Introduction

The subject of the case is a PSA spray exposed to cross-flowing air. A small low-pressure atomizer was used for the study. This atomiser was developed for spraying aviation fuel Jet A-1 (kerosene) into the combustion chamber of a small gas turbine (GT) engine. The here documented operation conditions of the atomiser and the flow velocity corresponding to the engine's low-power or steady-flight conditions. The airflow is forced perpendicularly to the main spraying axis, which is considered a cross-flow case. The flow is homogeneous, isothermal and with low turbulence intensity, Tu. Similar atomisers of this type and size used together with the operating pressure and cross-flow air velocity conditions cover many industrial spray applications ranging from small GT combustors to chemical spray reactors. The conditions are also relevant for agriculture and domestic sprayers. The processed results of the present case were published in [2], with work carried out in the frame of projects №. GA18-15839S and GA 22-17806S funded by Czech Science Foundation. The present case is one of several cases measured and studied in [2]. The data are relevant to CFD engineers and scientists. They can distinguish the crucial phenomena to be considered in their numerical simulations of that disperse two-phase flow case. The modellers can highlight the important features of the complex two-phase flows and provide data for validation purposes. It is as well as to interesting to engineers dealing with the processes where the gas–liquid energy transfer and droplet transport are important. The case data can be used for further processing to obtain new findings of the problem, derive empirical models and serve as benchmark data.

Main characteristics of the flow and spray

The PSA sprays water (which represents low viscosity liquid) into cross-flowing air with low turbulence. There are several forces relevant to the case. Cohesive and consolidating forces acting on the liquid film are the surface tension force Fσ and the viscosity force Fμ. These are counteracted with disruptive compressive and momentum forces Fp and Fm. Apart from those also the gravity force applies. We can neglect the other forces possibly acting on the droplets and other liquid structures, such as stochastic force that accounts for Brownian collisions of the droplet with surrounding fluid molecules, or Basset force. The case can be decomposed into several consequent stages with different relevant phenomena, due to the physical acting of these forces, as shown in Figure 2, left:

Liquid flow inside the atomiser, its discharge
Sheet formation and the primary break-up of the liquid sheet
Liquid secondary break-up and spray formation,
Interaction of the sprayed liquid with surrounding air: gas–liquid mixing, droplet collisions, droplet clustering, and droplet repositioning.

From a thermodynamic point of view, the case is isothermal and isobaric, except for possible evaporation which can modify the droplet size [3]. That can introduce thermal effects, such as the exchange of heat between the discharged liquid and the surrounding air, which are otherwise unimportant. For the purpose of numerical simulations, the case features a two-way to four-way coupling between the gas and liquid phases depending on the position in the spray [4].

Underlying flow physics which characterise this case

The four stages of this case are explained in the consequent subsections

Liquid flow inside the atomiser and its discharge

The formation of the liquid film and the resulting spray depend mainly on the internal flow, the geometry of the outlet and the interaction with the surrounding environment are other factors. The shape and stability of the air core inside the nozzle directly affect the geometrical characteristics of the liquid sheet and its stability [5]. Therefore, the flow field inside the swirl chamber is key for understanding these processes. The liquid, pumped under pressure through the tangentially oriented inlet ports, creates a swirling flow inside the swirl chamber. Its main purpose is to form a thin liquid film at the nozzle exit. The swirling momentum is determined by the swirl number, S, which indicates the ratio of the momentum from the swirl component of the velocity $w=Q/A_{in}$ to the axial component $u_{0}=Q/\pi r_{0}^{2}$ ‎
$S={\frac {wr_{c}}{u_{0}r_{0}}}={\frac {\pi r_{c}r_{0}}{A_{in}}}$ where $A_{i}$ is the cross-section of the inlet ports, $Q$ is the fluid flow rate, $r_{c}$ and $r_{0}$ are the radius of the swirl chamber and the exit orifice. Thus, the swirl number depends on the nozzle geometry and can be expressed by the nozzle dimension constant $k=A_{in}/4r_{c}r_{o}$ . The swirling flow reduces the pressure near the atomiser axis and when it drops below the air pressure at the exit, an air core (AC) establishes, which is crucial for the formation of a thin liquid sheet emerging from the nozzle. The sheet stability depends on the regularity of the AC [5].
The nature of the flow depends on Reynolds number, $Re$ (ratio of momentum and viscous forces), and insignificantly on Weber number, $We$ , (ratio of momentum and surface tension forces), see Equations (11, 12), where $v$ is considered as the velocity of the fluid entering the chamber and $D=D_{c}$ is its diameter. For larger nozzles and enlarged models, the Froude number:
$Fr={\frac {u}{\sqrt {gD}}}={\frac {Q_{l}}{2(\left(r_{o}^{2}-r_{ac}^{2}\right){\sqrt {r_{o}g}}}}$ which describes the effect of gravitational forces on the flow, is also important. Here $r_{g\varepsilon \varepsilon }$ is the $\mathrm {AC}$ radius, and $g$ is the gravitational acceleration. Earlier theoretical works assumed the internal flow as a non-viscous free vortex [6], and with the consideration of Bernoulli's equation for an ideal fluid neglecting the potential term and the radial velocity component, the continuity equation reads:

${\frac {Q_{l}^{2}}{2\pi ^{2}\left(r_{o}^{2}-r_{oac}^{2}\right)^{2}}}+{\frac {Q_{l}^{2}\left(r_{c}-r_{i}\right)^{2}}{2A_{in}^{2}r_{oac}^{2}}}={\frac {p_{in}}{\rho _{l}}}$ where $r_{i}$ is the radius of the inlet ports and $r_{0a}$ is the $\mathrm {AC}$ radius in the exit orifice. For the solution, the principle of maximum flow is assumed, i.e., $r_{\text{rog }}$ is adjusted so that the flow rate is always maximum: $\delta Q/\delta _{oxc}=0$ . The non-viscous description of the flow was revised in [6,7] and recognised as suitable only for understanding the nature of the flow or preliminary nozzle design. Discrepancies of experiments with the non-viscous theory and have led to corrections of this model [8] [9] [10] [11] [12] and more complex analytical approaches [13] [14] [15], which, however, do not reach the accuracy of CFD models. The liquid swirling inside the nozzle discharges from the exit orifice at a high velocity into the surrounding air (Figure 2, left). The annular liquid structure formed at the orifice features a relatively low discharge coefficient, $C_{D}$ , which is after Rizk and Lefebvre [8]: $C_{D}=0.35k^{0,5}\left({\frac {r_{c}}{r_{o}}}\right)^{0,25}=0.39$ for this case. This value well agrees with the experimental data in Table 2. The efficiency of the conversion of inlet potential energy into kinetic energy at the nozzle exit is $\eta _{n}=\rho _{l}u_{i}^{2}/2p_{in}$ . This so-called nozzle efficiency was estimated by several authors. Horvay with Leuckel [16] found </math>\eta_a=0.42-0.66</math>, Yule with Chinn [17] reported $\eta _{a}=0.73-0.86$ , and we [4] for similarly sized atomiser found $\eta _{a}=0.34-0.41$ .

Formation and primary break-up of the liquid sheet

The conical liquid sheet spreads in the axial and radial direction, attenuates downstream the nozzle and undergoes a dynamic liquid-gas interaction. That depends on the airflow conditions, which can be distinguished into simple categories of still, co-, counter- or crossflowing air. A high-velocity shear between the discharged liquid and the surrounding air produces the Kelvin-Helmholtz-type instabilities on the sheet. These add to turbulent perturbations induced by the swirling motion inside the chamber and deform the sheet. The disrupting gas forces and the consolidating surface tension forces of the liquid film are compared using the gas Weber number, $We_{g}=\rho _{Q}u_{l}^{2}\tau /2\sigma$ , where the indices $g$ and $l$ stand for air and liquid, respectively, and $\tau$ is the sheet thickness. A critical Weber number $We_{gcr}=27/16$ [18] distinguishes domination of long-wave or short-wave growth on the sheet; long waves prevail when $We_{g}<We_{gcr}$ and short waves in the opposite case. The actual We (see table 2), compared with $We_{gcr}$ , shows that long-wave growth appears at lower air velocity, and the transition to the short-wave happens at higher air velocity. The sheet thickness reduces to its critical value, and the surface tension forces perforate the perturbed sheet. The sheet then disrupts or tears into fragments at the break-up distance. The internal flow is complex, and the internal disturbances can turbulise the emerging liquid sheet and these disturbances may, depending on their frequency and intensity, reduce the break-up length. That is supported by (Sharief et al. [19] and Yule and Chinn [17], contradictorily to the numerical findings of Deng et al. [20]. The primary break-up features a contraction and ordering of detached sheet fragments into irregularly shaped filaments. These, due to the capillary instability [21], break down into single droplets that form a hollowcone spray. The relative importance of internal viscous and surface tension forces during the sheet disintegration is indicated by the ratio of $W_{e}$ and $R_{e}$ of the liquid phase at the discharge orifice after Yule and Dunkley [22]: $W_{e}/R_{e}=u_{o}\mu _{l}/\sigma$ . The originally two-dimensional sheet breaks down, and its oscillations and mixing with air result in a radial redistribution of the liquid fragments and droplets according to their size classes. The spray acquires the Gaussian velocity profile normal to the sheet surface [23]. The moving liquid film, fragments, and droplets experience mechanical interactions with the air through viscous drag. The droplets, moving with low $R_{e}$ , typically below 100, decelerate according to Stokes' law as ${\frac {du_{D}}{dt}}=-18\mu _{g}\left(u_{D}-u_{g}\right)/\rho _{l}d_{D}^{2}$ , and establish a positive size-velocity correlation which contribute to droplet collisions in the dense spray region [24]. The gas-liquid interaction is described in detail in [4].

Spray formation

The produced droplets cover a wide size range and form a single or double-peak size distribution (in Figure 2b), depending on the position in the spray. The droplets with $dD<20\mu m$ fast decelerate to the airflow velocity, medium-size droplets $(20\mu m<dD<50\mu m)$ feature a positive size–velocity correlation, and the largest droplets up to 100 µm keep the original velocity of the discharged liquid. The smallest droplets follow the local air velocity closely, and so the velocity of droplets sized below 5 µm can serve as the air velocity estimate.

Figure 2a: Internal and external flow with relevant phenomena and fluid structures
Figure 2b: Droplet size spectra and representative diameters

The size distribution of the droplets can be represented simply at each position with a suitable mean droplet diameter, for which a general expression is

$D_{ab}={\sqrt[{a+1}]{\sum _{i=1}^{n}d_{i}^{a}/\sum _{i=1}^{n}d_{i}^{b}}}$

where ${\underline {d}}$ is the diameter of individual droplet $i$ and $n$ is the total number of the droplets at the position. Thus, $D_{10}$ is for the arithmetic mean diameter and $D_{32}$ is the Sauter mean diameter (or volume/surface mean diameter). The spray itself, if sprayed into still air can be considered roughly axially symmetrical with large size and velocity variability in the radial direction. The radial profiles of the mean liquid velocity are self-similar along the axial locations with a peak close to the sheet position. The sprayed mass is mostly distributed along the sheet trajectory, and it forms a hollow-cone spray. The inner region contains only small droplets that are driven there due to the air drag. The main semi-conical spray region behind the disintegrated liquid sheet contains larger high-energetic droplets with high penetration ability. The outer spray periphery covers a small portion of droplets with velocity decreasing with radial distance. The liquid sheet, its fractions and larger droplets in the near-nozzle area follow the trajectory given by the discharge conditions, while the consequent flow and motion of small droplets in the far field are more influenced by the interaction with the surrounding gas. One of the PSA's main parameters is the break-up distance, $I_{b}$ , which determines the volume of the ligaments and the size of the resulting droplets. It can be determined from the empirical Equation (5) [25]:

$l_{\mathrm {h} }=0.123\tau ^{0.5}We^{-0.5}Re^{0.6},$

semi-empirical Equation (3) in [26], or analytically using linear instability analysis (LISA) [27]. Countless correlations have been developed to describe droplet size as a function of atomiser operating parameters. The one by Wang and Lefebvre [28] calculates $D_{32}$ considering relevant physical phenomena during atomisation:

$D_{32}=4.52\left({\frac {\sigma ul}{\rho _{g}p_{in}^{2}}}\right)^{0.25}\left(\tau _{0}\cos(SCA)\right)^{0.25}+0.39\left({\frac {\sigma \varrho _{l}}{\rho _{g}p_{in}}}\right)^{0.25}\left(\tau _{0}\cos(SCA)\right)^{0.75}$

Available correlations are not fully reliable or universal, so further experiments are required.

Interaction of the sprayed liquid with the surrounding air

All the above descriptions and most published studies have considered atomiser spraying in the absence of ambient flow. Though liquid spraying into still surrounding air is the most frequently investigated configuration, many applied atomisers work in a flowing environment. Cross-flow spray configuration is relevant, e.g. for Venturi scrubbers [29] where typical velocities range from $10m/s$ [30] or $20m/s$ [31] up to $30m/s$ and even $80m/s$ [32] [33] [34]. Droplet size at higher ambient flow velocities is affected by the secondary aerodynamic break-up. If we consider a maximum droplet size in the spray of 100 µm, then after [35]

$u=784/{\sqrt {d_{D}}}$

a velocity of about $78m/s$ is required for the aerodynamic break-up of a droplet. Thus, the ambient flow velocities tested here do not allow for the secondary break-up to apply. Their resulting size depends on the primary break-up and the design of the atomiser itself. The combination of the forces $F_{p}$ , $F_{\sigma }$ , $F_{m}$ and $F_{\mu }$ acting on the liquid sheet results in its disintegration [36]. The presence of transverse flow disturbs the flow field around the atomiser and can change $F_{p}$ and $F_{m}$ [37].

$F_{p}=n\rho _{g}\ (u_{1}^{2}+u_{2}^{2}\ )yzdx$

$F_{m}=-\rho _{l}\left(\tau {\frac {\partial ^{2}y}{\partial t^{2}}}+{\frac {\partial \tau }{\partial t}}{\frac {\partial y}{\partial t}}\right)zdx$

Figure 3: Schematic representation of a windward liquid sheet propagation: without cross-flow (left), with cross-flow(right); dimensions not to scale, the coordinate system does not correspond to that used for the experiment

Here $n$ is the wave number. With the presence of cross-flow, the air velocity $u_{1}$ around the liquid sheet changes, which then affects the $F_{p}$ acting on the liquid sheet, see Figure 3. The cross-flow represents an "additional resistance" of the surrounding environment that the liquid sheet must overcome. With this also, the rate of change of momentum of the liquid film and the ratios in Equation (9) change. The increased $F_{p}$ and $F_{m}$ in the cross-flow reduce $l_{b}$ and leads to the formation of larger droplets, as observed in [37]. The studies dealing with sprays in cross-flow express the effect of the ambient flow on the spray by the ratio of the liquid and air momentum (q), the aerodynamic Weber number (Wea) and the relative Weber number ( $We_{r}$ ) using Equations (10–12). The Weber number is defined in two ways here. The first $We$ definition, used in [38], incorporates the cross-flow velocity ( $u_{cf}$ ) and the diameter of the discharge orifice ( $d_{o}$ ). The second one, $We_{r}$ , contains the relative velocity of the liquid sheet ( $u_{l}$ ) to the cross-flow velocity ( $u_{cf}$ ), which is denoted $u_{r}$ . The calculation of $u_{r}$ is shown in Figure 4 and described by Equation (13).

$q={\frac {\rho _{l}u_{l}^{2}}{\rho _{g}u_{cf}^{2}}}$

${We}_{a}={\frac {\rho _{g}u_{cf}^{2}d_{o}}{\sigma }}$

${We}_{r}={\frac {\rho _{g\ }u_{r}^{2}\tau _{o}}{2\sigma }}$

$u_{r}=u_{l}+u_{cf}\sin {\left({\frac {SCA}{2}}\right)}$

Figure 4: Graphical representation of

u_{r}

in cross-flow

Main quantities of interest

The PDA measurements produce data allowing the calculation of droplet size and velocity statistics and, to some extent estimating the local airflow velocity. These data give detailed information on the velocity field of the sprayed liquid and surrounding air. The HSV provides photogrammetric information on the discharged liquid. The data can be used for the estimation of relevant dimensionless criteria that characterise the individual processes involved in the studied case, as summarised in Table 1. The table also contains information on experimental and simulation techniques and approaches used and applicable to study these processes by different researchers.

	Process	Output, parameters questioned	Relevant criteriag	Approaches
	Process	Output, parameters questioned	Relevant criteriag	Experiment		Simulation
1	Internal flow	Velocity field, air core properties	$Re_{l}$ , $Fr^{h}$ , S	$LDA^{a}$ , $HSV^{a}$		Laminar, URANS, LES
2	Discharge and liquid film formation	CD, SCA, velocity, stability, liquid film thickness	$Re_{l}$ , Wea, Fr, Boe	HSV		Laminar, URANS, LES
3	Break-up into smaller structures (primary)	break-up character, lb	$Re_{l}$ , Ohf, Wea, Wer, Boe	LIFa	HSV	(LES) DNS
4	Subsequent disintegration into droplets (secondary)	Droplet size, velocity, concentration	$Re_{l}$ , Oh, Wea, Wer	LIFa	PDAa, HSV	(LES) DNS
5	Interaction of droplets with the surrounding environment and with each other	Character of interaction, energy transfer, droplet collision, evaporation	Reg, Stki, Wecb, q, cc		PDAa, HSV	URANS, LES, Stat.d