} \right] $$
(3)
$$ \frac{{\partial \left({\bar{\rho}{\tilde{\it{u}}}_{\it{j}} {\tilde{\it{u}}}_{\it{j}}} \right)}}{{\partial {\it{x}}_{\it{j}}}} = \frac{{\partial {\bar{\it{p}}}}}{{\partial {\it{x}}_{\it{i}}}} + \frac{\partial}{{\partial {\it{x}}_{\it{j}}}}{\left({\mu \frac{{\partial {\tilde{\it{u}}}_{\it{i}}}}{{\partial {\it{x}}_{\it{j}}}} - \overline{{\rho {\it{u}}_{\it{i}}^{\prime\prime} {\it{u}}_{\it{j}}^{\prime\prime}}}} \right) + \left({\bar{\rho}_{\infty} - \bar{\rho}} \right){\it{g}}\delta_{\it{ij}}} $$
(4)

The introduction of the fluctuating functions and variables requires the use of a turbulence closure model. In the present work, we tested the k–ε standard turbulent model and the RSM second-order model. Whereas the first one is classically adopted for the modeling of the turbulent flows, the second presents the advantage of detailing the three-dimensional aspect of complex flows by computing the destruction of the turbulence through the use of the dissipation rate of the TKE. It is also able of considering separately each Reynolds stress term and its corresponding production which allows a better description of their corresponding anisotropy.

The introduction of the second order turbulence model leads to the resolution of the following equation:

$$ \underbrace{\frac{\partial}{\partial x_{\it k}}\left(\bar{\rho}\tilde{u}_{\it k} \overline{{u_{\it i}}^{\prime\prime}{{u}_{\it j}^{\prime\prime}}}\right)}_{{\it c}_{\it ij}}= \underbrace{\frac{\partial}{\partial {x}_{\it k}}\mu\frac{\partial}{\partial {x}_{\it k}}\left(\overline{{u}_{\it i}^{\prime\prime} {u}_{\it j}^{\prime\prime}}\right)}_{{\it D}_{\it ij}^{\it L}}- \underbrace{\bar{\rho}\left[\overline{{u}_{\it i}^{\prime\prime}{u}_{\it k}^{\prime\prime}}\frac{\partial\tilde{{u}_{\it j}}}{\partial {x}_{\it k}}+\overline{{u}_{\it j}^{\prime\prime}{u}_{\it k}^{\prime\prime}}\frac{\partial\tilde{{u}_{\it i}}}{\partial {x}_{\it k}}\right]}_{{\it P}_{\it ij}}+{{D}_{\it ij}^{\it T}}+{G}_{\it ij}+\phi_{\it ij}+\varepsilon_{\it ij} $$
(5)

C ij being the convective term and D L ij , P ij , D T ij , G ij , ϕ ij , ε ij , respectively, the molecular diffusion, the stress production, the turbulent diffusion, the buoyancy production, the pressure strain and the dissipation rate [22].

The equation of the TKE (k) and that of the dissipation rate of the kinetic energy (ε) which are relative to the second-order model are defined as follows:

$$ \frac{{\partial \left({\bar{\rho}{\tilde{\it{u}}}_{\it{j}} {\it{k}}} \right)}}{{\partial {\it{x}}_{\it{j}}}} = \frac{\partial}{{\partial {\it{x}}_{\it{j}}}}\left[{\left({\mu + \frac{{\mu_{\it{t}}}}{{\sigma_{\it{k}}}}} \right)\frac{{\partial {k}}}{{\partial {\it{x}}_{{\it{j}}}}}} \right] + \frac{1}{2}({\it{p}}_{\it{ii}} + {\it{G}}_{\it{ii}}) - \bar{\rho}\varepsilon $$
(6)
$$ \frac{{\partial \left({\bar{\rho}{\tilde{\it{u}}}_{\it{j}} \varepsilon} \right)}}{{\partial {\it{x}}_{\it{j}}}} = \frac{\partial}{{\partial {\it{x}}_{\it{j}}}}\left[{\left({\mu + \frac{{\mu_{\it{t}}}}{{\sigma_{\varepsilon}}}} \right)\frac{\partial \varepsilon}{{\partial {\it{x}}_{\it{j}}}}} \right] + {\it{C}}_{{\varepsilon {\text{1}}}} \frac{1}{2}\left({{\it{p}}_{\it{ii}} + {\it{C}}_{\varepsilon 3} {\it{G}}_{\it{ii}}} \right)\frac{\varepsilon}{\it k} - {\it{C}}_{\varepsilon 2} \bar{\rho}\frac{{\varepsilon^{2}}}{\it{k}} $$
(7)

For more information concerning the constants introduced in the different equations see Mahjoub et al. [23].

The resolution of the last equations’ system raises several problems when simulating numerically the problem since the topology of the flow requires a very fine meshing in a great part of the domain. In order to describe exactly any temperature or mass fraction variations, particularly near and immediately downstream of the jets, we adopted a non-uniform meshing, particularly tightened near the injection nozzles and between them.

The variation of the mesh size is described by the following relation: x i+1 = α x x i , where α x is the extension coefficient that adopts a different value relatively to the jets’ locations and top the directions. Table 1 summarizes the different adopted values within the different parts of the domain.

Table 1 Grid coefficients in the different locations of the domain
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The symmetry existing in the lateral direction is numerically reproduced on its corresponding grid. This is possible with the adoption of the extension rate (1.05) while leaving the central position (z = 0). In this way the obtained grid (Fig. 4) allows refining the calculations of the different variables around the jets’ nozzles and better capturing the slightest variations of the resulting flowfield.

Fig. 4
figure 4

Schematic presentation of the grid at the symmetry plane

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For all directions, the number of points is chosen so that the boundary conditions far from the interaction are satisfied. The adopted discretization in our case results in a total 131 × 43 × 41 grid number, respectively, in the x, y, and z directions. Herein we have to mention that this grid was chosen after the trial and the execution of much coarser grids.

The discretized equations are solved by means of a finite volume method. The numerical resolution uses the Patankar and Spalding’s algorithm “SIMPLE” [24] for the correction of the pressure and the convergence of the calculations is obtained when the sum of the normalized residues attains 10−3.

5 Results and discussion

Before detailing any of the obtained results, we need to choose a valid turbulence closure model for our simulation. For that, we propose to test both of the abovementioned models (standard k–ε and RSM) and confront their corresponding results to the experimentally measured data. The confrontation will be executed relatively to the longitudinal and the vertical velocity components. The variation of the latters was represented in the symmetry plane (z = 0) and more precisely within the first jet nozzle (x = 0). To reproduce as much as possible the experimental conditions, we adopted the same conditions; that is to say a velocity ratio of R = 1.29, a spacing of D = 3d between the jet nozzles and an initial inclination of the jets equivalent to α = 60°.

The confrontation of the calculated and measured results relatively to the longitudinal velocity component (Ũ) (Fig. 5a) gives a satisfying matching along all the variation of the feature. The slight difference we note in the vicinity of y = 20 mm may be due to the transition from the first jet plume to the surrounding transverse flow. At this location, the results processed with the RSM model are closer to the experimental data which supposes that the second order model is more adequate for the description of the experiments. Just downstream, the results coincide again which comforts the obtained validation.

Fig. 5
figure 5

Validation of the longitudinal and vertical velocity components in the symmetry plane (z = 0), within the upstream jet (x = 0 mm)

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In Fig. 5b, we proceeded the same with the vertical velocity component (\( \tilde{V} \)) by superposing the experimental and the numerical results. Here, we do not obtain a total matching like with the longitudinal velocity whatever the turbulent model is but they share the same trend. Just downstream of the jets’ exit, we note a slight lack of experimental points that extends approximately over 3 mm. That may originate from a non-uniformity in the jet seeding; even if the latter is regulated by a pumping system. Elsewhere we have a satisfying matching that enables us to presume a qualitatively acceptable agreement.

According to the above discussed confrontations of the velocity components, and in accordance with the better description of the experiments with the RSM model \( (\tilde{U}) \), we can then conclude that our numerical modeling is acceptable and models well the experimental configuration of the twin inclined jets issuing within an oncoming crossflow.

The next step consists in the introduction of a temperature gradient between the two flows (ΔT = 100 K). We are also going to inject a non-reacting fume mixed with air within the jet nozzles. These modifications will approach us to real and more concrete situations such as the discharge of effluents from chimney stacks or boat chimneys. These modifications will also allow us to consider the progression of the emitted fume: its evolution, its scaling in the different directions, the induced phenomena, its mixing process within the crossflow, etc.

The previously adopted conditions and the newly added ones are summarized in Table 2.

Table 2 Boundary conditions and composition of the introduced non-reacting fume
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These conditions will remain the same for the rest of the paper, apart from the injection ratio that will be taken R = 2; that’s simply an arbitrary choice that will allow us to track the development of each jet separately, their combination and finally the evolution of the combined plume. The second variation to introduce is that of the initial inclination of the twin emitted jets, and that’s the principle issue of this paper.

5.1 Thermal field

When handling a given application, we are generally confronted with the following dilemma on one hand we aim at reaching increasing temperatures and obtaining higher efficiencies either for cooling or heating whereas on the other hand, we want to control the temperature of the discharged effluents. This limitation is dictated by the environmental regulation laws since high temperature effluents would interact with the atmosphere and generate several drawbacks. It is also imposed by the destruction of the handled materials if we go beyond its critical temperature (its fusion temperature). These multiple requirements on the temperature show how the thermal exchanges acquire an essential importance and how the temperature limitation is primordial for the viability of the handled applications. In a trial to understand the different interfering mechanisms, we propose to dedicate the first part of this paper to the examination of the temperature distribution on the different directions, its evolution stages and its variation under the change of the initial streamwise inclination factor. To begin, we will consider the vertical distribution of the thermal field within the different streamwise locations of the domain (Fig. 6). The definition of these locations is only based upon the jet nozzles’ emplacement and does not depend on any other parameter which engenders the dividing of the processing domain into four main regions: the first corresponds to the rear jet, the second is contained within the twin jet columns, the third corresponds to the downstream jet and the last one covers the region that follows and that ends by the end of the domain. Within the first zone, we note that the inclination augmentation does not affect the global behavior of the temperature, we only note a delay of the temperature decrease and then of the homogenization of the resulting flowfield. This is due to the fact that a weak inclination doesn’t let the jets’ plumes expand far from the injection plate; they are rather trapped near it. So when we flee the plaque (for increasing y), we immediately go out from the high temperature zone (the jet’s core) to finally join the surrounding flow that stayed outside the interaction zone and where the temperature remained constant (T ∞). Beyond of the upstream jet, we progressively detect the presence of a peak that decreases as the jets are straightened. In the first injection case, however, we do not find a distinct maximum. In fact, since the jets are immediately flattened against the injection plaque, we directly start with the jet’s temperature. Whereas when they are straightened, their core develops progressively farther from the injection plate which justifies the delay of the accused peak. We also note a progressive widening of the temperature’s profile and a reduction of the decline slope. This is due to the vertical expansion of the jets’ plumes that is generated by the inclination rise. For the highest inclination case, we even find a temperature stage before its augmentation. This leads of course to a slower homogenization of the flowfield that is expressed by a farther homogenization position (y = 12, 17, 22 and finally 33 mm).

Fig. 6
figure 6

Vertical evolution of the temperature over the different streamwise locations of the domain and in the symmetry plane (z = 0)

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Within the second jet location, we no longer find a single peak; but two. The first corresponds to the crossing of the extension of the rear jet whereas the second is accused when passing through the second jet plume. Under the weakest injection case, the twin peaks are very close and attain the highest values (with reference to the other inclination cases) because of the jets’ trapping near the injection plaque and then the concentration of their “strength”. As the injection angle grows, the jets expand rather vertically before being tilted by the oncoming crossflow; meanwhile they lose some of their magnitude resulting in the weakening of the second peak; it’s so weak that vanishes in the last case where the inclination angle reaches its maximum (α = 90°). Of course the widening of the profiles is always present as the jets are straightened.

Downstream of the twin jets (Fig. 6d), we find back the same thermal behavior as between the jets’ nozzles, with, however, much lower attained temperatures and much wider profiles. In fact, in this location the jets have already lost an important part of their initial “strength” (temperature amplitude). We see then that the straightening of the injected jets enables them to go deep higher in the domain before being tilted which implies a deeper vertical diffusion of the their heat.

The consideration of the vertical variation of the temperature in the different streamwise locations showed then that the impact of the initial inclination angle does not appear directly since too little effect is noticed on the evolution of the temperature within the rear jet location. This impact begins really to be detected forward and is expressed by a later reaching of decreasing peaks and by wider temperature profiles as the inclination angle increases. That concretely signifies a higher expansion of the jets’ cores within the processing domain, a further vertical diffusion and a later homogenization (of the temperature) of the resulting flowfield.

Figure 7 comforts these results since it shows the higher expansion of both jets as the initial inclination factor is amplified. It also demonstrates the further expansion of the second jet relatively to the rear one whatever the inclination factor is. This is actually due to the blockage of the oncoming crossflow by the upstream jet which consequently shields the following one from a similar tilting. As a consequence, the initial “impulse” of the downstream jet is maintained longer and allows it to cross-deeper the neighboring flow. Finally we note that the variation of the inclination factor brings a decreasing effect since when moving from 30° to 45°, the expansion of the thermal potential core of both jets does not increase with the same trend on the symmetry plane.

Fig. 7
figure 7

Inclination impact on the trajectory of the potential cores of each of the twin emitted jets in the symmetry plan (z = 0)

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The examination of the combination of the twin emitted jets and the tracking of the different evolution stages of the jets and of their resulting plume, expressed here through their thermal potential core, is the following step of our consideration of the temperature field. For the matter, we propose to plot the temperature variation on two longitudinal positions relatively to the jet nozzles’ locations and on different vertical levels (y planes) (Fig. 8). The first plane is located at y = 2 mm, just downstream of the nozzles’ exit. The second plane is an intermediate one situated at y = 5 mm and the last one is situated at y = 8 mm. We did not go farther because beyond of this plane some profiles remain unchanged and then do bring no interest. This happens when the oncoming crossflow remains out of the interaction zone because of the limited expansion of the jets (under weak inclination angles) or of the fast or strong tilting imposed by the oncoming crossflow. In that zone the temperature remains equal to the crossflow’s uniform temperature T ∞ = 298 K.

Fig. 8
figure 8

Impact of the streamwise inclination on the lateral variation of the temperature within the twin jet nozzles and on different vertical levels

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Immediately downstream of the twin jet nozzles, there is no interesting difference on the temperature profiles: they are similar regardless of the inclination variation; the only slight discrepancy is observed in the case of α = 30° within the second jet, where the ascension and declination of the temperature happens a little bit slower. This is due to the rear jets’ plume that was rapidly tilted and then has rapidly joined the second nozzle and its corresponding plume; in this way we obtain a combined jet that is characterized by more pronounced dimensions (vertical and lateral). We also have to take into account the shielding effect brought by the upstream jet which allows a longer “conservation” of the heat contained within the following one and then prevents it from a fast decrease. In spite of this slight difference there is no significant variation in this plane (y = 2 mm) whereas in the following planes it becomes more evident and even very pronounced in some cases. In fact, it’s just downstream of the jet nozzles, that we are too close to feel the inclination impact; whereas when we flee the injection plate and then the nozzles’ exit (increasing y), the jets have enough time to evolve, to be affected by their initial inclination and to be tilted by the oncoming crossing flow. Here we insist particularly on the inclination impact that we clearly note in Fig. 8b and c. In fact, as the jets are straightened, they expand higher and do not lose much of heir amplitude; this is expressed by the high temperature peaks that we register along this feature’s vertical variation (increasing y). This same character is maintained within the second jet nozzle (x = 30 mm) with, however, some variations: the profiles are larger in the base and the registered peaks are higher for the weakest inclinations. In fact, under a weak transverse injection, the jets are immediately tilted by the crossflow and consequently flattened, leading to a larger base in the temperature profile. When this parameter grows, however, the jets have a higher impulse to cross the neighboring flow. As for the increase of the peak values, it is due to the combination of the twin jets’ plumes; the latter results in the reinforcing of the temperature plotted in the second jet location. This remark is comforted by the high discrepancy we detect within the highest plane (y = 8 mm) between what happens in the two jet locations. In fact, while the temperature profile relative to the weakest inclination angle has vanished in the first jet location, it became the most prominent in the downstream one. Globally speaking, we note that the temperature is maintained at a certain level (between 390 and 410 K) in the downstream position whereas it accuses an important decline in the upstream jet. In the second jet region, the inclination seems having lost its impact on the temperature evolution, but this is wrong. In fact, herein we are in presence of not only a single jet but the combination of two jets: one has just been emitted and the second has joined it after having crossed the neighboring flow and then after having lost some of its strength (temperature amplitude). Since a weak inclination means keeping the jets near the injection plate, so in the vicinity of the second jet we are in presence of a higher temperature amplitude than under the other inclination cases since we detect the entire “strength” of the second jet and the maximum that can come from the first one.

When we are located between the jet nozzles (Fig. 9I), the flow’s temperature adopts a behavior that differs at the three considered planes. In fact, although we detect a Gaussian temperature’s profile in the initial inclination case in approximately all the planes, the evolution of this behavior changes with the inclination and with the considered vertical position. In the closest plane for example (y = 2 mm), the temperature does no longer reach a single peak; but two. These maxima are first more distinct (α = 60°) but they tend to vanish under the highest inclination case (α = 90°) (Fig. 9Ia). The straightening of the jets has actually resulted in the generation of a trapped flow that is contained between the jets’ columns and which is progressively increasing. As we can see on the cartographies represented in Fig. 10, it’s like we have a furrow between the jets that is in fact responsible for the creation of the twin maxima and for the decline contained on their both sides. In the last inclination case, however, and when we are too close to the injection plate (y = 2 mm) we are literally out of the jets’ plume since that have been sufficiently fast and far before being tilted or flattened. As a consequence, we find back a single peak whose amplitude is very low (≈320 K). On the following plane (y = 5 mm), we note a more equilibrate organization of the profiles since they are approximately similarly distanced to one another and attain progressively declining peaks (especially between the jets). globally there is no significant change in the temperature behavior apart from the later apparition of the furrow’s impact (α = 60° rather than 45°) and the disappearing of the fading with the higher inclination angle (α = 90°): in this plane (y = 5 mm) we are a little farther from the injection plate and then, in almost all the injection cases, the jets have either been bent or began to do it; that’s why we detect higher temperatures between the jets’ columns. On the last plane (y = 8 mm), the global behavior is maintained even if the temperature profiles are much closer. Nevertheless, the highest variation of the temperature is no longer relative to the weakest inclination case; it’s rather relative to α = 45°. That means that the jets emitted at α = 30° are not provided with a sufficient amplitude to reach the plane y = 8 mm. In other words, the impulse given by this inclination angle begins to vanish at this level whereas the other inclinations are more sensed. That’s proved by the augmentation of the peaks registered at α = 45° (405 K rather than 380 K), α = 60° (385 K rather than 360 K) and α = 90° (345 K rather than 335 K). In this same plane, we also note that the inclination augmentation widens the temperature profiles since the jets are flattened when crossing the oncoming crossflow and that flattening is more important when the jets have already flee the injection plate.

Fig. 9
figure 9

Impact of the streamwise inclination on the lateral variation of the temperature between and downstream of the twin jet nozzles on the symmetry plane (z = 0)

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Fig. 10
figure 10

Contours of the temperature on different vertical levels and under the different inclination cases

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When we move to the last location (Fig. 9II); that’s to say downstream of the twin jets; we depict the same global behavior in the different y-plane: a Gaussian temperature profile that progressively adopts two peaks to finally turn back to the first trend. Nevertheless, this progression varies with the considered vertical level, in fact, while in the first plane (y = 2 mm) this decline occurs very quickly and the profiles are very close departing from α = 45°, they are more spaced when we flee the injection plate and the registered peaks attain progressively higher values. While between the jets the twin peaks appeared since the lowest plane, this apparition has been delayed when we went beyond the jets since they first developed on the intermediate plane (y = 5 mm). That’s due to the quick flattening of the jets at this level and that whatever the initial inclination is. Nevertheless, at this same height, the duplication of the attained peak occurs earlier: under α = 45 in stead of α = 60°. In fact, brought with a “stronger” initial “impulse”, the jets can cross further the environing flow. That also allows the jets to maintain longer a consistent thermal potential core which results in a more significant trapped flow. On the highest plane (y = 8 mm), finally, they do persist even under the highest inclination angle.

Figure 10 explains more explicitly the development process of the reached peaks, their amplitude, their vanishing, etc. We can see for example that the attained peaks are in fact engendered by the creation of a blocked flow which is trapped between the twin jet columns. Since the upstream jet is the first to cross the oncoming flow, it is the most damaged and the damage consists in a further lateral expansion of the jet as if it enlaces the trapped flow like a necklace. Consequently we obtain two peaks with a slightly variable value. The reaching of these maxima occurs earlier downstream of the jets since in this location we detect the combined effect of both jets that have already joined into a more prominent one. We also said that the reached peaks do not share the same value at a given position and under a given initial inclination: that expresses the asymmetry phenomenon that was described in the literature by several authors but only on the case of single jets in crossflow. Smith and Mungal [19] and Yuan and Street noted it on the vorticity contours and supposed it to exist for the mean structures and under specific conditions. Muppidi and Mahesh [26] could even explain it and that by stating that it actually originates from the influence of the oncoming crossflow that forces the jets to come back (partially) into the injection nozzles. The resulting flow is then disturbed and becomes asymmetric. Su and Mungal [27] stated that the difference of the momentum flux is a further element that generates this feature.

In addition to this, Fig. 10 specifies the spatial progression of the jet plumes under all the inclination cases and on different vertical levels (y = 2, 5, 8 mm). The most striking feature it signals is the thinning of the emitted jets as we flee the injection plate under the influence of two major factors. While on one side the crossflow tends to flatten the jets and direct them on its direction; we detect on the other side the high influence of the flow that has been trapped between the jet columns. This flow exerts a blockage effect that, together with the crossflow’s one result in the compression of the emitted jets. The shielding effect exerted by the first jet on the downstream one reduces the impact of the crossing flow on it and then lessens the blocking effect that follows the second jet: that engenders the diminution of the lateral expansion of the second jet (Fig. 10IIIc).

The different representations of the temperature profiles (vertical and lateral variations) show the high impact of the initial inclination angle on the evolution of the thermal field. They essentially state that this parameter helps straightening the emitted jets and then propelling the heat contained in their potential core. That generates a further vertical diffusion, a better mixing within the handled domain and then a better heating or cooling efficiency in this direction. The augmentation of the initial streamwise inclination also reinforces the asymmetry that develops at the different examined vertical levels. This asymmetry is particularly clear on the planes that cross the jet plumes at their intermediate stages of evolution since when we are too close from the injection plate the jet have not yet undergone the influence of the oncoming crossflow and when we are too far the jets have already lost their initial “strength”.

We move now to the examination of the mass transfer taking place between the emitted jets and the cooler environing crossflow.

5.2 Mass transfer

The dispersion of the different particles contained in any fume emitted in the atmosphere acquires a capital importance and raises a high interest due to its direct participation in the atmospheric pollution. In our case, the emitted fume is non-reactive but its examination may provide some further details on its dispersion process and on the influence of the initial jets’ inclination on it. To begin, we propose to consider the vertical expansion of the pollutants all over the domain and then within its four main characterizing zones and of course under the different inclination cases (Fig. 11). Herein, we have to mention that we followed one single pollutant: the carbon dioxide. This choice is arbitrary and not exclusive due to the non-reactivity of the considered fume. When we are located within the first jet location (Fig. 11a), all the profiles share the same initial mass fraction (21%) which is the imposed quantity on the emitted fume; as soon as we leave the injection plate, we begin to significantly detect the impact of the variation of the initial inclination angle. That consists in a slower decay rate of the corresponding profile with, however, an approximately constant decreasing slopes. That results in a wider profile as the jets are straightened which is understandable since a high injection angle promotes a further vertical expansion before regaining the final mass fraction. The latter is null and its reaching expresses that we attained the homogenization of the flow and then that we are in presence of the oncoming crossflow which is pure air. Going beyond this first location brings some significant changes consisting first in the variation of the initial mass fraction. Downstream of the rear jet, this feature is highly decided by the adopted initial inclination angle of the jet since a low angle keeps its corresponding plume near the injection plate; that justifies the little change in the value adopted under α = 30° (≈20% in stead of 21%). Whereas when the jets are straightened, they rapidly leave the injection wall before being deflected which does not bring too many changes on the constitution of the flow initially contained close to the wall. After leaving this initial value the mass fraction accuses a single peak reflecting the crossing of the extended plume of the rear jet. The magnitude of this peak is also decided by the initial inclination condition. In fact, the highest streamwise inclination allows a deeper crossing of the surrounding flow before being tilted; during this ascension its initial carbon dioxide mass fraction has been spread and then began to lose its “strength” before reaching the location where we will evaluate it. When the jet is slightly inclined, its initial “strength” is preserved and we could detect it without too major changes. The augmentation of the initial inclination, however, promotes a more significant homogenization that covers a larger region of the domain: y = 35 mm under α = 90°, instead of y = 18 mm under α = 30° (Fig. 11b). Leaving the injection plate at x = 30 mm (center of the downstream jet) allows us to cross the just emitted second jet and the expanded plume of the rear one which justifies the presence of a double peak on the profiles of the carbon dioxide mass fraction in this region. Nevertheless these peaks are less distinct as the jets are straightened and are totally fused under the highest inclination case (90°): the fusion of the peaks is simply expressing the combination of the plumes deriving from both jets. As a consequence the corresponding profile becomes wider and attains the total homogenization later but concerns once again a larger region of the domain. When we are located downstream of the twin jet nozzles we find back a similar behavior of the pollutant as between the nozzles with, however, wider profiles due to the further spreading of the jets before reaching the location where we are.

Fig. 11
figure 11

Vertical distribution of the CO2 mass fraction in the different streamwise characterizing zones of the domain and in the symmetry plane (z = 0)

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It comes from this examination that a higher initial inclination angle allows the jets to cross-deeper the oncoming crossflow which promotes a further vertical spreading of the pollutants.

Now that we saw how the initial inclination of the emitted jets affects the vertical dispersion of the pollutants they contain; we are going to consider the streamwise evolution of the carbon dioxide fraction at different vertical levels (Fig. 12). Different vertical planes are used for the matter: the first one is chosen close to injection nozzles’ exits (y = 2 mm) and the farthest one is placed at y = 20 mm where no more significant changes do occur. At a given plane, the closest one for example (y = 2 mm), the augmentation of the streamwise inclination promotes the distinction of the twin reached peaks that signal our crossing of the plumes spread, respectively, from the twin jet nozzles. This is understandable since the augmentation of the handled parameter, as mentioned before, allows the jets to go deeper within the domain before being tilted and deflected. As a consequence, between the twin jet nozzles, the flow is highly affected by the pollutant which explains the deep decrease contained between the twin registered peaks. Downward of the latter, the homogenization is quicker under a normal injection (α = 90°) since the jets have already left the considered plane. When we move to the upper planes (increasing y), the global behavior is preserved with, however, minor but constant and increasing changes that settle progressively: the reduction of the registered peaks and of the decline contained between them, the shifting of the peaks in the streamwise direction, the delay of the initial increasing and finally the widening of the global profile of the mass fraction. The shifting of the peaks is induced by the “impulse” brought by the initial inclination angle and is more pronounced under the lower inclination case. Together with the crossflow’s flattening and deflection they manage at y = 12 mm to fuse the twin peaks. At this level the jet plumes are in fact combined to result in one bigger jet. This does occur only under the first injection case because of the total deflection of the jets under the low inclination angle. This combination occurs later for the other cases: at y = 16 mm for α = 45. The two remaining ones haven’t reached this stage yet even at y = 20 mm but will certainly reach it in the next following planes. As for the reduction of the attained peaks, it’s simply due to the spreading of the initial concentration of the pollutant and its mixing within the oncoming crossflow. As a consequence the gradient of the carbon dioxide mass fraction between the jets and the environing flow highly decreases which justifies the reduction of the accused decline between the twin peaks as well as the lower homogenization downstream of the twin jet nozzles.

Fig. 12
figure 12

Influence of the initial inclination angle on the streamwise evolution of the mass fraction of the carbon dioxide on different vertical levels and in the symmetry plane (z = 0)

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Now we are going to select one vertical plane that is not too far from the main occurring interactions, y = 5 mm for example, and evaluate the impact of the initial inclination on the pollutant dispersion when we move laterally away from the injection nozzles (increasing z). The examination of Fig. 13 shows that under the lowest inclination case (α = 30°), the streamwise dispersion of the pollutant reaches two distinct peaks that progressively combine as we flee the injection nozzle laterally. This combination does no longer express the combination of the emitted jets on a single and more significant jet but rather our leaving of the lateral spreading of the twin jet plumes. As the inclination angle increases, the jets are emitted vertically deeper within the domain as already said; that’s also the case laterally which results in a further thinning of the accused peaks. The minimum contained between the latters becomes progressively constant under the augmentation of the initial inclination angle since it refers to the surrounding quantity of the carbon dioxide mass fraction. As for the homogenization downstream of the twin jets, it settles progressively earlier for the same reason: the faster leaving of the more straightened jets of the injection plate and then their less lateral spreading. When we are totally out of the injection nozzles (z > 5 mm), the augmentation of the jets’ inclination reduces the increasing slope of the peak attained by the carbon dioxide mass fraction. This peak scales the pollutant lateral dispersion which is higher for the low angles due to the important flattening and deflection of the jets in this case. When we reach the farthest location (z = 12 mm), the presence of the jets is hardly detected that’s why we note too few variations on the pollutant distribution in it.

Fig. 13
figure 13

Impact of the initial inclination of the emitted jets on the streamwise development of the pollutant on different spanwise planes and on the vertical plane y = 5 mm

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Figure 14 reconsiders the streamwise distribution of the carbon dioxide mass fraction but for a different purpose. We aim here at determining the influence of the initial streamwise inclination on the vertical location of the twin jets’ combination. This combination is expressed through the transition of the twin distinct peaks into a single one and takes place progressively later: in a further vertical location: y = 16, 20, 25 and 25 mm for an inclination angle, respectively, equivalent to α = 30°, 45°, 60° and 90°. That is equally due to the larger impulse brought to the jets and then their further lifting before undergoing the deflection imposed by the oncoming crossflow. The second consequence of the jets’ straightening consists in a global thinning of the streamwise distribution of the pollutants at the different tested levels. That reflects an earlier homogenization of the flow after the jets’ emission.

Fig. 14
figure 14

Impact of the initial streamwise inclination on the combination rate of the twin emitted jets

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We can then conclude that a higher initial inclination angle of the emitted jets enhances the mixing in the vertical direction and accelerates the homogenization in the streamwise one.

6 Conclusion

We have examined in this paper the thermal field and the mass transfer generated by the interaction of two elliptical inclined jets with an oncoming cooler crossflow. This study has been conducted both experimentally by means of the PIV technique and numerically by the introduction of the finite volume method together with a turbulent closure model. After confrontation with the measured data, we opted for the RSM second order model that could model well the slightest variations of the temperature and of the pollutant’s concentration. The new brand brought by our paper comes actually from the use of our model and from the detailed consideration of these fields. These handled variations are accompanied by the settlement of a complex vortical system mainly composed of the HSV, the upright vortices, the CVP and the SLV. The different plotted profiles and represented contours of the temperature showed that the increasing of the initial streamwise inclination angle highly affects the temperature. This impact is expressed through the development of a further vertical diffusion of the heat contained in the potential cores of the emitted jets. This diffusion allows a better mixing in that direction and results in a quicker homogenization of the global thermal field. According to the meant purpose, we would then rather straighten the jets (for a cooling or heating of the jets) or keep them close to the injection plate (cooling or heating the plate).

The same trend is adopted by the mass fraction of the fume contained within the jet nozzles since it is lifted when its initial inclination is increased. That also limits the propagation of the fume in the longitudinal direction and then controls its dispersion on it.

We can then conclude that the initial inclination angle highly affects both the thermal and mass transfer variations that result from the interactions of the twin tandem emitted jets within the crossflow. For a better exploration of these two fields, further parameters may be examined in other papers, like the jets’ height, the distance between the nozzles, etc.