2.1. Experimental facility

The experiments were conducted in a recirculation water tunnel (Zhu et al. Reference Zhu, Wang, Wang and Wang2017) with a Plexiglas test section of 3000 mm (length) × 700 mm (width) × 600 mm (height). The maximum free-stream velocity in the test section could be up to 0.5 m s⁻¹ with the turbulence level no more than 0.8 %. The synthetic jet actuator in this investigation was a piston-type one, as shown by the dashed box in figure 1(a). The length of the connecting rod was l_c = 300 mm, and the diameter of the piston was D_p = 29 mm. The synthetic jet actuator was connected with an L-shaped hollow circular cylinder to horizontally generate synthetic jets underwater. Note that the horizontal section of the L-shaped cylinder was 100 mm, and that was long enough to avoid the impact of the bent part on the formation of synthetic jets. A detailed introduction to this synthetic jet device could be found in Feng & Wang (Reference Feng and Wang2010) and Xu et al. (Reference Xu, Wang, Feng, He and Wang2018). The end of the L-shaped cylinder was flush-mounted in a plate with the jet orifice. The plate was mounted vertically to the bottom of the test section, and the cross-flow direction was parallel to the plate surface.

Figure 1.

(a) Isometric schematic view of experimental set-up for tomographic PIV in the test section of the water tunnel. (b) Top and side views of the plate in (a), and spanwise and streamwise configurations of the AR = 3 rectangular orifice.

During the experiments, the water height was kept at 550 mm, and the temperature was 26 ± 0.2 °C. The plate was made of anodized aluminium with 1500 mm (length) × 500 mm (width) × 10 mm (thickness), and this gave a blockage ratio of approximately 1.3 %. The leading edge of the plate was modified to be a 1 : 5 elliptic shape to minimize flow separation and boundary-layer transition. Two AR = 3 rectangular configurations with different orientations were investigated in this study. The rectangular long side was normal to, and aligned with, the free-stream direction, namely spanwise and streamwise configurations, respectively (figure 1b). The circular equivalent diameter of the orifice was D_e = 8 mm. The position of the jet orifice was 700 mm (87.5D_e) away from the leading edge of the plate. The plate surface was 400 mm (50D_e) away from the sidewall of the test section, which could guarantee less wall influence on the investigated field in the wall-normal direction. The coordinate system was defined such that the origin was located at the centre of the jet orifice with the x-axis along the free-stream direction, y-axis normal to the plate and z-axis along the spanwise direction.

2.2. Synthetic jet and cross-flow parameters

Throughout the experiments, the servo electromotor of the synthetic jet actuator was controlled by a constant sinusoidal wave signal, and the excitation frequency and amplitude of the synthetic jet actuator were fixed at f_e = 2 Hz (the excitation cycle T = 1/f_e = 0.5 s) and A_e = 1.38 mm. The circular equivalent diameter was used as the characteristic length by referring to the previous non-circular jet study (Gutmark & Grinstein Reference Gutmark and Grinstein1999). The instantaneous velocity at the jet exit u(t) can be simplified as follows based on mass conservation of the incompressible flow when ${l_c} \gg {A_e}\textrm{ }({{A_e}/{l_c} = 0.0046} )$:

(2.1)\begin{equation}u(t) \approx 2{\rm \pi} {A_e}{f_e}{({D_p}/{D_e})^2}\sin (2{\rm \pi} {f_e}t).\end{equation}

The averaged velocity at the jet exit U ₀ is expressed as follows (Smith & Glezer Reference Smith and Glezer1998; Glezer & Amitay Reference Glezer and Amitay2002), i.e. the time-averaged blowing velocity over the entire cycle:

(2.2)\begin{equation}{U_0} = \frac{1}{T}\int_0^{T/2} {u(t)\,\textrm{d}t} = 2{A_e}{f_e}{({D_p}/{D_e})^2}.\end{equation}

Thus U ₀ = 72.5 mm s⁻¹, and the two main dimensionless parameters of the synthetic jet, Reynolds number was $R{e_{{U_0}}} = 664$ and dimensionless stroke length was L = 4.5, defined as

(2.3)\begin{gather}R{e_{{U_0}}}\; = {U_0}{D_e}/\nu ,\end{gather}

(2.4)\begin{gather}L = {L_0}/{D_e}_{} = {U_0}T/{D_e},\end{gather}

where ν is the kinematic viscosity of water, and L ₀ is the stroke length.

The free-stream velocity over the plate in the test section was ${U_\infty } = 71.3 \pm 0.8\;\textrm{mm}\;{\textrm{s}^{ - 1}}$, and thus the synthetic-jet to free-stream velocity ratio $\textrm{VR} = {U_0}/{U_\infty } \approx 1$. This velocity ratio was used so as to compare the current results with previous studies for high-AR rectangular synthetic jets (Van Buren et al. Reference Van Buren, Beyar, Leong and Amitay2016a,Reference Van Buren, Beyar, Leong and Amitayb) under the same velocity ratio. Near the wall, the maximum streamwise velocity fluctuation upstream of the orifice did not exceed 4 % of the free-stream velocity, indicating the current noise level. The boundary-layer thickness ${\delta _{99}}$ at the jet exit was 16.12 mm (2.02D_e) determined by 99 % ${U_\infty }$ corresponding to the free-stream Reynolds number of $R{e_{{\delta _{99}}}} = 1317$. The boundary-layer shape factor is computed as

(2.5)\begin{equation}H = \; \frac{{{\delta ^\ast }}}{\theta } = \frac{{\int_0^\infty {\left( {1 - \frac{{{U_x}}}{{{U_\infty }}}} \right)\,\textrm{d}y} }}{{\int_0^\infty {\frac{{{U_x}}}{{{U_\infty }}}\left( {1 - \frac{{{U_x}}}{{{U_\infty }}}} \right)\,\textrm{d}y} }},\end{equation}

where ${\delta ^\ast }$ and $\theta $ denote the displacement thickness and the momentum thickness, respectively. At the jet exit, H was 2.60, indicating the laminar flow regime in the boundary layer. The comparison of the undisturbed velocity profile at the jet exit with the theoretical Blasius solution is shown in figure 2(a). Relevant parameters of the synthetic jet and boundary layer are summarized in table 1.

Figure 2.

(a) Comparison of undisturbed boundary-layer velocity profile at the jet exit with the theoretical Blasius solution. (b) Normalized velocity fluctuations ${u_{rms}}/{U_\infty }$, ${v_{rms}}/{U_\infty }$ and ${w_{rms}}/{U_\infty }$ at the vortex core near the jet exit versus cycle number for the spanwise case.

Table 1.

Major parameters of synthetic jet and boundary layer at the jet exit.

2.3. Tomographic particle image velocimetry measurements

Time-resolved tomographic PIV was applied to measure the 3-D velocity fields of rectangular synthetic jets interacting with a cross-flow (figure 1a). For comparison, the 3-D velocity field of a rectangular synthetic jet issued into a quiescent environment was also measured; and note that the y-axis corresponds to the streamwise direction for the free case. Hollow glass beads with diameters in the range of 5–20 μm and density of 1.05 g cm⁻³ were seeded to a particle image density of 0.05 particles per pixel. A high-frequency double-pulsed solid-state laser (Vlite-Hi-30k, 532 nm wavelength, 2 × 30 mJ pulse⁻¹ at 1 kHz) was used to produce the volumetric illumination source perpendicular to the plate through a combination of two optical lenses. The width of the volumetric illumination source and its location relative to the orifice centre in the x-axis direction were adjusted according to the cases with and without cross-flow.

Four high-speed complementary metal oxide semiconductor (CMOS) cameras (Photron Fastcam SA2/86K-M3) equipped with 85 mm Nikon lenses at f/11 were distributed under the bottom of the test section to capture the particle images in the investigated field (figure 1a). The viewing angle between CMOS 1 and CMOS 4 (CMOS 2 and CMOS 3) was approximately 60°, and that between CMOS 1 and CMOS 2 (CMOS 3 and CMOS 4) was approximately 30°. The distance between the lens and the measurement domain was around 700 mm, corresponding to a focus depth of 40 mm, guaranteeing the recording of all the particles in the measurement volume in focus. Before the experiment, the cameras, laser and data acquisition system were accurately synchronized by a synchronous controller. Each camera resolution was set to 2048 (x) × 1024 (y) pixels for the cases in cross-flow, and to 1024 (x) × 2048 (y) pixels for the free case.

The sequences of particle images were captured at 1 kHz ($ {\rm \Delta} t = 0.001 $ s). A pair of particle images with the interval of $ 3{\rm \Delta} t $ was used to calculate each velocity vector field. By doing this, the maximum particle displacement between adjacent frames was less than 10 pixels. In total, 10 914 particle images were recorded after the jet flow was adequately steady. To save calculation time, 50 instantaneous velocity fields were reserved in each excitation cycle, and thus the actual time resolution was 100 Hz (50 times the excitation frequency), which could effectively resolve the vortex dynamics. Using the spanwise configuration as an example, we performed a convergence examination of the velocity fluctuations at the vortex core near the jet exit (figure 2b). The results showed that the variation level of all the velocity fluctuations was less than 1 % of the free-stream velocity after the cycle number of 8. This result could meet the current comparable analysis of the flow fields of different orifice orientations.

In image preprocessing, after background subtraction and Gaussian smoothing with a grid size of 3 × 3 pixels, volume self-calibration was implemented to improve the accuracy of the mapping function (Wieneke Reference Wieneke2008) to achieve the reduction of the original disparity error to less than 0.1 pixel. Using the self-calibrated mapping function, the 3-D particle intensity distribution was reconstructed following the MART algorithm (Elsinga et al. Reference Elsinga, Scarano, Wieneke and van Oudheusden2006). The measurement volume size was 128 mm (x) × 56 mm (y) × 32 mm (z) (16D_e × 7D_e × 4D_e) for the cases in cross-flow (figure 1b), and 60 mm (x) × 104 mm (y) × 32 mm (z) (7.5D_e × 13D_e × 4D_e) for the free case, respectively. The digital image resolution was 14.3 pixels mm⁻¹ corresponding to the resolution of 13.2 voxels mm⁻¹ in the object space. The 3-D vector field was calculated by an iterative cross-correlation algorithm with multi-grid and window deformation (Scarano & Riethmuller Reference Scarano and Riethmuller2000), and the final interrogation volume size was 48 × 48 × 48 voxels with 75 % overlap. Then the space resolution was 3.64 mm (0.46D_e) and the vector spacing was 0.91 mm (0.11D_e) in each direction, enabling the large-scale vortical structures to be resolved in this study. The resulting 3-D vector field was spatially filtered using a divergence-free smoothing algorithm (Wang et al. Reference Wang, Gao, Wang, Wei and Wang2016).

The experimental uncertainty is mainly attributed to the entire data size and the cross-correlation analysis of PIV. Specifically, the uncertainties in the mean velocity and velocity fluctuation can be estimated as (Ye, Schrijer & Scarano Reference Ye, Schrijer and Scarano2016)

(2.6)\begin{gather} {\varepsilon _u} = \frac{{{\sigma _u}}}{{\sqrt N }},\end{gather}

(2.7)\begin{gather}{\varepsilon _{{u_{rms}}}} = \frac{{{\sigma _u}}}{{\sqrt {2(N - 1)} }},\end{gather}

where ${\sigma _u}$ is the typical level of normalized streamwise velocity fluctuations, and N is the number of uncorrelated samples. In this study, the maximum of the normalized streamwise velocity fluctuation is less than 0.6, corresponding to the maximum of ${\varepsilon _u}$ and ${\varepsilon _{{u_{rms}}}}$ being 2 % and 1 %, respectively. In addition, an error of 0.2 pixel is considered for the cross-correlation algorithm (Elsinga et al. Reference Elsinga, Scarano, Wieneke and van Oudheusden2006); the resulting velocity uncertainty is 2 % relative to the jet maximum velocity.

2.4. Data analysis methods

This section gives a brief outline of the methods applied in this study, including the vortex centre detection, the finite-time Lyapunov exponent and the Fourier mode decomposition.

2.4.1. Vortex centre detection

In the quantitative analysis of vortex dynamics, the vortex centre location (x_v, y_v, z_v) is defined as the centroid of the vorticity (Cantwell & Coles Reference Cantwell and Coles1983) as

(2.8)\begin{gather}{x_v} = \frac{1}{\varGamma }\int_\varOmega {x{\omega}\,\textrm{d}\varOmega } ,\end{gather}

(2.9)\begin{gather}{y_v} = \frac{1}{\varGamma }\int_\varOmega {y{\omega}\,\textrm{d}\varOmega } ,\end{gather}

(2.10)\begin{gather}{z_v} = \frac{1}{\varGamma }\int_\varOmega {z{\omega}\,\textrm{d}\varOmega } ,\end{gather}

(2.11)\begin{gather}\varGamma = \int_\varOmega {{\omega}\,\textrm{d}\varOmega } ,\end{gather}

where $\varOmega $ is the vortex region. To determine the vortex region, the ${\lambda _{ci}}$ criterion is first used to identify vortices, which is defined as the imaginary part of the complex eigenvalue of the velocity gradient tensor (Zhou et al. Reference Zhou, Adrian, Balachandar and Kendall1999). Here the velocity gradient tensor is computed using the finite difference scheme with second-order accuracy. Sensitivity analysis has been conducted to examine the vortical structures with the ${\lambda _{ci}}$ thresholds of 1 to 3.5 s⁻¹ with 0.5 s⁻¹ increment. The flow area with a ${\lambda _{ci}}$ above 1.5 s⁻¹ is recognized as the vortex area, which can display global characteristics of the 3-D vortical structures clearly. In addition, the main structures have no obvious change as ${\lambda _{ci}}$ increases from 1.5 to 3.5 s⁻¹, and thus a larger ${\lambda _{ci}}$ of 3.5 s⁻¹ is also used concurrently to characterize the main components in this study. Within the vortex area, the above integrations are calculated for the phase-averaged vorticity satisfying $|{\omega}|/|{\omega _{p}}|\ge 0.1$ (here ${\omega _{p}}$ denotes the local vorticity peak) to eliminate noise from the PIV data (Feng & Wang Reference Feng and Wang2010; Xu et al. Reference Xu, Wang, Feng, He and Wang2018). Based on sensitivity analysis, by selecting normalized vorticity levels of 0.05 to 0.2 with 0.05 increment, the circulation of the vortex ring is insensitive to this cut-off level with the difference less than 3 %.

2.4.2. Finite-time Lyapunov exponent

The ${\lambda _{ci}}$ criterion can effectively identify large-scale vortical structures. However, as vortices break into small scales in the downstream propagation, the results with a smaller threshold of ${\lambda _{ci}}$ could be contaminated by noise in the data, leading to a hard observation. The identification of Lagrangian coherent structures (LCSs) provides a robust method to deal with this issue, which can be achieved through computing the finite-time Lyapunov exponent (FTLE) field (Haller & Yuan Reference Haller and Yuan2000; Shadden, Dabiri & Marsden Reference Shadden, Dabiri and Marsden2006).

For a time-dependent flow map $\boldsymbol{\phi} _{{t_0}}^{{t_0} + {T_F}}({\boldsymbol{x}_0})\,:\boldsymbol{x}({t_0}) \mapsto \boldsymbol{x}({t_0} + {T_F})$ that maps fluid particles from their initial position ($\boldsymbol{x}_{0}$, t ₀) to their later position over a finite time T_F, the Cauchy–Green strain tensor is expressed as (Haller Reference Haller2015)

(2.12)\begin{equation}\boldsymbol{C}_{{t_0}}^{{t_0} + {T_F}}({\boldsymbol{x}_0}) = {[\boldsymbol{\nabla }\boldsymbol{\phi} _{{t_0}}^{{t_0} + {T_F}}({\boldsymbol{x}_0})]^\textrm{T}}\boldsymbol{\nabla }\boldsymbol{\phi} _{{t_0}}^{{t_0} + {T_F}}({\boldsymbol{x}_0}),\end{equation}

and the FTLE can be computed as (Green, Rowley & Haller Reference Green, Rowley and Haller2007; Xu et al. Reference Xu, Wang, Feng, He and Wang2018)

(2.13)\begin{equation}\textrm{FTL}{\textrm{E}_{{T_F}}}({\boldsymbol{x}_0},{t_0}) = \ln \sqrt {{\lambda _{\textit{max} }}(\boldsymbol{C}_{{t_0}}^{{t_0} + {T_F}}({\boldsymbol{x}_0}))} /|{T_F}|,\end{equation}

where ${\lambda _{\textit{max} }}(\boldsymbol{C}_{{t_0}}^{{t_0} + {T_F}}({\boldsymbol{x}_0}))$ represents the largest singular value of $\boldsymbol{C}_{{t_0}}^{{t_0} + {T_F}}({\boldsymbol{x}_0})$. The LCSs are indicated by ridges in the FTLE field corresponding to the regions of maximum fluid particle separation (${T_F} \gt 0$) or attraction (${T_F} \lt 0$). The ridge in the forward-time (${T_F} \gt 0$) FTLE field represents a repelling LCS, and that in the backward-time (${T_F} \lt 0$) FTLE field represents an attracting LCS. In this study, the 3-D FTLE is computed on a Cartesian grid with a resolution of 0.2 mm × 0.2 mm × 0.2 mm (0.025D_e × 0.025D_e × 0.025D_e) over a backward-time integration of ${T_F} ={-} 1.5\;\textrm{s}$, equal to three times the jet cycle (T = 0.5 s). This integration time enables one to identify details of the flow structures in the far-downstream field considering acceptable computational expense. The numerical integration adopts a fourth-order Runge–Kutta method with a time step of 0.01 s, and the space grid is interpolated by trilinear interpolation.

4.1. Identification of vortical structures

Figures 5 and 6 present the temporal evolution of the phase-averaged fields for the spanwise and streamwise cases, respectively. The vortical structures are visualized by isosurfaces of ${\lambda _{ci}}$ like those in the quiescent condition. In addition, the vertical velocity ${u_y}/{U_\infty } = 0.4$ (red) and −0.04 (blue) is also shown to reflect the local induced velocities of different vortical structures. Owing to the similar components of the vortical structures for the two cases, we give a representative description mainly based on the spanwise case.

Figure 5.

Temporal evolution of the spanwise case at t/T = 0.25 (a), 0.5 (b), 0.75 (c) and 1 (d). Isosurfaces of phase-averaged λ_ci = 1.5 (cyan) and 3.5 (green), and vertical velocity ${u_y}/{U_\infty } = 0.4$ (red) and −0.04 (blue, in the downstream near-wall region) are presented. Front views zooming-in on the black dashed box are shown on the right of (a) and (d). TVR, STV and TNV denote tilted vortex ring, secondary trailing vortex and tertiary near-wall vortex; h-SV and l-SV denote higher and lower shear-layer vortices, respectively.

Figure 6.

Temporal evolution of the streamwise case at t/T = 0.25 (a), 0.5 (b), 0.75 (c) and 1 (d). Identical isosurfaces with those in figure 5 are presented.

As shown in figure 5, the vortical structures for rectangular synthetic jets in cross-flow are composed of a tilted vortex ring (TVR) followed by a secondary trailing vortex (STV). The TVR exhibits downstream tilting, associated with the trailing structures, which establish an obstacle to the cross-flow and inhibit the momentum injection into the downstream branch of the vortex ring. In contrast, the upstream branch has higher momentum, and thus a larger propagation velocity than the downstream one. Consequently, the jet column as well as the vortex ring becomes bent downstream in the cross-flow direction, consistent with the results for pulse jets (Sau & Mahesh Reference Sau and Mahesh2008) and plasma synthetic jets (Zong & Kotsonis Reference Zong and Kotsonis2019) in cross-flow.

The STV is essentially a pair of counter-rotating vortices branching from the middle of the vortex ring. It has the same rotation sense as the corresponding longitudinal edge of the vortex ring. As a result, an upwash velocity is induced between the two legs of the STV, and inside the vortex ring (see ${u_y}/{U_\infty } = 0.4$). This upwash velocity can lift the low-momentum fluid away from the wall, leading to the local velocity deficit in the lee side of the jet. It is interesting to find that the STV does not extend to the wall, whereas the ends of the two legs are jointed in the spanwise direction by a shear-layer vortex named l-SV. Over the l-SV, another shear-layer vortex is detected to be seated on the STV underneath the upstream branch of the TVR, namely h-SV (see the zoom-in in figure 5a,d). These two shear-layer vortices will be further depicted in figure 7. In addition, a tertiary near-wall vortex (TNV) in a crescent shape is formed near the wall due to induction of the STV and l-SV. The TNV convects with a downwash velocity induced on its downstream side (see ${u_y}/{U_\infty } ={-} 0.04$). This downwash velocity can be advantageous for momentum injection near the wall.

Figure 7.

Contours of ${\lambda _{ci}}{\omega _z}/|{\omega _z}|\textrm{ }({\lambda _{ci}} \ge 1.5)$ in the jet symmetry plane for the (a–d) spanwise and (e–h) streamwise cases at t/T = 0.25 (a,e), 0.5 (b,f), 0.75 (c,g) and 1 (d,h). The upstream and downstream vortices inside the red dashed ellipse denote the windward and leeward vortices of the TVR. Solid and dashed lines represent positive and negative values, respectively. Contour levels are ±1.

The current flow scenarios exhibit vortex rings that convect downstream and penetrate the boundary layer (${\delta _{99}} \approx 2{D_e}$). These features have also been observed in the numerical results of Ravi et al. (Reference Ravi, Mittal and Najjar2004) for rectangular synthetic jets with ARs of 1, 2 and 4 in cross-flow. The difference is that their vortices penetrated out of the boundary layer with a longer streamwise distance due to the relatively low VR of 0.5. However, the vortex rings were not formed in previous studies of high-AR (AR > 5) rectangular synthetic jets in cross-flow (Van Buren et al. Reference Van Buren, Beyar, Leong and Amitay2016a,Reference Van Buren, Beyar, Leong and Amitayb). In addition, some differences in the vortical structures can be found in comparison with the tilted and distorted vortex rings for circular synthetic jets (Jabbal & Zhong Reference Jabbal and Zhong2010), which are summarized as follows: the legs of the present STV are connected in the spanwise direction by the two shear-layer vortices at different wall-normal heights, and the present TNV is a crescent-shaped spanwise vortex rather than a longitudinal streamwise vortex pair as in Jabbal & Zhong (Reference Jabbal and Zhong2010).

Contours of ${\lambda _{ci}}{\omega _z}/|{\omega _z}|\textrm{ }({\lambda _{ci}} \ge 1.5)$ in the jet symmetry plane for the two cases are compared in figure 7. The red dashed ellipse surrounding the windward and leeward vortices of the TVR can reflect the tilting angle. As the vortical structures move away from the wall, the TVR is gradually recovered to the wall-parallel orientation associated with the Biot–Savart induced effect under the cross-flow. As the STV becomes weak and breaks downstream, its inhibition to the cross-flow is diminished. According to the Kutta–Joukowski theorem, the windward vortex of the vortex ring experiences a downward force, while the leeward vortex experiences an upward force, leading to upstream tilting of the vortex ring (Sau & Mahesh Reference Sau and Mahesh2008). In particular, the vortex ring for the streamwise case exhibits faster recovery of the tilting angle and penetration through the boundary layer.

In addition, the shear-layer vortices are detected in the jet symmetry plane for the two cases that have a clockwise rotation for the h-SV, and an anticlockwise rotation for the l-SV. The l-SV is generated by the shear layer rolling-up in the windward side of the initial jet. However, the h-SV evidently differs from the l-SV, and has not been identified in previous research. It is inferred that the h-SV is resulted from the shear between high-momentum, outer cross-flow and low-momentum, near-wall fluid lifted by the STV, akin to the formation mechanism of the secondary and tertiary hairpin vortices in Zhou & Zhong (Reference Zhou and Zhong2010).

4.2. Axis switching in cross-flow

Axis switching of vortex rings is a crucial mechanism for non-circular jets in quiescent condition, by which they can significantly enhance entrainment and mixing relative to comparable circular and 2-D jets (Gutmark & Grinstein Reference Gutmark and Grinstein1999; Grinstein Reference Grinstein2001). However, for steady jets in cross-flow, axis switching no longer happens to non-circular orifices since the presence of a counter-rotating vortex pair inhibits the formation of the vortex ring (Lim, New & Luo Reference Lim, New and Luo2001; New et al. Reference New, Lim and Luo2003). For synthetic jets in cross-flow, vortex rings revert to primary structures as the form of stretched or tilted vortex rings. Hence, whether vortex rings can experience axis switching is of practical interest for non-circular synthetic jets in cross-flow. To our knowledge, no existing research has reported this.

The variation of the vortex ring diameter in the characteristic plane can reflect axis switching of vortex rings, which is defined as the distance between vortex centres corresponding to the positive and negative vorticities. Further, the vortex ring centre is obtained from the middle of the vortex ring diameter in the jet symmetry plane. Figure 8 shows the temporal evolution of the vortex ring diameter and vortex ring centre for the two cases, and the result for the free case is presented for comparison. For the vortex area identified by ${\lambda _{ci}} = 1.5\;{\textrm{s}^{ - 1}}$, the uncertainty of the vortex ring diameter is approximately 5 %. In addition, the influence of the vortex area on the vortex ring diameter has also been analysed by selecting the ${\lambda _{ci}}$ thresholds of 0.5 to 2 s⁻¹ with 0.5 s⁻¹ increment. The variation of the vortex area is found to have little effect on the vortex ring diameter, and the difference of the present result could be within 5 %.

Figure 8.

(a) Temporal evolution of vortex ring diameters D_rx/D_e and D_rz/D_e: D_rx (—+—) and D_rz (—×—) denote the vortex ring diameters in the y–x and y–z planes for the free case; D_rx (—□—, spanwise case; —◊—, streamwise case) and D_rz (—■—, spanwise case; —♦—, streamwise case) denote the vortex ring diameters in the x–y plane, and the plane through the vortex ring centre and normal to the vortex ring tilting plane. (b) Trajectory of vortex ring centre along the wall-normal direction for the three cases.

The diameter curves for the free case have two intersection points corresponding to two axis switchings. It is worth noting that the intersection also exists for the spanwise and streamwise cases, respectively, suggesting that the present rectangular synthetic jets in cross-flow can experience axis switching of vortex rings. This is an important difference between non-circular synthetic jets and steady jets in cross-flow. Specifically, the spanwise case undergoes once, while the streamwise case undergoes twice. In addition, the spanwise case has the latest first axis switching (t/T = 0.50), whereas the location (y/D_e = 1.88) is lower than that of the streamwise case (y/D_e = 2.04) due to slower vortex penetration (figure 8b).

Axis switching for the three cases can be intuitively displayed by the vortex filament through the vortex centre, as shown in figure 9. For the free case, the cylindrical coordinate system (r, θ, y) is used to determine the vortex centre location (r_v, y_v) in the radial plane by changing θ by 3° increments. The entire vortex centre locations are connected by a spline curve to obtain the vortex filament. For the case in cross-flow, rotation and translation of the coordinate are performed firstly to keep the vortex ring axisymmetric about the y-axis through the origin. Then the same method as the above is used to obtain the vortex filament, which is finally restored to the original spatial location.

Figure 9.

Temporal evolution of vortex filament through vortex centre for the (a) free, (b) spanwise and (c) streamwise cases, respectively.

Compared to self-induced deformation for the free case, the vortex deformation for the spanwise and streamwise cases can be affected by the cross-flow. Owing to the obstacle of the trailing structures to the cross-flow near the wall, the upstream section of the vortex filament is mainly affected by the cross-flow relative to the downstream one. For the spanwise case, the cross-flow has opposite direction to the azimuthal induced velocity of the upstream section, resulting in an unfavourable effect on the local vortex deformation. Conversely, the downstream section yields relatively larger streamwise stretching (see t/T = 0.25 in figure 9b). As the vortex filament moves away from the wall, the impact of the cross-flow on the downstream section increases. As a result, the vortex filament is evidently stretched in the streamwise direction after the first axis switching (see t/T = 1.25 in figure 9b), which can also be reflected by the vortex ring diameter in the jet symmetry plane (figure 8a). Hence, no new axis switching follows.

On the contrary, the initial azimuthal induced velocity of the upstream section has the same direction as the cross-flow for the streamwise case, leading to a favourable effect on the local vortex deformation. Consequently, the streamwise case has an earlier first axis switching than the spanwise case as depicted in figure 8. Additionally, the streamwise case has slower streamwise stretching under the cross-flow, and therefore the vortex filament can experience two axis switchings like the free case.

4.3. Vortex interactions

Owing to the periodic property of synthetic jets, interactions between different vortical structures are important for synthetic jets in cross-flow. The FTLE enables one to effectively identify the details of vortical evolution in the far-downstream field. Figure 10 exhibits contours of the backward-time FTLE in the jet symmetry plane for the two cases. The leeward vortex of the TVR is observed to interact with the downstream l-SV for the spanwise case. They approach each other and become a pair of contour-rotating vortices. Then the l-SV is entrained by the leeward vortex into the TVR as the vortical structures propagate downstream, as marked by white dashed circles in figure 10(a,b). However, such a process does not happen to the streamwise case, where the leeward vortex moves away from the l-SV with vortex penetration.

Figure 10.

Contours of normalized backward-time FTLE in the jet symmetry plane at t/T = 0.75 (a,c) and 1.25 (b,d) for the (a,b) spanwise and (c,d) streamwise cases. The + and × denote vortex centre locations of the leeward vortex and the l-SV, respectively.

The schematics in figure 11 can aid 3-D illustration of the difference between the two cases. Compared to the independent TVR and STV for the streamwise case, the spanwise case can form a chain of TVRs connected by STVs due to the above vortex interactions, resulting in slower vortex penetration and recovery of the vortex ring tilting. The axis switching properties of the two cases can account for this difference. For the spanwise case, the vortex ring expands its diameter in the jet symmetry plane during the first axis switching (see figure 8a), which reduces the distance between the leeward vortex and the downstream l-SV, and triggers the subsequent vortex interaction. In contrast, the vortex ring for the streamwise case shrinks its diameter in the jet symmetry plane during the first axis switching, and thus impedes the probable interaction.

Figure 11.

Schematic models of rectangular synthetic jets in cross-flow with (a) spanwise and (b) streamwise configurations. The black arrows denote the local rotation sense of vortical structures.

A quantitative description of the vortex interaction effect is presented in figure 12. The formation of a counter-rotating vortex pair can weaken the strengths of the leeward vortex and the l-SV. Thus, the leeward vortex for the spanwise case displays an earlier decay of the circulation to a lower magnitude at around x/D_e = 3. Furthermore, the circulation of the l-SV cannot be detected after x/D_e = 4 for the spanwise case, while that is maintained to the downstream field for the streamwise case (figure 12a). Similar behaviour is also observed in the normalized power spectrum of the vertical velocity along the vortex-ring-centre trajectory (figure 12b). The normalization is performed to exhibit the relative variation of the dominant frequency peak more clearly. Compared to the streamwise case, the peaks of the dominant frequency and high-frequency harmonics for the spanwise case decrease more rapidly from x/D_e = 0.5 to 4 (see the black arrow locations), suggesting faster jet transition under vortex interactions. Note that these flow-field characteristics are essentially the results of axis switching of vortex rings, which can be conducive to enhancing jet mixing and heat transfer for the spanwise case. On the other hand, the trailing structures for the streamwise case have greater streamwise persistency. They in turn can induce larger upwash velocity, leading to a larger velocity deficit in the free stream, which will be demonstrated later.

Figure 12.

(a) Circulations of the leeward vortex and the l-SV, and (b) normalized PSD of vertical velocity U_y by the maximum along the vortex-ring-centre trajectory at x/D_e = 0.5, 2, 4, 6 and 8 for the two cases. The black arrows in (b) mark the peak locations of the dominant frequency.

5.1. Mean flow field in the jet symmetry plane

This section pays attention to the flow-field characteristics of rectangular synthetic jets in cross-flow, and discusses the impact of the orifice orientation on the downstream boundary layer. Figure 13(a) presents the time-averaged streamwise velocity profiles at different streamwise locations in the jet symmetry plane. The local contour of ${\lambda _{ci}}{\omega _z}/|{\omega _z}|$ for the spanwise case is provided in figure 13(b) to help understanding. A prominent velocity deficit can be observed in the near-field profiles, and the magnitude decreases with the jet uplifting. This velocity deficit is attributed to the upwash induced between the legs of the trailing structures, and inside the tilted vortex ring. In particular, the maximum velocity deficit of the spanwise case is lower than that of the streamwise case. That is associated with vortex interactions, which restrict persistency of the trailing structures, and weaken the vortex strength, as depicted in figure 12. However, this comparison is opposite to the previous results for steady jets (Gutmark, Ibrahim & Murugappan Reference Gutmark, Ibrahim and Murugappan2008) and higher-AR synthetic jets (Van Buren et al. Reference Van Buren, Beyar, Leong and Amitay2016b) in cross-flow. The difference can be accounted for by no axis switching and following vortex interactions observed in those studies.

Figure 13.

(a) Time-averaged streamwise velocity profile ${U_x}/{U_\infty }$, (b) local contour of ${\lambda _{ci}}{\omega _z}/|{\omega _z}|$ (${\lambda _{ci}} \ge 1.5$, t/T = 0.25 for the spanwise case) superposed by velocity vector, and (c)boundary-layer shape factor H in the jet symmetry plane, for □ (red), the spanwise case, and ◊ (blue), the streamwise case. The black solid line in (a) represents the Blasius profile, and the black dashed line at x/D_e = 4 denotes the definition for the boundary-layer thickness δ.

In addition, velocity overshoot appears on both sides of the velocity deficit region. The upper velocity overshoot results from local induced velocity upon the leeward vortex, while that inside the boundary layer after x/D_e = 4 is due to momentum transfer by the near-wall vortex. With the merging between the jet and the boundary layer, the velocity in the boundary-layer profile does not recover to the free-stream value. Hence, the current boundary-layer thickness cannot be determined by the traditional 99 % ${U_\infty }$ criterion. According to the definition suggested by Sarkar & Sarkar (Reference Sarkar and Sarkar2009), we use the height of the local maximum of U_x between the wall and the velocity deficit region as the local boundary-layer thickness $\delta $, as shown by the black dashed line at x/D_e = 4. Thus, the boundary-layer shape factor can be calculated as

(5.1)\begin{equation}H = \frac{{\int_0^\delta {\left( {1 - \frac{{{U_x}}}{{{U_{x,\textit{max} }}}}} \right)\,\textrm{d}y} }}{{\int_0^\delta {\frac{{{U_x}}}{{{U_{x,\textit{max} }}}}\left( {1 - \frac{{{U_x}}}{{{U_{x,\textit{max} }}}}} \right)\,\textrm{d}y} }},\end{equation}

where ${U_{x,\textit{max} }}$ denotes the local maximum of U_x. The result in figure 13(c) indicates that the shape factor of the spanwise case is closer to the turbulent value (H = 1.4), corresponding to the larger velocity gradient inside the boundary layer.

The streamwise velocity fluctuation profiles in the jet symmetry plane are displayed in figure 14(a). The local peaks of velocity fluctuation in the jet region are caused by the tilted vortex ring and the shear-layer vortices. The corresponding maximum velocity fluctuation $u_{rms,\textit{max} }^{jet}$ decreases with the jet uplifting (figure 14b). Since the spanwise case is more bent downstream by the cross-flow, it can produce slightly larger $u_{rms,\textit{max} }^{\,jet}$ than the streamwise case before x/D_e = 4. However, as the trailing structures break down under vortex interactions, the spanwise case has a larger decay rate after x/D_e = 5.

Figure 14.

(a) Streamwise velocity fluctuation profile ${u_{rms}}/{U_\infty }$, and (b) maximum velocity fluctuation in the jet region $u_{rms,\textit{max} }^{jet}$, and the sheltering edge $u_{rms,\textit{max} }^{{\delta _s}}$ in the jet symmetry plane, for □ (red), the spanwise case, and ◊ (blue), the streamwise case. The black dashed line at x/D_e = 4 in (a) denotes the definition for the sheltering edge δ_s.

A new peak can be observed inside the boundary layer at around y/D_e = 0.3 (spanwise case) or 0.4 (streamwise case) downstream of x/D_e = 4 (figure 14a), which is contributed by the near-wall vortex. Following He, Wang & Pan (Reference He, Wang and Pan2013), an inflection point at the minimum of u_rms between the near-wall peak and the outside peak can be determined to define a sheltering edge ${\delta _s}$, as shown by the black dashed line at x/D_e = 4. The sheltering edge is slightly smaller than $\delta $, which can be used to exclude the jet's direct disturbance from the boundary layer in the later analysis. The spanwise case has a larger maximum velocity fluctuation inside the sheltering edge $u_{rms,\textit{max} }^{{\delta _s}}$ than the streamwise case (figure 14b), close to the value of a canonical turbulent boundary layer (Spalart Reference Spalart1988). This result is perceived to stem from the larger upstream jet disturbance which is transferred into the boundary layer through the near-wall vortex.

5.2. Mean flow field in the spanwise plane

Figure 15(a,c) presents the spanwise distribution of the velocity deficit and overshoot $({{U_{BL}} - {U_x}} )/{U_\infty }$ (here U_BL denotes the Blasius profile), superposed by the local time-averaged vorticity $|{\bar{\omega }_{xyz}}|{D_e}/{U_\infty }\textrm{ }(|{\bar{\omega }_{xyz}}|= \sqrt {\bar{\omega }_x^2\; + \; \bar{\omega }_y^2\; + \; \bar{\omega }_z^2} )$. A positive value of $({{U_{BL}} - {U_x}} )/{U_\infty }$ represents a velocity deficit and a negative value represents a velocity overshoot. Associated with weaker trailing structures and vortex ring (see $|{\bar{\omega}_{xyz}}|{D_e}/{U_\infty }$ in the jet region), the spanwise case exhibits lower peaks of the velocity deficit and overshoot than the streamwise case in the jet region. In contrast, the peak of the velocity overshoot inside the boundary layer is larger for the spanwise case.

Figure 15.

Spanwise distributions of (a,c) velocity deficit and overshoot $({U_{BL}} - {U_x})/{U_\infty }$, and (b,d) streamwise velocity fluctuation ${u_{rms}}/{U_\infty }$ for the (a,b) spanwise and (c,d) streamwise cases. The black solid line denotes the time-averaged vorticity $|{\bar{\omega }_{xyz}}|{D_e}/{U_\infty }$ with minimum value of 0.2 and interval of 0.2; the black dashed line in (b,d) denotes the contour of ${u_{rms}}/{U_\infty } = 0.05$; and the red dashed line denotes the sheltering edge δ_s.

The spanwise distribution of the streamwise velocity fluctuation is shown in figure 15(b,d). Owing to slower vortex penetration, the impact of the jet disturbance on the boundary layer can persist more downstream for the spanwise case. For instance, the disturbance of ${u_{rms}}/{U_\infty } = 0.05$ is jointed between the jet and the boundary layer at x/D_e = 8 for the spanwise case; however, that has been completely separated by the local sheltering edge for the streamwise case. Accordingly, the spanwise case has larger velocity fluctuation peak inside the boundary layer, displaying the merit of promoting boundary-layer transition.

Using (5.1), the spatial distribution of the boundary-layer shape factor is given in figure 16. Overall, the boundary-layer shape factors for the two cases decrease in the streamwise direction, reflecting the trend of boundary-layer transition to a turbulent one. Inparticular, the global shape factor of the spanwise case is lower than that of the streamwise case, consistent with the characteristics of larger velocity gradient and velocity fluctuation peak inside the boundary layer. This suggests that the spanwise orifice orientation could be more efficient to energize the boundary layer for resisting streamwise adverse pressure gradients.

Figure 16.

Spatial distribution of boundary-layer shape factor H for the (a) spanwise and (b) streamwise cases. Contour levels are 0.1.

5.3. Disturbance development inside boundary layer

Based on the above analysis, the boundary-layer transition in the lee side of the jet is associated with the near-wall vortex induced by synthetic jets. To further reveal the disturbance impact of the near-wall vortex on the boundary layer, the FMD method is used to extract the relevant dynamical information. The flow field in the range of [1, 14]D_e × [0, 4]D_e × [−1.5, 1.5]D_e in three directions is selected to perform the FMD, which contains the entire boundary-layer region downstream of the jet.

Figure 17 presents the global FMD power spectrum of the selected flow field for the three velocity components at each sampling point. The power spectra of the two cases both display profound peaks corresponding to the excitation frequency and its high-frequency harmonics. In addition, the peaks at lower frequencies of 0.23f_e for the spanwise case, and 0.46f_e for the streamwise case, can be detected corresponding to the low-frequency disturbance components.

Figure 17.

Global FMD spectrum of the selected region for the (a) spanwise and (b)streamwise cases.

The spatial FMD modes with the marked characteristic frequencies are examined, and figure 18 shows the streamwise components of these modes in the jet symmetry plane. Only the imaginary part of each spatial mode is given due to the conjugate relation between the real and imaginary parts. The modes upon the boundary layer reveal the jet structures with different spatial scales. The streamwise wavelength of these structures decreases with increased frequency. Inside the boundary layer, the near-wall modes with the dominant frequency of 1.0f_e characterize the downstream convection of the near-wall vortex (figure 18b,f).

Figure 18.

Streamwise components of FMD modes in the jet symmetry plane with different frequencies for the two cases: (a) 0.23f_e, (b) 1.0f_e, (c) 2.0f_e and (d) 3.0f_e for the spanwise case; (e) 0.46f_e, (f) 1.0f_e, (g) 2.0f_e and (h) 3.0f_e for the streamwise case. The red dashed line in (b,f) denotes the sheltering edge.

At a lower frequency of 0.23f_e, it is noted that an elongated structure with considerably large spatial scale appears at the downstream location for the spanwise case (figure 18a), similar to the result of boundary-layer transition induced by a circular cylinder wake in He et al. (Reference He, Wang and Pan2013). This suggests that the low-frequency disturbance of 0.23f_e can persist downstream independent of the external excitation frequency. In fact, similar structures with different spatial scales can also be found in some other low-frequency modes. For the streamwise case, the elongated structures are weaker (figure 18e), probably due to the relatively lower disturbance amplitude, which will be presented in figure 19. At higher frequencies of 2.0f_e and 3.0f_e, no near-wall modes with comparable amplitude relative to those in the jet region are observed.

Figure 19.

Streamwise evolution of (a) vorticity peak $|{\omega _{z,p}}|{D_e}/{U_\infty }$ of the near-wall vortex, and (b) averaged boundary-layer disturbance energy $E_{uu}^{{f_i}}$ of the FMD modes with different frequencies f_i for the two cases.

To quantitatively measure the disturbance amplitude of different frequencies, we compute the averaged boundary-layer disturbance energy $E_{uu}^{{f_i}}$ with a frequency f_i as

(5.2)\begin{equation}E_{uu}^{{f_i}} = {{\int {\int_0^{{\delta _s}} {(u_{rms}^{{\,f_i}})^2\,\textrm{d}y\,\textrm{d}z} } } \bigg / {\int {\int_0^{{\delta _s}} {U_\infty ^2\,\textrm{d}y\,\textrm{d}z} ,} }}\end{equation}

where $u_{rms}^{{f_i}}$ represents the streamwise velocity fluctuation of the flow field with the frequency f_i. Note that the integral upper limit here is selected as ${\delta _s}$ instead of $\delta $ to exclude the direct contribution of the jet disturbance. The boundary-layer disturbance energies of the selected frequencies (0.23f_e, 0.46f_e, 1.0f_e and 2.0f_e) for the two cases are compared in figure 19(b), and the vorticity peak of the near-wall vortex is given in figure 19(a) to aid interpretation.

As shown in figure 19(b), compared to the streamwise case, the disturbance energy of 1.0f_e for the spanwise case has significantly larger initial magnitude followed by a faster decay in the near flow field (x/D_e < 6). This result is ascribed to the stronger near-wall vortex for the spanwise case (figure 19a), which yields larger initial disturbance to trigger boundary-layer transition. Additionally, the short growth stage of 1.0f_e for the streamwise case could be related to the more downstream vorticity peak of the near-wall vortex. Since the current boundary-layer flow is in the early stage of the transition, the disturbance energies of the low-frequency components grow at a much slow rate. Even so, the magnitude of 1.0f_e has decayed to be lower than that of 0.23f_e after approximately x/D_e = 9.3 for the two cases. Particularly, 0.23f_e for the spanwise case has the largest disturbance magnitude corresponding to more turbulent characteristics. Such a result can help to understand the mechanism of boundary-layer transition induced by synthetic jets.