Figure 1 shows a schematic diagram of the experimental apparatus. Velocity measurement and flow visualization were carried out using a blow-down wind tunnel. An axial fan was used to supply airflow to the rectangular nozzle. The rectangular nozzle had a double contraction with dimension reduction ratios of 3.3:1 and 2.8:1. Figure 2 shows three-dimensional diagrams and cross-sections of rectangular nozzles with parallel plates and deflectors. The rectangular nozzle had a height H of 30 mm, a width B of 65 mm, and a length of 100 mm. The thickness of the deflector plate t was 0.8 mm. The angles between the parallel plates, the divergent deflector plates, and the convergent deflector plates were α=

0˚, 6˚, and −6˚, respectively. The mean and fluctuating velocities were measured using an X-type hot-wire probe and a constant-temperature hot-wire anemometer with a linearized output. The output signals of the hot-wire probe were converted by a 12-bit analog-to-digital (A/D) converter with a sampling frequency of 10 kHz, and 10⁵ data points were stored. A computer-controlled traversing mechanism was used for probe positioning. Experiments were performed under computer control for automated data acquisition. The mean velocity and turbulence intensity were obtained using a personal computer. The velocity profile for the rectangular jet at the nozzle exit exhibited a flat-topped shape. The mean bulk velocity U₀ and the turbulence intensity u ′ r m s / U 0 at x/H = 0.1 from the nozzle exit were 4.5 m/s and 0.7%, respectively. This wind speed corresponds to the low level of a car air conditioner. The Reynolds number Re_H (=U₀H/ν, where ν is the kinematic viscosity of air) was approximately 9000.

Flow visualization was carried out under the same conditions as the flow measurement experiments. To visualize the flow, an oil mist was injected into the wind tunnel. A pulsed Nd:YAG laser was used to visualize vortical structures, and images were recorded by a CCD camera.

Three-dimensional numerical simulations were performed using the large-eddy simulation (LES) of the commercial computational fluid dynamics (CFD) software ANSYS Fluent 17.2 to investigate the flow structures of a rectangular jet by solving the unsteady Navier-Stokes and continuity equations. Calculations were carried out at Re = 9000. The flow field was assumed to be unsteady, viscous, and incompressible. The working fluid was air (15˚C) with a density of 1.225 kg/m³ and a viscosity of 1.7894 × 10⁻³ Pa·s. The pressure implicit with splitting of operator (PISO) method was used as the calculation algorithm for the pressure-velocity coupling. The convection terms in the governing equation were discretized by the bounded central differencing scheme. Second-order central differencing was used for the diffusion term, whereas the second-order implicit scheme was used for the time marching.

The computational domain, mesh, and boundary conditions are shown in Figure 3. The streamwise and transverse lengths of the computational domain were set to 600 mm (=20H) and 450 mm (=15H), respectively. The nozzle geometry was defined to correspond to the experimental setup (H = 30 mm, B = 65 mm). Since the boundary layer in the deflectors occurs the flow separation on the walls easily and turbulence models must be quite sensitive in the vicinity of the mesh spacing near the wall and nozzle to yield accurate predictions, the computational meshes used for the LES were set finer than those for the laminar simulations. Therefore, the minimum cell size is 0.1 mm near the walls. The numbers of grid point in the x, y, and z directions were 231, 222, and 251, respectively. A total of approximately 1.1 × 10⁷ grid points were nonuniformly distributed in the computational domain. The Courant number is less than 0.1.

A uniform velocity profile corresponding to the nozzle exit velocity was specified at the inlet boundary. The velocity on the upper and lower inlet surfaces, which were outside of the rectangular nozzle inlet surface, was adjusted to 0.03U₀ to maintain a weak straight flow aligned toward the entrainment flow outlet [12]. The nonslip boundary condition was applied to the walls of the rectangular nozzle. The slip condition was applied at the periphery of the computational domain, where the gradients of all variables were set to be equal to zero. A pressure outlet condition of p = 0 was specified at the downstream boundary of the domain.

Figure 4(a) shows the profiles of the mean streamwise velocity ū/U₀ in the x-y plane along the minor axis at z = 0, and the x-z plane along the major axis side at y = 0 to compare the four jets, i.e., nozzles with no deflectors, parallel plates, divergent deflectors, or convergent deflectors. The streamwise velocities were normalized with respect to the mean bulk velocity U₀. With respect to the velocity distribution at x/H = 0.1 separated by deflectors, the jet on the central axis side is referred to as the inner jet, and the jets on both sides of the inner jet are referred to as the outer jets. The velocity profiles of the jets show good symmetry, in contrast with the central axis. For the jet with parallel plates, a velocity depression occurs between the inner and outer jets due to the wake of the parallel plates at x/H = 0.1. However, the ratio of the maximum velocity of the inner jet ū_imax to the outer jets ū_omax is ū_imax/ū_omax = 1.0, namely, each maximum velocity is approximately the same, since the angle between the parallel plates is 0˚. For the jet with divergent deflectors, the velocity profile at x/H = 0.1 exhibits a concave shape, and the velocity ratio is ū_imax/ū_omax = 0.8. Further downstream at x/H = 6, the concave shape disappears as the jet takes on a fully developed profile. In addition, the jet spread is larger than that of the other three jets. For the jet with convergent deflectors, the velocity ratio is ū_imax/ū_omax = 1.3. The velocity profile has a convex shape at x/H = 0.1, that is, the velocity of the inner jet is larger than that of the outer jets. The jet spread at x/H ≥ 3 is smaller than that of the other three jets. It can be seen that velocity profiles similar to that of a coaxial jet [12] are obtained by installing deflectors inside the rectangular nozzle. Some interesting features can be observed in the x-z plane. The jet spread for divergent deflectors is the smallest for these three jets at x/H = 6. In contrast, the jet spread for convergent deflectors is the largest for these three jets. These result in the direction opposite the mean streamwise velocity in the x-y plane. Figures 4(b)-(d) show the profiles of mean streamwise velocity ū/U₀ in the x-y and x-z cross sections for the three jets in order to compare the experimental results and the results of the numerical simulation. The experimental and numerical LES results show good agreement in the region from the nozzle exit at x/H = 0.1 to the jet downstream region at x/H = 6. These results demonstrate that the LES simulation can reproduce the actual flow field.

Figure 5(a) shows the profiles of the root mean square (RMS) values u ′ r m s / U 0 of the fluctuating streamwise velocity components in the x-y plane along the minor axis at z = 0, and the x-z plane along the major axis side at y = 0 to compare the three jets. In the x-y plane, the velocity fluctuation for the jet with divergent deflectors near the center of the nozzle at x/H = 0.1 is larger than that for the jet with convergent deflectors as a result of flow separation on the inner nozzle wall. At x/H = 6, the velocity fluctuation for the jet with divergent deflectors at the outer side of jet is larger than that for the jet with convergent deflectors.

This phenomenon is related to the larger amount of spreading near the nozzle exit for the jet with divergent deflectors. For the jet with convergent deflectors, the velocity fluctuation near the centerline is smaller than those for the other jets. However, the turbulence intensity near the centerline at x/H ≥ 5 increases more than that for the jet with divergent deflectors. In addition, for the x-z cross section, the velocity fluctuation at the nozzle exit is larger than that for the jet with convergent deflectors, and the growth rate for the velocity fluctuation near the centerline is low. Figures 5(b)-(d) show the profiles of the RMS values u ′ r m s / U 0 for the fluctuating streamwise velocity components in the x-y and x-z cross sections for each jet to compare the experimental results and the results of the numerical simulation. For the jet with no deflectors, the LES results show fair agreement with the experimental results from near the nozzle exit at x/H = 0.1 to downstream at x/H = 6. On the other hand, the agreement between the experimental and numerical results for the jets with divergent and convergent deflectors is poor in the upstream region. This difference may be caused that the transition flow from the flow of the relative low Reynolds number (Re = 9000) to the turbulent flow cannot simulate with accuracy by the LES.

Figure 6 shows the mean streamwise velocity along the jet centerline in the experiment. The local centerline velocity ū is normalized by the bulk velocity U₀ at the nozzle exit. The potential core length x_p, which is defined as the distance between the nozzle exit and the point at which the centerline velocity has decayed to 95% of its maximum value [13], is approximately 7H, 8H, 0.5H, and 10H for the jets with no deflectors, parallel plates (ū_imax/ū_omax = 1.0), divergent deflectors (ū_imax/ū_omax = 0.8), and convergent deflectors (ū_imax/ū_omax = 1.3), respectively. In short, the length of the potential core also increases as the velocity ratio increases. This result agrees with previous experimental studies performed by Warda et al. [14] [15] and Mergheni et al. [16]. For divergent deflectors, the centerline velocity ū_c/U₀ decreases from the vicinity of the nozzle exit and then

has a minimum value at x/H = 3. This is similar to the results of Durao and Whitelaw [17] and Buresti et al. [18] whereby the velocity decay ratio of the centerline velocity ū_c/U₀ decreases immediately after the jet issues from the nozzle exit and the potential core length becomes short. In the far field at x/H > 10, the centerline velocity decreases at ū_c/U₀ ~ x⁻¹ for all four jets, that is, this velocity decay ratio demonstrates behavior similar to that of a fully developed circular jet. Thus, this result shows that the rectangular jets with and without deflectors ultimately evolve into fully developed circular jets.

Figure 7 shows the RMS value for the streamwise velocity fluctuations along the jet centerline in the experiment. For the jet with divergent deflectors, the velocity fluctuation at x/H < 3 in the near field is larger than that for the other jets. However, the velocity fluctuation at x/H > 5 is small in comparison with that for the other three jets. In the far field at x/H > 20, there is little difference between the velocity fluctuations for the four jets.

Figure 8 shows the growth of the jet half-velocity width obtained experimentally in the x-y and x-z planes. For the jet with no deflectors, the jet spreads rapidly along the minor axis in the x-y plane and more slowly along the major axis in the x-z plane. The axis-switching location x_as/H, where the length of y_1/2/H and z_1/2/H is reversed as a result of axis switching, is at x_as/H ≈ 16 (Figure 8(a)). In the case of the jet with parallel plates, axis switching occurs closer upstream (x_as/H ≈ 8) than the jet with no deflectors because the value of z_1/2/H does not change much for x/H < 8 (Figure 8(b)). The jet with divergent deflectors spreads rapidly along the minor axis in the x-y plane and shrinks along the major axis in the x-z plane. As shown in Figure 8(c), the value of z_1/2/H shows a decrease in the range of x/H from 0 to around 8 in x increases because of the relative low velocity area near the jet centerline, i.e. the ratio of the maximum velocity of the inner jet ū_imax to the outer jets ū_omax is ū_imax/ū_omax = 0.8. As a result, the jet shrinks the z-direction. Axis switching also occurs closer to the nozzle exit (x_as/H ≈ 4)

than in the case of the jet with no deflectors. On the other hand, the jet with convergent deflectors spreads more slowly along the minor axis in the x-y plane than the jet with no deflectors (Figure 8(d)). As a result, axis switching occurs farther downstream (x_as/H ≈ 23) than in the case of the jet with no deflectors. Therefore, the deflectors installed inside the rectangular nozzle influence the axis-switching location of the rectangular jet.

Figure 9 shows the equivalent jet spread, i.e., the equivalent jet half-velocity width (y_1/2 × z_1/2)^0.5/H [19], along the x-axis non-dimensionalized by the height of the nozzle. For comparison, the figure also reports the data of Kumar et al. [9] for a rectangular jet with an aspect ratio of 2 and Quinn [20] for a planar jet with an aspect ratio of 20. The equivalent jet spread signifies the overall spread of the jet along the x-axis based on the given exit conditions. The equivalent jet spread for the nozzle with no deflectors is greater than that for the nozzle with divergent deflectors and less than that for the nozzle with the convergent deflectors. The jet spread along the minor axis is made dominant by the introduction of deflectors inside the nozzle. Although the jet half-velocity widths y_1/2/H and z_1/2/H in each plane shown in Figure 8 exhibit nonlinear variation with x/H, the equivalent jet half-velocity width (y_1/2 × z_1/2)^0.5/H varies linearly with x for x/H > 10. The spreading rate for the present jets shows close agreement with the data

of Kumar et al. [9] and Quinn [20]. Therefore, the jet spread data in Figure 9 were fitted by linear regression analysis using the function

( y 1 / 2 ⋅ z 1 / 2 ) 0.5 H = K s ( x H + C s ) (1)

where K_s is the spreading rate, and C_s is the geometric virtual origin. The values of these parameters obtained in the fitting are given in Table 1 for each of the cases plotted in Figure 9. There is a difference between the spreading rates K_s for the 2.17:1 and 20:1 rectangular jets. In general, rectangular jets with small aspect ratios generate three-dimensional flow fields, whereas those with large aspect ratios generate two-dimensional flow fields. Note that the spreading rate K_s for the jet with divergent deflectors is larger than those for the other jets.

Figure 10 shows the contours of mean streamwise velocity for the three jets obtained experimentally in the quarters of the y-z plane at x/H = 1, 4, 10, and 16.

These measurements of grid points were carried out for a step size of Δy = Δz = 1.0 mm for x/H = 1, Δy = Δz = 2.0 mm for x/H = 4, and Δy = Δz = 5.0 mm for x/H = 10 and 16, respectively. The contour interval is 0.1U₀. As shown in Figure 10(a), immediately downstream of the nozzle exit, i.e., x/H = 1, the rectangular jet with no deflectors retains the original shape of the rectangular nozzle, and the jet has a thin shear layer at the circumference of the nozzle. The jet deformation begins at the nozzle corners because of the nonuniform velocity induction (seen as an outward bump in the contours). As a result, the ambient fluid was brought in toward the jet centerline along the major axis side while the jet fluid was ejected along the minor axis side. Further downstream, the shear layer becomes thick and the shape of the jet changed as there was high growth along the minor axis side. The outline of the jet at x/H = 16 changes from rectangular to diamond-shaped, accompanied by axis switching. In the case of the jet with parallel plates (Figure 10(b)), the velocity defect occurs due to the wakes of the parallel plates at x/H = 1. However, the jet maintains a rectangular shape, as is the case for a jet with no deflectors. The axis switching occurred for x/H > 10. In the case of the jet with divergent deflectors (Figure 10(c)), at x/H = 1, the velocity in the outer mixing region is larger than that in the inner mixing region. The thickness of the shear layer at the long side of the rectangular nozzle at x/H = 4 is thinner than that at the short side. The jet outline is extended by high growth along the minor axis side for x/H > 4. As a result, the jet shape changes from a lateral rectangle to a vertical ellipse. On the other hand, as shown in Figure 10(d), the jet with convergent deflectors has a thin shear layer at the short side of the rectangular nozzle at x/H = 1, as compared with the long side. The jet extends more rapidly along the major axis side than along the minor axis side. Consequently, the jet outline at x/H = 16 roughly maintains the rectangular shape of the nozzle exit without axis switching occurring.

Figure 11 shows instantaneous vorticity contours in the x-y plane along the minor axis side at z = 0 for the three jets obtained by the large-eddy simulation. Vortex roll-up is observed in the shear layers at the nozzle exit, and vortices of opposite signs are positioned symmetrically about the jet centerline at x/H ≈ 1.2 for the jet with no deflectors (as shown in Figure 11(a)). For the case of the jet

with divergent deflectors (Figure 11(b)), vertex roll-up is observed in the outer shear layers at (x/H, y/H) ≈ (0.5, ±0.5). In the inner shear layer, the separated vortices on the inner wall surface of the deflectors interact with the vortices on the outer surfaces of the deflectors. As a result, the velocity fluctuation near the nozzle exit of the jet with divergent deflectors increased, as shown in Figure 5(c). In the case of the jet with convergent deflectors (Figure 11(c)), the periodical vortex shedding in the outer shear layers is suppressed, unlike in the jets with no deflectors or divergent deflectors. This phenomenon affects the separated flow generated on the outer surface of the deflectors in the rectangular nozzle, and then the mixing with the ambient fluid is suppressed. As a result, the rectangular jet does not spread along the y-direction, as shown in Figure 10(d). The flow structures can be changed by adjusting the deflectors. These results are consistent with the instantaneous flow visualization images qualitatively, as shown in Figure 12. For the jet with no deflectors, the periodic vortical structure, i.e., the vortex ring in the x-y cross section is arranged symmetrically with respect to the x-axis and changes to a small-scale vortical structure in the downstream. On the other hand, for the jets with divergent and convergent deflectors, the large-scale vortical structures are not clear in the shear layers.

Figure 13 shows the power spectra of the streamwise fluctuating velocity in the outer (x/H = 0.5, y/H = 0.5, z/H = 0) and inner (x/H = 0.22, y/H = 0.22, z/H = 0) mixing regions obtained by the numerical simulation. For the jet with no

deflectors (Figure 13(a)), f_p ≈ 132 Hz for the predominant spectrum peak in outer mixing region. In the experiment, the peak frequency of the streamwise fluctuating velocity in the outer mixing region (x/H = 1 and 2) was f_p ≈ 118 Hz. The Strouhal numbers St_H (=f_pH/U₀) for these peak frequencies in the LES and experiment are 0.88 and 0.79, respectively. Thus, the peak frequency obtained from the LES agrees approximately with the experimental value in the outer mixing region. For the jet with divergent deflectors (Figure 13(b)), a peak frequency of f_p ≈ 18 Hz is observed in the inner mixing region. Three clear peaks in the power spectra at 15, 74, and 121 Hz are observed in the outer mixing region near the nozzle of the jet with convergent deflectors (Figure 13(c)). The Strouhal numbers St_H for these frequencies are 0.1, 0.493, and 0.806, respectively.

Figure 14 shows instantaneous three-dimensional vortical structures for the three jets based on the second invariant isosurfaces of the velocity gradient tensor Q. In the case of the jet with no deflectors (Figure 14(a)), the vortex rings are generated and maintained periodically from the nozzle exit until x/H ≈ 3. The vortex ring collapses, and vortical structures, such as hairpin vortices, are

generated in the downstream region. In the case of the jet with divergent deflectors (Figure 14(b)), intermittent shedding of the vortex ring is observed, and its scale is smaller than that for the rectangular jet with no deflectors. In the downstream region, a large-scale vortical structure develops. In the case of the rectangular jet with convergent deflectors (Figure 14(c)), a vortical structure also develops in the same manner as for the rectangular jet with divergent deflectors. These vortical structures agree with the instantaneous images of rectangular jets in the x-y plane qualitatively, as shown in Figure 12.