3.1. Verification and validation

Two numerical simulations are performed to verify whether the model can provide the turbulent inlet condition under a $P_{b}$ of 4.0 MPa. The first simulation uses the geometry shown in figure 3($c$), whereas the second simulation uses a simplified geometry with the semicylinders and inlet region of the orifices removed. In both cases, water flows perpendicularly into the orifices from the round surfaces to the orifice axes. A constant velocity boundary condition is applied to the inlet surfaces in the second simulation to provide the same flow rate as in the first simulation.

These two simulations demonstrate the effects of the inlet conditions on the resulting liquid jet flow patterns and atomisation dynamics. Figure 5($a$) shows the geometric models and velocity fields at the cross-section for the two inlet geometries. For the full geometry, there is a complex velocity distribution with vortices inside the orifices and at their exits. By contrast, the velocity distribution is nearly parabolic for the simplified geometry.

Figure 5.

Effect of the inlet geometry on the impinging-jet dynamics. (a) Velocity fields at the cross-section for two different inlet geometries under a $P_{b}$ of 4.0 MPa. (b) Development of vortices inside the liquid before impingement of the two jets under a $P_{b}$ of 0.1 MPa. (c) Interfaces of the atomisation fields for the turbulent and laminar cases. (d) Interface profiles at the cross-section for the turbulent and laminar cases.

Figure 5($b$) shows the development and distribution of vortices for the first simulation case. The image in the upper left corner shows that vortices are formed at the inner sides of the two orifices. The enlarged view at the centre in figure 5($b$) shows the detailed development process of the vortices, which also represents the transition of the fluid flow from laminar to turbulent. There is a sudden contraction inside the nozzle. As the fluid passes through this area, the speed increases sharply, causing the fluid to become unstable and initiating oscillations that roll up into vortices. The inclined inlets introduce a pressure gradient at the orifice inlets that generates coherent structures from the boundary layer. The mean flow carries these vortices downstream and the vortices are stretched by the wall shear layer. This stretching process elongates the vortices into hairpin shapes (Theodorsen Reference Theodorsen1952) and generates secondary vortices. These vortex structures are similar to those observed by Robinson Robinson (Reference Robinson1991). These vortical structures breakdown, developing into small-scale streamwise vortices further downstream, eventually filling the jet interior. The coherent structures obtained from the numerical simulation demonstrate the ability of the model to resolve the turbulent characteristics of the impinging jets. Hairpin vortices have been observed in both experiments (Moin & Kim Reference Moin and Kim1982; Adrian Reference Adrian2007) and simulations (Moin & Kim Reference Moin and Kim1982; Wang, Huang & Xu Reference Wang, Huang and Xu2015). To the best of our knowledge, this is the first time that turbulence generation in a supply pipeline has been considered in the numerical simulations of impinging-jet atomisation. Our results demonstrate the importance of nozzle geometry modelling for capturing realistic turbulent conditions.

Figure 5($c$) shows significant differences in the results for the two inlet conditions. The first inlet condition results in turbulent liquid jets, which produce a complex atomisation field with spatially varying distributions of liquid sheets, ligaments and droplets. In contrast, the second inlet condition produces a laminar liquid sheet that generates a smooth liquid sheet near the impact plane. Unphysical breakup of the thin region is observed at the rim of the liquid sheet, mainly due to the insufficient grid resolution to resolve the thin film. Waves develop downstream on the liquid sheet. The two different flow patterns of the liquid sheet are consistent with previous observations of liquid sheets resulting from the impingement of laminar and turbulent liquid jets. In addition, the two simulations highlight the two different origins of the waves on the liquid sheets: aerodynamics for laminar jets and hydrodynamics for turbulent jets. Figure 5($d$) shows the interface profiles at the cross-section in the two cases. The wavy profiles in the turbulent case can be contrasted with the smooth profiles of the liquid jets and the liquid sheet in the laminar case. The impact wave quickly develops from the impingement point, whereas the aerodynamic wave develops much more slowly, resulting in a longer breakup length. This comparison also shows why the theoretical model based on infinitesimals can predict the aerodynamic wave for laminar jets. In contrast, a nonlinear model based on a finite initial distribution is required for the impact wave (Ibrahim & Outland Reference Ibrahim and Outland2008). However, this theoretical model mainly assumes two-dimensional flow, so that the complex 3-D effects shown in figure 5($c$) cannot be considered.

Based on the abovementioned understanding, a grid independence study was conducted using three distinct grid resolutions corresponding to AMR levels 2, 3 and 4, each set at the same $P_{b}$ of 4.0 MPa. The size of the background mesh is set to $80\ \mathrm {\mu }\textrm {m}$. With four levels of refinement, $D/\Delta x_{min}$ reaches 60, achieving the scale required for direct numerical simulation (Hasslberger et al. Reference Hasslberger, Ketterl, Klein and Chakraborty2019). The grid resolutions $D/\Delta x_{min}$ of 15, 30 and 60 correspond to the $\Delta x_{min}$ values of $20\ \mathrm {\mu }\textrm {m}$, $10\ \mathrm {\mu }\textrm {m}$ and $5\ \mathrm {\mu }\textrm {m}$, respectively. In this context, $\Delta x_{min}$ represents the smallest grid after AMR of the downstream fluid region. These resolutions can be named low, medium and high. When the simulations reach the steady state, the number of cells is approximately 5.5, 43.8 and 131.4 million for the three resolutions, respectively. A case was solved using a supercomputing platform equipped with 128 CPUs. The duration of the simulation for atomisation was 2 ms, with time steps of approximately $2 \times 10^{-8}$, $4 \times 10^{-8}$ and $8 \times 10^{-8}$, with approximately 100 000, 50 000 and 25 000 iterations, respectively. The computational process required 15 360, 61 440 and 184 320 CPU core hours, respectively, to reach a steady state. In comparison, equivalent uniform meshes would require approximately 54.0 million, 432.2 million and 3.5 billion cells, respectively. This presents a significant challenge with regards to the requisite computational resources and the time required for computation. The completion of numerical simulations using equivalent uniform meshes is quite challenging, highlighting the advantages of AMR. The results in figure 6 show that for the low resolution, the liquid jets and liquid sheets are smoother than those in the other two cases. In addition, the droplet generated in the axial downstream direction is larger. The increase in the grid resolution to the middle resolution results in distinct differences compared with the low-resolution results. More complex surface evolutions are observed in the jets and the sheet for the middle and high resolutions. Overall, the breakup length, which is measured from the impingement point to the breakup point of the sheet, increases with increasing resolution.

Figure 6.

Front views of the interfaces obtained under three different grid resolutions in the grid independence study. The resolutions are named as low, medium and high, represented by panels (a)–(c), respectively. These resolutions are analysed under a $P_{b}$ of 4.0 MPa.

The spreading angle from the front view ($\theta$), liquid sheet breakup length, droplet diameter probability density function (p.d.f.) and local Sauter mean diameter (SMD) are commonly used to quantitatively characterise spray fields. These parameters are measured for three different grid resolutions. Figure 7($a$) illustrates the method for measuring the $\theta$. As in the experiments, numerical simulations also determine the $\theta$ according to the droplet velocity, with $U_0 = \sqrt {U_z^2+U_y^2}/U_j$, where $U_0$ represents the velocity magnitude in the $zy$ plane, $U_z$ represents the droplet velocity in the $z$ direction and $U_y$ represents the droplet velocity in the $y$ direction. At a position 20 $D$ below the impact point, identify the furthest distances reached by droplets with a velocity of 0.1$U _0$ on either side. Connecting these points with the impact point forms an angle, which defines the $\theta$.

Figure 7.

($a$) Method for measuring the spreading angle from the droplet velocity, with $U_0 = \sqrt {U_z^2+U_y^2}/U_j$. ($b$) Measurement of liquid sheet breakup length.

After atomisation has stabilised, five different moments are selected, each representing five cycles, to measure the liquid sheet breakup length. For measuring the liquid sheet breakup length, the central interface of the liquid film at $y = 0$ is chosen and the distance from the impact point to the position of the first rupture of the liquid film is measured, as illustrated in figure 7($b$). The average values are calculated, with the error calculated as the difference between the maximum and minimum values. Figure 8($a$) illustrates the statistical region of the droplet diameter probability density function (p.d.f.), which describes the probability of different droplet sizes. The droplet diameters are analysed within the atomised field after transformation by atomizationFoam, with the statistical interval along the $z$-axis ranging from 8 mm to 12 mm. In numerical simulations using the VOF method, higher mesh resolution typically leads to the generation of a greater number of smaller droplets (Ling et al. Reference Ling, Fuster, Zaleski and Tryggvason2017). When evaluating the mesh independence of droplet diameter probability density, the focus is primarily on the independence behaviour of larger droplets. Accordingly, droplets with diameters greater than twice the minimum grid size at low resolution (i.e. droplets larger than $40\ \mathrm {\mu }\textrm {m}$) are chosen for analysis.

Figure 8.

($a$) Statistical region for the droplet diameter probability density function. ($b$) Spatial division of the statistical region for the local Sauter mean diameter.

The SMD is typically defined as

(3.1)\begin{equation} d_{32} =\frac{\sum d_{i}^3 n_{i}}{\sum d_{i}^2 n_{i}}, \end{equation}

where $d_{i}$ is the diameter of the $i{\mathrm {th}}$ type of droplet and $n_{i}$ is the number of droplets with diameter $d_{i}$. The numerator and denominator of the above equation are equal to $6V_0/{\rm \pi}$ and $A_0/{\rm \pi}$, respectively, where $A_0$ and $V_0$ are the total surface area and volume of the group of droplets, respectively. The above equation can be simplified to

(3.2)\begin{equation} d_{32} = \frac{6V_0/{\rm \pi}}{A_0/{\rm \pi}} = 6\frac{V_0}{A_0}. \end{equation}

Here, we employed the Integrate Variables function in ParaView to compute $A_0$ and $V_0$. To measure the SMD, one must select a specific region and calculate the droplets volume and surface area within it. As shown in figure 8($b$), the centreline of the atomisation field, originating at the impact point, is rotated $40^\circ$ to each side to define the atomisation region. This region is bounded by two concentric circles with radii of 8.5 and 10 mm and is further subdivided into $5^\circ$ angular segments. The area enclosed by the black dashed lines is designated for the local SMD calculation. During the breakup process, the liquid sheet exhibits a fan-shaped fragmentation. This shape was used to conduct a statistical analysis of the local SMD, to better understand the distribution of the local SMD from the centre to the edges during breakup. The axial distance for the analysis was chosen near the downstream region where the liquid filaments disintegrate.

Figure 9 illustrates the measured spreading angle, the breakup length of the liquid sheet, the p.d.f. distribution of droplet diameters and the local SMD, with varying grid resolutions. Figure 9($a$) indicates that increasing the grid resolution from low to medium leads to significant changes in the spreading angle and the liquid sheet breakup length. However, further increasing the resolution from medium to high results in relatively minor changes, suggesting that the grid has a weaker impact on the results at medium and high resolutions. Figure 9($b$) presents the droplet diameter p.d.f.s across different grid resolutions. The results indicate that droplets with diameters greater than $40\ \mathrm {\mu }\textrm {m}$ demonstrate excellent mesh independence. Figure 9($c$) presents the statistical distribution of local SMD across different grid resolutions. It is evident that the local SMDs for medium- and high-resolution images are very close, indicating good independence.

Figure 9.

Measurement of the ($a$) spreading angle and liquid sheet breakup length, ($b$) droplet diameter probability density function, and ($c$) local Sauter mean diameter based on different grid resolutions.

Based on the grid independence study, the spreading angle, liquid sheet breakup length, droplet diameter p.d.f. and local SMD obtained with medium and high grid resolutions are similar. Therefore, high resolution is employed for accurate near-field atomisation simulations, whereas medium resolution is carefully selected to balance computational efficiency and accuracy, and is used for feasible far-field simulations. This medium resolution ensures that key flow features are captured accurately without incurring excessively high computational costs, and it effectively resolves large-scale structures and droplet dynamics, enabling a detailed exploration of droplet behaviour over extended downstream distances. To validate this approach, simulations are carried out at both middle and high resolutions for five different $P_{b}$ values. Figure 10 illustrates the nearly identical atomisation morphologies observed with the two resolutions across various $P_{b}$. In total, 10 cases are examined, providing detailed insights into the near-field atomisation dynamics and the full far-field spray evolution. Each case is solved individually on a supercomputing platform with 128 CPUs. After reaching the steady state, the computational atomisation duration for each near-field atomisation case was 2 ms, with a time step of $2 \times 10^{-8}$ and 100 000 iterations, averaging approximately 168 960 CPU core hours. For each far-field atomisation case, the computational atomisation duration was 8 ms, with a time step of $4 \times 10^{-8}$ s and 200 000 iterations, averaging approximately 122 880 CPU core hours.

Figure 10.

Interfaces of the atomisation fields for different $P_{b}$ values of (a) 0.1, (b) 0.5, (c) 1.0, (d) 2.0 and (e) 4.0 MPa for Model I; and for different $P_{b}$ values of ( f) 0.1, (g) 0.5, (h) 1.0, (i) 2.0 and (j) 4.0 MPa for Model II.

Table 1 shows that the maximum error between the middle- and high-resolution simulations is 3.0 % at $P_{b} = 0.1$ MPa, which is within an acceptable range. This demonstrates that the use of these two resolutions is reasonable. Additionally, table 1 provides the grid numbers for different cases. The maximum number of cells is 131.4 million with $P_b = 4.0$ MPa and $D/\Delta x_{min} = 60$. Without AMR, the cell counts for uniform meshes with $D/\Delta x_{min} = 60$ and 30 would be approximately 3456 and 2731 million, respectively. The cell number ratio of the uniform to AMR meshes is approximately $26 \unicode{x2013} 46$.

Table 1.

Comparison of the SMD between two different grid resolutions. Cases 1, 3, 5, 7 and 9 for Model I with $P_{b}$ varying from 0.1 to 4.0 MPa , and Cases 2, 4, 6, 8 and 10 for Model II with $P_{b}$ varying from 0.1 to 4.0 MPa. The density ratio of the liquid phase to the gas phase is denoted by $\rho _{g}/\rho _{l}$. The Reynolds number of the liquid phase is represented by ${Re}_{l}$. The Weber number of the liquid phase is represented by ${We}_{l}$. The Weber number of the gas phase is represented by ${We}_{g}$. The ratio of the jet diameter to the smallest mesh is represented by $D/\Delta x_{min}$. The error is represented by $\textrm {err}_{SMD}$. The number of cells at the simulations reach the steady state in millions is represented by $N_{c}$ (M).

To validate our numerical simulations, experiments are conducted for the five different $P_{b}$ values. Figure 11 compares the front and side views of the two extreme cases of $P_{b}=0.1$ and 4.0 MPa. Good overall agreement is observed for both conditions. There is an evident transition of the flow patterns between these two conditions. Figure 11 compares the experimental and numerical results within the atomisation field, focusing specifically on the low- and high-$P_{b}$ scenarios. More specifically, figures 11($a$) and 11($b$) present the comparisons for $P = 0.1$ MPa, whereas figures 11($c$) and 11($d$) present comparisons at ${P} = 4.0$ MPa. Overall, the numerical simulation of the atomisation field closely replicates the experimental results, indicating the close agreement between the two sets of results.

Figure 11.

Comparison between the experimental and numerical simulation results of the overall atomisation field, with $D/\Delta x_{min} = 60$ and interface reconstruction using the isoAdvector method. (a) Front and (b) side views, with a $P_{b}$ of 0.1 MPa and a jet velocity of $18\ \textrm {m}\ \textrm {s}^{-1}$. (c) Front and (d) side views, with a $P_{b}$ of 4.0 MPa and a jet velocity of $18\ \textrm {m}\ \textrm {s}^{-1}$.

Many tiny liquid droplets were observed forming above the point of impact in the numerical simulation. However, such droplets are absent in the experimental observations, which may be due to insufficient imaging clarity that may have hindered the detection of such small features. Figure 12 illustrates the formation of droplets above the impact point under different backpressure conditions. This phenomenon occurs because the impinging jets create a small liquid film with upward velocity in that region that becomes unstable and breaks up into tiny upwards-moving droplets.

Figure 12.

Droplet formation above the impact point under different backpressure conditions for pressure values of (a) 0.1, (b) 0.5, (c) 1.0, (d) 2.0 and (e) 4.0 MPa.

Our findings differ considerably from those of previous numerical studies on impinging-jet atomisation, primarily due to the explicit consideration of the turbulent condition of the liquid jets. Upon close examination, differences in the detailed flow structures can be observed. There are two main reasons for these differences. The first is that the turbulent condition of the liquid jet cannot be exactly represented due to the complex nature of the turbulent flow and the possibility of a cavity at the entry of the orifices. The other reason may be that the grid resolution is still insufficient to reproduce the flow. Since the overall flow patterns of the simulations are similar to those of the experiments, we believe that our simulations capture most of the flow dynamics during the complex atomisation process.

Table 2 compares the SMD and $\theta$ obtained from experiments and simulations under different $P_{b}$ values. In the experiment, the SMD of the droplet was obtained by measuring the droplet 50 mm downstream of the impact point with the PDPA system. The $\theta$ was analysed using the PIV method with the shadow images captured by a high-speed camera. In the numerical simulations, the SMD measurement is also taken at 50 mm downstream of the impact point, and the measurement of the $\theta$ follows the same procedure as in the experiment. For both sets of results, the same trends of an initial decrease followed by an increase for SMD and a monotonic decrease pattern for $\theta$ are observed. Overall, the largest deviation between the two sets of measurements for the SMD is approximately $-12.1\,\%$ when $P_{b} = 4.0$ MPa. Moreover, the largest deviation for $\theta$ is $8.6\,\%$ under $P_{b} = 4.0$ MPa. This discrepancy may be attributed to the fact that when $P_{b}$ is 4.0 MPa, the dense spray can fill the container with many droplets, leading to significant measurement errors in these two parameters. Additionally, when $P_{b}$ is 0.1 MPa, the low pressure may induce a cavitation effects in the nozzle, leading to relatively large deviations in the SMD ($-8.0\,\%$) between the experiment and the simulation, which cannot consider cavitation. Furthermore, the maximum error in the spreading angle is $8.6\,\%$. Both the SMD and the spreading angle, which are critical parameters of the atomisation field, exhibit good agreement between the experimental and numerical results, validating the precision of the numerical simulation method.

Table 2.

Comparison of the SMD and spreading angle between experiments and numerical simulations, with $\textrm {err}_{SMD}$ and err $\textrm {err}_{\theta }$ representing the differences in SMD and spreading angle between the two methods.

3.2. Atomisation dynamics

Figure 11 presents a side-by-side comparison of the atomisation field in the near field for the two distinct limits of $P_{b}$ from different views. Figures 11($a$) and 11($c$) display front views of the liquid sheets. The sheet configuration and breakup behaviour noticeably change with increasing $P_{b}$. The collapse of the lower portion of each sheet within the steady-sheet regime is an evident result of the heightened $P_{b}$ and leads to the reduced size and minimal surface fluctuations of the liquid sheet, as well as a shorter breakup length. Post-breakup, the liquid ligaments evolve into droplets, intensifying the atomisation. Moreover, at an intermediate $P_{b}$, the sheet undergoes periodic disintegration into ligaments. However, this periodicity decreases with increasing pressure, leading to a decrease in the number of ligaments and an increase in droplet formation. The transition of the spray field from a sparse to a denser state suggests an increase in the number of overlapping water droplets.

Figure 13 compares the details of the interfaces at different locations at the two limits of $P_{b}$. To enhance the observation of atomisation characteristics, droplets within the liquid sheet, at its boundaries and in the breakup region have been removed. Panels  (a)–(e) correspond to a $P_{b}$ of 0.1 MPa, while panels ( f)–( j) correspond to a $P_{b}$ of 4.0 MPa. Many flow scenarios occur during the atomisation process. Figures 13($a$) and 13($\,f$) illustrate how an increase in the pressure promotes the inception and axial growth of a disturbance in a liquid jet. Notably, the liquid jets are significantly turbulent, with the development of surface structures from velocity fluctuations. Similar resulting surface structures are observed at the nozzle exits, while the surface structures develop significantly faster under a $P_{b}$ of 4.0 MPa than under a $P_{b}$ of 0.1 MPa. Rounded convex structures become lobe patterns for a $P_{b}$ of 4.0 MPa prior to the impingement of the liquid jets.

Figure 13.

Detailed images for (a–e) 0.1 MPa and ( f–j) 4.0 MPa, with interface reconstruction using the isoAdvector method, illustrating various atomisation regions. The interfaces are coloured according to the $x$ coordinate. These images depict (from left to right) the impingement jets, liquid sheet, boundaries of the liquid sheet, breakup region of the liquid sheet and droplet region.

As shown in figures 13($b$) and 13($g$), the flow structures of the liquid jets result in different patterns of the liquid sheet. The liquid sheet is relatively smooth for a $P_{b}$ of 0.1 MPa, whereas larger fluctuations are observed on the liquid sheet for a $P_{b}$ of 4.0 MPa. The flapping motions of the liquid sheet are more pronounced for a $P_{b}$ of 0.1 MPa. The large fluctuations can even generate finger structures on the surface of the liquid sheet, highlighting the stronger shear stresses under a $P_{b}$ of 4.0 MPa than under a $P_{b}$ of 0.1 MPa. The side boundaries of the liquid sheet also contract inwards under the abovementioned effect.

Owing to the inline impact of the two liquid jets, the fluctuation at the top of the formed liquid sheet propagates downstream along the boundary. The thickness of the liquid sheet can fluctuate along its boundary, leading to the formation of structures such as liquid fingers that connect to the sheet. These structures can grow downstream and contribute to the shape of the sheet. The liquid sheet can also break up into liquid ligaments in two ways. The first is through the development of the aforementioned flow structures, which can tear the sheet at the root region or rupture it at the centre region due to the collapse of structures of different sizes. In this scenario, the flapping motions of the liquid sheet contribute to the formation of liquid fingers. The second is through the breakup of the liquid sheet at the centre region due to the nonlinear development of thinning and stretching caused by flapping motions.

Figure 13($c$) shows the rupture of the liquid sheet at the side boundary for a $P_{b}$ of 0.1 MPa. In a typical flow structure, ligaments connect to the liquid sheet at locations influenced by the flapping motion of the liquid sheet. The breakup of liquid ligaments into shorter ligaments or droplets is observed at the side boundary.

As the backpressure increases, the lengths of the liquid fingers decrease and the flapping motion becomes irregular due to the increase in the amplitude of the surface fluctuations. Downstream, long liquid filaments are not observed. Meanwhile, the liquid sheet breaks up more randomly, leading to the formation of liquid ligaments and droplets. Subsequently, circumferential propagation occurs, with the breakups recurring at regular intervals. As shown in figure 13($h$), these recurring events result in progressively shorter liquid ligaments as the backpressure increases. Additionally, the boundary of the liquid sheet exhibits more intact ligaments at lower backpressures than at higher backpressures. Figures 13($d$) and 13($i$) show that the downstream portion of the flow tends to detach from the liquid sheet at a backpressure of 0.1 MPa. With increasing backpressure, this section transitions from liquid sheet breakup to a more pronounced liquid necking phenomenon, ultimately leading to droplet formation. Upon further examination of the downstream flow, at lower backpressures, the liquid clearly transforms into droplets. Conversely, under higher backpressures, droplet collisions cause the droplet volume to expand, as demonstrated in figures 13($e$) and 13($j$).

3.3. Impact wave

The resulting flow structures in the liquid sheets are predominantly influenced by the laminar or turbulent conditions of the liquid jets before impingement. When the liquid jets are laminar, they produce a relatively flat liquid sheet at the impingement point, and a long downstream distance is required for infinitesimal disturbances to develop into a wave pattern. Conversely, turbulent liquid jets introduce natural disturbances with a finite amplitude to the liquid sheet, causing it to become wavy immediately after the impingement.

Turbulence in liquid jets leads to disturbance of the finite amplitude in the liquid sheet formed by the impinging jets. Comparison of the atomisation process of impinging jets for laminar and turbulent liquid jets under a constant backpressure of 4 MPa reveals noticeable differences in the resulting atomisation morphology and liquid sheet characteristics. Laminar liquid jets are obtained by maintaining the diameter of the jet inlet nozzle, which allows the jet flow to remain in the laminar regime. In this case, the impinging laminar jets form a large and coherent liquid sheet with relatively small fluctuations on the impact surface. By contrast, turbulent liquid jets are obtained by progressively decreasing the diameter of the jet inlet nozzle, inducing turbulence in the jet flow. The impinging turbulent jets generate a significantly smaller and more unstable liquid sheet with intense and chaotic fluctuations. This distinct behaviour suggests that turbulence in the liquid jets plays a key role in inducing violent disturbances and fluctuations in the liquid sheet formed during impinging-jet atomisation. The intense fluctuations caused by turbulence lead to the enhanced breakup of the liquid sheet into small droplets. The comparisons reveal that the transition from laminar to turbulent jet flow can dramatically affect the liquid sheet morphology and stability during impinging-jet atomisation, resulting in substantially different droplet sizes and spray patterns. The ability of turbulent jets to stimulate intense fluctuations is a critical factor in achieving effective atomisation performance.

To investigate the droplet generation process, the droplets that are removed from the atomisation field must be analysed. Figure 14 shows the structures of jets, sheets and ligaments under various backpressures. To facilitate the visualisation of the complex 3-D interface, the $y$-coordinate perpendicular to the impact plane of the liquid sheet is used to colour the interface. The colour of both the liquid sheets and ligaments changes in space. Under a backpressure of $P_{b} = 0.1$ MPa, we observe a zigzag distribution of the sheets and ligaments downstream. This phenomenon allows the impinging jets to rapidly mix with the liquid from the two jets. Finger structures appear from the top of the liquid sheet, and propagate and grow along the periphery of the liquid sheet. The development of liquid fingers is caused by both the local capillary effect and the flapping motion of the liquid sheet. The impact wave causes the liquid sheet to break into liquid fragments. These fragments connect to the liquid fingers, forming a liquid complex. The sheet-like shape of the complex induces rapid contraction, resulting in long liquid ligaments with complicated shapes downstream. The time required for the fingers on the two sides to collapse at the centre of the liquid sheet and the time required for the liquid sheet to break up at the centre are essential for the droplet generation process. As $P_{b}$ increases, the ligaments attached to or separated from the liquid sheet become shorter. The breakup of the liquid sheet becomes more complex and chaotic, with more holes appearing in the downstream breakup region, and the long liquid ligaments gradually transform into a reticular structure, with the liquid ligaments becoming more aggregated. This phenomenon is attributed mainly to the increase in the shear stress of the gas phase.

Figure 14.

Three-dimensional structures of liquid sheets and ligaments generated by impinging jets, with the droplets removed from the flow field. The interfaces are coloured according to the $x$ coordinates. Shown are data for the pressures of (a) 0.1, (b) 0.5, (c) 1.0, (d) 2.0 and (e) 4.0 MPa.

Heidmann et al. (Reference Heidmann, Priem and Humphrey1957) classified the liquid sheet formed at the impinging jets with increasing turbulence intensity into four modes: closed rim, periodic rim, open rim and fully developed sheet models. The case studied in this paper corresponds to the fully developed sheet model, for which the experiments and modelling do not provide a good explanation of the fragmentation mechanism due to the complex fragmentation phenomenon of the liquid sheet. The perforation of a liquid sheet through hole formation and hole nucleation in a turbulent single-phase sheet has not been studied in detail and is the subject of the present work. Two mechanisms can be identified in figure 14 that affect the fragmentation of the liquid sheet: fragmentation of the edge openings or centre hole due to oscillation of the liquid sheet itself and disturbance of the sheet spreading direction due to turbulence, which eventually develops into spreading direction breakup. For the backpressure of $P_{b} = 0.1$ MPa, the fragmentation caused by liquid sheet oscillation plays a dominant role, and when the backpressure is $P_{b} = 4.0$ MPa, the spreading breakup caused by turbulence plays the dominant role.

Different cross-sections are observed to gain a clearer understanding of the liquid sheet breakup process, as shown in figure 15. The time interval is $10\ \mathrm {\mu }\textrm {s}$. The liquid sheet formed is indicated by the transparent grey area, whereas the red surface represents the cross-section of the liquid film at $y$ values of 0.3, 0 and $-0.3$ mm. At $P_{b} = 0.1$ MPa, bubbles are entrained into the liquid sheet in the impact region. As the edges of the liquid sheet swing and fold, the bubbles grow and expand. As the liquid sheet flows, the bubbles penetrate into the interior of the liquid sheet, ultimately leading to its rupture. Figure 15($a$) illustrates the processes of bubble formation, development and eventual breakup of the liquid sheet, as indicated by the orange and blue arrows. At $P_{b} = 4.0$ MPa, fewer bubbles are observed within the liquid sheet, and the fracture of the liquid sheet is caused primarily by its unstable development. The thickness of the liquid at the trough continues to decrease, ultimately resulting in rupture.

Figure 15.

Process of the development of liquid sheet breakup under $P_{b}=0.1$ MPa and $P_{b}=4.0$ MPa, with a time interval of $10\ \mathrm {\mu }\textrm {s}$. The transparent grey areas indicate the formed liquid sheet, whereas the red surfaces represent the cross-sections of the liquid sheet located at $y = -0.3$ mm, $y = 0$ mm and $y = 0.3$ mm: ($a$) $P_{b}=0.1$ MPa; ($b$) $P_{b}=4.0$ MPa.

Figure 16($a$) displays the centre section at $y = 0$ mm of the liquid sheet outer contour for five consecutive moments. The initial impingement wave, identified as the wave with distinct crests and troughs downstream of the impact point, closely matches the sinusoidal wave. However, as it develops, the impinging wave rapidly destabilises and becomes a nonlinear fluctuation. The measurements of the initial impact wave crest velocity and wavelength are presented in figure 16($b$). Once the flow stabilises, the measurements are averaged over five instances with a time interval of $10\ \mathrm {\mu }\textrm {s}$, with the error defined as the difference between the maximum and minimum values. This interval of $10\ \mathrm {\mu }\textrm {s}$ roughly corresponds to the oscillation period of the liquid sheet. The results indicate a significant decrease in the wave velocity with increasing backpressure, whereas the wavelength remains relatively constant. Importantly, the change in the wave velocity does not induce a change in the wavelength. This result is in good agreement with the findings of Anderson et al. (Reference Anderson, Ryan, Pal and Santoro1993) and Ryan et al. (Reference Ryan, Anderson, Pal and Santoro1995). Ryan et al. (Reference Ryan, Anderson, Pal and Santoro1995) argued that the change in the wavelength of turbulent jets is independent of the velocity and depends mainly on the diameter of the jet. The formation of impact waves is attributed to the nature of the turbulence within the jet and the surface tension interaction. Figures 17($a$) and 17($b$) display the superposition of the outer contours at five moments of the $P_{b}=0.1$ MPa and $P_{b}=4.0$ MPa lateral liquid sheets. The two red dashed lines indicate the edges of the fluctuations on the two sides of the liquid sheet. A high backpressure clearly causes more violent disturbances on the surface of the jet, resulting in complex fluctuations in and breakup of the liquid sheet. At a low backpressure, the magnitude of the liquid sheet fluctuations is greater. Figure 17($c$) shows the variations in the front- and side-view spreading angles with increasing backpressure. Both the frontal and lateral spray nozzle angles decrease as the backpressure increases. The decreasing trend is stronger as the backpressure increases from 0.1 MPa to 2.0 MPa. This suggests that as the backpressure increases, the spray field space is compressed, resulting in denser spray field droplets.

Figure 16.

Measurement of the wavelength and wave velocity of the impact wave. ($a$) Measurement of the wavelength under $P_{b}=0.1$ MPa as an example. ($b$) Variations in the wavelength and velocity of the impact wave with $P_{b}$.

Figure 17.

Envelope of the vibrating liquid sheet and measured spreading angles. ($a$) Envelope of the side-view vibrating liquid sheet at $P_{b}=0.1$ MPa, ($b$) envelope of the side-view vibrating liquid sheet at $P_{b}=4.0$ MPa and ($c$) measurement of the front- and side-view spreading angles.

3.4. Flow topologies

Turbulent atomisation of impinging jets is a very complex process. Local flow topology analysis is an effective approach for understanding the characteristic flow features and regimes of turbulent atomisation. Local flow topology analysis is based on the three invariants ($P, Q, R$) of the second-order tensors of velocity fields, as described in the pioneering work of Chong, Perry & Cantwell (Reference Chong, Perry and Cantwell1990) and Perry & Chong (Reference Perry and Chong1987). Regardless of the large-scale flow geometry, these general topological features govern a wide variety of turbulent flows, such as turbulent jets (da Silva & Pereira Reference da Silva and Pereira2008), turbulent boundary layers (Elsinga & Marusic Reference Elsinga and Marusic2010), turbulent mixing layers (Soria et al. Reference Soria, Sondergaard, Cantwell, Chong and Perry1994), isotropic turbulence (Ooi et al. Reference Ooi, Martin, Soria and Chong1999) and turbulence channel flows (Blackburn, Mansour & Cantwell Reference Blackburn, Mansour and Cantwell1996).

This methodology has recently been applied in the context of multiphase flows. For example, Hasslberger et al. (Reference Hasslberger, Ketterl, Klein and Chakraborty2019) investigated the behaviour of flow topologies in the primary atomisation of liquid jets and introduced a simple model for the size distribution of coherent structures. Dodd & Jofre (Reference Dodd and Jofre2019) focused on the flow structures near finite-size droplets in decaying isotropic turbulence. The results indicated that a flow topology similar to that found in canonical homogeneous isotropic turbulence is favoured near the interface of droplets, whereas the flow topology of low-entropy/high-dissipation structures with a strong tendency towards boundary-layer turbulence is favoured inside the viscous layer at the droplet surface.

Hasslberger, Klein & Chakraborty (Reference Hasslberger, Klein and Chakraborty2018) and Hasslberger et al. (Reference Hasslberger, Cifani, Chakraborty and Klein2020) analysed the flow topology in a bubble-induced channel flow considering the two cases of freely moving and deforming air bubbles in liquid water. The strong vorticity budget therein was linked to the regions of high fluid velocity and high interface curvature. The inclusion of bubbles inwards acted as a mixing element, fragmenting the large topology structures and sustaining a similarity to laminar creeping flow in the carrier fluid.

According to Chong et al. (Reference Chong, Perry and Cantwell1990) and Perry & Chong (Reference Perry and Chong1987), the local flow topologies can be characterised by the invariants of the velocity-gradient tensor $A_{i j}=\partial u_i / \partial x_j=S_{i j}+W_{i j}$, where $S_{i j}=0.5(A_{i j}+A_{j i})$ and $W_{i j}=0.5(A_{i j}-A_{j i})$ are the symmetric and antisymmetric components, respectively. The three eigenvalues $\lambda _1$, $\lambda _2$ and $\lambda _3$ of $A_{i j}$ are the solutions of the characteristic equation $\lambda ^3+P \lambda ^2+Q \lambda +R=0$, where $P$, $Q$ and $R$ are the first, second and third principal invariants of $A_{i j}$ and are given by

(3.3) \begin{equation} \left.\begin{gathered} P={-}\operatorname{trace}(A_{i j})={-}\boldsymbol{S}_{i i}=0, \\ Q=(\operatorname{trace}(A_{i j})^2-\operatorname{trace}(A_{i j}^2)) / 2={-}\boldsymbol{S}_{i j} \boldsymbol{S}_{j i} / 2+\boldsymbol{W}_{i j} \boldsymbol{W}_{j i} / 2, \\ R={-}\operatorname{det}(A_{i j})=(-\boldsymbol{S}_{i j} \boldsymbol{S}_{j k} \boldsymbol{S}_{k i}-3 \boldsymbol{W}_{i j} \boldsymbol{W}_{j k} \boldsymbol{W}_{k i}) / 3. \end{gathered}\right\} \end{equation}

The rate-of-strain tensors for incompressible flows are restricted to the $P=0$ plane. Therefore, the local flow topologies are analysed mainly in the $Q\unicode{x2013} R$ plane, and the value of the discriminant

(3.4)\begin{equation} \varDelta=\left(\frac{Q}{3}\right)^3+\left(\frac{R}{2}\right)^2 \end{equation}

divides the $Q\unicode{x2013} R$ phase space into two regions: $\varDelta >0$ gives one real and two complex conjugate eigenvalues, and $\varDelta <0$ gives three distinct real values, while on the lines $r_{1 a}=-2(-3 Q)^{3 / 2} / 27$ and $r_{1 b}=+2(-3 Q)^{3 / 2} / 27$, for which $\varDelta =0$, three real eigenvalues are obtained, two of which are equal. In the region where $\varDelta >0, A_{i j}$ has purely imaginary eigenvalues on the line $r_2=0$. Lines $r_{1 a}, r_{1 b}$ and $r_2$ divide the $Q\unicode{x2013} R$ phase space into four flow topologies in the incompressible case, named $\textrm {S} 1\unicode{x2013} \textrm {S} 4$.

Here, $Q>0$ indicates rotational flow, whereas $Q<0$ signifies deformation. Additionally, $R>0$ corresponds to compression, while $R<0$ indicates expansion. Therefore, the $\textrm {S} 1$ topological region is primarily influenced by rotational compression mechanisms, whereas the $\textrm {S} 2$ region is driven by compressive deformation. The $\textrm {S} 3$ region is mainly governed by tensile deformation mechanisms and the $\textrm {S} 4$ region is predominantly affected by rotational tensile forces. Different mechanisms control each topological region. Figure 18 shows the invariant topology in the cross-section of the spray field. The field contains focal topologies S1 and S4, and nodal topologies S2 and S3. Here, $\mathrm {S} 1\unicode{x2013} \mathrm {S} 4$ are depicted in yellow, red, light blue and dark blue, respectively. This result verifies the findings obtained from the vorticity field results with decreasing structure size and increasing density as the backpressure increases. Under all conditions, jet impingement features dominant S2 topologies. This observation aligns with the flow characteristics near the impingement point, where the flow converges as fluid from the two jets merges towards the impingement area. This convergence creates a topological region dominated by compressive deformation, corresponding to the S2 topologies.

Figure 18.

Slices of the flow topology field, where the colour bar refers to S1–S4. The black line indicates the phase interface. The pressures are (a) 0.1, (b) 0.5, (c) 1.0, (d) 2.0 and (e) 4.0 MPa.

Figures 19 and 20 illustrate the variations in the volume fractions of the different topological structures in the liquid and gas phases as a function of the downstream distance in the atomisation field under different backpressure conditions, analysed at a single time point with fully developed flow. In the gas phase, a change in the backpressure results in significant changes in the proportions of the different topological structures in the jet, liquid film and droplet region. Overall, the proportions of the S1 and S4 structures tend to increase, and the proportions of the S2 and S3 structures tend to decrease.

Figure 19.

Gas phase volume fraction distributions of focal topologies S1 and S4, as well as nodal topologies S2 and S3, for the pressures of (a) 0.1, (b) 0.5, (c) 1.0, (d) 2.0 and (e) 4.0 MPa.

Figure 20.

Liquid phase volume fraction distributions of focal topologies S1 and S4, as well as nodal topologies S2 and S3, are presented. The distributions are shown for the pressures of (a) 0.1, (b) 0.5, (c) 1.0, (d) 2.0 and (e) 4.0 MPa.

In the liquid phase, the proportions of the different topological structures in the jet region remain relatively constant, whereas in the liquid sheet region, the proportion of the S2 structure increases sharply. In the droplet region, increasing backpressure leads to a gradual increase in the proportions of the S1 and S4 structures, and a gradual decrease in the proportions of the S2 and S3 structures. Overall, only small variations can be observed in the topological fractions in the jet and droplet regions in the axial direction. Their average values remain relatively constant. In all cases, the volume fractions are approximately 30 % and 40 % for focal topologies S1 and S4, respectively, and 20 % and 10 % for nodal topologies S2 and S3, respectively. Similar results were reported by Hasslberger et al. (Reference Hasslberger, Ketterl, Klein and Chakraborty2019) for the primary atomisation of liquid jets. However, the proportion of topological structures changes dramatically in the liquid sheet region. In contrast to the relatively stable jet and droplet regions, the liquid sheet region is subject to complex fluid fluctuations and interfluid separation, giving rise to drastic changes in its topology.

The initial distribution induced by the turbulent jets causes folding of the liquid sheet at the jet interface. To illustrate the above phenomenon, the initial turbulent flow pattern of the liquid jets must be considered in the numerical simulations. Neglecting the finite disturbance caused by turbulence can lead to relatively flat liquid sheets in simulations, whereas experiments show violently flapping liquid sheets. The coupling of turbulent flow and liquid/gas interactions is highly important.

At the exit plane, as illustrated in figure 3($a$), strong gradients are imposed on the boundary layer as the transition from a no-slip wall to a free-surface condition occurs at the edge of the boundary layer. The boundary layer thickness at the exit plane is highly important for high-velocity injectors because the vorticity convected from the boundary layer leads to unbalanced radial forces on the free surface immediately downstream of the orifice exit.

The complex processes of vortex generation, development and breakdown are described in this section. This discussion highlights the importance of numerical simulations in the study of these phenomena. The defining characteristic of turbulence is its chaotic nature, marked by complex and seemingly unpredictable fluctuations in both time and space. This chaotic behaviour stems from the strong nonlinearity of inertial flows as they evolve over time and space. As a result, specialised techniques are necessary to investigate turbulence effectively.

Figure 21 shows the vorticity distributions at the flow field cross-section under different $P_b$ values. The vorticity is defined as $\omega _y=\partial u_z / \partial x-\partial u_x / \partial z$ in figure 21. The generation of vorticity in shear layers is well established (Brown & Roshko Reference Brown and Roshko1974). The vorticity level is much lower in the liquid phase than in the gas phase, and a clear jump at the interface is observed for all $P_b$ values. Farsoiya, Mayya & Dasgupta (Reference Farsoiya, Mayya and Dasgupta2017) reported that the jump of the vorticity field at the interface depends on the viscosity and density ratio of the two fluids. As the density ratio increases, the vorticity field jump becomes more pronounced. According to our numerical simulations, the jump in the vorticity from the gas to the fluid is significantly reduced as $P_b$ increases from 0.1 MPa to 4.0 MPa, consistent with observations reported in the literature. Vortex pairs interact with changes in the surface structure of the liquid. Higher vorticity primarily arises in the liquid at the impingement point where the jet first contacts the surface, likely due to jet-gas shear. More intense vortices occur mainly along the spreading sheet surface after impingement in the gas. The significantly lower vorticity intensity observed in the liquid phase compared with the gas phase across all $P_b$ values can be attributed to the continuity of shear stress at the interface. Due to the continuity of shear stress at the gas–liquid interface, combined with the gas–liquid density ratio and viscosity ratio mentioned in this paper, a discontinuity in the velocity gradient occurs at the interface, leading to a jump in vorticity at the gas–liquid interface. In the liquid phase, higher vorticity primarily arises at the impingement point where the jet first contacts the surface, likely due to jet-gas shear. In the gas phase, more intense vortices occur mainly along the spreading sheet surface after impingement. As $P_b$ increases, the intensities of the vortices around the sheet and droplets noticeably decline overall, potentially due to the lower gas–liquid velocity differential at higher $P_b$. However, the spatial distribution of the vorticity field becomes more complex and irregular with increasing $P_b$, particularly where the thin sheet breaks up into ligaments and droplets. The increased gas flow introduces more turbulence, disrupting the smooth patterns observed at lower $P_b$. Furthermore, as illustrated in figures 21($a$), 21($b$) and 21($e$), the entrainment of air bubbles around the impingement point is observed more clearly, which is consistent with the similar observations in figure 15.

Figure 21.

Normalised plane-normal component of the vorticity slices, $\omega _yD/{U_0}$ for the pressures of (a) 0.1, (b) 0.5, (c) 1.0, (d) 2.0 and (e) 4.0 MPa.

Figure 22 shows the 3-D vorticity distributions at $P_b = 0.1$ MPa and 4.0 MPa, obtained using the 3-D $Q=4U^2/D^2$ iso-contour. At 0.1 MPa, when the impinging jets hit the surface, a thin radial sheet formed. The velocity difference between the liquid and gas induces hydrodynamic instabilities, resulting in sinuous patterns that roll into larger vortical structures. At 4.0 MPa, the velocities of the sheet and gas become more balanced, leading to the generation of intense small-scale vortices due to the violent impact. These vortices dynamically interact and stimulate turbulent multiphase mixing. This mixing process is crucial for the transfer of momentum, mass and energy across the sheet. The vortices entrain fluid from the jet core and ambient gas, resulting in complex interactions between the different streams. This influx intensifies the flow complexity as more vortices amplify the mixing process.

Figure 22.

Three-dimensional isocontours of interface reconstruction using the isoAdvector method (semitransparent red) showing the phase interface and $Q=4U^2/D^2$ (green) indicating prominent focal structures in the $(a)$ ${P} = 0.1$ MPa and $(b)$ ${P} = 4.0$ MPa cases.

3.5. Droplet statistics

Statistical analyses of the droplet diameters reveal the atomisation characteristics under varying $P_{b}$. Figure 23 shows the SMD distribution in the near field under different $P_{b}$ values, following the region division method shown in figure 8($b$). As illustrated in figure 23($a$), the SMD distribution along the circumferential direction was analysed within the fan-shaped region, which is bounded by two concentric circles with radii of 8.5 and 10 mm. The SMD peaks in the middle of the sector and transitions from a pattern of low sides/high middle to high sides/low middle as $P_{b}$ increases. At the backpressures ranging from 0.1 MPa to 1 MPa, the droplets separated from the edge of the liquid sheet are smaller than the droplets formed downstream of the liquid surface. However, at backpressures ranging from 2 MPa to 4 MPa, the droplets separated from the edge of the liquid sheet are larger than the droplets formed downstream of the liquid surface. Previous analyses indicate that the separation mechanism at the edge of the liquid film changes as the backpressure increases, resulting in a change in the spatial distribution of the droplet size in the near field. As illustrated in figure 23($b$), the SMD was statistically analysed within the fan-shaped region along the radial direction. This region has a width of 1 mm, with the statistical range extending from 6 mm to 11 mm. In the same near-field region, the SMD initially decreases and then increases as the backpressure rises. Furthermore, as the radial distance extends, the SMD consistently exhibits an increasing trend across different backpressure conditions.

Figure 23.

Statistics for the local Sauter mean diameter in the near field under different backpressures: ($a$) Sauter mean diameter distribution along the circumferential direction; ($b$) Sauter mean diameter distribution along the radial direction.

The variation in the downstream average diameter provides insights into the far field. The region between the two vertical red dashed lines in figure 24 shows an initial increase in the droplet diameter that then fluctuates around a constant value with movement further downstream. The increase in the droplet diameter during the initial stages can be attributed to the breakup of liquid filaments/clusters. However, at a certain distance, a state of stability is reached, where the growth due to breakup is balanced by other factors, resulting in a constant average diameter. As the backpressure increases, the region of droplet diameter growth expands, indicating that a stable atomisation field can form soon after the liquid sheet breaks up at a low backpressure.

Figure 24.

Variation in the average droplet diameter in the far field of impinging-jet atomisation with the downstream distance.

The droplet size distribution is an important statistical property for characterising the atomisation of impinging jets. When impinging-jet atomisation develops and sufficiently stabilises, the statistical distribution of the droplet sizes eventually stabilises. The stable numerical simulation results of the atomisation field obtained using the interIsoFoam solver are further calculated using the atomizationFoam solver. In this process, the VOF representation of droplets is transformed into Lagrangian particles, as illustrated in figure 25. The two grey areas indicate the upstream and downstream statistical regions, and the velocity distributions of the droplets are indicated by different colours.

Figure 25.

Combination of VOF simulation conversion and Lagrangian particle tracking via atomizationFoam.

This transformation enables us to obtain the size distribution of individual droplets in the atomisation field under various backpressure conditions. Notably, the gamma distribution function suitably describes all backpressure conditions, as revealed by the numerical simulation results:

(3.5)\begin{equation} P(X = d/d_0) = \frac{n^n}{\varGamma(n)}X^{n-1} {\rm e}^{{-}nX}, \end{equation}

where $n$ is a constant and $\varGamma (n)$ is the gamma function. This function is expected to provide a good description of the droplet size during atomisation (Bremond & Villermaux Reference Bremond and Villermaux2006; Villermaux Reference Villermaux2007). The gamma function is currently widely employed to characterise the droplet size distribution in atomisation processes (Bremond & Villermaux Reference Bremond and Villermaux2006; Villermaux & Bossa Reference Villermaux and Bossa2011; Kooij et al. Reference Kooij, Sijs, Denn, Villermaux and Bonn2018; Sijs, Kooij & Bonn Reference Sijs, Kooij and Bonn2021).

Figure 26($a$) shows the p.d.f.s of the droplet diameters at various backpressures. With increasing backpressure, the exponent $n$ initially increases and then decreases. This pattern suggests that the distribution of the droplet sizes first narrows and then widens. Figure 26($b$) shows the p.d.f.s of the droplet diameters upstream and downstream for backpressures of 0.1 and 4 MPa. The upstream and downstream droplet diameter distributions at 4.0 MPa show an opposite trend to those at 0.1 MPa. In the impingement atomisation process under low backpressure, upstream droplets continuously break up as they move downstream, reducing the number of larger droplets; this leads to an increase in the $n$ value when the droplet size distribution is fit with the Gamma function. Conversely, under high backpressure, increased droplet collisions and coalescence result in a greater number of larger droplets, causing the $n$ value to decrease.

Figure 26.

P.d.f. of the droplet diameter with respect to the average diameter: ($a$) p.d.f. for various backpressures; ($b$) p.d.f. upstream and downstream for the 0.1 MPa and 4.0 MPa cases.

The droplet velocity reflects the flow in the atomisation field. Figure 27 shows the p.d.f. distribution of the droplet velocity at various backpressures. In the atomisation process, the air resistance experienced by the liquid during atomisation varies with backpressure. As a result, even when the injected momentum remains constant, changes in the backpressure significantly impact the distribution of droplet velocities. Figure 27($a$) presents the velocity distribution of initially atomised droplets upstream. Under low-backpressure conditions, the initial droplet velocity distribution is typically broader, as the liquid sheet retains more kinetic energy, resulting in the formation of more high-velocity droplets after breakup. Owing to their reduced air resistance, droplets can sustain higher velocities in the initial stage. With increasing backpressure, the air resistance intensifies, leading to a lower initial droplet velocity and a more confined distribution range. Figure 27($b$) show the velocity distribution of the droplets downstream after atomisation has stabilised. Under low backpressure, the droplet velocity distribution remains relatively broad during the stable phase. Although droplet velocities begin to decline due to ongoing air resistance, the overall distribution remains relatively high. As the backpressure increases, the droplets velocities decrease rapidly, with the distribution becoming more sharper and ultimately concentrating in a lower velocity range. Overall, droplet velocities exhibit significant differences across varying backpressure conditions: higher speeds and broader distributions occur under low backpressure, whereas lower speeds and more concentrated distributions are observed under high backpressure. As droplets progress downstream, their velocities gradually diminish and tend to stabilise, with higher backpressure leading to lower velocities and more concentrated distributions.

Figure 27.

P.d.f. of the droplet velocity for various backpressures: ($a$) p.d.f. of the upstream droplet velocity for various backpressures and ($b$) p.d.f. of the downstream droplet velocity for various backpressures.