5.1. Lobe stretching, bending and perforation

The early deformation pattern for ${We}_G <1000$ involves lobes stretching downstream into the gas phase above the liquid surface. For higher thermodynamic pressures, the lobes become thinner and their dynamics are easily influenced by the gas flow around them. In particular, they bend upward and are eventually perforated (e.g. cases B1 and C1). Here, case C1 is analysed in detail since it presents the strongest real-fluid thermodynamics and intraphase mixing effects. Figure 8 conveys the differences between cases A2, B1 and C1, all of which have a similar ${We}_G$.

The evolution of the initial vortex (KH vortex) forming from the shear between the fluids is presented in figure 5. This vortex is responsible for the early lobe deformation. Here $\lambda _{\rho,t}=-2.5\times 10^{15}$ has been chosen to represent this vortex, as it is a good compromise between vortex identification and visualisation noise. Note that during the deformation of the vortex, some regions weaken and the magnitude of $\lambda _\rho$ falls below the chosen threshold $\lambda _{\rho,t}$. This roller vortex rotates clockwise (i.e. toward $-z$) as seen from the $xy$ planes in figure 5, and rapidly deforms into a hairpin vortex between $t^*=3$ and $t^*=4.2$. During this process, the vortex head is always located downstream of the lobe's tip. Initially, the vortex induces the stretching of the lobe in the streamwise direction, which is easily deformed due to the extreme transcritical environment at 150 bar. The dissolution of the ambient gas into the liquid phase and the heating decrease the liquid density and generate a liquid mixture with a gas-like viscosity near the interface (see figure 9 in PS). Thus, this fluid is easily perturbed by the dense and compressed gaseous phase at 150 bar. As the vortex pins (or legs) align with the streamwise direction, the vortex head moves upward into the oxidiser stream. As a consequence, the lobe stretches and bends sharply, facing the incoming gas before being perforated between $t^*=4.05$ and $t^*=4.2$. This combined motion of lobe and vortex is caused by the velocity induced by the counter-rotating vortex pins as the gas flows underneath them and into the bottom of the lobe. Such self-induction mechanism is described in Zandian et al. (Reference Zandian, Sirignano and Hussain2018). Two other vortices are identified: a vortex along the lobe's flank and a vortex within the liquid phase. Following the nomenclature presented in figure 3, the first vortex is henceforth called rim vortex. A similar vortex is identified as a hairpin vortex in Zandian et al. (Reference Zandian, Sirignano and Hussain2018) because of its shape. The other vortex in the liquid phase is identified via the density-weighting effect of $\lambda _\rho$. Both vortices rotate clockwise when viewed toward $-z$.

Figure 5. Vortex dynamics of the lobe extension, bending and perforation mechanism for case C1: (a) $t^*=3$; (b) $t^*=3.75$; (c) $t^*=4.05$; and (d) $t^*=4.2$. The top figures show the side view at $z=35\ \mathrm {\mu }\textrm {m}$ and the bottom figures show the top view of the liquid. The liquid surface is identified by the translucent blue isosurface with $C=0.5$ and the vortex structures are the red isosurface with $\lambda _{\rho,t}=-2.5\times 10^{15}$.

The velocity and vorticity fields explain the deformation process. Figure 6 presents contours of relevant components of the velocity and vorticity vectors in two flow cross-sections at $t^*=4.05$. The $xy$ plane at $z=25\ \mathrm {\mu }\textrm {m}$ and the $yz$ plane at $x=26\ \mathrm {\mu }\textrm {m}$ cut the lobe near its tip. The initial evolution of the lobe looks spanwise symmetric across its centreline, but over time symmetry is broken and the liquid structures behave independently. The rotation of the vorticity field is highlighted for completeness. The vortex head is aligned with the spanwise direction, while the vorticity vector in the vortex pins is predominantly in the streamwise direction (see figure 5). The vorticity around the rim vortex is also noteworthy.

Figure 6. Vortex dynamics of the lobe stretching, bending and perforation mechanism for case C1 at $t^*=4.05$: (a) $u$ at $z=25\ \mathrm {\mu }\textrm {m}$; (b) $v$ at $z=25\ \mathrm {\mu }\textrm {m}$; (c) $\omega _z$ at $z=25\ \mathrm {\mu }\textrm {m}$; (d) $v$ at $x=26\ \mathrm {\mu }\textrm {m}$; (e) $w$ at $x=26\ \mathrm {\mu }\textrm {m}$; and (f) $\omega _x$ at $x=26\ \mathrm {\mu }\textrm {m}$. The interface is identified by the solid isocontour with $C=0.5$ and the vortex cross-sections are identified by the dashed isocontour with $\lambda _{\rho,t}=-2.5\times 10^{15}$.

Figure 6 captures well the gas flow underneath the hairpin vortex and how it is ejected upward and toward the lobe, causing the described lobe–vortex deformation and hole formation. The hairpin vortex stretches and thins the lobe in both $x$ and $z$, leading to perforation. Liquid-phase stretching is enhanced at high pressures and defines a dominant deformation mechanism where liquid stretching and folding precedes the formation of liquid sheets or layers. This is a result of the low surface tension and the strong variation of liquid properties caused by the dissolution of the ambient gas (see PS).

The complex coupling between vorticity dynamics, thermodynamics and surface deformation is a key characteristic of transcritical flows. Not only the variation of fluid properties within each phase caused by mixing are determinant, but also the local interface state (e.g. surface tension, ambient gas dissolution and liquid vaporisation). Figure 7 shows the lobe at $t^*=4.05$ from above the liquid surface. In each sub-figure, the surface is coloured by the local interface temperature, fuel mass fraction in the gas phase, oxygen mass fraction in the liquid phase and surface-tension coefficient. Various features become immediately clear, such as the enhanced oxygen dissolution in the liquid phase at 150 bar or the strong correlation between surface temperature and composition given by phase equilibrium (see figure 2). For the observed temperature range, the amount of oxygen dissolved in the liquid does not vary much with temperature. However, considerably more fuel is vaporised as the interface temperature increases. Altogether, higher local temperatures result in lower surface tension. Here, the vortical motion responsible for the lobe stretching and bending carries the lobe's tip toward hotter regions in the oxidiser stream. Thus, a temperature rise occurs. Moreover, locally thin liquid regions heat faster. That is, the heat coming from the hotter gas cannot diffuse into colder liquid regions. This effect is observed along the lobe's edge and at the perforation location.

Figure 7. Solution of the LTE along the lobe for case C1 at $t^*=4.05$. A top view of the lobe from above the liquid surface is shown. The liquid surface is coloured by the local interface value for: (a) temperature, $T$; (b) surface-tension coefficient, $\sigma$; (c) fuel vapour mass fraction, $Y_F$; and (d) dissolved oxygen mass fraction, $Y_O$.

The local heating of the liquid enhances all the above-mentioned transcritical features (e.g. reduced liquid density and viscosity and denser gas), explaining the vorticity-induced enhanced lobe deformation. In addition, PS showed that, at very high pressures, the interface may undergo net condensation or net vaporisation depending on the local thermodynamic state and the species mass and energy balances. Despite the constant fuel evaporation caused by the hotter gaseous environment, the dissolution of the ambient gas increases substantially. Consequently, there are interfacial regions where the liquid internal energy exceeds the gas internal energy, thereby resulting in net condensation (Poblador-Ibanez & Sirignano Reference Poblador-Ibanez and Sirignano2018). For the analysed configuration, net condensation dominates at interface temperatures around 460–470 K. At higher temperatures, n-decane evaporates faster than the dissolution rate of oxygen (i.e. net vaporisation occurs).

The changes in ambient or thermodynamic pressure affect strongly the evolution of the lobe and the initial roller vortex. Although the evolution of vorticity is coupled to the highly varying properties of the transcritical fluid, we only focus on certain dynamical aspects and summarise the observed features. Figure 8 shows how cases A2, B1 and C1i evolve differently than C1 despite having similar ${We}_G$. Case C1i consists of an incompressible simulation without real-fluid thermodynamics and where the fluid properties are set equal to those in table 1 for case C1. The goal of this simulation is to understand the influence of liquid intraphase mixing in the deformation process.

Figure 8. Pressure and mixing effects on the vorticity dynamics of the lobe extension, bending and perforation mechanism: (a) case A2 at $t^*=4.2$ with $\lambda _{\rho,t}=-3\times 10^{15}$; (b) case B1 at $t^*=4$ with $\lambda _{\rho,t}=-5\times 10^{15}$; and (c) case C1i at $t^*=4.05$ with $\lambda _{\rho,t}=-1\times 10^{15}$. The top figures show the side view at $z=35\ \mathrm {\mu }\textrm {m}$ and the bottom figures show the top view of the liquid. The liquid surface is identified by the translucent blue isosurface with $C=0.5$ and the vortex structures are identified by the red isosurface of $\lambda _\rho$.

Figure 8 presents the liquid lobe and the deformed roller vortex right before the hole forms in cases B1 and C1 around $t^*\approx 4$. The formation of a clear hairpin vortex is visible for the 100-bar and 150-bar cases at the chosen $\lambda _{\rho,t}$ values. At 50 bar (i.e. case A2), the vortex is subject to higher velocities with the $70\ \textrm {m}\ \textrm {s}^{-1}$ gas freestream velocity, resulting in rapid streamwise stretching and weakening of the vortex head compared with the cases with higher pressures and smaller velocities. How each vortex affects the lobe depends on the properties of the liquid phase and how they vary at a given ambient or thermodynamic pressure. At lower pressures, lobes are thicker and rounder because of higher surface tension. That is, smaller curvatures are required for similar pressure jumps across an interface. Moreover, less oxygen dissolves into the liquid, and the density and viscosity remain more similar to the pure liquid properties. Coupled with the lower gas inertia, the lobes do not deform as easily as at higher pressures.

Further insights are obtained by looking at the relative position between the hairpin vortex and the lobe (see figures 5 and 8). As pressure decreases, the relative distance between the vortex and the lobe increases. For case C1 at 150 bar, the lobe's tip is closer to the vortex head than for case B2 at 100 bar. Similarly, this distance increases between case B2 and case A2 at 50 bar. Disregarding the differences in gas freestream velocity between cases, the separation between the lobe's tip and the vortex head can be understood as a measure of the lobe's streamwise stretching (i.e. influenced by the vortex induced velocity). Similarly, the separation between the vortex pins and the lobe's lateral edges can be understood as a measure of the lobe's spanwise stretching. For case C1, the lobe easily overlaps the vortex pins, but it barely occurs for case B2. For case A2, the vortex pins and the lobe are separated. As a result, case A2 does not undergo hole formation whereas in cases B2 and C1 the lobe stretches enough to become sufficiently thin and be perforated. Figure 8 also illustrates how intraphase mixing affects the evolution of the lobe. Even though the hairpin vortex evolves similarly in cases C1 and C1i and the freestream fluid properties are the same, the lobe is not stretched in the same manner. The constant fluid properties across the lobe due to lack of ambient gas dissolution prevent the lobe's stretching and a hole cannot develop, resulting in a surface deformation consistent with the incompressible studies by Zandian et al. (Reference Zandian, Sirignano and Hussain2017, Reference Zandian, Sirignano and Hussain2018).

5.2. Lobe and crest corrugation

Another early deformation mechanism is observed for configurations with ${We}_G > 1000$, which is characteristic of cases B2, C2 and C3 (i.e. above ambient pressures of 100 bar and 150 bar). Initially, the lobes extend over the liquid surface, similar to the deformation mechanism detailed in § 5.1. However, instead of experiencing a spanwise rotation, a streamwise corrugation of the lobe and the crest of the main perturbation is observed. The corrugation of the lobe generates a very thin liquid film that expands and bursts into droplets, similar to a bag-breakup mechanism.

Case C2 is chosen for the analysis of this deformation mechanism. Similar to § 5.1, one of the 150-bar cases is chosen to illustrate the transcritical environment. Figure 9 presents the evolution of the vortical structures during the early times, which are identified using a value of $\lambda _{\rho,t} = -9 \times 10^{15}$. Note that higher gas freestream velocities translate into higher vorticity as highlighted by the magnitude of $\lambda _{\rho,t}$. The initial roller vortex, which rotates clockwise when seen towards $-z$, also deforms into a hairpin vortex. However, now the vortex remains attached to the lobe's edge everywhere, as seen in figure 9(a,d,g). This is consistent with the stronger vorticity inducing further lobe stretching than in case C1. Comparing against figure 5, the rim vortex identified at lower ${We}_G$ is not observed (i.e. it merges with the roller vortex).

Figure 9. Vortex dynamics of the lobe and crest corrugation mechanism for case C2: (a,d,g) $t^*=2.5$; (b,e,h) $t^*=3$; (c,f,i) $t^*=3.5$; (j,m,p) $t^*=4$; (k,n,q) $t^*=4.5$; and (l,o,r) $t^*=5$. The top figures show the side view at $z=35\ \mathrm {\mu }\textrm {m}$, the middle figures show the front view from the respective sub-figure domain, and the bottom figures show the top view of the liquid. The liquid surface is identified by the translucent blue isosurface with $C=0.5$ and the vortex structures are identified by the red isosurface with $\lambda _{\rho,t}=-9\times 10^{15}$.

As the roller vortex deforms into a hairpin, the stronger stretching causes the vortex pins to move underneath the lobe between $t^*=2.5$ and $t^*=3.5$ or, equivalently, the gas-like liquid phase wraps around the vortex. Once the vortex pins are fully captured by the lobe, the spanwise stretching is enhanced and the gas flow underneath the lobe is intensified. This process rapidly inflates the lobe, which becomes a very thin sheet, and then bursts into droplets. The bursting occurs first near the lobe's tip, which is stretched both in $z$ by the vortex pins and in $x$ by the head of the hairpin vortex. That is, the thinning occurs similar to the results shown in figure 6 for case C1. After the gas perforates the lobe, the hole rapidly recedes under the action of surface tension. The strong recirculation in the region promotes the formation of tiny droplets leading to a highly perturbed flow. From $t^*=3.5$ to $t^*=5$, liquid ligaments and droplets accelerate into the oxidiser stream. The hairpin vortex stretches rapidly and additional vortical structures emerge. For instance, a small secondary roller vortex above the hairpin vortex is seen at $t^*=5$. More details about the formation of this vortex are given in § 5.5.2.

The deformation process is visualised in figures 10(a)–10(d) where contours of $\omega _x$ are shown in $yz$ planes at various times and streamwise locations. As the liquid phase wraps around the vortex pins, the gap between the lobe's sides and the liquid surface below is reduced, accelerating the gas entrainment and intensifying the vortex streamwise strength. In addition, some vorticity appears under the vortex pins with the opposite rotation. Previous incompressible works by Jarrahbashi et al. (Reference Jarrahbashi, Sirignano, Popov and Hussain2016) and Zandian et al. (Reference Zandian, Sirignano and Hussain2018) do not show such a vortex-capturing mechanism. Lobes overlapping vortices are observed, but these vortices are secondary. Therefore, the results presented here differ substantially and the capturing process of the initial roller vortex is attributed to the transcritical environment. The sides of the lobe eventually merge with the liquid below. This ends the lateral gas entrainment and traps the vortex pins in the gaseous cavity underneath the lobe. The remaining vortical motion is then responsible for lifting the liquid surface below and flattening the lobe as continuous spanwise stretching occurs.

Figure 10. Vortex dynamics of the lobe corrugation mechanism for case C2: (a) lobe at $x=17 \ \mathrm {\mu }\textrm {m}$ and $t^*=3$; (b) lobe at $x=23\ \mathrm {\mu }\textrm {m}$ and $t^*=3.5$; (c) lobe at $x=27\ \mathrm {\mu }\textrm {m}$ and $t^*=4$; (d) lobe at $x=30\ \mathrm {\mu }\textrm {m}$ and $t^*=4.5$; (e) crest at $x=12.5\ \mathrm {\mu }\textrm {m}$ and $t^*=3$; and (f) crest at $x=17\ \mathrm {\mu }\textrm {m}$ and $t^*=3.5$. The contours of $\omega _x$ are shown on $yz$ planes at various $x$ following the lobe and the perturbation crest over time. The interface is identified by the solid isocontour with $C=0.5$ and the vortex cross-sections are identified by the dashed isocontour with $\lambda _{\rho,t}=-9\times 10^{15}$.

Figures 10(e) and 10(f) show the crest corrugation mechanism, which is caused by the induced flow from the upstream-facing side of the hairpin vortex. The upstream head of the hairpin vortex can be observed in figure 9(b,e,h) under the liquid wave at $z=15\ \mathrm {\mu }\textrm {m}$ or $z=35\ \mathrm {\mu }\textrm {m}$ between $x=10\ \mathrm {\mu }\textrm {m}$ and $x=15\ \mathrm {\mu }\textrm {m}$. Vorticity deforms the thicker liquid sheet in that region. First, the induced velocity field brings together the sides of the liquid sheet. This displacement corrugates or folds the liquid around $z=15\ \mathrm {\mu }\textrm {m}$, much like a sheet of paper folds when you bring together the two opposite edges. The deformation process continues with the liquid sheet weakly wrapping around the vortex pins, ejecting the gas in between the vortical structures, the liquid-core surface and the liquid sheet. Altogether, the wave crest corrugates, presenting a characteristic nose-like shape highlighted in PS. Figure 11 presents a sketch of the lobe and crest corrugation mechanisms to emphasise the main features and how both deformations are directly caused by the hairpin vortex. The lobe corrugation sketch is representative of the numerical solution for case C2 depicted in figure 10(b), whereas the crest corrugation sketch follows figure 10(f).

Figure 11. Sketch of the lobe and crest corrugation mechanisms and generated by the deformed initial roller or hairpin vortex. The gaseous side of the phase interface is marked with a triangle.

The crest corrugation mechanism is clearly observed in cases B2, C2 and C3. It is also apparent in other analysed configurations. The main reason why this mechanism becomes more relevant at high ambient pressures is directly linked to the increased liquid layer formation at highly transcritical conditions. As discussed previously, the thickness of the initial lobes depends on the surface tension and the dissolution of the ambient gas. Similarly, the liquid sheet near the wave crest becomes thinner at higher pressures. Thus, the thinner layers in the 100-bar and 150-bar configurations are affected more by the deformation mechanism discussed here.

As done in § 5.1, figure 12 presents the lobe in case C2 at $t^*=3.5$ viewed from an $xz$ plane above the liquid surface, showcasing the LTE solution at the interface. Compared with case C1 with a lower freestream velocity of $30\ \textrm {m}\ \textrm {s}^{-1}$, further lobe stretching and thinning are observed. This results in a faster heating of the liquid, a higher surface temperature and a lower surface-tension coefficient. Therefore, the edge of the lobe in case C1 is rounder than in case C2. The change in composition as the interface temperature increases is not shown here, and the reader is referred to figures 2 and 7 for more details.

Figure 12. Solution of the LTE along the lobe for case C2 at $t^*=3.5$. A top view from an $xz$ plane located above the liquid surface is provided. The liquid surface is coloured by the local interface value for: (a) temperature, $T$; and (b) surface-tension coefficient, $\sigma$.

Other features are observed. Once the lobe bursts, smaller liquid structures are generated that heat very fast, resulting in an increased vaporisation of the fuel. In addition, the corrugation of the perturbation crest submerges it into a hotter gaseous region, significantly raising the surface temperature compared to the cooler liquid below. Since crest corrugation is less pronounced in case C1, such a large temperature increase is not observed in figure 7. The heating of the perturbation crest and subsequent decrease in surface tension might cause a substantial change in the surface dynamics (see PS). As the crest submerges into the hotter gas, it also moves into a faster gaseous stream. This increase in relative velocity between liquid and gas, coupled with the decrease in $\sigma$, results in the emergence of short wavelength surface instabilities that quickly grow, leading to ligament shredding and the formation of small droplets (see PS).

5.3. Formation of liquid sheets or layering

Similar to the deformation cascade process that the liquid jet undergoes, vortices form, deform and break up over time. Understanding this process is the key since it defines the evolution of the liquid surface, its atomisation, and the mixing within each phase. For instance, vorticity is directly responsible for the formation of high temperature regions and fuel-rich blobs in the gas phase. Therefore, we examine the evolution of vortex structures to identify regions of vortex generation and how these vortices evolve toward smaller scales. In particular, we focus on the impact of the layering of liquid sheets characteristic of very high-pressure transcritical environments.

Major events in the evolution of the vorticity field for case C1 are presented in figures 13 and 14. The 150-bar lower velocity configuration has been chosen to clearly show the layering. In the following, we describe the major differences between case C1 and higher velocity cases. Initially, the shear between liquid and gas generates a roller or KH vortex along the perturbed surface, downstream of the lobe (see figure 13 at $t^*=2.25$). We name this vortex structure R1 (i.e. the first roller). Usually, a distinct rim vortex also appears along the lobe's edge. The roller vortex evolves into a hairpin vortex, whose impact on the early lobe deformation has been described in §§ 5.1 and 5.2. R1 is advected into the oxidiser stream, subject to continuous stretching and streamwise alignment (i.e. $\omega _x$ generation). Eventually, the hairpin head breaks from the hairpin legs, and then it reorients along the streamwise direction before breaking again. This vortex breakup process must be understood as the weakening of the local vorticity; thus, the value of $\lambda _{\rho,t}$ cannot capture the complete vortex structure anymore. The subsequent deformation of R1 can be observed in figure 13 from $t^*=2.25$ to $t^*=8.4$.

Figure 13. Evolution of vortex structures for case C1. An oblique view shows the liquid surface identified by the translucent blue isosurface with $C=0.5$; the vortex structures are identified by the red isosurface with $\lambda _{\rho,t}=-2.5\times 10^{15}$.

Figure 14. Evolution of vortex structures for case C1: (a) side view at $z=20\ \mathrm {\mu }\textrm {m}$, $t^*=8.4$; (b) front view at $x=30\ \mathrm {\mu }\textrm {m}$, $t^*=9.9$; and (c) $t^*=15$, top view from above the liquid surface. Each snapshot corresponds to a different $t^*$. The liquid surface is identified by the translucent blue isosurface with $C=0.5$ and the vortex structures are identified by the red isosurface with $\lambda _{\rho,t}=-2.5\times 10^{15}$.

Over time, new roller vortices form downstream of the growing perturbation. As seen in figure 13, vortex R2 forms between $t^*=5.4$ and $t^*=6$, and vortex R3 forms between $t^*=7.5$ and $t^*=8.4$. This vortex formation can be understood as a vortex pairing process. During the growth of the surface perturbation, the vortex sheet between liquid and gas detaches at intervals. The resulting spanwise vorticity downstream of the wave's edge tends to concentrate, forming the vortex tubes or rollers we identify as R2 and R3. Compared to R1, these new roller vortices remain very close to the liquid surface and further contribute to the liquid stretching that leads to the folding and layering of the liquid sheets. The wave growth and subsequent surface area growth enhance the heat and oxygen mass transfer into the liquid, accelerating the features that facilitate the liquid phase stretching. The rapid wave growth also causes strong gas entrainment, which swallows the roller vortices underneath the wave (see figure 14a,b). Such strong gas entrainment can be visualised at $t^*=8.4$, where the roller vortex is stretching under the liquid and resembles more a vortex sheet.

This vortex-capturing mechanism defines the local formation of small liquid sheets, ligaments and droplets under the liquid wave and, most importantly, explains the layering mechanism that dominates the transcritical liquid jet deformation. Figure 15 shows this process, highlighting the formation of the various roller vortices (i.e. R1, R2 and R3). After the onset of layering (i.e. for $t^*>10$), generation of strong vortices above the liquid layers is reduced, and most of the vortices visualised with $\lambda _{\rho,t}=-2.5\times 10^{15}$ are confined within the various liquid layers (see figure 14c).

Figure 15. Vortex capturing mechanism in case C1 and how it leads to the layering of liquid sheets. The interface is represented by the isocontour of $C=0.5$ along $xy$ planes at $z=5\ \mathrm {\mu }\textrm {m}$.

At this deformation stage, the liquid–gas mixing mechanisms are more complex, having a larger variability in interface properties and fluid properties within each phase. Figure 16 shows the interface temperature, liquid density and liquid viscosity for case C1 at $t^*=8.4$ before layering becomes dominant. During the growth of the main wave, the liquid region facing the hotter oxidiser stream is heated, raising the interface temperature. Once the roller vortices stretch the wave and cause its rolling, hotter liquid blobs move underneath the wave due to gas entrainment. As seen in figure 16(a), various liquid structures (e.g. sheets, ligaments and droplets) at different temperatures start to interact with the colder liquid in the wave trough. This non-uniform temperature's effect on the interface composition and surface-tension coefficient was presented in figures 2, 7 and 12. The liquid density and viscosity in figure 16 emphasise the variation of dynamical properties along the liquid surface and, consequently, within the liquid phase. Note that due to the dissolved oxygen at 150 bar, the liquid density and viscosity in the cold interface already differ substantially from the freestream properties detailed in table 1.

Figure 16. Solution of the LTE along the liquid for case C1 at $t^*=8.4$. A three-dimensional view and a side view from an $xy$ plane located at $z=20\ \mathrm {\mu }\textrm {m}$ are provided. The liquid surface is coloured by the local interface value for: (a) temperature, $T$; (b) liquid density, $\rho _l$; and (c) liquid viscosity, $\mu _l$.

An extreme case of the heating effects on smaller liquid structures is seen in figure 16. An isolated droplet is immersed in the hotter gas above the liquid, formed during the early lobe breakup described in § 5.1. Due to the smaller liquid volume, the surface temperature increases substantially (i.e. above 500 K). The increased temperature results in more fuel vapour and more dissolved oxygen than in other liquid regions. Thus, the liquid density, viscosity and surface-tension coefficient are substantially smaller than anywhere else in the liquid.

Some common vorticity patterns are observed in the layering deformation regime. The tearing of the liquid sheets and frequent hole formation may result in the formation of rim vortices, similar to the rim vortex identified during the early lobe deformation in case C1. As the hole expands, these vortices tend to weaken. Moreover, additional roller vortices are occasionally generated, whose importance is limited but continuing to contribute to the layering process. However, layering mitigates the gas entrainment into the two-phase mixture and flow recirculation overall, with little spanwise vorticity generation. As shown in PS, layering tends to uniformise the momentum mixing layer and the velocity field. Thus, the remaining vortices within the liquid sheets tend to align with the streamwise direction.

All configurations at 100 and 150 bar show the layering mechanism with a qualitative description similar to that discussed in previous lines. Nonetheless, the interface is more easily perturbed at higher velocities, with short-wavelength perturbations causing frequent ligament shredding and droplet formation. This behaviour is discussed in § 5.2 and in more detail in PS. This scenario is critical in cases B2, C2 and C3. The short-wavelength perturbations generate smaller roller vortices that are responsible for the formation and stretching of smaller lobes. Eventually, these lobes break up into numerous ligaments and droplets, hence multiphase turbulence increases and flow visualisation becomes difficult for the purposes of this study. As the smaller ligaments and droplets heat faster, the variations in surface properties become larger.

Although not shown here, the long-term evolution of vortex structures in the flow at 50 bar (i.e. cases A1 and A2) resembles more that of subcritical atomisation. Layering occurs rarely and the liquid phase cannot be so easily stretched due to the higher surface tension and the mitigation of intraphase mixing (PS). The vortex structures in the gas phase do not weaken as fast as in the higher-pressure cases, which emphasises the following: (a) layering is a deformation process that reduces vorticity generation by stratifying the flow; (b) a gaseous phase with lower density and viscosity reduces viscous diffusion and dissipation; and (c) other vorticity generation mechanisms, such as baroclinicity, may play a key role in the lower ambient pressure.

5.4. Formation of fuel-rich gas blobs

The fuel–oxidiser mixing at trans- and supercritical conditions is commonly assumed to be driven by mass diffusion and the rapid formation of a nearly homogeneous mixture beneficial for combustion purposes (Mayer & Tamura Reference Mayer and Tamura1996; Roy & Segal Reference Roy and Segal2010; Hickey & Ihme Reference Hickey and Ihme2013). That is, the fluid is characterised by a gas-like diffusivity and the mixing front displays fewer irregularities, promoting a more uniform combustion. Such behaviour is key for combustion stability and a cleaner fuel burning.

However, the computations by PS show that the emergence of fuel-rich gaseous blobs is common during the early atomisation stages where a two-phase regime is sustained, even at pressures up to 150 bar with n-decane as a fuel. These exist due to the non-uniform fuel vaporisation along the surface and the breakup of liquid structures. Similar to the liquid deformation, the fluid dynamics (i.e. vorticity) also determine the evolution of these blobs, rather than mass diffusion alone. Thus, the dimension and shape of these blobs relate to the size of the liquid structures (i.e. micrometres) and the vorticity. Because the results presented by PS do not extend beyond a few microseconds, it is unclear whether these fuel-rich pockets diffuse quickly. Nonetheless, the existence of multitude of cool liquid structures which take time to vaporise suggests that fuel-rich blobs will appear as long as the liquid mixture remains transcritical.

Some examples of these fuel-rich regions are shown in figure 17 for case C1 and in figure 18 for case C2, which depict the isosurface of the fuel mass fraction with $Y_F=0.05$. First, figure 17 shows a fuel-rich blob around the isolated vaporising droplet at $t^*=6$, while the roller vortex R2 (see figure 14) promotes more fuel vaporisation along the edge of the growing wave. In particular, the blob is influenced strongly by the fluid dynamics. It presents a streamwise-elongated shape, and easily wraps around the head of the hairpin vortex R1 because of the induced flow. Then, figure 18(a) highlights the fuel-rich regions at $t^*=8.25$ (i.e. $t=3.3\ \mathrm {\mu }\textrm {s}$ for case C2) around the train of liquid structures shown evolving in figure 3. Later, these blobs interact with the vortices similar to figure 17 by wrapping around them (see figure 18(b) at $t^*=10$). Another example of fuel-rich and fuel-lean regions is figure 27 discussing the vorticity and mixing within the liquid layers, which shows large variations of the fuel vapour concentration.

Figure 17. Interaction between fuel-rich gaseous blobs and vortex structures for case C1 at $t^*=6$. The liquid surface is identified by the blue isosurface with $C=0.5$, the vortex structures are identified by the red isosurface with $\lambda _{\rho,t}=-2.5\times 10^{15}$ and the fuel-rich regions are identified by the translucent black isosurface with $Y_F=0.05$.

Figure 18. Interaction between fuel-rich gaseous blobs and vortex structures for case C2: (a) $t^*=8.25$; and (b) $t^*=10$. The liquid surface is identified by the blue isosurface with $C=0.5$, the vortex structures are identified by the red isosurface with $\lambda _{\rho,t}=-9\times 10^{15}$ and the fuel-rich regions are identified by the translucent black isosurface with $Y_F=0.05$.

Although we only show the isosurface with $Y_F=0.05$, the blobs typically have a much higher concentration of fuel vapour inside, especially when they surround broken liquid structures such as those depicted in figures 17 and 18. These small droplets and ligaments are observed to reach temperatures of 520 K, resulting in a fuel vapour concentration up to $Y_F=0.33$ around them. This is also highlighted by the phase-equilibrium diagrams shown in figure 2.

5.5. Vorticity generation in the transcritical flow

Sections 5.1 to 5.4 provided details of the deformation of vortical structures and how their interaction with the transcritical liquid mixture defines the fuel–oxidiser mixing and atomisation processes. For example, the growth of three-dimensional instabilities during the atomisation of the liquid is directly related to the generation of streamwise vorticity, as described in previous works such as Jarrahbashi & Sirignano (Reference Jarrahbashi and Sirignano2014), Jarrahbashi et al. (Reference Jarrahbashi, Sirignano, Popov and Hussain2016) and Zandian et al. (Reference Zandian, Sirignano and Hussain2017, Reference Zandian, Sirignano and Hussain2018). We impose an initial sinusoidal perturbation along the spanwise direction with a wavelength equal to the domain size (i.e. $L_z=20\ \mathrm {\mu }\textrm {m}$), but towards the end of our computations (e.g. figure 14c) two or more spanwise wavelengths are seen over the same distance (see PS). This process occurs because the initial roller vortex deforms into a hairpin where the pins align with $x$ (i.e. $\omega _x$ is generated). Thus, $\omega _x$ perturbs the surface and generates additional waves in $z$. Vortex tilting and streamwise alignment occur frequently (see, e.g., figures 13 and 14), defining both the liquid deformation and the fuel–oxidiser mixing (e.g. deformation patterns and fuel-rich gas blobs).

The vorticity equation is obtained by taking the curl of the momentum equation (3.2). The compressible form of the vorticity equation becomes

(5.1)\begin{equation} \frac{{\rm D}\boldsymbol{\omega}}{{\rm D}t}=(\boldsymbol{\omega}\boldsymbol{\cdot}\boldsymbol{\nabla})\boldsymbol{u}-\boldsymbol{\omega} (\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{u})+\frac{\boldsymbol{\nabla}\rho\boldsymbol{\times}\boldsymbol{\nabla} p}{\rho^2} + \boldsymbol{\nabla}\boldsymbol{\times}\left(\frac{1}{\rho}\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{\tau}\right)+\boldsymbol{\nabla}\boldsymbol{\times} \boldsymbol{F}_{\boldsymbol{\sigma}}, \end{equation}

where $\boldsymbol {F}_{\boldsymbol {\sigma }}$ is the body force term used to model surface tension (i.e. CSF approach). In the analysis that follows, the surface tension term and the viscous term $\boldsymbol {\nabla }\boldsymbol {\times } (({1}/{\rho })\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {\tau })$ are neglected (Zandian et al. Reference Zandian, Sirignano and Hussain2017). This approximation is also done for the $\lambda _\rho$ post-processing, as discussed in § 4. Hence, only the vortex stretching and tilting term, $(\boldsymbol {\omega }\boldsymbol {\cdot }\boldsymbol {\nabla })\boldsymbol {u}$, the vortex stretching due to volume dilatation, $-\boldsymbol {\omega }(\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {u})$, and the baroclinic term, $({\boldsymbol {\nabla }\rho \boldsymbol {\times }\boldsymbol {\nabla } p})/{\rho ^2}$, are analysed to study the vorticity of the large-scale dynamics. At smaller scales, viscosity may become important. Since (5.1) is a vector equation, each component of the vorticity field is given by

(5.2)\begin{gather} \frac{{\rm D}\omega_x}{{\rm D}t} = \omega_x\frac{\partial u}{\partial x} + \omega_y\frac{\partial u}{\partial y} + \omega_z\frac{\partial u}{\partial z} - \omega_x (\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{u}) + \frac{1}{\rho^2}\left(\frac{\partial \rho}{\partial y}\frac{\partial p}{\partial z} - \frac{\partial \rho}{\partial z}\frac{\partial p}{\partial y}\right), \end{gather}

(5.3)\begin{gather}\frac{{\rm D}\omega_y}{{\rm D}t} = \omega_x\frac{\partial v}{\partial x} + \omega_y\frac{\partial v}{\partial y} + \omega_z\frac{\partial v}{\partial z} - \omega_y (\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u}) + \frac{1}{\rho^2}\left(\frac{\partial \rho}{\partial z} \frac{\partial p}{\partial x} - \frac{\partial \rho}{\partial x}\frac{\partial p}{\partial z}\right), \end{gather}

(5.4)\begin{gather}\frac{{\rm D}\omega_z}{{\rm D}t} = \omega_x\frac{\partial w}{\partial x} + \omega_y\frac{\partial w}{\partial y} + \omega_z\frac{\partial w}{\partial z} - \omega_z (\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{u}) + \frac{1}{\rho^2}\left(\frac{\partial \rho}{\partial x}\frac{\partial p}{\partial y} - \frac{\partial \rho}{\partial y}\frac{\partial p}{\partial x}\right). \end{gather}

Fluid compressibility and variable density are important for vortex generation in transcritical flows. Vortices submerged in the mixing layers are subject to density and pressure gradients. Volume dilatation affects vorticity, and the baroclinicity now extends beyond the interfacial region. Nonetheless, baroclinicity might be negligible compared with other terms in dense fluids due to the scaling with $\rho ^{-2}$ (Zandian et al. Reference Zandian, Sirignano and Hussain2017, Reference Zandian, Sirignano and Hussain2018), such as in the liquid phase or the gas phase at very high pressures.

Similar to the $\lambda _\rho$ post-processing, the filtered velocity field (see § 4) is used to quantify vorticity and the relevant terms in (5.1), (5.2), (5.3) and (5.4) to minimise the influence of spurious currents around the liquid–gas interface in the visualisation. This filtering process does not affect significantly the results shown in this section; however, the differences in $\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {u}$ become important in the incompressible case C1i shown in figure 8 (i.e. filtered velocity field is not divergence free). Although we could gain more insight into the vorticity behaviour by just looking at the full terms in (5.1), splitting the vorticity vector equation into components allows us to focus on the specific vorticity generation mechanisms relevant to atomisation (e.g. streamwise vorticity).

In the discussion that follows, we refer to streamwise ($\partial /\partial x$), transverse ($\partial /\partial y$) and spanwise ($\partial /\partial z$) terms to describe the vortex stretching and tilting for each vorticity component. Stretching occurs along the direction of the analysed vorticity component (e.g. streamwise stretching for $\omega _x$) and tilting takes places in the other two orthogonal directions (e.g. transverse and spanwise tilting for $\omega _x$). The vortex stretching and tilting in (5.1) has three stretching components and six tilting terms (i.e. two per direction). Here, we define $\dot {\omega }_{x\rightarrow x}=\omega _x({\partial u}/{\partial x})$, $\dot {\omega }_{y\rightarrow y}=\omega _y({\partial v}/{\partial y})$ and $\dot {\omega }_{z\rightarrow z}=\omega _z({\partial w}/{\partial z})$ as the three vortex stretching contributions to vorticity generation. Then, transverse tilting refers to the reorientation of transverse vorticity or $\omega _y$. The two terms are defined as $\dot {\omega }_{y\rightarrow x}=\omega _y({\partial u}/{\partial y})$ and $\dot {\omega }_{y\rightarrow z}=\omega _y({\partial w}/{\partial y})$. Similarly, streamwise tilting terms are $\dot {\omega }_{x\rightarrow y}=\omega _x({\partial v}/{\partial x})$ and $\dot {\omega }_{x\rightarrow z}=\omega _x({\partial w}/{\partial x})$ and spanwise tilting terms are $\dot {\omega }_{z\rightarrow x}=\omega _z({\partial u}/{\partial z})$ and $\dot {\omega }_{z\rightarrow y}=\omega _z({\partial v}/{\partial z})$.

The following sections present a detailed description of the vorticity generation mechanisms of some of the most relevant features observed in §§ 5.1, 5.2 and 5.3. Section 5.5.1 focuses on the $\omega _x$ generation that explains the alignment with $x$ of the initial hairpin vortex legs. Then, § 5.5.2 describes the $\omega _z$ generation mechanisms of the secondary roller vortex shown in figure 9. Lastly, § 5.5.3 addresses the early stages of the layered flow to understand the sources of $\omega _x$ that promote the streamwise alignment of vortical structures trapped between the liquid layers.

5.5.1. Stretching of the hairpin vortices

The initial velocity distribution described in § 2 results in a vortex sheet with only spanwise vorticity ($\omega _z<0$), and a single non-zero velocity gradient ($\partial u/\partial y >0$). With the initial interface displacement, the major source of $\omega _x$ at the start of the simulation is the baroclinic term. After some time, vorticity is generated in the other directions and $\partial u/\partial x$ and $\partial u/\partial z$ are, in general, non-zero. Given the relative strength of $\omega _z$ and $\partial u/\partial y$ with respect to the other quantities, $\dot {\omega }_{y\rightarrow x}$ and $\dot {\omega }_{z\rightarrow x}$ become important $\omega _x$ generators. After the initial transient, Zandian et al. (Reference Zandian, Sirignano and Hussain2017, Reference Zandian, Sirignano and Hussain2018) show that, for low density ratios (i.e. $\rho _G/\rho _L$), the streamwise vorticity generation mechanism responsible for the formation of hairpin vortices in the gas phase is the baroclinic term. As the density ratio increases (i.e. the gas phase becomes denser), the relative importance of the baroclinic term diminishes due to its $\rho ^{-2}$ scaling in favour of $\dot {\omega }_{x\rightarrow x}$. In general, $\dot {\omega }_{z\rightarrow x}$ and $\dot {\omega }_{y\rightarrow x}$ are stronger than $\dot {\omega }_{x\rightarrow x}$, but they tend to cancel each other. A similar behaviour is observed in round jets (Jarrahbashi & Sirignano Reference Jarrahbashi and Sirignano2014).

This description explains the initial deformation of the KH roller vortex into a hairpin; thus, let us look at $t^*>2$ to understand the additional stretching the vortex pins undergo. Zandian et al. (Reference Zandian, Sirignano and Hussain2017, Reference Zandian, Sirignano and Hussain2018) visualised the baroclinic term because their computations used a more diffuse interface capturing method (i.e. level-set). The density gradient between liquid and gas extends a few cells beyond the interface, allowing a smoother visualisation. However, our sharp interface VOF method concentrates the density gradient at the interface cells. Together with the pressure oscillations near the interface due to the generation of spurious currents, the baroclinic contribution of the two-phase flow cannot be properly visualised during the early times (see figure 19d). Once the mixing layers grow sufficiently, we can capture the baroclinic term.

Figure 19. Terms for generating $\omega _x$ during the stretching of the initial hairpin vortex for case C1 at $t^*=3$: (a) $\omega _x$; (b) vortex stretching and tilting; (c) compressible stretching; and (d) baroclinic term. The front view from a $yz$ plane at $x=16\ \mathrm {\mu }\textrm {m}$ is shown. The interface is identified by the solid isocontour with $C=0.5$, and the cut vortex structures are identified by the dashed isocontour with $\lambda _{\rho,t}=-2.5\times 10^{15}$.

To understand the continuous stretching of the hairpin vortex pins, figure 19 shows the streamwise vorticity and its generation terms from (5.2) for case C1 at $t^*=3$ in a $yz$ plane at $x=16\ \mathrm {\mu }\textrm {m}$. The $\omega _x$ contours in figure 19(a) resemble those in figure 6(f), which depicts $\omega _x$ at a later time. The vorticity vector is predominantly aligned with $x$ inside the vortex pins. Another region of strong $\omega _x$ appears between the hairpin and the liquid below. Despite the variable density in the transcritical flow, the vortex stretching and tilting terms are at least an order of magnitude larger than the compressible stretching or baroclinic terms. This characteristic repeats for all analysed cases. In our study, lower pressures are associated with larger characteristic velocities and milder density variations across the mixing regions. Thus, vortex stretching and tilting terms continue to dominate, even if the lower density in the gas phase favours a stronger baroclinic term.

The combined effect of vortex stretching and tilting increases the $\omega _x$ magnitude in the vortex pins, effectively stretching the vortical structure. Compressible stretching opposes this deformation, but it has a weak contribution. Baroclinicity inside the vortex pins is not strong. Figure 20 presents the individual components of the vortex stretching and tilting terms from (5.2), and shows the partial cancellation of $\dot {\omega }_{y\rightarrow x}$ and $\dot {\omega }_{z\rightarrow x}$ reported in Zandian et al. (Reference Zandian, Sirignano and Hussain2017, Reference Zandian, Sirignano and Hussain2018). A closer look into figure 19(b) reveals that $\dot {\omega }_{x\rightarrow x}$ is relatively weak and does not justify the strong streamwise vorticity generation inside the vortex pins. Despite the partial cancellation, $\dot {\omega }_{y\rightarrow x}$ is larger than $\dot {\omega }_{z\rightarrow x}$ in some regions near the vortex pins, strengthening the combined effect of stretching and tilting. Although not presented here, a slice through $x=26\ \mathrm {\mu }\textrm {m}$ at $t^*=4.05$ shows that $\omega _x$ inside the vortex pins is still strong. Nonetheless, none of the terms of the streamwise vorticity equation significantly increase or decrease $\omega _x$ inside the vortex.

Figure 20. Terms for generating $\omega _x$ during the stretching of the initial hairpin vortex for case C1 at $t^*=3$: (a) $\dot {\omega }_{x\rightarrow x}$; (b) $\dot {\omega }_{y\rightarrow x}$; and (c) $\dot {\omega }_{z\rightarrow x}$. The front view from a $yz$ plane at $x=16\ \mathrm {\mu }\textrm {m}$ is shown. The interface is identified by the solid isocontour with $C=0.5$, and the cut vortex structures are identified by the dashed isocontour with $\lambda _{\rho,t}=-2.5\times 10^{15}$.

All the terms in (5.2) contribute to increasing the magnitude of the $\omega _x$ observed in figure 19(a) between the liquid surface and the hairpin vortex. Vortex stretching and tilting still dominate, but the contributions from volume dilatation and baroclinicity cannot be neglected. Figure 21 shows the gas-phase density, the pressure field relative to the ambient pressure and the divergence of the velocity field or volume dilatation rate. This snapshot's flow patterns reveal the vorticity generation mechanisms that occur throughout the computations in the variable-density transcritical flow. Note the local pressure minima inside the vortex pins in figure 21(b), which reflects the rationale behind the $\lambda _\rho$ vortex identification method. Moreover, the snapshot captures the interfacial pressure jump due to surface tension in the high curvature region near the lobe's tip.

Figure 21. Density in the gas phase, relative pressure, $p_{rel}=p-p_{amb}$, and volume dilatation rate during the stretching of the initial hairpin vortex for case C1 at $t^*=3$: (a) gas density, $\rho$; (b) relative pressure, $p-p_{amb}$; and (c) volume dilatation rate, $\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {u}$. The front view from a $yz$ plane at $x=16\ \mathrm {\mu }\textrm {m}$ is shown. The interface is identified by the solid isocontour with $C=0.5$ and the cut vortex structures are identified by the dashed isocontour with $\lambda _{\rho,t}=-2.5\times 10^{15}$.

The sign of the streamwise component of the baroclinic torque, ${1}/{\rho ^2}(({\partial \rho }/{\partial y})({\partial p}/{\partial z}) - ({\partial \rho }/{\partial z})({\partial p}/{\partial y}))$, can be justified by looking at the density and pressure fields in figure 21. In general, the two-dimensional pressure gradient around the vortex pins (i.e. pressure gradient on the illustrated slice) is oriented radially outwards from each local pressure minimum. In contrast, the two-dimensional density gradient not only depends on the density jump across the interface, but also on the temperature and composition variations in the gas phase. Thus, the baroclinic term in (5.2) changes sign multiple times around the vortex pins, increasing and decreasing $\omega _x$ locally. For example, the sign of the baroclinic term in the region under the vortex pin centred around $z=22\ \mathrm {\mu }\textrm {m}$ in figure 19(d) is explained as follows. For $z<22\ \mathrm {\mu }\textrm {m}$, the baroclinic term is positive since ${\partial \rho }/{\partial y}<0$, ${\partial p}/{\partial z}<0$, ${\partial \rho }/{\partial z}>0$ and ${\partial p}/{\partial y}<0$. That is, both $({\partial \rho }/{\partial y})({\partial p}/{\partial z})>0$ and $-({\partial \rho }/{\partial z})({\partial p}/{\partial y})>0$ are positive and increase $\omega _x$ as depicted in figure 19(a). Then, the baroclinic term becomes negative for $z>22\ \mathrm {\mu }\textrm {m}$ because of the two-dimensional pressure gradient. ${\partial \rho }/{\partial y}<0$, ${\partial \rho }/{\partial z}>0$ and ${\partial p}/{\partial y}<0$ remain unchanged, but ${\partial p}/{\partial z}>0$. This change results in $({\partial \rho }/{\partial y})({\partial p}/{\partial z})<0$ and $-({\partial \rho }/{\partial z})({\partial p}/{\partial y})>0$, where the negative term dominates because the transverse density gradient and the spanwise pressure gradient are larger in magnitude than the spanwise density gradient and the transverse pressure gradient, respectively.

In contrast, the compressible vortex stretching sign is dictated by the local volume dilatation rate. If the fluid expands, the rotation of a fluid element is distributed over a larger volume, decreasing the vorticity magnitude. On the other hand, fluid compression increases vorticity. Here, the combination of the Stefan flow generated around the vaporising or condensing interface and the vortical flow determine the sign of $\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {u}$. Typically, the evaporation of the fuel results in a volume expansion near the interface. However, the transcritical environment introduces various changes in fluid behaviour. As discussed in § 5.1, the enhanced oxygen dissolution at 150 bar may result in net condensation. Therefore, volume compression occurs despite the evaporation of the fuel. In addition, density variations in the gas phase are more significant, and the increase in fuel vapour results in higher gas densities that could limit the expansive behaviour of evaporating flows.

Away from the recirculation region caused by the vortex pins, figure 21(c) reveals the local volume compression above the liquid surface, where weaker evaporation of the fuel results in net condensation. In contrast, the volume dilatation around the vortex structures behaves differently. On the one hand, the entrainment of the hot gas enhances the evaporation of the fuel, increasing the gas mixture density below the vortex pins. Thus, the lighter gas flows into a denser gas region, undergoing compression. On the other hand, the gas phase expands above the vortex pins. Fuel evaporation is stronger near the lobe's tip, substantially increasing the gas density in the surroundings. However, the vortical motion rapidly advects the gas into fluid regions of lower density.

The described $\omega _x$ generation mechanisms for the stretching of the initial hairpin vortex do not change significantly across the analysed configurations that show lobe stretching, bending and perforation. Other deformation mechanisms show deviations in the evolution of the initial hairpin vortex. For example, figure 9 describing the lobe corrugation mechanism for case C2 shows a stronger stretching of the hairpin legs. Figure 22 depicts, for case C2, the streamwise vorticity and the combined contribution of vortex stretching and tilting in the generation of $\omega _x$. Two different $yz$ planes are shown: one at $x=12\ \mathrm {\mu }\textrm {m}$ and $t^*=2.5$ and the other at $x=28\ \mathrm {\mu }\textrm {m}$ at $t^*=4$. During the coupled deformation process between lobe and hairpin described in § 5.2, $\omega _x$ generation occurs inside the vortex pins, increasing the strength of each vortex. Here, $\dot {\omega }_{x\rightarrow x}$ is the main vorticity generation mechanism, having a similar magnitude as the vortex tilting terms that dominate in other flow regions. The behaviour of the compressible vortex stretching and the baroclinic terms is very similar to what has been described previously.

Figure 22. Terms for generating $\omega _x$ during the stretching of the initial hairpin vortex for case C2: (a) $\omega _x$ at $x=12\ \mathrm {\mu }\textrm {m}$ and $t^*=2.5$; (b) vortex stretching and tilting at $x=12\ \mathrm {\mu }\textrm {m}$ and $t^*=2.5$; (c) $\omega _x$ at $x=28\ \mathrm {\mu }\textrm {m}$ and $t^*=4$; and (d) vortex stretching and tilting at $x=28\ \mathrm {\mu }\textrm {m}$ and $t^*=4$. The contours of the different terms are shown in $yz$ planes at various $x$ locations and times. The interface is identified by the solid isocontour with $C=0.5$, and the cut vortex structures are identified by the dashed isocontour with $\lambda _{\rho,t}=-9\times 10^{15}$.

The importance of the coupled deformation between the liquid surface and the hairpin vortex is revealed by comparing cases B1 and C2, both having a gas freestream velocity $u_G=50\ \textrm {m}\ \textrm {s}^{-1}$. Case B1 shows a weaker contribution of $\dot {\omega }_{x\rightarrow x}$ in $\omega _x$ generation, and the hairpin deformation is consistent with the previous discussion of case C1. Not surprisingly, both cases B1 and C1 involve the same early deformation mechanism. Arguably, the term ${\partial u}/{\partial x}$ has a similar strength in cases B1 and C2. Thus, the amplified contribution of $\dot {\omega }_{x\rightarrow x}$ in case C2 must come from an $\omega _x$ increase in the vortex pins. As described in § 5.2, the lobe wraps around the vortex pins, enhancing the gas entrainment under the lobe as the gaps between the sides of the lobe and the liquid surface are reduced. This deformation process results in the strengthening of the $\omega _x$ (see figures 10 or 22).

5.5.2. The secondary roller vortex

New roller vortices form during the jet development, particularly in front of the main perturbation wave as it grows (see figure 13). These vortical structures are expected to emerge from the stretching of the vortex sheet between the liquid wave and the faster-moving gas. Along a sharp wave edge (i.e. a sharp flow turn), the vortex sheet separates and is rapidly affected by the gas entrainment underneath the liquid. Thence, a roller vortex forms. However, the formation of smaller roller vortices occurs without such a clear mechanism. For example, figure 9 shows that during the early lobe deformation in case C2, a small secondary roller vortex is observed on top of the hairpin vortex pins. No clear flow feature seems to explain its occurrence. Here, we describe the nature of the strengthening of the spanwise vorticity in that region, leading to the formation of the vortex.

The vortex formation mechanism originates in the destabilisation of the vortex sheet between liquid and gas. Figure 23 depicts the deformation of the early lobe as it corrugates between $t^*=3$ and $t^*=4$, showing the vortex sheet (represented by the green isosurface with $\omega _z=-1.5\times 10^{7}\ \textrm {s}^{-1}$) and the hairpin vortex (red isosurface with $\lambda _{\rho,t}=-9\times 10^{15}$). The vortex sheet forms due to the shear between the two phases, and it follows the liquid surface along with the stretching and deformation of the lobe (i.e. the vortex sheet is also stretched and deformed). At $t^*=3$, no distinctive perturbation is seen. However, once the lobe bursts into small ligaments and droplets, the vortex sheet detaches from the liquid surface and is suddenly perturbed by the increased multiphase turbulence. This triggers a destabilisation of the vortex sheet and the formation of the secondary roller vortex, as explained in the following lines.

Figure 23. Destabilisation of the vortex sheet caused by the lobe bursting in case C2. The liquid surface is identified by the translucent blue isosurface with $C=0.5$, the vortex structures are identified by the red isosurface with $\lambda _{\rho,t}=-9\times 10^{15}$ and the vortex sheet is identified by the translucent green isosurface with $\omega _z=-1.5\times 10^{7}\ \textrm {s}^{-1}$.

Figure 24 shows the evolution of the vortex sheet between $t^*=4.25$ and $t^*=5$ after the lobe bursts. The contours of $\omega _z$ are shown in an $xy$ plane at $z=25\ \mathrm {\mu }\textrm {m}$, together with a three-dimensional view of the liquid surface and the vortex structures. To enhance the visualisation of the vortices in this figure, $\lambda _{\rho,t}=-5\times 10^{15}$ instead of $\lambda _{\rho,t}=-9\times 10^{15}$ used in previous figures for case C2 and the vortex isosurface is coloured with $\omega _z$. Pockets of stronger spanwise vorticity occur in the detached vortex sheet, as seen at $t^*=4.25$, arguably due to the interactions between ligaments and droplets. Over time, these vortical regions come together via mutual induction and subsequent pairing/merging, and some small roller vortices can be observed at $t^*=4.5$. In particular, one of these regions of vortex pairing grows considerably between $t^*=4.25$ and $t^*=5$, resulting in the formation of the secondary roller vortex seen in figure 9. Moreover, the strengthening of $\omega _z$ and the vortex pairing are aided by the induced flow coming from the hairpin vortex pins or legs. Once the lobe bursts, the vortex sheet is not only perturbed by the smaller liquid structures, but it is also under the direct influence of the hairpin vortex. Before, the presence of the liquid lobe between the vortex pins and the vortex sheet has a shielding effect.

Figure 24. Vortex pairing mechanism resulting in the formation of the secondary roller vortex in case C2. Contours of $\omega _z$ are shown in an $xy$ plane at $z=25\ \mathrm {\mu }\textrm {m}$. The liquid surface is identified by the solid isocontour or the translucent blue isosurface with $C=0.5$, and the vortex structures are identified by the dashed isocontour or isosurface with $\lambda _{\rho,t}=-5\times 10^{15}$.

Figure 25(a) manifests the implications of this relative position between the vortex sheet and the hairpin vortex. The contours of $\dot {\omega }_{z\rightarrow z}$ from (5.4) are shown in the same $xy$ plane from figure 24 at $t^*=4.25$. Due to the presence of $\omega _z$ and a spanwise stretching (i.e. $\partial w/\partial z>0$), there is $\omega _z$ generation that tends to strengthen the small vortices during the pairing process. Other terms in (5.4) have a negligible effect in the detached vortex sheet. As seen in figure 25(b), the hairpin legs (red vortex) are responsible for the spanwise stretching of the perturbed vortex sheet. Note that $\lambda _{\rho,t}=-5\times 10^{15}$ in this figure too. Moreover, the secondary roller vortex (green) wraps around the hairpin vortex pins, similar to how the liquid phase wraps around the vortical structures.

Figure 25. Spanwise vortex stretching affecting the formation of the secondary roller vortex in case C2: (a) $\dot {\omega }_{z\rightarrow z}$ in the $xy$ plane at $z=25\ \mathrm {\mu }\textrm {m}$ and $t^*=4.25$; and (b) induced flow patterns from the hairpin vortex at $t^*=5$. The liquid surface is identified by the solid isocontour or the translucent blue isosurface with $C=0.5$, and the vortex structures are identified by the dashed isocontour or isosurface with $\lambda _{\rho,t}=-5\times 10^{15}$.

The mechanisms described in this section resemble the findings in previous works. The initial destabilisation or bursting of the vortex sheet is tightly coupled to the coiling of vortex lines and the turbulence cascade (Stout Reference Stout2021), here involving the multiphase flow. The later vortex pairing, emergence of the secondary roller vortex, and wrapping around the hairpin legs is a clear example of vortex reconnection (Hussain & Duraisamy Reference Hussain and Duraisamy2011; Stout & Hussain Reference Stout and Hussain2023; Yao & Hussain Reference Yao and Hussain2022).

5.5.3. Streamwise vortex alignment in the layered flow

Lastly, we look at the initial stages of the vortex alignment in the streamwise direction inside the layered flow. As seen in figure 14(c), which depicts a late snapshot of case C1 at $t^*=15$ once various liquid layers have formed, some of the major structures contained in between the layers are aligned with $x$. In § 5.3, we describe how the vorticity within the layered flow primarily comes from the various roller vortices that form during the growth of the main perturbation wave. Other minor vortical structures also emerge in the layered flow, such as rim vortices or small roller vortices.

Figures 26 and 27 present the contour plots of different variables in a $yz$ plane at $x=16\ \mathrm {\mu }\textrm {m}$ and $t^*=8.4$ for case C1. Figure 26 shows the various terms in (5.2), and figure 27 shows the vorticity components, the fuel vapour mass fraction, gas density and temperature. This non-dimensional time corresponds to the snapshot in figure 14(a), where the remnants of the roller vortex R2 are found underneath the liquid. At $x=16\ \mathrm {\mu }\textrm {m}$, the $yz$ plane cuts through the region where most of the vortex structures are. Despite being an early snapshot of the layered flow, the results in figure 26 provide relevant details of the $\omega _x$ generation mechanisms as the liquid layers form. Of course, other events impact the evolution of the vortical structures, such as the entrainment of the roller vortex R3, overlapping layers and frequent tearing and hole formation.

Figure 26. Terms for generating $\omega _x$ in between the liquid layers for case C1 at $t^*=8.4$: (a) vortex stretching and tilting; (b) $\dot {\omega }_{x\rightarrow x}$; (c) $\dot {\omega }_{y\rightarrow x}$; (d) $\dot {\omega }_{z\rightarrow x}$; (e) compressible stretching; and (f) baroclinic term. The front view from a $yz$ plane at $x=16\ \mathrm {\mu }\textrm {m}$ is shown. The interface is identified by the solid isocontour with $C=0.5$ and the cut vortex structures are identified by the dashed isocontour with $\lambda _{\rho,t}=-2.5\times 10^{15}$.

Figure 27. Vorticity components, fuel vapour mass fraction, gas density and temperature in between the liquid layers for case C1 at $t^*=8.4$: (a) $\omega _x$; (b) fuel vapour mass fraction, $Y_F$; (c) $\omega _y$; (d) gas density, $\rho$; (e) $\omega _z$; and (f) temperature, $T$. The front view from a $yz$ plane at $x=16\ \mathrm {\mu }\textrm {m}$ is shown. The interface is identified by the solid isocontour with $C=0.5$ and the cut vortex structures are identified by the dashed isocontour with $\lambda _{\rho,t}=-2.5\times 10^{15}$.

The contribution of vortex stretching and tilting to $\omega _x$ generation mainly comes from $\dot {\omega }_{y\rightarrow x}$ and $\dot {\omega }_{z\rightarrow x}$. As reported in § 5.5.1, these two terms partially cancel each other. Note that $\dot {\omega }_{x\rightarrow x}$ has a weak contribution except in a region above the liquid core near the symmetry plane of the computations. Vorticity is predominantly aligned in the spanwise direction coming from the roller vortex R2, but streamwise vorticity is not negligible (see figure 27a,e). Thus, $\dot {\omega }_{x\rightarrow x}$ is not a major source of $\omega _x$, at least at this stage, because of a negligible $\partial u /\partial x$. In contrast to previous §§ 5.5.1 and 5.5.2, the contributions from variable-density effects become more important in the layered flow. Despite being an order of magnitude lower, compressible vortex stretching and baroclinicity affect flow regions where vortex stretching and tilting are weak. Therefore, these terms can be substantial drivers of $\omega _x$ and, in general, vorticity.

The increased importance of compressible terms is not unexpected. A closer look at the variations of composition, temperature and density in the gas phase is given in figures 27(b), 27(d) and 27(f). Multiple fuel-lean pockets are observed with higher temperatures and lower densities. These regions exist because of the entrainment of hotter ambient gas, as detailed in § 5.3 and visualised in figures 13–16. This increased complexity in the scalar mixing patterns results in a coupled mixing-deformation mechanism where vorticity defines the scalar mixing, which, in turn, affects vorticity generation and the vortex deformation due to the variable-density and compressible effects seen in (5.1). Scalar mixing patterns are also affected by vorticity at low ambient pressures. Nevertheless, the gas mixture density variations are often negligible at low pressures (Poblador-Ibanez & Sirignano Reference Poblador-Ibanez and Sirignano2018; Poblador-Ibanez et al. Reference Poblador-Ibanez, Davis and Sirignano2021), and a significant two-way coupling may not exist. Further analysis of this feature is beyond the scope of this work. Future studies of transcritical atomisation should address three key aspects together: (a) the liquid–gas mixing process (i.e. atomisation), (b) the mechanisms of scalar mixing (e.g. fuel mixing) and (c) the coupling with vorticity dynamics.