EXP 1-1 Introduction

Pressure-swirl spray in a low-turbulence cross-flow

Front Page

Introduction

Review of experimental studies

Description

Experimental Set Up

Measurement Quantities and Techniques

Data Quality and Accuracy

Measurement Data and Results

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 atomizer 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 atomizer and the flow velocity correspond 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 atomizers 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 , with work carried out in the frame of projects №. GA18-15839S and GA 22-17806S funded by Czech Science Foundation and project “Computer Simulations for Effective Low-Emission Energy Engineering” No. CZ.02.1.01/0.0/0.0/16_026/0008392 funded by Operational Programme Research, Development and Education, Priority axis 1: Strengthening capacity for high-quality research. The present case is one of several cases measured and studied in . The data are relevant to CFD engineers and scientists. They can differentiate 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 use the data for validation purposes. The case is equally 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 $F_{\sigma }$ and the viscosity force $F_{\mu }$. These are counteracted by disruptive compressive $F_{p}$ and momentum forces $F_{m}$. Apart from these also the gravity force acts. 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 actions of these forces, as shown in Figure 2a:

• Liquid flow inside the atomizer, 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 and 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 .

Underlying flow physics which characterise this case

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

Liquid flow inside the atomizer and its discharge

The formation of the liquid film and the resulting spray depend mainly on the internal flow in the atomizer shown in Figure 7,the geometry of the outlet and the interaction with the surrounding environment being additional factors. The liquid is forced into rotational motion inside the swirl chamber due to the tangentially oriented inlet ports.The swirling flow reduces the pressure near the atomizer axis, and when it drops below the air pressure at the exit, an air core establishes along the main atomizer axis. The shape and stability of the air core inside the nozzle directly affect the formation, geometrical characteristics and stability of the liquid sheet that emerges from the nozzle . 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, see Figure 7. Its main purpose is to form a thin liquid film at the nozzle exit. The swirl momentum is determined by the swirl number, $S$, which represents the ratio of the momentum from the swirl component of the velocity $w=u_{in}=Q_{l}/A_{in}$ to the axial component $u_{o}=Q_{l}/\pi r_{o}^{2}$ ‎

$S={\frac {wr_{c}}{u_{o}r_{o}}}={\frac {\pi r_{c}r_{o}}{A_{in}}}$

 

 

 

 

(1)

where $A_{in}$ is the cross-section of the inlet ports, $Q_{l}$ is the fluid flow rate, $r_{c}$ and $r_{o}=d_{o}/2$ are the radii of the swirl chamber and the exit orifice respectively (as shown in Figure 7). 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 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 $u=u_{in}$ 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:

$Fr={\frac {u}{\sqrt {gd}}}={\frac {Q_{l}}{2\pi \left(r_{o}^{2}-r_{ac}^{2}\right){\sqrt {r_{o}g}}}}$

 

 

 

 

(2)

which describes the effect of gravitational forces on the flow, is also important. Here $r_{ac}$ is the air core radius, and $g$ is the gravitational acceleration. Earlier theoretical works assumed the internal flow as a non-viscous free vortex , and with the consideration of Bernoulli's equation for an ideal fluid neglecting the potential term and the radial velocity component, the resulting equation reads:

${\frac {Q_{l}^{2}}{2\pi ^{2}\left(r_{o}^{2}-r_{oac}^{2}\right)^{2}}}+{\frac {Q_{l}^{2}\left(r_{c}-r_{i}\right)^{2}}{2A_{in}^{2}r_{oac}^{2}}}={\frac {p_{in}}{\rho _{l}}}$

 

 

 

 

(3)

where $r_{i}$ is the radius of the inlet ports and $r_{oac}$ is the air core radius in the position of the exit orifice. For the solution, the principle of maximum flow is assumed, i.e., $r_{\text{oac }}$ is adjusted so that the flow rate is always maximum: $\delta Q_{l}/\delta r_{oac}=0$. The non-viscous description of the flow was revised in , and recognised as suitable only for understanding the nature of the flow or preliminary nozzle design. Discrepancies of experiments with the non-viscous theory led to corrections of this model and more complex analytical approaches , 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 2a). The annular liquid structure formed at the orifice features a relatively low discharge coefficient, $C_{D}$, which according Rizk and Lefebvre is: $C_{D}=0.35k^{0.5}\left({\frac {r_{c}}{r_{o}}}\right)^{0.25}=0.39$ for this case. This value agrees well with the experimental data in Table 2. The efficiency of the conversion of inlet potential energy into kinetic energy at the nozzle exit is $\eta _{n}=\rho _{l}u_{in}^{2}/2p_{in}$. This so-called nozzle efficiency was studied and estimated by several authors. Horvay with Leuckel found $\eta _{a}=0.42-0.66$, Yule with Chinn reported $\eta _{a}=0.73-0.86$, and we for similarly sized atomizer 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 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, $We_{g}=\rho _{l}u_{l}^{2}\tau /2\sigma$, where the indices $g$ and $l$ stand for air and liquid, respectively, and $\tau$ is the sheet thickness. A critical Weber number $We_{gCr}=27/16$ 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. This scenario is supported by Sharief et al. and Yule and Chinn , in contradiction to the numerical findings of Deng et al. . The primary break-up features a contraction and ordering of detached sheet fragments into irregularly shaped filaments. These, due to the capillary instability , break down into single droplets that form a hollow-cone spray. The relative importance of internal viscous and surface tension forces during the sheet disintegration is indicated by the ratio of $We$ to $Re$ of the liquid phase at the discharge orifice after Yule and Dunkley  : $We/Re=u_{o}\mu _{l}/\sigma$. 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 a Gaussian velocity profile normal to the sheet surface . The moving liquid film, fragments, and droplets experience mechanical interactions with the air through viscous drag. The droplets, moving with low $Re=\left(u_{D}-u_{g}\right)d_{D}\rho _{g}/\mu _{g}$, 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 . The gas–liquid interaction is described in detail in .

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 $d_{D}<$ 20 μm decelerate fast to the airflow velocity, medium-size droplets (20 μm $<d_{D}<$ 50 μm) 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.

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  Figure 2a: Internal and external flow with relevant phenomena and fluid structures (without cross flow, i.e. $u_{cf}=$ 0 m/s)

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  Figure 2b: Droplet size spectra and representative diameters

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

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

 

 

 

 

(4)

where $d_{i}$ is the diameter of individual droplet $i$ and $n$ is the total number of droplets at the position. The most often used mean droplet diameters are: $D_{10}$, which is the arithmetic mean diameter (here $a=1$ and $b=0$, this diameter is used for comparison of disperse systems), $D_{20}$ (surface mean diameter for vaporization studies) and $D_{32}$ is called the Sauter mean diameter (or volume/surface mean diameter) and this one is used for mass and heat transfer evaluations. 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 ensuing 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, $l_{b}$, which determines the volume of the ligaments and the size of the resulting droplets. It can be determined from the empirical Equation (5) :

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

 

 

 

 

(5)

semi-empirical Equation (3) in , or analytically using linear stability analysis (LISA) . Countless correlations have been developed for swirl atomizers spraying into the steady environment to describe droplet size as a function of atomizer operating parameters. The one by Wang and Lefebvre calculates $D_{32}$ considering relevant physical phenomena during atomization:

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

 

 

 

 

(6)

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 atomizer spraying in the absence of ambient flow. Though liquid spraying into still surrounding air is the most frequently investigated configuration, many applied atomizers work in a flowing environment. Cross-flow spray configuration is relevant, e.g. for Venturi scrubbers where typical velocities range from 10 m/s or 20 m/s up to 30 m/s and even 80 m/s . 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 $d_{D}=$ 100 µm, then after

$\Delta v=784/{\sqrt {d_{D}}}$

 

 

 

 

(7)

a relative velocity between the droplet and surrounding gas, $\Delta v$, of about 78 m/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 atomizer itself. The combination of the forces $F_{p}$, $F_{\sigma }$, $F_{m}$ and $F_{\mu }$ acting on the liquid sheet results in its disintegration . The presence of transverse flow disturbs the flow field around the atomizer and can change compressive force $F_{p}$ and momentum force $F_{m}$ which are derived in detail in .

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

 

 

 

 

(8)

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

 

 

 

 

(9)

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  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, $u_{l}$ is the velocity of the liquid sheet, $u_{1}$ and $u_{2}$ is the velocity of the air on the front and rear side of the sheet respectively. 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 the break-up distance, $l_{b}$, and leads to the formation of larger droplets, as observed in . 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 ($We_{a}$) and the relative Weber number ($We_{r}$) using Equations ((10) – (12)). The $u_{l}$ in Equation (10) denotes the velocity of the liquid sheet at the discharge point and it is calculated after . The Weber number is defined in two ways here. The first $We$ definition, used in , 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 determination of $u_{r}$ is shown in Figure 4 and its calculation given by Equation (13), where $SCA$ is the spray cone angle.

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

 

 

 

 

(10)

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

 

 

 

 

(11)

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

 

 

 

 

(12)

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

 

 

 

 

(13)

Figure 4: Graphical representation of $u_{r}$ in cross-flow, the right image is a magnified view of the velocity vectors

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.

Table 1: Processes involved in PSA spraying

	Process	Output, parameters questioned	Relevant criteria ${^{g})}$	Approaches
				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	$C_{D}$, $SCA$, velocity, stability, liquid film thickness	$Re_{l}$, $We_{a}$, $Fr$, $Bo^{e)}$	HSV
3	Break-up into smaller structures (primary)	break-up character, $l_{b}$	$Re_{l}$, $Oh^{f)}$, $We_{a}$, $We_{r}$, $Bo^{e)}$	LIF$^{a)}$	HSV	(LES) DNS
4	Subsequent disintegration into droplets (secondary)	Droplet size, velocity, concentration	$Re_{l}$, $Oh$, $We_{a}$, $We_{r}$		PDA$^{a)}$, HSV
5	Interaction of droplets with the surrounding environment and with each other	Character of interaction, energy transfer, droplet collision, evaporation	$Re_{g}$, $Stk^{i)}$, $We_{c}^{b)}$, $q$, $c^{c)}$			URANS, LES, Stat$^{d)}$

$^{a)}$Laser Doppler anemometry, phase Doppler anemometry, high-speed visualisation, laser-induced fluorescence, $^{b)}$collision $We$ (for definition see ), $^{c)}$concentration of droplets in spray, ${}^{d)}$statistical approaches, ${}^{e)}$Bond number (also called Eötvös number) $Bo=\Delta \rho gD^{2}/\sigma$, can only be significant at very low discharge velocity, $\Delta \rho$ is the difference in density between the liquid and the gas, $^{f)}$Ohnesorge number $Oh={\sqrt {We}}/\operatorname {Re} =\mu /{\sqrt {\rho \sigma D}},D$ is the characteristic dimension, ${}^{g)}$ $Re_{g}$, $Re_{l}$ and $We_{g}$ are defined similarly to $We_{a}$, $We_{r}$ and $Re_{in}$ in Equations (11) (12) (14) with appropriate characteristic lengths and velocities and the index $g$ and $l$ denote the gas and liquid respectively. ${}^{h)}$Froude number according to Equation (2). ${}^{i)}$Stokes number $Stk=\rho _{l}{\bar {D}}_{p}^{2}\Delta {\bar {v}}/18\mu _{g}L,\Delta v$ is the difference between the gas and droplet velocity, $L$ is characteristic distance.

References

Contributed by: Ondrej Cejpek, Milan Maly, Ondrej Hajek, Jan Jedelsky — Brno University of Technology

Front Page

Introduction

Review of experimental studies

Description

Experimental Set Up

Measurement Quantities and Techniques

Data Quality and Accuracy

Measurement Data and Results

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