2.1. Experimental facility and operating conditions

Experiments were conducted in an open circuit blow-down wind tunnel at the Interdisciplinary Center for Energy Research, Indian Institute of Science, Bangalore. All of the experiments were performed with a free stream Mach number ( $M_\infty$ ) of 2.5. The test section has a cross-section of 15 cm x 15 cm with a length of 1 m. Transparent windows on both the side and top of the test section allowed for optical access. The streamwise, cross-stream and spanwise directions are denoted by $x$ , $y$ and $z$ , respectively, with $x$ being the direction of cross-stream air and $y$ the direction of injection of the liquid jet. The liquid nozzle was flush mounted on the tunnel wall, at a streamwise location of 100 mm from the C-D nozzle exit. The free stream (cross-stream) velocity ( $U_\infty$ ) and the incoming boundary-layer thickness ( $\delta$ ) measured at this injection location using PIV were 585 m s⁻¹ and 8.85 mm, respectively. The turbulence level measured in the free stream was lower than 1.5 % $U_\infty$ (Munuswamy Reference Munuswamy2020). The schematic of the wind tunnel, liquid injection system and sharp-edged nozzle are depicted in figures 2(a) and 2(b), respectively.

Figure 2. (a) Schematic of supersonic wind tunnel with liquid injection facility. (b) Schematic of the sharp-edged injector. $L$ and $D$ represent the length of the injector and the equivalent diameter of the orifice, respectively. Blue coloured arrow denotes the direction of water flow inside the nozzle.

An elliptical and a circular orifice with an equivalent diameter ( $D$ ) of 1.2 mm and a length-to-diameter ratio ( $L/D$ ) of 2 were selected for the current study. Using the two orifice shapes, three configurations were investigated based on the orientation of the orifice axis with respect to the cross-flow, as listed in table 1. It is important to note that when the $\textit{AR}$ is changed, the centre of the orifice ( $O$ ) is kept at the same streamwise location (100 mm). This ensures that the injected liquid jet experiences a similar boundary layer thickness for all $\textit{AR}$ . Furthermore, the orifice exit area remains constant for all three cases. The details of the orifice shape, dimensions and its $\textit{AR}$ for all three configurations are listed in table 1.

Table 1. Jet orifice geometric details used in the present study. The arrows denote the cross-stream direction. The streamwise and spanwise dimensions of the elliptical orifice are denoted as $a$ and $b$ , respectively.

The key dimensionless groups characterising the flow include:

1. (i) free stream (cross-flow) Mach number, $M_\infty = U_\infty /\sqrt {\gamma R T_\infty }$ ;

2. (ii) momentum flux ratio, defined as the ratio of momentum flux of the jet to momentum flux of the cross-flow, expressed as $J = (\rho _j U_j^2)/(\rho _\infty U_\infty ^2)$ ;

3. (iii) aerodynamic Weber number, defined as the ratio of aerodynamic force to surface tension force, expressed as We $_\infty =(\rho _\infty U_\infty ^2 D)/\sigma$ ;

4. (iv) Reynolds number of the liquid jet expressed as Re $_j=(\rho _j U_j D)/\mu _j$ ;

5. (v) Orifice aspect ratio ( $\textit{AR}$ ),

where $\rho _j$ , $U_j$ and $\rho _\infty$ , $U_\infty$ denote the density and velocity of the liquid jet and the cross-stream (free stream) air, respectively. The mean velocity of the liquid jet is estimated from the volume flow rate and the orifice exit area. The surface tension of the air–water interface and the dynamic viscosity of water are represented by $\sigma$ and $\mu _j$ , respectively. The static temperature of the air, specific heat ratio and gas constant are denoted by $T_\infty, \gamma$ and  $R$ , respectively. The values or range of values of experimental parameters in the present investigation are summarised in table 2.

Table 2. Values of experimental parameters considered in the present study during jet injection.

2.2. Experimental methods

Figure 3. Schematic showing the main components and arrangements of pulsed laser shadowgraphy (PLS), particle/droplet image analysis (PDIA) and particle image velocimetry (PIV) used in the present work. Visualisation plane in these experiments is the mid-span plane ( $z$ = 0), as indicated with a green thin coloured sheet. The laser head is connected to a diffuser in the case of PLS and PDIA, and to the sheet optics for PIV.

Atomisation of a liquid jet in a supersonic flow is a multiscale phenomenon that demands different imaging systems operated at different spatial and temporal resolutions (Fdida et al. Reference Fdida, Mallart-Martinez, Le Pichon and Vincent-Randonnier2022). Therefore, in the present investigation, we have employed different experimental techniques to visualise the flow field interactions. The pulsed laser shadowgraphy (PLS) technique is employed to identify the differences in spray morphology and the global breakup behaviour for different values of orifice $\textit{AR}$ . This technique comprises a fluorescent diffuser plate illuminated by a nanosecond pulsed laser (Litron 200 mJ pulse⁻¹, 532 nm dual cavity Nd-YAG laser), a 4MP (2360 × 1776 pixels) microsecond exposure CCD camera (Imager SX) and a programmable timing unit (PTU), which acts as a synchroniser between the laser and the camera, as shown schematically in figure 3. Images are acquired near the jet exit, focusing on a narrow field of view of approximately 30 mm x 23 mm (25 $D$ x 19 $D$ ) using a 105 mm Nikon lens at an acquisition rate that was limited to the low pulse rate of 15 Hz. Therefore, to reveal the temporal dynamics as well as the complex shock structures involved during the jet cross-flow interaction, high-speed shadowgraphy was employed, where the framing rate was 10 000 Hz. The major differences between the PLS and high-speed shadowgraphy are the light source, camera and acquisition rate. In this technique, instead of a diffused laser light source, a collimated light beam was produced using a spherical concave mirror (20 cm in diameter) and a halogen lamp (150 W). A high-speed camera (Photron, FASTCAM SA5) with a microsecond exposure setting was used to acquire the images. These images were acquired at 10 000 Hz with a pixel resolution of 30 pixels mm⁻¹ using a 105 mm Nikon lens in front of the camera. This allows visualisation of a very narrow field of view of approximately 15 $D$ x 20 $D$ in the streamwise and transverse directions, respectively. In both techniques, the spray was illuminated through transparent windows on the back side and the density gradients were captured by a camera viewing through the transparent front-side window.

The microscopic details of the downstream spray droplets, viz. droplet size and velocities, were determined by using particle/droplet image analysis (PDIA). This is a well-established technique and has been applied to a variety of two-phase flow scenarios, ranging from ambient sprays (Kashdan et al. Reference Kashdan, Shrimpton and Whybrew2003, Reference Kashdan, Shrimpton and Whybrew2004; Kourmatzis, Pham & Masri Reference Kourmatzis, Pham and Masri2015) to liquid jets in subsonic cross-flow (Sinha et al. Reference Sinha, Prakash, Mohan and Ravikrishna2015; Prakash et al. Reference Prakash, Sinha, Tomar and Ravikrishna2018). Recently, Medipati et al. (Reference Medipati, Deivandren and Govardhan2023) also used the technique for a transverse circular jet in a supersonic flow case and shown that the droplet size data across the plume are in good agreement with the well-established phase Doppler particle analyser (PDPA) measurements in a similar flow configuration (Lin et al. Reference Lin, Kennedy and Jackson2004). The experimental set-up is similar to PLS except that the lens connected to the camera head was replaced with a long-distance microscope (LDM) (Questar, QM-100). This helped visualise the flow down to a very small field of view of approximately 2 mm x 1.5 mm. The chosen laser pulse duration (4 ns), along with a microsecond exposure camera, enabled us to obtain high-resolution instantaneous spray images with negligible motion blur. By operating the laser in a double exposure mode with a known interframe time of 0.5 $\unicode{x03BC}$ s, time delayed pairs of images were captured. Particle tracking velocimetry (PTV) was employed on these time delayed pairs of images to obtain the velocity achieved by the spray droplets. It may be noted that the recent droplet velocity measurements obtained by digital inline holography (Johnson et al. Reference Johnson, Marsh, Douglas, Ochs, Hammack, Menon and Mazumdar2024) for a similar circular jet in the supersonic cross-flow configuration were found to be in good agreement with the velocities obtained by particle tracking through PDIA measurements (Medipati et al. Reference Medipati, Deivandren and Govardhan2023). At each station in the spray plume, a large number of instantaneous images (approximately 500) were captured during a single run. This resulted in the collection of many droplets for the estimation of long time averaged statistics of droplet size and velocities. The process was repeated at various stations in the spray plume to determine the effect of $\textit{AR}$ on the dispersion of droplets in the spray plume. It is important to note that this technique captures and quantifies the sizes as well as velocities of both spherical and non-spherical droplets (Kashdan et al. Reference Kashdan, Shrimpton and Whybrew2003, Reference Kashdan, Shrimpton and Whybrew2004). A concern with PDIA is that the technique has a limitation in resolving smaller droplet sizes below 4–5 $\unicode{x03BC}$ m, due to diffraction limits (Sinha et al. Reference Sinha, Prakash, Mohan and Ravikrishna2015; Prakash et al. Reference Prakash, Sinha, Tomar and Ravikrishna2018). Hence, in the present work, droplet sizes below 5 $\unicode{x03BC}$ m are not considered and are not used for the calculation of drop size statistics. Its effect on the values of SMD is minimal as the SMD mean is closer to larger droplets in the distribution. The uncertainty in drop size measurement with this technique, as calculated using the procedure of Sinha et al. (Reference Sinha, Prakash, Mohan and Ravikrishna2015), was found to be below 2 $\unicode{x03BC}$ m.

To understand the source of unsteadiness in jet penetration height and shock motion, PIV was used upstream of the jet exit. The experimental test set-up for PIV remained the same as PLS and PDIA, but with minor changes on the laser side. The fluorescent diffuser plate connected to the laser head during PLS and PDIA was replaced with a 1.5 mm thick laser sheet with sheet optics ( $f$ = −10 mm). The laser sheet was allowed to enter the test section transversely through the transparent top window (spanwise central plane, $z = 0$ ) to illuminate the plane normal to the bottom wall ( $x$ – $y$ ). The camera (Imager SX), equipped with a Nikon 105 mm f/2.8D lens along with a 532 nm bandpass filter, was used for viewing the flow through the transparent side window. These experiments were performed during liquid jet injection with a field of view of approximately 25 $D$ x 15 $D$ . In this case, the free stream air was seeded with olive oil particles of 1 $\unicode{x03BC}$ m size, which acted as tracer particles and was introduced into the flow upstream of the settling chamber. To reduce the reflections from solid surfaces for the near-wall PIV measurements, a fluorescent paint was prepared, which contained a mixture of 3 g rhodamine 6G (C28H31N2O3Cl), 10 mL ethanol and 500 mL transparent acrylic paint (water soluble), and was coated on the wall surfaces. The images were acquired in a double exposure mode with an interframe time of 0.5 $\unicode{x03BC}$ s similar to PDIA. These instantaneous images were processed using Davis 8.4.0 software (LaVision GmbH) by adaptive correlation with multiple passes to obtain the PIV velocity fields. The final window size used was 32 x 16 pixels with 75 % overlap. The low/high momentum streaks present inside the boundary layer extend up to 40 $\delta$ in streamwise and 0.5 $\delta$ in spanwise directions (Ganapathisubramani et al. Reference Ganapathisubramani, Clemens and Dolling2007), which implies that we have an adequately large number of vectors (approximately 18) in the streamwise direction. Detailed information about the PIV used in the present study can be found in our previous studies (Murugan & Govardhan Reference Murugan and Govardhan2016; Munuswamy & Govardhan Reference Munuswamy and Govardhan2022). It is important to note that in the present investigation, all the measurements were carried out on the mid-spanwise plane ( $z$ = 0), as indicated by the green colour plane in figure 3.

3.1. Initial breakup mechanism of liquid jet

Figure 4. High-resolution instantaneous visualisations of the water jet in the supersonic cross-flow of $M_\infty$  = 2.5, captured using pulsed laser shadowgraphy highlighting the differences in the evolution of the surface waves in the windward side of the jet. These are acquired for (a) $\textit{AR}$ = 0.3, (b) $\textit{AR}$ = 1 and (c) $\textit{AR}$ = 3.3, and $ J$ = 3.7. Zoomed-in visualisations near the jet exit on the windward side are shown as insets on the left side. $\lambda$ and $\delta$ represent the surface wavelength and mean boundary-layer thickness, respectively. The column breakup location is indicated with a red coloured dot, and its instantaneous positions from the orifice centre in the streamwise and transverse directions are $x_{b}$ and $y_{b}$ , respectively.

Short-exposure (18 $\unicode{x03BC}$ s) visualisations of the injected water jet into the supersonic cross-flow acquired using pulsed laser shadowgraphy are shown in figure 4. A large set of such instantaneous images revealed a significant influence of $\textit{AR}$ on the formation of surface waves on the windward side of the liquid jet. In each $\textit{AR}$ case, as the jet traverses into the supersonic cross-flow, it is observed that near the jet exit (windward side), the injected jet is relatively free from surface deformations, presumably because of the wall boundary layer. A little further into the cross-flow, surface waves are seen on the windward side, as expected (Sallam et al. Reference Sallam, Aalburg and Faeth2004). These are likely due to the large accelerations experienced by the injected liquid jet when suddenly exposed to high-speed air (lighter) cross-flow. This accelerative destabilisation mechanism is the well-known Rayleigh–Taylor instability (RTI), which occurs when a low-density fluid accelerates over a high-density fluid (Taylor Reference Taylor1950). For $\textit{AR}$ = 0.3 (figure 4 a), it is evident that the surface waves formed have a larger wavelength ( $\lambda)$ when compared with those for $\textit{AR}$ = 1 (figure 4 b), with the liquid column deformations in both these cases being relatively smooth. In contrast, the AR = 3.3 case (figure 4 c) has a much smaller windward surface wavelength and the liquid column deformations are not as smooth as the previous cases. These differences seen in the observed surface waves between the different $\textit{AR}$ cases studied is indicative of significant variations in the primary destabilisation mechanism with orifice AR, as discussed now.

The primary instability mechanism of the liquid jet in the present experiments shares common features with other well-studied, two-phase flow scenarios, namely, atomisation of a droplet in high-speed cross-flow (Joseph, Belanger & Beavers Reference Joseph, Belanger and Beavers1999) and atomisation of a liquid jet when injected parallel to the high-speed gas stream (Varga, Lasheras & Hopfinger Reference Varga, Lasheras and Hopfinger2003; Maarmottant & Villermaux Reference Maarmottant and Villermaux2004). When a liquid drop is placed in a supersonic flow along with the initial flattening of the droplet perpendicular to the air stream (pressure difference), surface waves are formed on the windward side of the drop surface due to the strong accelerations experienced by the liquid droplet perpendicular to the interface and directed from gas to liquid. Such a droplet–air interaction will make the droplet unstable due to the RTI. These surface waves are driven from the stagnation point towards the equator of the drop by the shear flow of the gas, which can in turn also lead to Kelvin–Helmholtz instability (KHI). Close to the stagnation region of the drop, it is the RTI that will be dominant, as the KHI is negligible due to the fact that the tangential velocity is close to zero in this region. When these RT waves around the stagnation region are sufficiently amplified, the droplet can undergo catastrophic breakup into finer droplets (Joseph et al. Reference Joseph, Belanger and Beavers1999). However, further away from the stagnation region, the shear is significant and the KHI can also be important.

Figure 5. Schematic showing the variation in jet and cross-flow interaction and atomisation mechanisms for different ARs and at different transverse heights. Panels (a,b,c), (d,e,f) and (g,h,i) represent the cross-section of the jet at the orifice exit, the deformed jet slice and the spray core in the transverse plane (x–z), respectively. The arrows in panels (a,b,c) indicate the free stream direction. The liquid surface participating in the shear breakup is highlighted with red colour.

A similar scenario may also be anticipated in the present study, where instead of the flattened drop, we consider the flattened/deformed liquid jet slice at a given height from the wall. This is illustrated schematically in figure 5 for each of the three different orifice $\textit{AR}$ cases studied here, with each row corresponding to a given $\textit{AR}$ . In the figure, the first column (panels a,b,c) indicates the orifice geometry, the second column (panels d,e,f), the flattened or deformed liquid jet slice, and the third column (panels g,h,i), the associated flow field with spray. As seen from the schematics, there are considerable differences between the three $\textit{AR}$ cases, and this is discussed further now, starting from the circular orifice ( $\textit{AR}$ = 1) case, and moving onto the $\textit{AR}=3.3$ and $\textit{AR}=0.3$ cases.

The jet slice in $\textit{AR}$ = 1 deforms in the lateral direction into a cupcake shape and flattens, which is attributed to the pressure difference between the windward (high pressure) and leeward (low pressure) sides of the jet slice (Joseph et al. Reference Joseph, Belanger and Beavers1999; Chai et al. Reference Chai, Iyer and Mahesh2015; Behzad, Ashgriz & Karney Reference Behzad, Ashgriz and Karney2016). Along with the strong acceleration of the liquid slice due to RTI, the cross-flow fluid accelerates around the low horizontal momentum jet resulting in high levels of shear on the lateral sides of the jet leading to KHI (Joseph et al. Reference Joseph, Belanger and Beavers1999; Chai et al. Reference Chai, Iyer and Mahesh2015), which results in liquid stripping from the jet slice (figure 5 e). As the liquid slice traverses further, the flattening of the liquid slice continues and the increased projected frontal area results in a significant rise in aerodynamic drag forces (Wu et al. Reference Wu, Kirkendall, Fuller and Nejad1997). This eventually disintegrates the bulk liquid mass into liquid clumps due to RTI, which has been referred to as column breakup (Wu et al. Reference Wu, Kirkendall, Fuller and Nejad1997) (see figure 4). Simultaneously, there is the formation of ligaments and droplets from the jet surface due to shear instability caused by the KH waves (Xiao et al. Reference Xiao, Wang, Sun, Liang and Liu2016). Together, they result in the formation of the spray plume core (figure 5 h). These liquid clumps, ligaments and droplets will further undergo fragmentation (secondary atomisation) to produce finer droplets because of the continuous interaction with the surrounding fluid.

In the case of $\textit{AR}$ = 3.3 (figure 5 c), the cross-flow experiences the jet slice, which is an already deformed one in the lateral direction due to the orientation of the major axis perpendicular to the cross-flow. This results in a much stronger acceleration of the jet slice (amplified RTI) along with an enhanced flattening in the lateral direction (figure 5 f) compared with the $\textit{AR}$ = 1 case. Hence, the liquid jet can undergo catastrophic breakup with the formation of a fine spray mostly due to the amplified RTI. This results in greater liquid jet deflection, early formation of droplets and extended deformation of the liquid column. Hence, the bulk liquid mass undergoes intense atomisation within the spray core (figure 5 i). These enhanced jet cross-flow interactions cause the jet to undergo primary atomisation at a shorter streamwise distance.

In contrast, the atomisation mechanism for the $\textit{AR}$ = 0.3 case is strikingly different due to the vastly different form of the jet slice in this case (figure 5 a), which is reminiscent of the coaxial jet case (Varga et al. Reference Varga, Lasheras and Hopfinger2003). The reduced frontal area in this case would lead to a smaller acceleration of the liquid slice (from (3.2)), and result in larger RT wavelengths, as discussed later in this sub-section. However, these RT waves are unlikely to be dominant in this relatively thick (streamwise) liquid jet breakup. The main mechanism for breakup of the liquid jet in this case would be expected to be similar to the coaxial jet case and be related to the large velocity differences between the relatively high-speed cross-flow air and the lower-speed liquid jet on the lateral surfaces. This velocity difference would lead to the formation of substantial KH waves on the lateral sides of the injected jet and the formation of liquid tongues as in coaxial jet studies. A key difference between the coaxial case (Varga et al. Reference Varga, Lasheras and Hopfinger2003) and the present case is that the liquid jet is now in a form of a vertical sheet (two-dimensional) with higher-speed cross-flow air on both sides, rather than the relatively axisymmetric nature of the liquid jet and high-speed air in the coaxial jet case. Further, the windward surface of the present $\textit{AR}$ = 0.3 jet is also free to deform in the present case due to the high pressure in the stagnation region of the cross-flow (as shown in figure 5 d). This deformed jet slice will then be subjected to KHI along the lateral sides of the liquid jet to form liquid tongues, which when exposed to the high-speed cross-flow air leads to mass-stripping and atomisation, possibly through RT as discussed by Varga et al. (Reference Varga, Lasheras and Hopfinger2003) (figure 5 d). Hence, in the $\textit{AR}=0.3$ case, there is vigorous mass stripping from the lateral surfaces of the liquid jet, leading to the formation of liquid sheets/ligaments of irregular shape and size in the core of the spray (figure 5 g).

To summarise, the primary instability mechanism near the stagnation region on the windward side of the liquid jet when exposed to the high-speed cross-flow in all the orifice $\textit{AR}$ cases is the RTI. In the $\textit{AR}=3.3$ case, this RTI is also expected to amplify and result in catastrophic breakup of the flattened jet into drops. However, as the AR decreases to 1 and 0.3, the primary destabilisation mechanism will be mainly governed by KHI on the lateral sides. This is especially clear in the $\textit{AR}=0.3$ case, where the KH waves on the lateral sides can result in large mass stripping, with the tongues of the resulting surface deformation being amenable to breakup by an RTI caused by the exposed liquid tongues in the high-speed cross-flow. We now proceed to present the most amplified surface wavelength values determined from experimental visualisations for each of the different orifice $\textit{AR}$ cases and then compare them with theoretical estimates.

Using the instantaneous visualisations shown in figure 4 for different AR, the difference between two troughs formed on the windward side of the liquid jet was determined from image analysis, which is also denoted as surface wavelength ( $\lambda$ ) in the respective images of figure 4 (left side, zoomed-in insets). For each orifice AR and J, the average value of the surface wavelength was obtained from a large set of images (approximately 200) like figure 4. Figure 6 shows the variation of the long-time-averaged surface wavelength against the cross-flow Weber number on a log–log scale. The surface wavelength here is normalised by the corresponding frontal dimension ( $b$ ) of the orifice exit for each of orifice $\textit{AR}$ cases studied. It is important to note that the effective cross-flow Weber number ( $We_{eff}= \rho _2 U_2^2 b/\sigma$ ) used here is similar to that of Xiao et al. (Reference Xiao, Wang, Sun, Liang and Liu2016) and Kuhn & Desjardins (Reference Kuhn and Desjardins2022), and is defined based on the frontal dimension ( $b$ ) of the orifice for each $\textit{AR}$ , where $\rho _2$ , $U_2$ and $\sigma$ represent the density and free stream velocity of the air behind the normal shock (corresponding to $M_\infty$ = 2.5), and the surface tension of the water–air interface, respectively. As seen in the figure, when the liquid jet is injected with $\textit{AR}$ = 0.3 (filled black square), the value of $\lambda /b$ (0.5) is significantly larger than the other cases because of the relatively smaller acceleration of the liquid jet due to the lower frontal frontal dimension (and drag) of the jet exposed to high-speed air. In contrast, in the $\textit{AR}$ = 3.3 case (filled black triangle), the $\lambda /b$ is much smaller (0.06) because of the larger frontal dimension of the jet with the supersonic cross-flow (larger drag) leading to an intense acceleration of the liquid jet. This results in an enhanced accelerative destabilisation of the liquid jet (RT), which ultimately leads to the catastrophic breakup of the liquid jet and the early formation of a fine spray, as evident from the visualisations (leeward side darker regions of figure 4 c). Hence, a clear decreasing trend in $\lambda /b$ is observed with an increase in $\textit{AR}$ , as seen in the figure. The circular orifice case with $\textit{AR}=1$ , as expected, lies in between the two extreme orifice $\textit{AR}$ cases. It may be noted that the surface wavelength seen in the present $\textit{AR}=1$ case lies along the data of Sallam et al. (Reference Sallam, Aalburg and Faeth2004) for a circular liquid jet over a large range of $We_{eff}$ . The present surface wavelength data for different orifice $\textit{AR}$ cases thus clearly show that the wavelength decreases rapidly with $\textit{AR}$ .

Figure 6. Variation of measured dimensionless surface wavelength with effective cross-flow Weber number for different orifice $\textit{AR}$ . Also shown are the data for a circular orifice ( $\textit{AR}=1$ ) over a wide $We_{eff}$ range from Sallam et al. (Reference Sallam, Aalburg and Faeth2004). The theoretical value of the most unstable RT wavelengths calculated for the different cases are also shown.

Having seen the experimental measurements, we shall now present predictions of surface wavelength corresponding to the most unstable wave using RT linear stability analysis. The theoretical expression of surface wavelength with maximum growth rate responsible for the transverse destabilisation of a liquid jet due to the acceleration ( $a$ ) of a low-density cross-flow fluid over a high-density liquid ( ${\rho }_l$ ) surface is given by (Joseph et al. Reference Joseph, Belanger and Beavers1999; Varga et al. Reference Varga, Lasheras and Hopfinger2003; Chandrasekhar Reference Chandrasekhar2013; Xiao et al. Reference Xiao, Wang, Sun, Liang and Liu2016; Varkevisser et al. Reference Varkevisser, Kooij, Villermaux and Bonn2024)

(3.1) \begin{equation} \lambda = 2\pi \sqrt {\frac {3\sigma }{\rho _l a}}. \end{equation}

This indicates that for a given liquid volume, the most unstable wavelength depends strongly on the acceleration ( $a$ ) of the liquid volume by the surrounding gas. We now proceed to estimate the acceleration ( $a$ ) of the liquid element in our case. We consider a volume of the liquid jet ( $V_l$ ), which is accelerating in the presence of the cross-flow fluid. In this case, we can write acceleration ( $a$ ) following Joseph et al. (Reference Joseph, Belanger and Beavers1999) as

(3.2) \begin{equation} a = \frac {F_D}{m_l} = \frac {0.5C_D \rho _{2} U_{2}^2 A_f}{\rho _l V_l}, \end{equation}

where $C_{D}$ is the drag coefficient, $\rho _l$ is the density of the liquid and $A_f$ is the surface area/frontal area exposed to the high-speed air. Equation (3.2) strongly emphasises the fact that for a given cross-flow condition and $V_l$ , the acceleration of the liquid volume strongly varies with the product of the exposed surface area/frontal area and the value of $C_{D}$ , as discussed by Varga et al. (Reference Varga, Lasheras and Hopfinger2003). It is important to note that in the present study, an increase in AR leads to an increase in frontal/exposed surface area as well as $C_{D}$ , leading to a significant increase in acceleration. The estimated value of a is of the order of $10^4$ times the acceleration due to gravity. This is inline with the calculations of Joseph et al. (Reference Joseph, Belanger and Beavers1999) for a single drop. Hence, the liquid jet when subjected to strong acceleration will be highly susceptible to RTI.

Substituting (3.2) into (3.1) by replacing $V_l$ as a product of cross-section area $A_c$ and a unit thickness, as well as $A_f$ as a product of frontal dimension and a unit thickness, yields the expression for the most amplified $\lambda /b$ as a function of $We_{eff}$ as

(3.3) \begin{equation} \frac {\lambda }{b} = \frac {2\pi }{b}\sqrt {\frac {6 A_c}{C_D We_{eff}}}. \end{equation}

Finally, to determine the theoretical values of $\lambda /b$ from (3.3), $C_{D}$ is needed. For the subsonic case (Sallam et al. Reference Sallam, Aalburg and Faeth2004), a $C_{D}$ value of 1.69 is assumed (Wu et al. Reference Wu, Kirkendall, Fuller and Nejad1997; Mashayek, Jafari & Ashgriz Reference Mashayek, Jafari and Ashgriz2008), whereas for the present case, values are taken from the supersonic flow over solid bodies of different aspect ratios (Gowen & Perkins Reference Gowen and Perkins1953; Heddleson, Brown & Cliffe Reference Heddleson, Brown and Cliffe1957), as shown in Appendix A.

The predicted $\lambda /b$ values for different orifice AR cases studied here (unfilled red square) are shown in figure 6 along with the predicted $\lambda /b$ values corresponding to the experimental conditions of Sallam et al. (Reference Sallam, Aalburg and Faeth2004) (unfilled orange right triangles). It is evident from the figure that in both the studies, the theoretical predictions (qualitative as well as quantitative trends) show a very good agreement with the experimentally measured ones, which is inline with earlier studies (Joseph et al. Reference Joseph, Belanger and Beavers1999; Varga et al. Reference Varga, Lasheras and Hopfinger2003; Sharma et al. Reference Sharma, Singh, Rao, Kumar and Basu2021). As seen in the experimental measurements, as the orifice $\textit{AR}$ increases, the $\lambda /b$ rapidly decreases, confirming that the acceleration-based destabilisation (RTI) is significantly amplified in this case. Further, in the $\textit{AR}=3.3$ case, we would expect this RT wavelength to amplify and lead to catastrophic breakup with the early formation of fine droplets, whose mean droplet size can be predicted to be a fraction of $\lambda /b$ (Varga et al. Reference Varga, Lasheras and Hopfinger2003).

To highlight the orifice $\textit{AR}$ effect further, it should be noted that the surface wavelength trend would indicate that a further increase in $\textit{AR}$ (say double from $\textit{AR}=3.3$ ) would lead to substantially lower $\lambda /b$ (approximately 0.02 instead of 0.06), which should lead to even smaller final droplet sizes.

Figure 7. Variation of time-averaged (a) $x_b$ and (b) $y_b$ with $We_{eff}$ for different J. (c) Variation of normalised breakup time of the liquid column with $\textit{AR}$ for a jet in supersonic cross-flow. The inset shows the comparison of the present values of $t_b$ / $t^{*}$ with those of the existing studies of single drops in supersonic flow. The dashed horizontal line indicates $t_b$ = 5 $t^{*}$ . , Engel (Reference Engel1958); , Nicholls & Ranger (Reference Nicholls and Ranger1969); , Reinecke & Waldman (Reference Reinecke and Waldman1970); , Hsiang & Faeth (Reference Hsiang and Faeth1992); , Reinecke & McKay (Reference Reinecke and McKay1969).

Further, it is evident from figure 4 that the initial windward-surface waves ( $\lambda /b$ ) developed on the jet surface grow rapidly due to the acceleration by the high-speed air (Varkevisser et al. Reference Varkevisser, Kooij, Villermaux and Bonn2024), and eventually results in the jet column fracture whose instantaneous location is identified and shown as a red coloured dot in figure 4(b) (Wu et al. Reference Wu, Kirkendall, Fuller and Nejad1997; Sallam et al. Reference Sallam, Aalburg and Faeth2004). The instantaneous breakup distance in the streamwise ( $x_b$ ) and the cross-streamwise ( $y_b$ ) directions from the orifice centre are calculated and the corresponding time-averaged normalised breakup distances, with $We_{eff}$ for the jets with different $\textit{AR}$ and $J$ are shown in figures 7(a) and 7(b), respectively. It is found that ( $\overline {x_b}$ /D) has changed significantly with $\textit{AR}$ . For a given $J$ , higher ( $\overline {x_b}$ /D) values are exhibited by the $\textit{AR}$ = 1 case, although $\textit{AR}$ = 0.3 has the least frontal surface. This is due to the rapid stripping of liquid mass from the large lateral surfaces of the jet (for $\textit{AR}$ = 0.3) leading to a decrease in penetration height, as discussed earlier. As anticipated, $\textit{AR}$ = 3.3 has the least ( $\overline {x_b}$ /D) due to the dominant RT-driven accelerative destabilisation mechanism resulting in an enhanced jet breakup. Figure 7(b) shows the variation of time-averaged normalised cross-streamwise breakup distance, ( $\overline {y_b}$ /D) with $We_{eff}$ . Unlike ( $\overline {x_b}$ /D), it is found that, irrespective of the $\textit{AR}$ , a higher penetration height (higher $J$ ) of the jet (upward arrow in figure 7 b) results in a larger ( $\overline {y_b}$ /D), which is consistent with the subsonic literature (Broumand & Birouk Reference Broumand and Birouk2016a ). In the present study, the higher penetrating case is $\textit{AR}$ = 1, and hence, it has a higher ( $\overline {y_b}$ /D) and vice versa for the $\textit{AR}$ = 3.3 case.

The quantitative measurements of the streamwise distance of column breakup, $\overline {x_b}$ , enable us to determine the time required for the jet breakup, $t_b = \sqrt { {2 \overline {x_b}}/{a}}$ , as in previous studies of droplet breakup in supersonic flows (Engel Reference Engel1958; Nicholls & Ranger Reference Nicholls and Ranger1969; Hsiang & Faeth Reference Hsiang and Faeth1992) as well as the jet in subsonic cross-flows (Wu et al. Reference Wu, Kirkendall, Fuller and Nejad1997; Sallam et al. Reference Sallam, Aalburg and Faeth2004; Broumand & Birouk Reference Broumand and Birouk2016a ). Figure 7 shows the variation of normalised $t_b$ with $\textit{AR}$ for the liquid jet in supersonic cross-flow with $M_\infty$ = 2.5. The data of $t_b$ are normalised with their characteristic inertial time scale, $t^* = { ({b}/{U_\infty }})\sqrt ({ {\rho _j}/{\rho _\infty }})$ , of the cross-flow (Wu et al. Reference Wu, Kirkendall, Fuller and Nejad1997; Sallam et al. Reference Sallam, Aalburg and Faeth2004). It is observed that as $\textit{AR}$ increases, the normalised breakup time decreases, indicating the enhanced atomisation of the liquid jet due to pure RT instability. The inset in figure 7(c) shows the variation of normalised $t_b$ with $We_{eff}$ for liquid drops encountering supersonic cross-flow as open symbols and for the present liquid jets in Mach 2.5 cross-flow as filled symbols. The general rule reported in previous studies (Engel Reference Engel1958; Nicholls & Ranger Reference Nicholls and Ranger1969; Hsiang & Faeth Reference Hsiang and Faeth1992) is that $t_b$ / $t^*$ of a single drop when placed in a supersonic cross-flow is roughly equal to 5, which is indicated as a dashed line in the inset plot of figure 7(c). The data shown in the plots suggest that the atomisation mechanism of the liquid jet is similar to the atomisation of a drop when encountering a supersonic flow.

We shall discuss this further in § 5, how the AR influences the actual measured droplet sizes, as well as the width of the droplet size distribution in connection with the difference in the mechanism of the instability and the jet breakup time.

3.2. Shock structures

Having seen the effect of orifice AR on the surface waves and destabilisation mechanism of a liquid jet, we shall now present the effect of these surface waves on the shock structures upstream of the injected liquid jet obtained using high-speed shadowgraphy.

Figure 8 shows the instantaneous side view images for $\textit{AR} = 0.3$ , 1 and 3.3 at two different instants (left and right), respectively. Each side view image spans 15 jet diameters in streamwise and cross-stream directions (18 mm x 18 mm), and both streamwise and cross-stream coordinates are normalised with the equivalent orifice diameter ( $D$ ), which is the same in all the three orifice $\textit{AR}$ cases studied here. Also, these instantaneous images are marked with the normalised mean boundary-layer thickness ( $\delta /D$ ) measured using PIV in the wall-normal plane at the injection location and in the absence of a jet. These instantaneous visualisations are helpful to understand the shock structures present and their interactions with windward surface waves and with boundary layer structures.

Figure 8. Instantaneous visualisations highlighting the shock structure, the jet windward spray edge and the leeward side wake region between the instants ( $t_1$ and $t_2$ ) for (a) $\textit{AR}$ = 0.3 (b) $\textit{AR}$ = 1 and (c) $\textit{AR} = 3.3$ , and $J = 3.7$ for $M_\infty = 2.5$ . White, green and yellow colour dashed rectangular boxes highlight the near-field jet bending behaviour for $\textit{AR} = 0.3$ , 1 and 3.3 respectively.

As seen in figure 8, a three-dimensional bow shock is formed in front of the liquid jet. It is evident from the figure that the windward surface wavelength is much larger and rounded in the case of $\textit{AR}$ = 0.3, compared with the $\textit{AR} = 1$ and 3.3 cases, as discussed previously. The differences in the windward surface wavelength seen in the different $\textit{AR}$ cases lead to significant changes in the corresponding shock structures formed ahead of the liquid jet. These surface deformations and associated large-scale coherent structures play a pivotal role in the near-field mixing (Ben-Yakar, Mungal & Hanson Reference Ben-Yakar, Mungal and Hanson2006).

In the case of $\textit{AR}$ = 0.3 (figure 8 a), the formation of larger windward surface wavelength with larger liquid structures leads to the formation of highly corrugated/bumpy complex shock structures. This results in large variations in the local curvature/strength of the bow shock. In contrast, as the $\textit{AR}$ increases, the windward surface wavelength reduces, leading to a reduction in corrugations, as evident from figure 8(c). Also, for a given $\textit{AR}$ (and $J$ ), the comparison of instantaneous images between instants elucidates the dynamic nature of the flow. For instance, in figure 8(a), the lower penetration of the instantaneous jet in the left image compared with the right image results in a downstream shock motion (in the left image) as well as more jet fluid (darker region) coming closer to the (injection) wall in the wake in comparison to the right image of figure 8(a). Also, the presence of large-scale liquid deformation at $y/D = 7$ in figure 8(a) (right image) causes the upstream shock displacement with higher strength, forming a bumpy shock.

Figure 9. Instantaneous (a) shock traces and (b) variation of shock-induced velocity along the corresponding shock traces with $(x/D)$ to emphasise the effect of corrugations on the velocity variation around the liquid structure for $\textit{AR} = 0.3$ , 1 and 3.3, and $J = 3.7$ . The shock-induced velocity is normalised with free stream velocity.

The variations in shock structures with orifice $\textit{AR}$ seen in the visualisations here can be better seen by plotting sample instantaneous shock structures for different $\textit{AR}$ , as shown in figure 9(a) (all for $J = 3.7$ ). As seen in the visualisations, the corrugations in the shock are larger for the $\textit{AR}=0.3$ case compared with the higher orifice $\textit{AR}$ cases. To further highlight the importance of the formation of corrugated shock structures, the corresponding post-shock velocity of free stream air ( $U_2$ ) variation along the streamwise direction is shown in figure 9(b), with this being calculated using simple inviscid shock relations similar to Ben-Yakar et al. (Reference Ben-Yakar, Mungal and Hanson2006). It is observed that in the cases of $\textit{AR} = 0.3$ and 1, the instantaneous value of $U_2$ fluctuates considerably between 200 m s⁻¹ and 400 m s⁻¹. However, in the $\textit{AR} = 3.3$ case, the velocity variation is smooth with no such large-scale variations. These differences in air velocity impacting the liquid jet eventually affect the downstream spray characteristics. Further, as seen in the visualisations, the shock corrugations and, hence, the associated fluctuations in $U_2$ can vary substantially with time in the $\textit{AR}$ = 0.3 and 1 cases. This ultimately leads to large-scale unsteadiness in the downstream spray characteristics. A similar observation was made when a sonic circular gaseous jet was injected into supersonic cross-flow (Gruber et al. Reference Gruber, Nejad, Chen and Dutton1997; Murugappan, Gutmark & Carter Reference Murugappan, Gutmark and Carter2005; Ben-Yakar et al. Reference Ben-Yakar, Mungal and Hanson2006), wherein these vortical structures at the interface were named ‘braided regions’. Hence, it is evident that the local curvature or the shock strength and its unsteadiness are strongly related to the formation of the irregular liquid structures in the windward side of the jet. This is more intense in the case of $\textit{AR}$ = 0.3, due to the presence of higher $\lambda /b$ (figure 6) and the resulting larger coherent liquid structures compared with other cases.

Apart from the above mentioned effects that are caused by surface waves, which are in turn affected by the orifice $\textit{AR}$ , there is another cause for unsteadiness that is related to the boundary layer low- and high-speed streaks affecting the shock foot (Gruber et al. Reference Gruber, Nejad, Chen and Dutton1996; Munuswamy & Govardhan Reference Munuswamy and Govardhan2022). This can lead to substantial variations in jet penetration height and shock position between the instants, for all orifice $\textit{AR}$ , and will be discussed further in the next section on the liquid jet penetration height.

4.1. Mean penetration height

Quantitative measurement of instantaneous spray/jet penetration height ( $h$ ) and shock location are of prime importance in any jet in cross-flow studies (Mahesh Reference Mahesh2013). As in our previous study (Medipati et al. Reference Medipati, Deivandren and Govardhan2023) for a circular jet, jet penetration height, shock location and their unsteadiness are evaluated using instantaneous images, as shown in figures 8(a), 8(b) and 8(c) for $\textit{AR} = 0.3$ , 1 and 3.3, respectively. These measurements are taken by capturing the intensity jump across the liquid jet and shock location using an edge detection algorithm (Medipati et al. Reference Medipati, Deivandren and Govardhan2023). Approximately one thousand images are used for statistical convergence. Figures 10(a), 10(b) and 10(c) represent the variation of ensemble averaged penetration ( $\overline {h}$ ) trajectories along the streamwise direction ( $x/D$ ) for $\textit{AR} = 0.3$ , 1 and 3.3, and for different $J$ . As expected, with an increase in $J$ , the mean penetration height increases for all the $\textit{AR}$ cases studied. A closer look at the data also shows that there is significant differences in the penetration data with $\textit{AR}$ , as discussed further now.

There have been many scalings used for jet penetration trajectories in cross-flow literature (Mahesh Reference Mahesh2013; Chai et al. Reference Chai, Iyer and Mahesh2015; Fries, Ranjan & Menon Reference Fries, Ranjan and Menon2021). The scaling for the jet penetration height is derived from experimental data (for $x/D$ = 0–5) by identifying the exponent of $J$ for which the recorded trajectories at different $J$ collapse onto a single curve. It is found that for circular jets, the penetration trajectories collapse when normalised with $J^{0.5}$ , as shown in figure 10(e). It is important to note that the $J^{0.5}$ scaling exhibits an excellent match with the liquid jet in the subsonic cross-flow study (Wu et al. Reference Wu, Kirkendall, Fuller and Nejad1997), where the same scaling is derived theoretically and validated experimentally. This is also inline with Munuswamy & Govardhan (Reference Munuswamy and Govardhan2022) when a sonic gaseous circular jet is injected into supersonic cross-flow. The existing supersonic literature on liquid jets (Lin et al. Reference Lin, Kennedy and Jackson2004; Ghenai et al. Reference Ghenai, Sapmaz and Lin2009; Sathiyamoorthy et al. Reference Sathiyamoorthy, Danish, Iyengar, Srinivas, Harikrishna, Muruganandam and Chakravarthy2020; Medipati et al. Reference Medipati, Deivandren and Govardhan2023) discusses the scaling for circular jets only. The present study proposes penetration height scalings for elliptical jets, which are found to be significantly different. We find that for elliptical jets, $J^{0.75}$ and $J^{0.55}$ lead to a good collapse of the mean penetration trajectories for $\textit{AR} = 0.3$ and 3.3, as shown in figures 10(d) and 10(f), respectively (the corresponding $R^2$ values are 0.96 and 0.98). In the cases of $\textit{AR}$ = 0.3 and $\textit{AR}$ = 3.3, we find that the estimated $J$ exponents are significantly different (0.75 for $\textit{AR}$ = 0.3 and 0.55 for $\textit{AR}$ = 3.3), indicating that some of the assumptions for the theoretical predictions in the case of $\textit{AR}$ = 1, namely, negligible mass stripping and no catastrophic breakup (due to RTI) are no longer valid for these $\textit{AR}$ cases. As discussed in § 3.1, in the $\textit{AR}$ = 0.3 case, there is significant mass stripping, while the $\textit{AR}$ = 3.3 case has a catastrophic breakup due to RTI. These results indicate that the scaling of mean penetration trajectories strongly depends on the orifice shape and its orientation with respect to the cross-flow.

Figure 10. Variation of mean penetration height along the streamwise direction for (a) $\textit{AR}$ = 0.3, (b) $\textit{AR}$ = 1 and (c) $\textit{AR}$ = 3.3, and different $J$ . Scaling of mean penetration heights with a scaling factor of $J^{0.75}$ , $J^{0.5}$ and $J^{0.55}$ , both in the streamwise and cross-stream directions, for (d) $\textit{AR}$ = 0.3, (e) $\textit{AR}$ = 1 and (f) $\textit{AR}$ = 3.3. The vertical bars indicate the standard deviation in measured penetration height.

Figure 11. (a) Variation of mean penetration height and (b) shock position with $\textit{AR}$ for different $J$ at $x/D$ = 3.5.

The near-field penetration behaviour of liquid jets is dramatically different among the $\textit{AR}$ values, as evident from figures 8 and 4. Since $\textit{AR} = 0.3$ possesses the least frontal area, it is intuitive that the jet penetration would be highest. However, the circular jets are found to penetrate farther into the supersonic cross-flow. This can be attributed to the higher mass stripping in the case of $\textit{AR} = 0.3$ due to intense shear breakup from the lateral surfaces of the liquid jet slice when compared with the circular jet, as explained in the previous section. This is in line with the observations made by Gruber et al. (Reference Gruber, Nejad, Chen and Dutton2000), when a sonic gaseous jet was injected into a supersonic cross-flow using elliptical ( $\textit{AR} = 0.26$ ) and circular ( $\textit{AR} = 1$ ) orifices. Figures 11(a) and 11(b) show the variation in mean penetration height ( $\overline {h}$ ) and the corresponding shock position ( $y_s/D$ ) with $\textit{AR}$ for different $J$ . For all $J$ values, $\textit{AR} =1$ shows the highest penetration height and the shock is most upstream, while $\textit{AR} = 3.3$ shows the least penetration and the shock is pushed downstream, as in this case, the liquid jet is deflected more downstream due to the large drag on it. Between $\textit{AR}$ = 0.3 and 3.3, better penetration was exhibited by $\textit{AR} = 0.3$ . It may be noted that although the results are shown at a streamwise distance of $x/D = 3.5$ , the trends reported here are independent of $x/D$ locations. This indicates that $\textit{AR}$ plays a distinct role in controlling the mean trajectories and their shock position, in addition to $J$ . To validate the present penetration height measurements and the proposed scaling, we compare them with the existing circular jet in cross-flow studies in the literature. The key details, such as upstream Mach number ( $M_\infty$ ), momentum flux ratios ( $J$ ), injectant fluid, incoming boundary-layer thickness-to-diameter ratio ( $\delta /D$ ) and the experimental technique employed to determine the mean penetration heights are tabulated in table 3.

Table 3. Data sets for time-averaged circular jet penetration trajectory.

The mean penetration heights for circular jets from the above-cited literature, along with the current study, are plotted by scaling with $J^{0.5}$ , as shown in figures 12(a) and 12(c) for the similar $M_\infty$ = 2.5 cases and varied $M_\infty$ cases (1.5–2.5), respectively. It is evident that these trajectories lead to a strong collapse within the respective data set when scaled with $J^{0.5}$ , with $R^2$ values ranging from 0.93 (Ghenai et al. Reference Ghenai, Sapmaz and Lin2009; Munuswamy & Govardhan Reference Munuswamy and Govardhan2022) to 0.99 (Portz & Segal Reference Portz and Segal2006), as shown in table 4. This indicates that the proposed scaling is valid for a wide range of $J$ values (0.34–23) and Mach numbers (1.5–2.5), although we can observe variations between the similar $M_\infty$ cases as well as different $M_\infty$ cases, which is further discussed later. Broadly, this indicates that the scaling for penetration height is well captured by $J$ , which may be varied by either varying the cross-flow velocity or the jet velocity. From the similar $M_\infty$ comparison plot (figure 12 a), it is observed that the scaling is independent of the injectant used, which is an important outcome of this comparison. This indicates that the penetration height of the jet is independent of the jet phase and is completely governed by $J$ when the cross-flow Reynolds number and the Weber numbers are very high. A strong scatter seen for the similar $M_\infty$ comparison plot (figure 12 a) is due to the incoming boundary-layer thickness ( $\delta /D$ ) being significantly different between the cases. A similar effect was observed by Muppidi & Mahesh (Reference Muppidi and Mahesh2005) for the jet in subsonic cross-flow, and by Portz & Segal (Reference Portz and Segal2006) and Fries et al. (Reference Fries, Ranjan and Menon2021) for the sonic jet in supersonic cross-flow. Figure 12(a) indicates that a higher $\delta /D$ results in higher penetration of the jet and vice versa. Hence, it suggests that consideration of the $\delta /D$ in the cross-stream direction scaling is vital when the comparison is drawn among different works in the literature. We find that a $(\delta /D)^{0.5}$ scaling leads to a good collapse of the trajectory data across different $(\delta /D)$ cases, as shown in figure 12(b).

Figure 12. Comparison of mean penetration trajectory of the circular jet for different $J$ from the present study and the data available from the literature. For a similar $M_\infty$ = 2.5, (a) $J^{0.5}$ is used as a scaling factor in both streamwise and cross-stream directions, and (b) $J^{0.5}$ and $J^{0.5}$ $(\delta /D)^{0.5}$ are used as scaling factors in the streamwise and cross-stream directions. For varying $M_\infty$ cases, (c) $J^{0.5}$ is used as a scaling factor, and (d) $J_2^{0.5}$ is used as a scaling factor in both streamwise and cross-stream directions, where $J_2$ is based on the conditions downstream of the shock.

In the varying $M_\infty$ comparison case (figure 12 c), the data do not collapse well and show that an increase in $M_\infty$ leads to higher penetration for the same $J$ . This is likely due to the presence of shocks upstream of the injected jet, with the shock strength being dependent on $M_\infty$ , which would result in different velocities and densities of air downstream of the shock that interacts with the injected jet. This would suggest that we can obtain a better collapse of the penetration data at different $M_\infty$ , if we define a modified momentum flux ratio ( $J_2$ ) based on the conditions downstream of the shock. Although the shock angle and strength vary with height, we calculate $J_2$ using a representative velocity ( $U_{2\infty }$ ) and density ( $\rho _{2\infty }$ ) downstream of a normal shock that occurs close to the injection point. A comparison of the penetration data with a $J_2^{0.5}$ scaling is shown in figure 12(d), indicating a significantly better collapse of the data across different $M_\infty$ cases (compared with that in figure 12 c). This indicates that $J_2^{0.5}$ is a better scaling when comparing penetration height data from varying supersonic $M_\infty$ cases. The remaining small variability between the studies seen in figure 12(d) may be attributed to the differences in the incoming boundary layer thickness experienced by the jet, which is not given in the literature, and to experimental uncertainties in the delineation of the penetration height and the diagnostics used. This analysis clearly suggests that both $\delta /D$ and shock strength play a significant role in altering the penetration height and have to be considered while the comparison of penetration height is drawn among the studies.

Table 4. Coefficient of determination ( $R^2$ ) for collapse of penetration height for different studies when scaled with $J^{0.5}$ .

4.2. Mass stripping

One of the aspects that appears to significantly affect penetration is the stripping of liquid mass away from the injected jet. This, as discussed in the previous section, occurs more in the $\textit{AR}=0.3$ case, due to the high shear on the relatively large deformed lateral surfaces of the liquid jet (see figure 5) that leads to relatively large chunks or ligaments breaking away. Such mass stripping from the liquid jet has many consequences as it leads to a rapid reduction in the mass associated with the liquid jet.

An indirect way of estimating the amount of mass stripping from the jet surface is by determining the increase in width of the spray plume in the streamwise plane (y–z) (Mashayek et al. Reference Mashayek, Jafari and Ashgriz2008). For this, planar laser Mie scattering (PLMS) is employed in the streamwise plane by firing the laser sheet perpendicular to the free stream direction at three chosen streamwise positions (x/D = 8.3, 30 and 60). A sample instantaneous Mie scattering image is depicted in figure 13(a) along with the outer spray edge contour (shown with red colour). For each instantaneous visualisation, the outer spray edge is determined using a simple threshold, and then the instantaneous maximum spray width (W) is calculated as marked in figure 13(a). Using a large number of instantaneous images (500 images) at each streamwise position and for a constant experimental condition, its long time-averaged value ( $\overline {W}$ ) is deduced. Figure 13(b) illustrates the variation of $\overline {W}$ /b with AR at x/D = 8.3, 30 and 60 for J = 9.7. It may be noted that since the spray spreading dimension in the $z$ -direction is determined by the frontal geometry scale $b$ , the measurement of $\overline {W}$ is normalised using $b$ . The measurements indicate that in the case of AR = 0.3, $\overline {W}$ /b is higher (50 % more than $\textit{AR}$ = 1 at x/D = 8.3) than its counterparts, due to the vigorous mass stripping from the lateral sides of the jet surface. This is consistent with x/D as well as with J. This is most likely due to the higher mass stripping from its lateral surfaces, which results in a decrease in penetration height (Mashayek et al. (Reference Mashayek, Jafari and Ashgriz2008); Broumand & Birouk (Reference Broumand and Birouk2016b ) when compared withAR = 1. It may be noted that this decrease in penetration height for the $\textit{AR}=0.3$ case occurs even though AR = 0.3 has the least frontal area indicating that there is considerable mass stripping from the lateral sides of the injected jet, which leads to a rapid decrease of the vertical injected liquid jet momentum.

Figure 13. (a) A sample planar laser Mie scattering image acquired at x/D = 60 for AR = 1 and J = 9.7 to depict the spray plume cross-section when a liquid jet is injected into a supersonic cross-flow. The outer edge of the spray plume cross-section is shown in red. W and $A_p$ represent the instantaneous maximum spray width and spray plume area in the spanwise direction, respectively. (b) Variation of mean spray and (c) variation of mean spray plume area with $\textit{AR}$ for $J$ = 9.7 at $x/D$ = 8.3, 30 and 60. The mean spray width and plume area are normalised with the initial frontal dimension ( $b$ ) and the orifice exit area ( $A_e$ ) of the orifice, respectively (Gruber et al. Reference Gruber, Nejad, Chen and Dutton2000).

Similar to the mean spray width calculation, a time-averaged spray plume area ( $\overline {A_p}$ ) (see figure 13 a) is computed, whose variation with $\textit{AR}$ at different $x/D$ is shown in figure 13(c). For a given $\textit{AR}$ , as $x/D$ increases, the plume area grows rapidly from the initial orifice exit area ( $A_e$ ). This is due to the rapid atomisation of the jet, along with the increase in jet penetration height and the entrainment of the cross-flow fluid into the jet core. In the case of $\textit{AR}$ = 1, at $x/D$ = 8.3, the plume area has increased to 290 $A_e$ . This is in good agreement with the value (251.37 $A_e$ ) predicted by the correlation,

(4.1) \begin{equation} \frac {\overline {A_p}}{A_e}=31.9J^{0.21}\left(\frac {x}{D}\right)^{0.75}, \end{equation}

proposed by Lin et al. (Reference Lin, Kennedy and Jackson2004). As $\textit{AR}$ increases from 1 to 3.3, the enhanced drag force acting on the increased frontal area of the jet results in a significant reduction in the jet penetration height, which in turn leads to a significant decrease in the value of $\overline {A_p}$ , since the major contribution to the spray plume area is from the jet penetration height (Wu et al. Reference Wu, Kirkendall, Fuller and Nejad1998; Lin et al. Reference Lin, Kennedy and Jackson2004). In contrast, as the $\textit{AR}$ reduces to 0.3, the vigorous mass stripping from the large lateral surfaces leads to an early column breakup with a consequent decrease in jet penetration height, and an increase in spray width, with an overall drop in plume area.

This mass stripping can be further seen by looking at a simple model for the injected jet trajectory. This may be done using a simple force balance to describe the near-field jet trajectories by adopting the procedure as well as the assumptions followed in the subsonic literature (Wu et al. Reference Wu, Kirkendall, Fuller and Nejad1997; Ashgriz Reference Ashgriz2011). In our case, the effect of the formation of the shock wave on the variation of free stream velocity and density in the transverse direction as encountered by the jet is taken into account in comparison to subsonic studies, as depicted in figure 14. The complete details about the methodology, as well as the mathematical formulations, are given in Appendix A. The differences between the trajectories predicted from the simple force balance model and the experiments are presented next.

Figure 14. Schematic illustrating the forces acting on the fluid element. $O$ (0,0) is the centre of the orifice/fluid element in the present analysis.

Figure 15. Comparison of mean penetration trajectory obtained from experiments and simple theoretical model for (a) $\textit{AR} = 1$ , (b) $\textit{AR} = 3.3$ and (c) $\textit{AR} = 0.3$ , and different $J$ .

Figures 15(a), 15(b) and 15(c) show the comparison of mean windward edge trajectories derived from experiments and the model for $\textit{AR} = 1$ , 3.3 and 0.3, and different $J$ values. It is evident from figures 15(a) and 15(b) that the predicted trajectories are in close agreement with the experimental measurements for $\textit{AR} = 1$ and 3.3 for all $J$ . This implies that the assumptions made in the modelling are reasonable for these cases. It may be noted that the computed trajectory from the model is for the centrelines of the plumes. The determination of the centreline of the spray plume from the experiments is complex; hence, the outer spray edge obtained from the experiments is used for comparison. However, the centreline and the upper boundary are very close as the jet deforms into a sheet-like shape (Ashgriz Reference Ashgriz2011). This results in the close match between the experimental and modelling spray edges. However, for $\textit{AR}=0.3$ , a large deviation is observed between the experimentally determined and the analytically predicted jet trajectories, as illustrated in figure 15(c). This can be attributed to the fact that the model does not take into account the mass stripping of liquid from the lateral sides of the jet surface, which can be expected to be much higher for the $\textit{AR}=0.3$ case, as discussed earlier in connection with figure 13. The simple model that does not account for mass stripping has much larger inertia, which implies lower jet deflection in the model. Comparatively, the experiment has high mass stripping in the $\textit{AR}=0.3$ case leading to lower jet inertia, which results in much larger jet deflection in the experiment for this case, as seen in the figure. This is consistent with the observation of Mashayek et al. (Reference Mashayek, Jafari and Ashgriz2008)and Broumand & Birouk (Reference Broumand and Birouk2016b ) that neglecting the mass stripping phenomenon can over predict the penetration height significantly. In summary, this simple force balance based model indicates that the trajectory (deflection) of the injected (core) liquid jet does not match well with the experimental trajectory only for the $\textit{AR}=0.3$ case, due to the large mass stripping that occurs only for this $\textit{AR}=0.3$ case.

4.3. Unsteady behaviour of the liquid jet

From the analysis of ensemble averaged penetration and shock data, it is observed that the $\textit{AR}$ , $J$ and $\delta /D$ play a very important role in controlling the jet penetration and shock strength. The instantaneous visualisations depicted in figure 8 (left and right) indicate that substantial unsteadiness is present in the jet penetration height and shock motion, irrespective of the $\textit{AR}$ . We shall quantify the unsteadiness in penetration height in this sub-section for different orifice $\textit{AR}$ , and discuss the source of this unsteadiness.

Figure 16. Schematic highlighting the instantaneous flow field quantities. $u_l$ is the line averaged velocity from PIV measurements. $x_s$ , $y_s$ and $x_j$ , $h$ represents the streamwise, and cross-stream positions of the shock and jet, respectively.

The key instantaneous features during the interaction viz. incoming boundary-layer fluctuations ( $u_l$ ), bow shock position ( $x_s/D$ and $y_s/D$ ) and windward spray edge ( $x_j/D$ and $h/D$ ) are illustrated in figure 16. To highlight the inherent unsteadiness present, a large number of instantaneous traces of bow shock wave (left) and the corresponding windward spray edge (right) captured during a single test run for $\textit{AR}$ = 0.3 (orange), 1 (blue) and 3.3 (green) are superimposed along with their mean traces (thick black), at a given $J$ , as depicted in figures 17(a), 17(b) and 17(c). Also, two sample instants of shock traces and corresponding windward edges are shown in figure 17(a), these two being chosen so that they correspond to upstream and downstream extremes of the bow shock. As observed in figure 17(a), an instant with the liquid jet penetrating more (dashed black line) into the cross-flow, results in an upstream displacement of the shock and vice versa for the lower penetrating instant (thin black line). This emphasises the correlation between the windward edge of the liquid jet and the corresponding shock position between the instants, as may be expected. This is discussed in detail through the correlation analysis performed for different $\textit{AR}$ cases presented next.

Figure 17. Instantaneous and mean shapes of bow shock (left) and windward spray edge (right) of liquid jet for (a) $\textit{AR} = 0.3$ , (b) $ \textit{AR}= 1$ and (c) $\textit{AR} = 3.3$ , and $J = 9.7$ when a liquid jet is injected transversely into a supersonic cross-flow. In all the plots, instantaneous shock traces and windward trajectories are shown in orange, blue and green colours along with their time-averaged shape in black colour for $\textit{AR} = 0.3$ , 1 and 3.3. The differences in shock structure and jet penetration between the instants in time are highlighted as black long dash and continuous lines in panel (a).

Figure 18. Scatter plot between the instantaneous $x_s$ and $x_j$ at $y/D = 12$ for (a) $\textit{AR} = 0.3$ , (b) $\textit{AR} = 1$ and (c) $\textit{AR} = 3.3$ , and $J = 9.7$ . (d) Variation of $r$ with $\textit{AR}$ .

The interaction dynamics between the bow shock wave and windward jet position are characterised for different $\textit{AR}$ values by representing their instantaneous positions in a scatter plot, figure 18. These scatter plots shows the typical distance between the instantaneous positions of the bow shock wave ( $x_s$ ) and windward edge ( $x_j$ ) in the streamwise direction for $\textit{AR} = 0.3$ , $\textit{AR} = 1$ and $\textit{AR} = 3.3$ , respectively, at a cross-stream location $y/D = 12$ . It is apparent from this figure that a significant unsteadiness is present in both the shock and windward edge positions, irrespective of the $\textit{AR}$ . As an example, in the case of $\textit{AR} = 0.3$ , the shock position oscillates between 0.6 $D$ and 2 $D$ , during which the windward edge fluctuates between 1 $D$ and 3.5 $D$ . These fluctuations are significant compared with the variation in mean penetration height due to the variation in $\textit{AR}$ and  $J$ . The correlation between them can be quantified using a correlation coefficient ( $r$ ). The correlation coefficient between variations in instantaneous shock location and jet penetration for different $\textit{AR}$ are shown in figure 18(d). It is found that the value of $r$ remains at 0.8 for all values of $\textit{AR}$ . Thus, the instantaneous bow shock and liquid jet positions are strongly correlated. This confirms our earlier qualitative observation that when a liquid jet penetrates more into the cross-flow, the shock is displaced upstream and vice versa. It is important to note that the value of $y/D = 12$ is chosen only for representation; a similar behaviour is seen at other locations and $J$ values.

Figure 19. (a) Variation of $\sigma _{{h}}$ along the streamwise direction for different $\textit{AR}$ and $J$ = 9.7. (b) Comparison of $\sigma _{{h}}/\overline {h}$ at $x/D = 2$ for different $\textit{AR}$ and $J$ .

The effect of $\textit{AR}$ on the fluctuation/unsteadiness in the jet penetration is characterised by estimating the standard deviation of $h$ ( $\sigma _{{h}}$ ) along the streamwise direction $x/D$ , and then normalising it with the corresponding mean penetration height ( $\overline {h}$ ). Figure 19(a) illustrates the variation of unsteadiness in jet penetration height ( $\sigma _{{h}}/\overline {h}$ ) for different $\textit{AR}$ along the streamwise direction. It is observed that along the streamwise direction, the unsteadiness reduces monotonically until the jet reaches its maximum penetration height, after which it flattens, this trend being independent of the $\textit{AR}$ . The highest and lowest unsteadiness are found for $\textit{AR} = 3.3$ and 1, respectively. For example, at $x/D = 2$ , the unsteadiness significantly reduces from approximately 7 % to 3 % when the $\textit{AR}$ has changed from 3.3 to 1, as can be seen from figure 19(b). The difference in unsteadiness with $\textit{AR}$ arises due to the fact that the interaction frontal area of the jet with the oncoming turbulent boundary layer is significantly different among the $\textit{AR}$ cases, as we shall discuss next.

Figure 20. Instantaneous boundary-layer velocity fields near the jet exit in the wall-normal plane to emphasise the cross-flow momentum fluctuations present between the instants (a) thicker and (b) thinner due to the presence of low and high momentum streaks, respectively. The horizontal dashed line indicates the location of line averaged velocity. (c) Probability density function (p.d.f.) of the instantaneous line average boundary-layer velocities at $y/D = 2.2$ or $y/\delta = 0.3$ .

To gain insight into the cause for the difference in unsteadiness, as well as the source of unsteadiness, PIV measurements were done upstream of the jet. The key difference among the $\textit{AR}$ cases is the difference in their area of interaction with the incoming boundary layer and then the cross-flow. Hence, a narrow field of view of approximately 25 $D$ by 12 $D$ is chosen upstream of the jet exit for detailed measurements. Sample instantaneous velocity fields obtained here are shown in figures 20(a) and 20(b). These figures highlight that there is a substantial unsteadiness in the incoming boundary-layer thickness experienced by the jet, with panel (a) corresponding to an instant with a low-speed streak (and thicker boundary layer), while panel (b) corresponds to a high-speed streak (and thinner boundary layer). To characterise this unsteadiness, a line averaged velocity in the streamwise direction ( $u_l/U_\infty$ ) between $x/D$ = −10 and −35, at $y/D = 2.2$ (or $y/\delta = 0.3$ ) is chosen within the boundary layer, as shown schematically in figure 16. The same quantity has been used by Ganapathisubramani et al. (Reference Ganapathisubramani, Clemens and Dolling2007), Murugan & Govardhan (Reference Murugan and Govardhan2016)and Munuswamy & Govardhan (Reference Munuswamy and Govardhan2022) to identify the existence of low- and high-speed streaks in the incoming turbulent boundary layer, which can influence the shock position. In their studies (Ganapathisubramani et al. Reference Ganapathisubramani, Clemens and Dolling2007; Murugan & Govardhan Reference Murugan and Govardhan2016), the shock is formed due to obstruction caused by the supersonic flow of the solid body. However, in the current study and Munuswamy & Govardhan (Reference Munuswamy and Govardhan2022), the presence of boundary-layer streaks can influence the jet deflection and, hence, the shock foot position and jet penetration height. It is observed from figure 20(b) that the line averaged velocity varies between 0.65 $U_\infty$ and 0.9 $U_\infty$ . This suggests that the total cross-flow momentum (boundary-layer momentum + free stream momentum) is substantially different between the instants. Hence, an instant with a thicker boundary layer (existence of a lower momentum streak as in panel a) can help the jet to penetrate more into the cross-flow and vice versa for the thinner boundary layer (in panel b), which is consistent with the findings of Munuswamy & Govardhan (Reference Munuswamy and Govardhan2022).

The next question to be addressed is how the jet issuing out of the different orifice $\textit{AR}$ cases are influenced by these boundary-layer fluctuations. It is well known that the momentum streaks present inside the turbulent boundary layer span approximately 40 $\delta$ and 0.5 $\delta$ in the streamwise and spanwise directions, respectively (Ganapathisubramani et al. Reference Ganapathisubramani, Clemens and Dolling2007). The frontal injected jet dimension ( $b$ ) to streak width in the present study are 0.15, 0.27 and 0.5 for $\textit{AR}$ = 0.3, 1 and 3.3, respectively. Between $\textit{AR}$ = 1 and 3.3, since AR = 3.3 possesses higher frontal to streak width, it results in stronger interaction of the liquid jet with boundary-layer streaks. Hence, the unsteadiness in the penetration height is higher when compared with the circular case. Whereas, in the case of $\textit{AR}$ = 0.3, despite having the least frontal to streak width, higher unsteadiness is seen in comparison to the circular case possibly due to the higher interaction of the liquid jet along the streak length (40 $\delta$ ) on the lateral side.

Also, there is a considerable increase in unsteadiness with decrease in $J$ for all the $\textit{AR}$ values studied, as shown in figure 19(b). This is due to the fact that injected jets are relatively weaker, with less penetration into the cross-flow. Hence, a larger portion of the liquid jet is within the boundary-layer thickness, resulting in stronger interaction with incoming boundary-layer streaks. To conclude, the naturally occurring boundary layer streaks are likely to be the source of the observed unsteadiness in the penetration height. At an instant with a thicker boundary layer (low-speed streak), the jet penetrates more into the cross-flow, leading to the upstream displacement of the bow shock wave, while at an instant with a thinner boundary layer (high-speed streak), the opposite happens with lower penetration and more downstream shock location.

5.1. Time-averaged droplet size distribution

A sample instantaneous image depicting the transverse locations at which droplet sizing and velocity measurements are performed is shown in figure 27 of Appendix B. We shall particularly focus our attention on station C, corresponding to the spray core (figure 27), which is representative of the bulk atomisation, and station E, at the spray edge (figure 27), where large temporal variations exist. We define the spray edge and hence the penetration height $h_d$ based on the droplet count, in this section, with the edge being defined as the location where there are fewer than 50 drops over 500 images, as the measurement station is moved from the core of the spray plume towards the outer supersonic flow (Medipati et al. Reference Medipati, Deivandren and Govardhan2023).

Figures 28(a) and 28(b) (Appendix B) show zoomed-in (2 mm x 1.5 mm view) high-resolution instantaneous spray images for different $\textit{AR}$ values at stations C and E, respectively. The dark spots observed in these images are the spray droplets. It is evident from these visualisations that the injected liquid jet has atomised into relatively fine droplets by this $x/D=60$ location. As discussed in § 3.1, this is dominantly through the KH route in the $\textit{AR}=0.3$ and $\textit{AR}=1$ cases, and more directly through the RT route in the $\textit{AR}=3.3$ case. Due to the supersonic cross-flow, this results in the formation of relatively fine droplets even at small $x/D$ locations, unlike liquid jets in subsonic studies (Inamura & Nagai Reference Inamura and Nagai1997; Wu et al. Reference Wu, Kirkendall, Fuller and Nejad1998; Prakash et al. Reference Prakash, Sinha, Tomar and Ravikrishna2018). One can also see that there are distinct differences with orifice $\textit{AR}$ (figures 28 a), with the $\textit{AR}=0.3$ image on the left showing relatively large structures, while the $\textit{AR}=3.3$ case shows finer droplets. We shall discuss this in more detail next in relation to the earlier discussions on initial breakup mechanisms in § 3.1.

In the case of $\textit{AR} = 0.3$ , because of the lower frontal area of interaction with the cross-flow, the surface wavelengths due to RTI near the stagnation region are larger, and this leads to the formation of a corrugated bow shock as discussed previously. The dominant breakup mechanism in this case, as discussed previously, occurs on the already deformed lateral surfaces by the windward stagnation region through KHI that eventually results in the large number of ligaments leading to the generation of larger as well as non-spherical droplets at $y/h_d = 0.5$ , as evident from figure 28(ai). A reason for the presence of larger droplets could be the generation of ligaments and subsequent breakup due to the delayed breakup time (figure 7) (there is a delay in forming spherical droplets). In contrast, the $\textit{AR} = 3.3$ case is subjected to stronger accelerations, leading to amplified RTI resulting in the formation of lower surface wavelengths followed by smaller breakup time along with a bow shock relatively free from corrugations. All these conditions favour atomisation of the liquid jet directly through the amplification of the RTI leading to catastrophic breakup. Hence, the droplet sizes produced are smaller and more spherical in nature (figure 28 aiii). These droplet images are thus consistent with our earlier observations on different breakup mechanisms as well as breakup times for varying orifice $\textit{AR}$ , and they show that the differences seen earlier in surface wavelengths do lead to clearly observable differences in the droplet shape and sizes much further downstream. Traversing the measurement station from C to D of figure 27 (not shown here), continuous interaction with the supersonic cross-flow results in the formation of droplet mist filled with very fine droplets. This is independent of the $\textit{AR}$ . Furthermore, the droplet count decreases considerably as the measurement station advances to the spray edge (E) (figure 28 b).

Figure 21. Probability density function (p.d.f.) plots of (a) droplet size based on number, (b) droplet size based on the number in semi-log scale to highlight the exponential fall-off at larger diameters, (c) droplet size weighted with volume and (d) droplet streamwise velocity normalised with free stream velocity. These measurements correspond to $y/h_d$ = 0.5 (station ‘C’) and $x/D$ = 60 for different AR, and $J = 9.7$ .

Figure 22. Probability density function (p.d.f.) plots of (a) droplet size based on number, (b) droplet size based on the number in semi-log scale, (c) droplet size weighted with volume and (d) droplet streamwise velocity normalised with free stream velocity. These measurements correspond to $y/h_d$ = 1 (station ‘E’) and $x/D$ = 60 for different AR, and $J = 9.7$ .

Atomisation of a liquid jet under very high cross-flow Weber number conditions, as in the present case, leads to the formation of smaller droplets with higher number density. Hence, the smaller droplets dominate the sample collected at a given station, as expected. This can be understood from figure 21(a), which shows the droplet number probability density function (p.d.f.) for the three orifice $\textit{AR}$ cases at station C; the peak of the droplet number p.d.f. occurs at approximately 10 microns in all the three cases. To better interpret the tail portion of the p.d.f. (40–70 microns), the data of figure 21(a) are shown in semi-log scale in figure 21(b) to capture the effect of $\textit{AR}$ on the distribution of larger droplets. It is found from analysis of droplet size distribution measured in the spray core region that the droplet distribution function, p( $d$ ), displays an exponential fall-off at larger diameters, which is represented by

(5.1) \begin{equation} p(d) \approx e^{-nd}, \end{equation}

with a significant difference in the exponent n for the different orifice aspect ratios $\textit{AR}$ . The parameter, $n$ , captures the width of the droplet size distribution whose value increases from 0.08 to 0.15 with an increase in $\textit{AR}$ from 0.3 to 3.3. A faster jet breakup (figure 7 c) resulting from the enhanced RT instability in the case of $\textit{AR}$ = 3.3 develops a narrow droplet size distribution (figure 21 b). In contrast, in the case of $\textit{AR}$ = 0.3, the delayed breakup (KH followed by RT) results in a wider droplet size distribution with a lower value of exponent ( $n$ ). A similar correlation with $n$ for the exponential (droplet distribution) fall has been reported by Maarmottant & Villermaux (Reference Maarmottant and Villermaux2004) for the coaxial jet breakup as the annular air velocity is increased.

Although the differences in large droplet sizes between $\textit{AR}$ cases seem small, they actually represent a considerable volume of liquid as these few large drops contain relatively very large amounts of liquid. Hence, to better differentiate between the $\textit{AR}$ cases, it is better to compare the volume weighted p.d.f. distributions, as has been done by Guildenbecher et al. (Reference Guildenbecher, Gao, Chen and Sojka2017) and Jackiw & Ashgriz (Reference Jackiw and Ashgriz2022). This volume p.d.f. enables better visualisation of the large-sized droplets present in the sample, as it represents the actual liquid volume present within drops of a given size range. Figures 21(c) and 22(c) represent the volume weighted p.d.f.s at stations C and E, respectively, for different $\textit{AR}$ . It may be seen from figure 21(c) that the volume of total liquid in the large drops (40–70 $\unicode{x03BC}$ m) at station C is significantly higher for $\textit{AR} = 0.3$ , when compared with $\textit{AR} = 1$ and 3.3. This is due to the fact that in the $\textit{AR}=0.3$ case, there are larger fragments within the spray core (station C) from the mass stripping and KH related breakup of liquid from the laterally deformed surfaces of the liquid jet. This indicates that at this station, a significant portion of the total volume of the droplets collected is occupied by larger droplets. The corresponding streamwise droplet velocity p.d.f. shown in figure 21(d) exhibits a lower velocity peak, attributed to the higher inertia associated with the larger droplets at this station.

In contrast to the spray core discussed before, the drop number and volume p.d.f.s at the spray edge (E) are similar for all the orifice $\textit{AR}$ cases as seen in figure 22, where panels (a) and (b) correspond to the droplet number p.d.f. in linear and semi-log scale and panels (c) and (d) correspond to the volume p.d.f. and the velocity p.d.f., respectively. This behaviour suggests that more direct interaction of the liquid jet with the supersonic cross-flow results in a similar distribution of smaller droplets in all $\textit{AR}$ cases, with the peak of the volume weighted droplet size occurring at approximately 20 microns and a velocity of approximately 0.8 $U_\infty$ .

Figure 23. Diameter–velocity correlation plots for circular and elliptical jets in the supersonic cross-flow with different aspect ratios at (a) $y/h_d = 0.5$ (station ‘C’), and (b) $y/h_d = 1$ (station ‘E’) for $x/D = 60$  and  $J = 9.7$ .

Figure 23 shows the diameter-velocity correlation plots for the atomisation of liquid jets with different $\textit{AR}$ in the supersonic cross-flow at stations C and E along the transverse direction at $x/D = 60$ and $J = 9.7$ . For a given sample, they are determined by averaging the velocities corresponding to each droplet bin size (1 $\unicode{x03BC}$ m). In the plots, the value of $u$ is normalised using $U_\infty$ . It is observed from figure 23 that, as expected, larger diameter droplets move with lower velocities due to their higher inertia. At station C in the spray core, considerable variation in the droplet diameters and velocities are present among the different $\textit{AR}$ cases due to the differences in breakup mechanisms, as discussed earlier. Among the three $\textit{AR}$ cases, it may be seen that the $\textit{AR}=3.3$ case shows higher $u/U_\infty$ at station C (figure 23 a), which suggests a better momentum exchange between the droplets and free stream air as opposed to the other $\textit{AR}$ cases. Also, a higher velocity gain by the spray droplets suggests an early completion of primary atomisation. This, as discussed earlier, is likely due to the direct catastrophic breakup of the flattened liquid jet slice by the RT waves in this case, which leads to early atomisation. In contrast, in station E at the spray edge, all the $\textit{AR}$ cases show similar behaviour (figure 23 b) as may be expected from the corresponding p.d.f.s seen earlier. The maximum and minimum velocity range at both station C and E for different $\textit{AR}$ is given in table 5. It should be noted that the observed variation in the velocity ranges in station C at the spray core between different orifice $\textit{AR}$ cases is considerable, while there is no change at station E.

Table 5. Droplet velocity range at stations C and E for different $\textit{AR}$ .

Figure 24. Variation of (i) SMD and (ii) $ U$ along the transverse direction for the spray with $J = 9.7$ and different $\textit{AR}$ at different streamwise positions: (a) $x/D = 60$ ; (b) $x/D = 90$ ; and (c) $x/D = 115$ .

Figure 24(i) shows the evolution of SMD in the transverse direction of the spray plume for different $\textit{AR}$ and $J = 9.7$ . The value of SMD is calculated from the drop size distribution using the following expression:

(5.2) \begin{equation} \mathrm{SMD} = \frac {\sum _{i=1}^{m}n_id_i^{3}}{\sum _{i=1}^{m}n_id_i^{2}}, \end{equation}

where $m$ is the number of droplet size classes present in the distribution and $n_i$ is the number of droplets in the droplet size class with mean diameter $d_i$ . For a given $x/D$ , the spray from $\textit{AR}$ = 0.3 and 1 both exhibit an ‘S’-shaped profile of SMD variation in the transverse direction whereas $\textit{AR} = 3.3$ significantly deviates from the ‘S’-shaped profile. The reasons behind the observed ‘S’-shaped profile of SMD variation were already discussed in our previous study (Lin et al. Reference Lin, Kennedy and Jackson2004; Medipati et al. Reference Medipati, Deivandren and Govardhan2023). Also, it is noteworthy to mention that similar ‘S’-shaped profiles for SMD were observed in circular liquid jets at higher subsonic cross-flow ( $U_\infty$ = 103 m s⁻¹) studies (Inamura & Nagai Reference Inamura and Nagai1997; Wu et al. Reference Wu, Kirkendall, Fuller and Nejad1998). This strongly suggests that the atomisation mechanism is similar in both high subsonic and supersonic cross-flows, except that the atomisation phenomena are quite intense in the latter which is manifested as a significant drop in the SMD range i.e. from (50–100) $\unicode{x03BC}$ m to (15–25) $\unicode{x03BC}$ m.

Empirical correlations to predict SMD of circular ( $\textit{AR}=1$ ) jet-in-cross-flow spray have been reported in the literature (Becker & Hassa Reference Becker and Hassa2002; Song et al. Reference Song, Cary Cain and Guen Lee2015) and their details are given in table 6. It is important to mention that these correlations are derived at high-We $_\infty$ with subsonic cross-flow, where shock waves are absent in the flow ahead of the liquid jet. In contrast, in the present work of jet in supersonic cross-flow, a bow shock is present ahead of the liquid jet, and in this case, we use a mean effective Weber number, $\overline {We}_{{eff}}= \overline {\rho _{\infty 2}U_{\infty 2}^2} D/\sigma$ , that is calculated based on the average density ( $\overline {\rho _{\infty 2}}$ ) and free stream velocity ( $\overline {U_{\infty 2}}$ ) downstream of the shock. It may be noted that in the present flow, the shock angle varies with height, and we use representative values of ( $\overline {\rho _{\infty 2}}$ ) and ( $\overline {U_{\infty 2}}$ ) that are calculated as the average of two extreme conditions of the bow shock, namely, the normal shock (near the stagnation point) and the Mach wave (at the maximum penetration height). The predicted values of SMD from these correlations for the present $\textit{AR}$ = 1 jet conditions are shown in table 6. It is seen that the predictions are comparable with the present experimentally determined SMD, with the differences likely due to the fact that the correlations were derived for subsonic cross-flow conditions that are free from shock waves.

Table 6. SMD correlations for circular jet-in-cross-flow spray reported in the literature. For comparison, the present circular ( $\textit{AR}$ = 1) jet case corresponds to a mean effective $We$ after the shock of ( $\overline {We}_{{eff}}$ ) = 1594.5.

Though the SMD data of the spray from $\textit{AR}$ = 0.3 and 1 follow a similar profile shape, significant differences exist in the droplet sizes at the core of the spray plume. The spray from $\textit{AR} = 0.3$ exhibits higher SMD because of larger droplets than for $\textit{AR} = 1$ and 3.3. The plots reveal that in the core region of the spray plume ( $y/h_d = 0.5$ ), the SMD of the spray with $\textit{AR} = 3.3$ is smaller than that with $\textit{AR} = 1$ and 0.3. It can be concluded quantitatively that the spray from $\textit{AR} = 3.3$ exhibits a better quality of atomisation, i.e. finer droplets. Also, the detailed study on the atomisation of liquid jet in cross-flow by Amighi & Ashgriz (Reference Amighi and Ashgriz2019) showed that $\mathrm{SMD} \propto D^{-0.18}$ . Since varying $D$ is equivalent to varying the transverse dimension of the jet in the present study, i.e. $\textit{AR}$ , while maintaining similar jet flow conditions. From the present experimental spray core data, it is observed that $\mathrm{SMD} \propto D^{-0.17}$ , indicating that an increase in $\textit{AR}$ leads to lower SMD, which is in line with the correlation of Amighi & Ashgriz (Reference Amighi and Ashgriz2019).

Moreover, irrespective of the streamwise distance, the measured SMD for the spray with $\textit{AR} = 3.3$ near the tunnel floor is significantly higher than that for $\textit{AR} = 0.3$ and $\textit{AR} = 1$ . This is most likely due to the early breakup (figure 7), and large bending of the $\textit{AR} = 3.3$ jet by the cross-flow. The larger droplets/liquid ligaments entrain into the low-velocity boundary layer near the tunnel floor. Subsequently, these droplets undergo a less intense secondary atomisation along the streamwise direction. Smaller droplets (more susceptible to the surrounding airflow) reach the tunnel floor for the sprays with $\textit{AR} = 0.3$ and $\textit{AR} = 1$ due to the delayed jet breakup and lifting of the spray plume from the tunnel wall. At downstream locations ( $x/D = 90$ and 115), the earlier larger droplets undergo secondary atomisation, resulting in the observed reduction in the SMD. It is observed from the instantaneous images (figure 4) that the lower part of the spray plume region in the case of $\textit{AR}$ = 1 (figure 4 b) contains relatively smaller droplets compared with those in the case of $\textit{AR}$ = 0.3 (figure 4 a). This is due to the difference in atomisation mechanism near the leeward side of the jet exit, as already discussed. The formation of denser spray on the leeward side ( $y/h_d$ < 0.5) due to the vigorous mass stripping in the case of $\textit{AR}$ = 0.3 can lead to early fragmentation of ligaments or droplets in the spray plume. These fragmented droplets or ligaments are in continuous interaction with the low-speed boundary layer fluid near the tunnel floor as well as the lower momentum cross-flow fluid (caused by a stronger shockwave). Such delayed breakup in spray droplets causes the presence of larger droplets (higher SMD) at these heights ( $y/h_d$ < 0.5) of the downstream measurement stations (figure 24 i). In contrast, reduced mass stripping in the case of $\textit{AR}$ = 1 jet results in the formation of a dilute spray region in the downstream measurement locations at lower transverse heights ( $y/h_d$ < 0.5) without significant presence of larger droplets.

The variation of mean streamwise velocity ( $U$ ), measured using the PTV technique, along the transverse direction for the sprays with different $\textit{AR}$ at different locations in the spray plume ( $x/D$ = 60, 90 and 115), is shown in figure 24(ii). The cross-stream variation of $U$ exhibits a C-shaped curve with a minimum of approximately 0.35 $U_\infty$ and 0.4 $U_\infty$ at $y = 0.5 h_d$ for $\textit{AR} = 0.3$ and 1, respectively. The larger droplets present at this region (see figures 21 i and 21 ii) have higher inertia; hence, move with reduced velocities. However, for $\textit{AR} = 3.3$ , the minimum mean streamwise velocity (0.35 $U_\infty$ ) is observed near the tunnel floor. For all jets, beyond $y = 0.5h_d$ , the value of $U$ rises with the increase in $y$ inside the spray plume and reaches the maximum $U$ of approximately 0.85 $U_\infty$ at the top spray edge. From this, it can be concluded that the entire spray plume experiences a non-uniform velocity in the transverse direction, which most likely arises from the bow shock formation in the supersonic cross-flow.

To summarise, we see substantial differences in the droplet number, volume and velocity p.d.f.s in the core of the spray as the orifice $\textit{AR}$ is varied, with some of the key points being shown in table 7. In the case of $\textit{AR}=0.3$ , although we see RT waves near the stagnation region of the injected jet, the primary atomisation likely is initiated by KH waves on the deformed lateral sides, leading to tongues of liquid that protrude into the cross-flow (as reported by Varga et al. (Reference Varga, Lasheras and Hopfinger2003)) that are broken by RTI. This results in a substantial increase in the contribution of liquid volume in the larger droplets after being fragmented due to mass stripping from the lateral sides as seen in the volume p.d.f. (see figure 21 c). Consequently, the volume p.d.f. in this case is significantly different from the other $\textit{AR}$ cases with it having a broad peak centred at approximately 35 $\unicode{x03BC}$ m along with a higher SMD (27 $\unicode{x03BC}$ m) in the spray core. In the $\textit{AR}=1$ case, although RT waves are again present near the stagnation region of the injected jet, the primary atomisation would again result from KHI on the deformed small lateral sides (see figure 5) of the injected jet slice with RT that could occur subsequently (on the tongues) as in the $\textit{AR}=0.3$ case. However, it is important to note that there are large quantitative differences in the volume p.d.f. for the $\textit{AR}=1$ case compared with the $\textit{AR}=0.3$ case. In particular, the volume p.d.f. peak in this $\textit{AR}=1$ case is much narrower and the peak shifts substantially to smaller values (25 $\unicode{x03BC}$ m), with the SMD in the core consequently also being reduced significantly reduced to 21 $\unicode{x03BC}$ m compared with the 27 $\unicode{x03BC}$ m for the $\textit{AR}$ = 0.3 case. In contrast, in the case of $\textit{AR}=3.3$ , the atomisation occurs directly through RT instability in a catastrophic manner as observed by Joseph et al. (Reference Joseph, Belanger and Beavers1999). This results in the peak of the volume p.d.f. reducing to even lower droplet diameters (18 $\unicode{x03BC}$ m), and the relatively direct clear atomisation by RT results in the SMD (18 $\unicode{x03BC}$ m) also being close to the peak in the volume p.d.f. (18 $\unicode{x03BC}$ m). Further, it may be noted that the catastrophic break up directly by RT in this $\textit{AR}=3.3$ case also results in early breakup of the liquid jet as seen in the relatively large droplet velocities as seen in the velocity p.d.f. (see figure 21 d).

Table 7. Effect of AR on $\lambda /b$ , peak in the volume PDF, SMD and the destabilisation mechanism.

5.2. Temporal variation of spray characteristics

Having obtained the long-time averaged information on droplet size distribution, we shall now focus our attention on the unsteadiness involved in the spray characteristics which are significantly altered by AR. These are caused by surface wave-induced shock corrugations in the spray core for the $\textit{AR}=0.3$ case and at the spray edge for all $\textit{AR}$ cases primarily due to the unsteadiness in jet penetration height that is mainly driven by the boundary layer velocity fluctuations as discussed in § 4.3.

Figure 25. Temporal variation of SMD and $U$ at the (i) core ( $y/h_d = 0.5$ and $x/D = 60$ ) and (ii) edge ( $y/h_d = 1$ and $x/D = 60$ ) of the spray in supersonic cross-flow with (a) $\textit{AR} = 0.3$ , (b) $\textit{AR} = 1$ and (c) $\textit{AR} = 3.3$ , and $J = 9.7$ .

Figures 29(a) and figures 29(b) (Appendix B) show a sample pair of droplet images acquired at the core of the spray plume ( $y/h_d = 0.5$ ) and top edge ( $y/h_d = 1$ ) of the spray, respectively, for $\textit{AR} = 0.3$ , and $J = 9.7$ at $x/D = 60$ . These pairs of images are separated by a very large time difference of 66 ms, which essentially means that these are completely independent realisations of the spray. Note that the PDIA technique is able to size also non-spherical drops and for these, an area equivalent diameter is used. Furthermore, the PDIA exhibits a high dynamic range in drop size. Figures 29(a) (i and ii) clearly distinguish the number of droplets, droplet size and shape between the time instants at $y/h_d = 0.5$ for the spray with $\textit{AR} = 0.3$ . A similar pattern but at a reduced scale is observed for $\textit{AR} = 1$ and 3.3. Although highly nonlinear spray dynamics in the form of intense ligaments/droplets formation, which undergo stretching followed by a breakup, is taking place by the time the spray arrives at the present measurement station, one of the possible reasons for this peculiar unsteady behaviour observed in the case of the $\textit{AR} = 0.3$ jet could be due to the increased fluctuations of instantaneous formation of large-scale windward surface wavelengths resulting in variations in curvature/strength of the stronger bumpy shocks with time (see figures 8 ai and 8 aii). The quantified difference in this unsteady spray behaviour between the $\textit{AR}$ s will be discussed in detail next.

At the spray edge ( $y/h_d = 1$ ), due to the continuous interaction of the spray plume with the oncoming supersonic air, the droplet size (narrower) and shape are similar between the instants, but there is a wide range in the number of droplets between the instants (figures 29 biii and 29 biv). Irrespective of AR, the unsteadiness in jet penetration heights governs the variations in droplet shape, size and number at all cross-streamwise directions.

To quantify the unsteady behaviour showcased by the previous images, SMD and mean streamwise velocity ( $U$ ) for each instance are determined by identifying the droplet size and its velocity at each instant. Figures 25(i) and 25(ii) illustrate the temporal variation of SMD and $U/U_\infty$ at $y/h_d = 0.5$ and 1 for $\textit{AR} = 0.3$ , 1 and 3.3. In the core of the spray plume (figure 25 i) for $\textit{AR} = 0.3$ , the SMD (10–45) $\unicode{x03BC}$ m and $U/U_\infty$ (0.2–0.45) vary over a larger range when compared with other $\textit{AR}$ cases. Also, these variations are much higher when compared with variations in long-time averaged SMD and $U/U_\infty$ among the stations, as well as with $\textit{AR}$ . A substantial reduction of instantaneous difference in maximum and minimum SMD values was found with an increase in $\textit{AR}$ i.e. from 35 $\unicode{x03BC}$ m (AR = 0.3) to 20 $\unicode{x03BC}$ m (AR = 3.3) because of the significant reduction in the variations in size of the windward liquid structures; hence, the shock bumpiness. At the spray edge, the temporal variation in SMD (10–25) $\unicode{x03BC}$ m and $U/U_\infty$ (0.7–0.9) exhibits significant reduction irrespective of the $\textit{AR}$ , as shown in figure 25(ii). These fluctuations are attributed to the variations in the spray edge position/penetration height of the liquid jet between the instants (see figure 17). This is likely related to shock boundary layer interactions and the resulting low-frequency fluctuations that are known to occur with typical frequency ( $f$ ) represented by a Strouhal number, St = $f\delta /U_\infty \sim$ 0.01 (Clemens & Narayanaswamy Reference Clemens and Narayanaswamy2014), where $\delta$ and $U_\infty$ represent the incoming boundary-layer thickness and the free stream velocity, respectively. In the case of liquid jet injection into supersonic cross-flow, Medipati et al. (Reference Medipati, Deivandren and Govardhan2023) have shown that this can lead to naturally induced pressure and mass flow fluctuations in the liquid line at $f\delta /U_\infty \sim$ 0.01, which can lead to significant unsteadiness in the penetration height and drop sizes.

A quantitative correlation analysis is performed between the instantaneous values of SMD, and $U/U_\infty$ for different aspect ratios and spray plume stations. The correlation values fall in the range of -0.5 to -0.7. It indicates that an instant with larger droplets leads to higher SMD, resulting in lower streamwise droplet velocity ( $U$ ). This strongly suggests that the values of SMD and $U/U_\infty$ are highly correlated.

In summary, the unsteadiness in droplet sizes in the spray core, particularly seen at $\textit{AR}=0.3$ case, is seen to be related to the large wavelength surface waves and the associated shock corrugations. However, large variations in droplet sizes in time seen at the spray edge in all the $\textit{AR}$ cases are seen to be related to the unsteadiness in jet penetration height induced by low- and high-speed streaks within the boundary layer interacting with the shock foot.