2.1. Actuator design

The synthetic jet actuator (SJA) consists of three main components: the driver (a loudspeaker), a cavity assembly and the orifice (figure 2a). The cavity was pancake shaped and had a volume, $V_c$ of $2.63056\times 10^{-5}\ \textrm {m}^3$. The geometrical properties of the actuator, excluding the aspect ratio ($AR = l/d$) of the orifice, were kept constant throughout the study. The orifice had a rectangular slot with a fixed width, $d$ of 1.5 mm. Cases with $AR$ of 3, 6 and 12 were achieved by simply altering the length, $l$ along the major axis of the slot. The SJA was driven by a Visaton SC 8N speaker with an impedance of $8{\rm \Omega}$, a rated power of 30 W, a frequency response between 70 and 20 000 Hz and diaphragm resonance occurring around 110 Hz. Additional technical information about the actuator design and modus operandi can be found in Jankee & Ganapathisubramani (Reference Jankee and Ganapathisubramani2019).

Figure 2. A schematic of the test facility is presented. (a) A cross-sectional view of the actuator design is shown with the CVP generated during the blowing phase, pinching-off and moving under its self-induced velocity. The components of the set-up are: (1) the driver, (2) the cavity assembly and (3) the false floor which is flush with the orifice. (b) A view of the wind tunnel configuration with a zigzag trip at the leading edge, the major axis of the orifice aligned with the incoming flow, a flap at the trailing edge to control the position of the stagnation point and laser sheets and cameras for planar particle image velocimetry measurements (not to scale). (c) Boundary layer profiles of the unperturbed flow at $x/\delta =-0.5$ (blue diamonds) and at $x/\delta =6$ (orange circles) compared to a log law with coefficients $\kappa =0.384$ and $\varPi =4.4$ (green line). (d) Variation of the turbulent intensity with wall-normal location at $x/\delta =-0.5$ (blue diamonds) and at $x/\delta =6$ (orange circles).

The actuator was calibrated and characterised using hot-wire anemometry under quiescent conditions. A single-wire traversing hot-wire probe with a diameter, $d_w$, of $2.5\ \mathrm {\mu } \textrm {m}$ and prong spacing of 2.5 mm was operated in constant temperature mode with an overheat ratio of 0.8. The wire resistance was measured to be $4.5{\rm \Omega}$. The probe was aligned such that the sensing bit was positioned at the centre of the orifice exit and placed at a height of $1d$, away from the orifice exit. Following rectification of the signal, the velocity of a synthetic jet can be expressed by (2.1)

(2.1)\begin{equation} u_0(t) = u_{max}\sin\left(\frac{2 {\rm \pi}t}{T}\right), \end{equation}

where $u_0$ ($\textrm {m}\ \textrm {s}^{-1}$), $u_{max}$ ($\textrm {m}\ \textrm {s}^{-1}$), $t$ (s) and $T$ (s) represent the variation in centreline jet exit velocity over time, the maximum velocity during the blowing cycle and the time and period of the cycle, respectively. Based on Smith & Glezer (Reference Smith and Glezer1998), the mean blowing velocity, $\overline {u_j}$ can then be derived as follows:

(2.2)\begin{equation} \overline{u_j} = \frac{1}{T} \int_{0}^{T/2} u_o(t) \,\textrm{d}t = \frac{u_{max}}{\rm \pi}. \end{equation}

2.2. Description of facility

The experiment was carried out in a $3\times 2\ \textrm {ft}$ suction wind tunnel facility at the University of Southampton. The test section has a length of 4.5 m, width of 0.9 m and a height corresponding to 0.6 m. A false floor made up of smooth aluminium plates is fitted in order to generate a zero pressure gradient turbulent boundary layer. The false floor has a sharp leading edge with a zigzag trip 0.03 m downstream and an adjustable flap to control the position of the stagnation point at the leading edge. The SJA is embedded in the false floor such that the major axis of the orifice is aligned with the incoming flow and is flush with the surrounding surface.

The free-stream velocity was kept constant at approximately $8\ \textrm {m}\ \textrm {s}^{-1}$ for all test cases. This resulted in a turbulent boundary layer with thickness, $\delta$, of 52.1 mm based on $U(\delta )=U_{99}$. Using typical values of $\kappa = 0.384$ and $A = 4.3$ from law of the wall analysis, results in a friction velocity of $0.314\ \textrm {m}\ \textrm {s}^{-1}$. This results in a flow Reynolds number, $Re_{\tau }$, of 1044. Normalising the region of interest with the boundary layer thickness culminates in a flow field which spans from $x/\delta = -1$ to $x/\delta = 8$ in the streamwise direction and up to $y/\delta = 2$ in the wall-normal direction. This provides sufficient area to capture the interaction of the jet with the cross-flow in both the near-field and far-field regions.

2.3. Particle image velocimetry measurements

Particle image velocimetry (PIV) is the main measurement technique used to acquire the velocity fields. Planar PIV measurements were performed for all cases in table 1 in the $x\text {--}y$ plane along the centreline of the orifice. The coordinate system used has $x$, $y$ and $z$ in the streamwise, wall-normal and spanwise directions, respectively. Two aligned Litron 200 mJ dual-pulse Nd:YAG lasers were used to create a single laser sheet with a thickness of approximately 1 mm. Seeding is provided by a Martin Magnum 1200 smoke machine ejecting smoke particles with a mean diameter of $1\ \mathrm {\mu } \textrm {m}$. Three Lavision imager pro LX 16 MP cameras, each fitted with a 200 mm focal length lens, were placed on the left side of the test section and positioned next to the one another facing the region of interest (figure 2b).

The laser and the cameras were phase locked to the driving signal at $12$ equidistant phases. For each phase, $200$ image pairs per camera were recorded. Vector fields were determined by GPU processing using Lavision Davis 8.3.0 software. An initial step with a window size of $64\times 64\ \textrm {px}$ with an overlap of $50\,\%$ was applied followed by two passes of $24\times 24\ \textrm {px}$ with a $75\,\%$ overlap. This resulted in a resolution of 0.21 mm per vector in both the $x$ and $y$ directions. The complete flow field was obtained by stitching the frames from each camera.

The uncertainty associated with such PIV measurements depends on a range of factors which include the laser and optical set-up, the camera settings, seeding particles, background noise, calibration procedure and cross-correlation algorithm used amongst others. Although there exist multiple approaches to determine the uncertainty in the derived velocity fields, a simplified version (2.3) is adopted in this study based on the guidelines of Raffel et al. (Reference Raffel, Willert, Wereley and Kompenhans2007) and Sciacchitano & Wieneke (Reference Sciacchitano and Wieneke2016). The instantaneous velocity component, in this example the streamwise component, $u_i$, is obtained by taking into account the displacement of the seeding particles in pixels ($\Delta x$) over a certain time interval ($\Delta t$)

(2.3)\begin{equation} u_i = M_o \frac{\Delta x}{\Delta t}. \end{equation}

The magnification factor, $M_o$, is computed using the pixel pitch and the digital resolution. The latter represents the ratio of a known length scale ($l_c$) in the object plane and its subsequent dimension in pixels in the image plane ($n_c$). The uncertainty in magnification factor is thus

(2.4)\begin{equation} \frac{\delta M_o}{M_o} = \sqrt{\left(\frac{\delta l_c}{l_c}\right)^2+\left(\frac{\delta n_c}{n_c}\right)^2}. \end{equation}

Since the uncertainty in pixel length is negligible, the uncertainty in instantaneous velocity simplifies to

(2.5)\begin{equation} \frac{\delta u_i}{u_i} = \sqrt{\left(\frac{\delta M_o}{M_o}\right)^2+\left(\frac{\delta \Delta x}{\Delta x}\right)^2}. \end{equation}

The final ensemble-averaged vector field is obtained by averaging over all the recorded snapshots, $N$, which should be accounted for during the calculation of the final uncertainty (2.6). In this experiment, $\delta M_o/M_o$ was 0.025 and $\delta \Delta x/ \Delta x$ was 0.0063. For 200 image pairs and in the free stream where the streamwise velocity was $8\ \textrm {m}\ \textrm {s}^{-1}$, this results in an uncertainty in the streamwise velocity component of $\delta u/u=\pm 0.025$

(2.6)\begin{equation} \frac{\delta u}{u} = \sqrt{\left(\frac{\delta M_o}{M_o}\right)^2+\frac{1}{N}\left(\frac{\delta \Delta x}{\Delta x}\right)^2}. \end{equation}

3.1. Extracting synthetic jet trajectory

Jet trajectories can be determined by identifying and tracking the movement of CVPs. Other means include the position of the local velocity maximum, local scalar maximum or the time-averaged streamline originating from the orifice (Mahesh Reference Mahesh2013). These concepts, coupled with vorticity-based low-order models, have been used to predict the global trajectories of transverse jets at high velocity ratios (Karagozian Reference Karagozian2014). However, the derived scaling laws have shown singular sensitivity to the method used to extract the trajectory. For example, a jet trajectory based on the centreline streamline penetrates deeper into the cross-flow in comparison to vorticity-based trajectories (Fearn & Weston Reference Fearn and Weston1974). Smith & Mungal (Reference Smith and Mungal1998) corroborated these findings and reported that trajectories based on the maximum local velocity penetrates 5 %–10 % deeper into the cross-flow compared to trajectories extracted using the maximum scalar concentration.

Although many methods have been proposed, using the velocity-deficit approach is favoured in this study. As the jet evolves and interacts with the cross-flow, streamwise momentum is transferred from the cross-flow to the jet and results in a velocity deficit being observed in the streamwise velocity component. Such a velocity deficit has been attributed to viscous blockage, up-wash of low momentum fluid and the velocity induced by the vortical structures which is in opposition to the cross-flow (Berk & Ganapathisubramani Reference Berk and Ganapathisubramani2019). Under conditions of vortex interaction, the vortical structures generated by the synthetic jet are disintegrated and no longer remain coherent in the far field. Thus, the vortex tracking method is lacking in such circumstances while the velocity deficit can still be computed. The trajectory is obtained by extracting the coordinates at which the maximum velocity deficit occurs. Such an example is given in figures 3(a) and 3(b) for case $7$ (table 1), for which the boundary layer has been subtracted from the phase-averaged flow field. It should be highlighted that the trajectory is time averaged, unless otherwise specified.

Figure 3. Illustration of the use of the velocity deficit to extract the trajectory for case $7$. (a) Variation of the streamwise velocity-deficit profile, $\Delta u$, with normalised wall-normal distance, $y/\delta$ for streamwise locations, $x/\delta$, of $0.29$, $0.97$, $2.90$ and $4.84$. (b) Contour plot of velocity deficit obtained by subtracting the boundary layer from the streamwise velocity flow field. The dotted lines correspond to the streamwise locations in (a) and the orange scatter represents the trajectory of the jet as extracted from the raw data.

3.2. Nonlinear fitting process

This study adopts the same concept of nonlinear regression as that used by Berk et al. (Reference Berk, Hutchins, Marusic and Ganapathisubramani2018) to identify the scaling parameters for a synthetic jet trajectory. Although it is common practice to fit scaling parameters against pre-defined non-dimensional groups, such a method is accompanied by limitations as the results are confined to these assumed groups. Instead, a more sensible approach is to fit the scaling parameter against dimensional quantities and use the subsequent coefficients as a guidance to establish the relevant non-dimensional numbers. In the current investigation, the dimensional quantities relevant for the interaction of the synthetic jet with a cross-flow are the mean blowing velocity of the jet, $\overline {u_j}$, the actuation frequency, $f$, and the orifice length along the major axis, $l$, for constant free-stream velocity, $u_{\infty }$, and corresponding cross-flow properties. The kinematic viscosity, $\nu$, is also assumed to be constant for conditions of standard temperature and pressure throughout the data acquisition process. This results in the general form of (1.1) adopting the following arrangement:

(3.1)\begin{equation} \frac{y}{(\overline{u_j}^{\alpha_1}f^{\alpha_2}l^{\alpha_3}u_{\infty}^{\alpha_4}L_1^{\alpha_5})D} = A\left(\frac{x}{(\overline{u_j}^{\alpha_1}f^{\alpha_2}l^{\alpha_3}u_{\infty}^{\alpha_4}L_1^{\alpha_5})D}\right)^n. \end{equation}

The values of the constants $A$ and $n$ and the coefficients $\alpha _{1-3}$, for each configuration described in figure 1, are determined through nonlinear regression of the trajectories extracted from the flow field based on the lowest root-mean-square (r.m.s.) value of the residuals (table 3). Here, $D$ represents a quantity used to normalise the wall-normal and streamwise locations of the jet ($x,y$) and can correspond to any unvarying length scale such as $d$ or $\theta$. The choice for $D$ has no impact on the exponents of the dimensional parameters and only alters the value of $A$, a scaling factor dependent on $d/\theta$. Thus, in the current study, $D$ is assumed to have a value of 1 m for simplicity. Further, $L_1$ is considered to be a fixed length scale or combination of fixed length scales ($d,\theta$) used to normalise the dimensional parameters varied. Since the free-stream streamwise velocity, $u_{\infty }$ and length scale(s), $L_1$, were constant throughout, $\alpha _4$ and $\alpha _5$ cannot be directly obtained from the fitting process and need to be derived from the values of $\alpha _{1-3}$ based on dimensional analysis. As $g(\ldots )$ is non-dimensional, it follows that

(3.2)\begin{equation} [g(\ldots)]=\overline{u_j}^{\alpha_1}f^{\alpha_2}l^{\alpha_3}u_{\infty}^{\alpha_4}L_1^{\alpha_5}=[\textrm{ms}^{{-}1}]^{ \alpha_1}[\textrm{s}^{{-}1}]^{\alpha_2}[\textrm{m}]^{\alpha_3}[\textrm{ms}^{{-}1}]^{\alpha_4}[\textrm{m}]^{\alpha_5}=[\textrm{ms}]^0. \end{equation}

This inequality can be solved to determine the values of $\alpha _4$ and $\alpha _5$

(3.3)\begin{gather} s: \alpha_4={-}\alpha_1-\alpha_2. \end{gather}

(3.4)\begin{gather}m: \alpha_5={-}\alpha_1-\alpha_3-\alpha_4. \end{gather}

Following Buckingham-pi theorem, the number of independent non-dimensional groups which can be formed out of the five dimensional parameters is determined. Grouping of the non-dimensional terms is then performed based on educated assumptions which rely on previous literature. The exact form of the non-dimensional group describing the scaling parameter and the physical interpretation for each corresponding category are provided in § 4.

3.3. Near-field and far-field delimitations

There is considerable ambiguity associated with existing thresholds to distinguish between the near field and the far field. Indeed, applying definitions by Broadwell & Breidenthal (Reference Broadwell and Breidenthal1984) or Smith & Glezer (Reference Smith and Glezer1998) does not delineate the data obtained in this study, possibly due to the fact that velocity ratios are less than 1.0 and the cross-flow has a turbulent boundary layer profile. Hence, there is a need to redefine this feature for the current cases. The use of the gradient of the trajectory is deemed a suitable alternative as it is representative of the ratio of the rate of increase in the wall-normal direction and the rate of increase in the streamwise direction. Over a cycle, this is in fact a function of the velocity ratio (3.5)

(3.5)\begin{equation} \frac{\delta y}{\delta x} = f\left(\frac{\overline{u_j}}{u_{\infty}}\right). \end{equation}

For each value of the gradient, in the range between $0.05$ and $0.35$ the time-averaged trajectory for each case is separated into near-field and far-field regions. Following concatenation of all points in the near field, nonlinear regression can be performed against the variable parameters and a residual value of this fit can be determined. The same process can be applied to points in the far field and repeated for all gradient values in the specified range. The threshold is selected as the point at which the value of the error of the fit remains invariant (figure 4b). Further, as observed from figure 4b, the far field occurs when the gradient of the trajectory is $0.25$, which corresponds to an angle of $14\,^{\circ }$ with respect to the horizontal. For all cases in table 1, the streamwise location ($x_t$) at which the trajectory is deflected at an angle of $14^{\circ }$ can be determined (figure 5a). By fitting the $x$-coordinate of these streamwise locations against the dimensional parameters of the jet (table 2), it is possible to determine an empirical equation describing the threshold for the onset of the far field which has a general form of

(3.6)\begin{equation} \frac{x_t}{\theta} = A\left(\frac{u_j}{u_{\infty}}\right)^{0.5}\left(\frac{fl}{u_j}\right)^{0.1}\left(\frac{l}{L_1}\right)^{0.5}, \end{equation}

where $L_1$ is representative of a length scale and $A$ is a constant.

Figure 4. (a) Time-averaged trajectory of a synthetic jet overlaid with the gradient of the time-averaged trajectory for case 7. The dark grey region represents the near field, the lighter grey corresponds to the transition region and the lightest grey area represents the far field. (b) A plot of the normalised r.m.s. error against gradient of the trajectory for nonlinear regression of near-field (NF) and far-field (FF) data against dimensional parameters of the jet. The dotted line represents the value of the gradient at which convergence in the normalised r.m.s. error occurs.

Figure 5. (a) Scatter plot of the coordinates $(x_t,y_t)$ of the onset of far field for cases in table 1. (b) Scatter plot of the coordinates $(x_t,y_t)$ normalised by the boundary layer thickness, $\delta$. (c) Plot of normalised $x_t$ obtained experimentally against normalised $x_t$ from newly derived empirical model.

Table 2. Fitted coefficients and normalised residual for the scaling of the streamwise location corresponding to the onset of the far field where the gradient of the time-averaged trajectory is 0.25. The 95 % confidence interval limits ($CI_{95}$) for the fitted parameters are also provided.

From the coefficients obtained through nonlinear regression, it is sensible to normalise the jet exit velocity with the free-stream velocity, leading to the velocity ratio. Further based on the lowest r.m.s. value of the residual from the fit, the non-dimensional form of the actuation frequency comprises the orifice slot length and the jet exit velocity. Further, $L_1$ can adopt any parameter between the orifice slot width ($d$) or the momentum thickness ($\theta$) since these remained unchanged throughout the experiment and both combinations resulted in the lowest residual. Selecting any of these only changes the constant, $A$, which is a scaling factor

(3.7)\begin{equation} \frac{x_t}{\theta} = 2.96\left(\frac{u_j}{u_{\infty}}\right)^{0.5}\left(\frac{fl}{u_j}\right)^{0.1}\left(\frac{l}{d}\right)^{0.5}. \end{equation}

(3.8)\begin{equation}\frac{x_t}{\theta} = 2.96\,St^{0.1}\sqrt{r\,AR}. \end{equation}

When $L_1$ is the orifice slot width, $d$, the $x$-location defining the far field depends on the aspect ratio of the orifice in addition to the velocity ratio and the jet-based Strouhal number (3.8). These results show a certain degree of similarity with Broadwell & Breidenthal (Reference Broadwell and Breidenthal1984) ($\propto rd$) and Smith & Mungal (Reference Smith and Mungal1998) ($\propto r^2d$) in that the far field depends on the velocity ratio and the dimensions of the orifice. However, a contrasting exponent is achieved for $r$ and is believed to arise from the low velocity ratios used in this study. This implies that, as the jet reaches the far field, the influence of the jet exit velocity is considerably less than the free-stream velocity and the jet moves mostly in the streamwise direction. The emergence of the aspect ratio is not surprising as Van Buren et al. (Reference Van Buren, Beyar, Leong and Amitay2016) has previously reported that the size of the flow field exhibiting unsteadiness relies on the velocity ratio and the aspect ratio.

When $L_1$ corresponds to the momentum thickness of the boundary layer, (3.9) can be rearranged to include the momentum coefficient (3.10). Thus, apart from the Strouhal number, the onset of the far field can be defined by the square root of the ratio of the momentum coefficient and the velocity ratio. This alternate arrangement illustrates the momentum transfer between the jet and the cross-flow and its importance in delimiting the flow field. In fact, it shows that the ratio of momentum per mass flux shapes the flow field and the far field is achieved when an equilibrium state is reached, analogous to Van Buren et al. (Reference Van Buren, Beyar, Leong and Amitay2016), where the flow attains a steady state in the far field. Applying this newly derived criterion to the dataset of Berk et al. (Reference Berk, Hutchins, Marusic and Ganapathisubramani2018) results in approximately $50\,\%$ of the data points being in the near-field region, thus contradicting the previous belief, based on the definition of Broadwell & Breidenthal (Reference Broadwell and Breidenthal1984), which stipulated that all the data points were located in the far-field region

(3.9)\begin{gather} \frac{x_t}{\theta} = 6.03\left(\frac{u_j}{u_{\infty}}\right)^{0.5}\left(\frac{fl}{u_j}\right)^{0.1}\left(\frac{l}{\theta}\right)^{0.5}. \end{gather}

(3.10)\begin{gather}\frac{x_t}{\theta} = 6.03~St^{0.1}\sqrt{\frac{C_{\mu}}{r}}. \end{gather}

3.4. Vortex pair interaction

To validate the vortex interaction criterion defined by Johari (Reference Johari2006), we consider cases 22, 23 and 24, which have the same momentum coefficient, $C_{\mu }=0.73$, but actuation frequencies of 100 Hz, 200 Hz and 400 Hz, respectively (figure 6). At a low actuation frequency $(\,f = 100\ \textrm {Hz},\lambda _i=1.17)$, independent vortical structures can be observed convecting downstream into the far field without any interaction. The substantial tilting of the vortex ring and stretched legs are clear signs of coherent structures. Berk & Ganapathisubramani (Reference Berk and Ganapathisubramani2019) attributed the tilting aspect to the slight self-induced motion of the ring which is in opposition to the incoming flow. These coherent structures are perturbed by an increase in actuation frequency as the vortex interaction index falls below 1.0 ($f = 200\ \textrm {Hz},\lambda _i=0.57$ and $f = 400\ \textrm {Hz},\lambda _i=0.28$). Under such circumstances, the spatial separation between subsequent pulses abates, leading to vortex interaction. These observations (figure 6b,c) agree with Eroglu & Breidenthal (Reference Eroglu and Breidenthal2001), where the main flow structures consisted of distorted vortex rings which have the stretched leeward portion merged with the windward part of the ring formed during the previous cycle. Additionally, the location at which the interaction occurs shifts further upstream and closer to the orifice exit with a decreasing value of the vortex interaction index.

Figure 6. Plots of the boundary layer subtracted streamwise velocity field, $\Delta u$, normalised with the free-stream velocity when the jet is at peak blowing, i.e. at phase-locked value of $\phi = 90^{\circ }$, for (a) case 22, (b) case 23 and (c) case 24. These cases have the same aspect ratio and velocity ratio, resulting in the same momentum ratio per cycle of actuation, but differing actuation frequency.

The interaction of vortical structures is also present in the similarity analysis of the phase-locked streamwise velocity field. Keffer & Baines (Reference Keffer and Baines1963) stated that the transverse jet does not achieve self-similarity, as the flow at $20d$ downstream is still dominated by the twin vortex which prevents any asymptotic tendency towards self-similarity. However, it is clear now that such conclusion emanated from the limited flow field region investigated. In the case of pulsed jets or synthetic jets, the self-similar behaviour of the jet is dependent on the vortex interaction phenomenon. From figure 7, it can be observed that the velocity profile of the jet tends to similarity for cases with vortex interaction. Under such configurations, the coherent structures have already broken down and the flow is analogous to turbulent continuous jets. The interaction of subsequent pulses alters the distribution of vorticity within the structures which undergo disintegration and achieve a quasi-steady jet-like state. These results are consistent with the findings of Amitay et al. (Reference Amitay, Kibens, Parekh and Glezer1999), who observed a reduction in turbulent kinetic energy of the spectrum and the disappearance of any peak corresponding to the passage frequency of coherent structures. The emergence of the spectral band having a $-5/3$ slope was also noted, indicating enhanced dissipation. A similar analysis can be adopted for the interaction of coherent structures in the current study. Interaction and break down of coherent structures lead to a turbulent jet which carries similar properties as a continuous jet in the far field.

Figure 7. Self-similarity analysis for the cases presented in figure 6. (a) $f = 100\ \textrm {Hz}$, (b) $f = 200\ \textrm {Hz}$ and (c) $f = 400\ \textrm {Hz}$. The velocity deficit $\Delta u$, following subtraction of the unperturbed flow field, is normalised by the maximum velocity-deficit value of the profile ($\Delta u_{min}$) while $y$, $y_0$ and $y_{1/2}$ represent the wall-normal location, wall-normal location at which the maximum velocity deficit occurs and the jet half-width, respectively.

4.1. Near field with vortex interaction

Based on the coefficients found in table 3, the non-dimensional groups can be formed following (4.1)–4.4, which result in reasonable collapse of the trajectories (figure 8). The jet exit velocity can be coupled with the free-stream velocity to form the velocity ratio. Based on the remaining coefficients, the Strouhal number is observed to depend on the free-stream velocity and a fixed length scale. It is sensible that $L_1$ takes the form of the momentum thickness, consistent with the notion that the exchange of momentum is important in defining the jet trajectory. The velocity ratio and ratio of length scales ($l/\theta$) can be rearranged as $C_{\mu }/r$. This quantity represents the ratio of momentum transferred per mass of fluid over a certain area

(4.1)\begin{gather} g(\ldots) = \left(\frac{u_j}{u_{\infty}}\right)^{0.85}f^{{-}0.17}l^{1.11}u_{\infty}^{0.17} \theta^{{-}1.28}. \end{gather}

(4.2)\begin{gather}g(\ldots) = \left(\frac{u_j}{u_{\infty}}\right)^{0.85}\left(\frac{f \theta}{u_{\infty}}\right)^{{-}0.17}\left(\frac{l}{\theta}\right)^{1.11}. \end{gather}

Rounding off these values results in

(4.3)\begin{equation} g(\ldots) = \left(\frac{u_j}{u_{\infty}}\right)\left(\frac{f \theta}{u_{\infty}}\right)^{{-}0.2}\left(\frac{l}{\theta}\right), \end{equation}

which can be rearranged as

(4.4)\begin{equation} g(\ldots) =\frac{C_{\mu}}{r}St_1^{{-}0.2}. \end{equation}

Figure 8. (a) Trajectories for all cases listed in table 1 which satisfy vortex interaction, in the near field. (b,c) The trajectories have been scaled by the corresponding $g(\ldots )$ as defined in (4.4) based on the respective coefficients in table 3. (d) Comparison of the time-averaged trajectories between the experimental data ($y_{expt}$) and the newly derived model ($y_{mod}$).

Figures 8(a)–8(d) show the scattered time-averaged jet trajectories pre- and post-scaling with the group of non-dimensional parameters derived in (4.4). The dependence of $g(\ldots )$ on $C_{\mu }/r$ and a cross-flow-based Strouhal number highlights the importance of momentum transfer between the cross-flow and the jet in determining the near-field trajectory. Such an approach reflects the ideology of Muppidi & Mahesh (Reference Muppidi and Mahesh2005), who emphasised the importance of considering the distribution of momentum when scaling jet trajectories and is further discussed in § 5.

4.2. Far field with vortex interaction

By manipulating the exponents for vortex interaction cases in the far field (4.5)–(4.7), it can be observed that the same non-dimensional numbers as in the near field remain important in collapsing the trajectory

(4.5)\begin{gather} g(\ldots) = \left(\frac{u_j}{u_{\infty}}\right)^{0.52}f^{{-}0.2}l^{0.7}u_{\infty}^{0.2} \theta^{{-}0.9}. \end{gather}

(4.6)\begin{gather}g(\ldots) = \left(\frac{u_j}{u_{\infty}}\right)^{0.52}\left(\frac{f \theta}{u_{\infty}}\right)^{{-}0.2}\left(\frac{l}{\theta}\right)^{0.7}. \end{gather}

Rounding off these values and rearranging the terms results in

(4.7)\begin{equation} g(\ldots) =\sqrt{\frac{C_{\mu}}{r}}St_1^{{-}0.2}\left(\frac{l}{\theta}\right)^{0.2}. \end{equation}

Alternatively, the momentum thickness could be replaced by the orifice slot width to account for the aspect ratio

(4.8)\begin{equation} g(\ldots) =\sqrt{\frac{C_{\mu}}{r}}St_1^{{-}0.2}\left(\frac{l}{d}\right)^{0.2}. \end{equation}

Figures 9(a)–9(d) show the scattered time-averaged jet trajectories pre- and post-scaling with the group of non-dimensional parameters derived in (4.8). The most appropriate parameter to non-dimensionalise the orifice length, $l$, cannot be established since both the momentum thickness of the boundary layer and the slot width remained unchanged throughout the experiment. However, the inclusion of $d$ agrees with the findings of Van Buren et al. (Reference Van Buren, Beyar, Leong and Amitay2016). The aspect ratio of the orifice is key in reorienting the vorticity in the flow field upon interaction between the coherent structures and bears relevance in the subsequent trajectory followed by the jet. Determining the exact form of this parameter is not within the scope of this study due to the limitation in the dataset. Nonetheless, $g(\ldots )$ reveals that the ratio of momentum per mass of fluid ($C_{\mu }/r$) is no longer to the leading order. This implies that the bulk of the momentum transfer between the cross-flow and the jet transpires in the near field. The influence of the cross-flow predominates in the far field as the jet convects primarily in the streamwise direction and the flow field echoing characteristics of a quasi-steady flow.

Figure 9. (a) Trajectories for all cases listed in table 1 which satisfy vortex interaction, in the far field. (b,c) The trajectories have been scaled by the corresponding $g(\ldots )$ as defined in (4.8) based on the respective coefficients in table 3. (d) Comparison of the time-averaged trajectories between the experimental data ($y_{expt}$) and the newly derived model ($y_{mod}$).

4.3. Near field with independent vortex rings

If the ejected vortical structures are independent of one another, the near-field trajectories collapse when the non-dimensional groups are formed using (4.9)–(4.11). Formation of the velocity ratio leaves three dimensional parameters ($f$, $l$ and $u_{\infty }$) with the same exponents, implying that these could be grouped together as the Strouhal number. The exact form of the Strouhal number is found to depend on the length of the orifice aligned with the cross-flow and the free-stream velocity. Although the coefficients/exponents are marginally different, this result is consistent with the findings of Berk et al. (Reference Berk, Hutchins, Marusic and Ganapathisubramani2018), who identified the same non-dimensional numbers. The discrepancy in the values of the exponents is believed to arise from the lack of consideration of the constraints influencing the jet trajectory. Applying the predefined constraints to the dataset from Berk et al. (Reference Berk, Hutchins, Marusic and Ganapathisubramani2018) reveals that $15\,\%$ of the cases in that study incur vortex interaction and $50\,\%$ of the remaining data points are in the near field. Applying nonlinear regression on these points results in the exponents matching the ones in this study (table 3). Any deviations in the values of $A$ and $n$ can be attributed to the number of data points used for the nonlinear regression process

(4.9)\begin{gather} g(\ldots) = \left(\frac{u_j}{u_{\infty}}\right)^{1.4}f^{{-}0.4}l^{{-}0.4}u_{\infty}^{0.4}. \end{gather}

(4.10)\begin{gather}g(\ldots) = \left(\frac{u_j}{u_{\infty}}\right)^{1.4}\left(\frac{fl}{u_{\infty}}\right)^{{-}0.4}. \end{gather}

Rounding off these values and rearranging the terms result in

(4.11)\begin{equation} g(\ldots) =r^{1.4}St_2^{{-}0.4}. \end{equation}

Alternatively, (4.9) can be rearranged such as to include the stroke length, $L_o$

(4.12)\begin{align} g(\ldots) = \left(\frac{u_j}{u_{\infty}}\right)^{1.0}f^{{-}0.4}l^{{-}0.4}u_j^{0.4} = \left(\frac{u_j}{u_{\infty}}\right)^{1.0}\left(\frac{f}{u_j}\right)^{{-}0.4}l^{{-}0.4} = \left(\frac{u_j}{u_{\infty}}\right)^{1.0}\left(\frac{l}{L_o}\right)^{{-}0.4}. \end{align}

Figures 10(a)–10(d) shows the scattered time-averaged jet trajectories pre- and post-scaling with the group of non-dimensional parameters derived in (4.11). The non-dimensional parameters constituting $g(\ldots )$ indicate that, in the absence of vortex interaction, synthetic jet trajectories in the near field only rely on the initial impulse of the CVP and the spatial separation between subsequent vortical structures, akin to the findings of Berk et al. (Reference Berk, Hutchins, Marusic and Ganapathisubramani2018). The possibility of including the stroke length, important in vortex formation, in (4.12) shows the importance of this parameter in dictating the trajectory of the synthetic jet in the near field. The non-dimensional groups derived in this case show similarities with the scaling derived by Johari (Reference Johari2006) for pulsed jets in cross-flow.

Figure 10. (a) Trajectories for all cases listed in table 1 where there is no vortex interaction in the near field. (b,c) The trajectories have been scaled by the corresponding $g(\ldots )$ as defined in (4.11) based on the respective coefficients in table 3. (d) Comparison of the time-averaged trajectories between the experimental data ($y_{expt}$) and the newly derived model ($y_{mod}$).

4.4. Far field with independent vortex rings

For individual vortical structures in the far field, the nonlinear fitting process reveals that the actuation frequency of the jet is insignificant in determining the trajectory. The velocity ratio and the ratio between the orifice slot length and the momentum thickness of the boundary layer are the pertinent non-dimensional groups which generate the lowest r.m.s. value of the residual. Analysis of the exponents of these parameters indicates that they can be combined together to form the momentum coefficient (1.5)

(4.13)\begin{gather} g(\ldots) = \left(\frac{u_j}{u_{\infty}}\right)^{0.58}f^{{-}0.03}l^{0.29}u_{\infty}^{0.03}\theta^{{-}0.32}. \end{gather}

(4.14)\begin{gather}g(\ldots) = \left(\frac{u_j}{u_{\infty}}\right)^{0.58}\left(\frac{f \theta}{u_{\infty}}\right)^{{-}0.03}\left(\frac{l}{\theta}\right)^{0.32}. \end{gather}

Rounding off these values and omitting the non-dimensional actuation frequency results in

(4.15)\begin{equation} g(\ldots) =C_{\mu}^{0.3}. \end{equation}

Figures 11(a)–11(d) show the scattered time-averaged jet trajectories pre- and post-scaling with the group of non-dimensional parameters derived in (4.15). For this configuration, $g(\ldots )$ is defined by the momentum coefficient only. It can thus be conjectured that in the absence of any form of interaction, the path followed by independent vortical structures in the far field solely depends on the initial ratio of total momentum between the jet and the cross-flow over one cycle of actuation.

Figure 11. (a) Trajectories for all cases listed in table 1 where there is no vortex interaction in the far field. (b,c) The trajectories have been scaled by the corresponding $g(\ldots )$ as defined in (4.15) based on the respective coefficients in table 3. (d) Comparison of the time-averaged trajectories between the experimental data ($y_{expt}$) and the newly derived model ($y_{mod}$).