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| = Introduction = | | === Spray formation === |
| 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. | | 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 <math> dD < 20 \mu m </math> fast decelerate to the airflow velocity, medium-size droplets <math> (20 \mu m < dD < 50 \mu m) </math> 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. |
| 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.
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| 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].
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| 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.
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| The case data can be used for further processing to obtain new findings of the problem, derive empirical models and serve as benchmark data.
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| == Main characteristics of the flow and spray ==
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| 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.
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| 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:
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| *Liquid flow inside the atomiser, its discharge
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| *Sheet formation and the primary break-up of the liquid sheet
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| *Liquid secondary break-up and spray formation,
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| *Interaction of the sprayed liquid with surrounding air: gas–liquid mixing, droplet collisions, droplet clustering, and droplet repositioning.
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| 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].
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| == Underlying flow physics which characterise this case ==
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| The four stages of this case are explained in the consequent subsections
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| === Liquid flow inside the atomiser and its discharge ===
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| 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.
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| 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 <math>w =Q/A_{in}</math> to the axial component <math>u_{0}=Q/\pi r_{0}^{2}</math> <br />
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| <math>
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| S = \frac{wr_{c}}{u_{0}r_{0}} = \frac{\pi r_{c} r_{0}}{A_{in}}
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| </math>
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| where <math> A_{i} </math> is the cross-section of the inlet ports, <math>Q</math> is the fluid flow rate,<math> r_{c} </math> and <math> r_{0} </math> 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 <math> k = A_{in} / 4r_{c}r_{o} </math>. 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]. <br />
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| The nature of the flow depends on Reynolds number, <math> Re </math> (ratio of momentum and viscous forces), and insignificantly on Weber number, <math> We </math>, (ratio of momentum and surface tension forces), see '''Equations (11, 12)''', where <math> v </math> is considered as the velocity of the fluid entering the chamber and <math> D = D_{c} </math> is its diameter. For larger nozzles and enlarged models, the Froude number: <br />
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| <math>
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| Fr=\frac{u}{\sqrt{gD}}=\frac{Q_l}{2(\left(r_o^2-r_{ac}^2\right)\sqrt{r_og}}
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| </math>
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| which describes the effect of gravitational forces on the flow, is also important. Here <math> r_{g \varepsilon \varepsilon} </math> is the <math> \mathrm{AC}</math> radius, and <math>g</math> is the gravitational acceleration.
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| 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: <br/>
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| <br/>
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| <math>
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| \frac{Q_l^2}{2 \pi^2\left(r_o^2-r_{o a c}^2\right)^2}+\frac{Q_l^2\left(r_c-r_i\right)^2}{2 A_{i n}^2 r_{o a c}^2}=\frac{p_{i n}}{\rho_l}
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| </math>
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| where <math>r_i</math> is the radius of the inlet ports and <math>r_{0 a}</math> is the <math>\mathrm{AC}</math> radius in the exit orifice. For the solution, the principle of maximum flow is assumed, i.e., <math>r_{\text {rog }}</math> is adjusted so that the flow rate is always maximum: <math>\delta Q / \delta_{o x c}=0</math>. 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.
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| 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, <math>C_D</math>, which is after Rizk and Lefebvre [8]: <math>C_D=0.35 k^{0,5}\left(\frac{r_c}{r_o}\right)^{0,25}=0.39</math> 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 <math>\eta_n=\rho_l u_i^2 / 2 p_{i n} </math>. 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 <math>\eta_a=0.73-0.86</math>, and we [4] for similarly sized atomiser found <math>\eta_a=0.34-0.41</math>.
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| ===Fomration and primary break-up of the liquid sheet ===
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| 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.
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| 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, <math>W e_g=\rho_Q u_l^2 \tau / 2 \sigma</math>, where the indices <math>g</math> and <math>l</math> stand for air and liquid, respectively, and <math>\tau</math> is the sheet thickness. A critical Weber number <math>W e_{g c r}=27 / 16</math> [18] distinguishes domination of long-wave or short-wave growth on the sheet; long waves prevail when <math>W e_g<W e_{g c r}</math> and short waves in the opposite case. The actual We (see table 2), compared with <math>W e_{g c r}</math>, 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.
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| 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 <math>W_e</math> and <math>R_e</math> of the liquid phase at the discharge orifice after Yule and Dunkley [22]: <math>W_e / R_e=u_o \mu_l / \sigma </math>. 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 <math>R_e</math>, typically below 100, decelerate according to Stokes' law as <math>\frac{d u_D}{d t}= -18 \mu_g\left(u_D-u_g\right) / \rho_l d_D^2</math>, 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].
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| === Spray formation ===
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