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Surface wave generation via a gas-jet penetration into a liquid sheet
Accepted Manuscript Surface wave generation via a gas-jet penetration into a liquid sheet Qi Liu, Zhe Lin, Xiao-ping Chen, Zu-chao Zhu, Bao-ling Cui PII: DOI: Reference:
S0894-1777(18)30073-6 https://doi.org/10.1016/j.expthermflusci.2018.03.035 ETF 9432
To appear in:
Experimental Thermal and Fluid Science
Received Date: Revised Date: Accepted Date:
26 January 2018 27 March 2018 28 March 2018
Please cite this article as: Q. Liu, Z. Lin, X-p. Chen, Z-c. Zhu, B-l. Cui, Surface wave generation via a gas-jet penetration into a liquid sheet, Experimental Thermal and Fluid Science (2018), doi: https://doi.org/10.1016/ j.expthermflusci.2018.03.035
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Surface wave generation via a gas-jet penetration into a liquid sheet
Qi Liu, Zhe Lin*, Xiao-ping Chen, Zu-chao Zhu, Bao-ling Cui Key Laboratory of Fluid Transmission Technology of Zhejiang Province, Zhejiang Sci-Tech University, Hangzhou, 310018, China
Abstract The mechanism and frequency of surface wave generated by normal wall-jet impingement at a vertical gas–liquid interface were experimentally characterized for a gas-jet penetrating into a liquid sheet. The oscillatory gas–liquid interface accompanied by surface wave generation and propagation were captured with a high-speed camera. Periodic oscillations of the vertical gas–liquid interface were revealed in the context of energy transfer from the wall-jet to surface waves. In addition, wave generation was considered as a means of releasing the maximum interfacial elastic energy. The radial distance between the waist and neck of the wavy interface was defined as the amplitude and the characteristic amplitude was adopted to reveal periodic deformation of the wavy interface during dynamic oscillations. Furthermore, the characteristic wavelength, i.e. vertical distance between waist and neck, was about one third of the liquid sheet thickness, irrespective of penetration parameters, such as gas flow rate, jet height, and liquid sheet thickness. The effects of those parameters on the surface wave generation frequency were analyzed. A combined dimensionless parameter of Froude number and the ratio of jet height to nozzle diameter was used to normalize the surface wave generation frequency with all the penetration parameters. It was found that the frequency could be expressed by a natural exponential function with the combined dimensionless parameter for normalization via fitting analysis. This work should be helpful to characterize interfacial motion in actual industrial applications. Keywords: Surface wave, Wall-jet, Liquid sheet, Oscillation, Generation frequency
1. Introduction The behavior of gas–liquid interface in two-phase flow is an important issue in many scientific fields. One well-known is the bubble formation by desorbing gas in solution when increasing the heat flux. Due to bubble formation, the interaction between gas and liquid became greatly stronger and accordingly intensified the convection effect in solution, which served as the major mechanism of enhancing heat transfer [1-3]. And, the physical characteristics of nucleate and rate of bubble formation were often focused on to reveal its relationship with the heat flux in diverse liquids [4-6]. When increasing the heat flux during the flow boiling in nanofluids, thermal performances of that were also significantly intensified due to the fast bubble formation and agitation in nucleate boiling region [7-9]. In addition, the performance of bubble formation can be improved by the deposition of the nano-particles in nanofiluds [10]. However, the behavior of bubbles formed in nanofluids can negatively affect the stability of nanofluids [11]. The effect of mixed convection of nanofluid on heat
*
Corresponding author. Tel.: +86 0571-86843348 E-mail address: [email protected]
transfer was explained by the Lattice Boltzmann method [12, 13], and physical and thermal characteristics were experimentally studied under the effect of solid volume fraction of various nanofluids [14-17]. With the good thermal performance, nanofluids were investigated as the potential coolant for actual applications [18, 19]. Besides, another classical behavior of gas–liquid interface is surface waves. As an interfacial flow, the wavy motion was also related to the heat transfer in fluids as well as the stability of the gas–liquid interface [20-22]. Surface waves generated by a gas at a gas–liquid interface, especially shearing effects, have been widely studied by many researchers in fluid mechanics. Phillips [23] proposed a resonance mechanism for surface wave generation in which pressure fluctuations initiated waves by selective amplification. Miles [24, 25] used the Orr-Sommerfeld equation to study surface wave generation in terms of momentum transfer from the gas phase to the liquid phase via shearing. Benjamin [26] extended the work of Miles by theoretically analyzing normal and tangential stress variations over the wavy interface during wavy flow generation. Many have attributed interfacial stability to surface wave generation induced by shearing gas flow parallel to the liquid surface. The Kelvin-Helmholtz instability modelled by Miles [27] was adequate to interpret the formation of surface waves in relative motional fluids with less viscosity. Katsis and Akylas [28] revealed that wavy generation sheared by high turbulent speeds was related to the Kelvin-Helmholtz instability mechanism, which predicted a minimum gas speed necessary to trigger wavy growth against the stabilizing effects of viscosity. Shtemler et al. [29] compared the wavy generation mechanisms of the Kelvin-Helmholtz instability and Miles theory by using an asymptotic model and found that wavy generation flow speed differed quantitatively in the two regimes. Young and Wolfe [30] considered linear stability in the generation of surface waves and reported that two unstable modes existed in the interaction between the waves and the air, or in the water for inviscid air parallel-shearing over water with gravity and capillarity. Furthermore, the characteristics of a wavy interface, like the evolution of wave amplitude, wave shape, and wave speed were also fascinating. Sutherland [31] found that the spectral growth of a wave train driven by parallel shear gas flow, characterized by power spectral density measurements, was a function of wave frequency. Wilson et al. [32] experimentally determined the relationship between surface wave amplitude attenuation or growth rate with air-flow shearing velocity. Jurman and McCready [33] derived a nonlinear wave equation using boundary-layer approximations to quantitatively describe surface waves induced by a turbulent gas flow shearing a horizontal liquid film. In addition, a reduced wave equation was derived for predicting periodic wave amplitudes and shapes. Ottens et al. [34] analyzed video images for accurate measurements of wave speed and length during gas–liquid flows. Frank [35, 36] used Navier-Stokes equations to analyze the evolution of the shape, amplitude, and speed of long nonlinear waves traveling on a liquid film that was sheared by a laminar gas flow. Lin et al. [37] developed an air-water couple model to initiate a direct simulation on wind-wave generation processes which consisted of linear and exponential growth stages at low wind speed. Moisy et al. [38] characterized an optical method of free-surface synthetic schlieren for the potential measurement of instantaneous wavy topography. Trifonov [39] theoretically analyzed the flooding of the wavy film in the context of counter-current gas shearing the liquid film between vertical plates. Tseluiko et al. [40] modified the Kuramoto-Sivashinsky equation by adding an additional term due to the presence turbulent gas in order to investigate the dynamics of nonlinear waves in counter-current gas-liquid film flowing under an inclined channel. Grare et al. [41] investigated the growth and dissipation of wing-forced, deep-water waves by obtaining the tangential stresses in the
water and air adjacent to the interface through the microphysical measurement techniques, in which the input of energy by wind shearing over water based on the measured tangential stresses was analyzed comparatively with the previous theory. Tian and Choi [42] numerically investigated on the evolution of deep-water waves under wind shearing and the breaking effects by introducing a wave prediction model based on pseudo-spectral method, and the numerical results were verified by the detailed experiments. Vellingiri et al. [43] theoretically found that waves propagating in a liquid film due to co-flowing turbulent gas shearing had increased speed by enhancing the gas and liquid flow rates. Zonta et al. [44] used direct numerical simulation to analyze the dynamics of interface waves, including initial generation process, amplitude and wavenumber spectra in the context of air shearing countercurrent water. Paquier et al. [45] used the free surface synthetic schlieren to accurately determine the spatio-temporal structure of the surface deformation that was for two regimes of wave generation induced by a turbulent wind shearing over a viscous liquid surface. They also revealed that different types of surface deformation, such as disorganized wrinkles and regular gravity-capillary waves, were dependent on the wind speed, which also determined the growth of deformations. Finally, Paquier et al. [46] examined the effects of liquid viscosity on wave thresholds and the corresponding characteristic amplitudes in different generation regimes. Buckley and Veron [47] presented the novel experimental techniques of PIV and LIF to detect the movement of surface waves and measure the airflow velocity field on the side of air-water interface. Kofman et al. [48] carried out a comprehensive investigation on the characteristics including shape, amplitude and velocity of waves on a falling liquid film sheared by a turbulent counter-current gas flow. However, Very few studies have examined wavy flow driven by normal impingement instead of shearing. An experimental setup was developed in our laboratory to study the mechanism and frequency of surface waves induced by the normal wall-jet impingement on a vertical gas-liquid interface via a gas-jet penetration into a liquid sheet. We hope to have a good understand for behavior of the liquid film lateral surface under the sealing effect of a gas wall-jet in immersion lithography through this experimental setup. It was found in our experiments that the surface wave generation was the notable behavior of the vertical gas-liquid interface. Due to the normal wall-jet impingement, the generating mechanism was considered to be different from the traditional surface wave caused by the shearing effect. And the generation frequency of surface waves was the dominant factor of reflecting the stability of the vertical gas-liquid interface, which was confirmed as a key role to trigger the Rayleigh instability of gas-liquid interface [49]. Thus, these were attracted us to study in depth the generating mechanism and frequency characteristics of surface wave at a vertical gas-liquid interface under the normal impingement of wall-jet via a gas-jet penetration into a liquid sheet. In this paper, the generating mechanism and frequency of surface waves experimentally generated at a vertical gas–liquid interface by normal wall-jet impingement were characterized for a gas-jet penetration into a liquid sheet. Interfacial oscillations accompanied by surface wave generation and propagation were captured with a high-speed camera. Periodic oscillations of the vertical gas–liquid interface were revealed and regarded as external energy transferred from the wall-jet to surface waves. In addition, wave generation was considered as a way of releasing the maximum interfacial elastic energy. The characteristic amplitude was defined by normalizing the amplitude of the radial distance between the waist and neck of the wavy interface, and was used to reveal the periodic deformation of the wavy interface during dynamic oscillations. The characteristic wavelength was defined as the vertical distance between the waist and neck of the wavy interface. It was independent of the penetration parameters and was always about one third of the liquid sheet thickness. The effects on the
frequency of surface wave generation by penetration parameters such as nozzle diameter, gas flow rate, jet height, and liquid sheet thickness were analyzed. To normalize that frequency with all those parameters, a combined dimensionless parameter of the Froude number and the ratio jet-height/nozzle-diameter was used. The frequency could be expressed by a natural exponential function with the dimensionless normalization parameter via fitting. Overall, this work will be helpful concerning interfacial motion in industrial applications.
Nomenclature A
amplitude (mm)
N2
fitting coefficient of the power function (Hz)
A
*
characteristic amplitude
n
times of repeating each experimental case
△A
amplitude variation (mm)
Q
flow rate (l/min)
Amax
maximum amplitude (mm)
V0
mean jet velocity at outlet of the nozzle (m/s)
Anin
minimum amplitude (mm)
Vs
sound speed (m/s)
d
nozzle diameter (mm)
R
correlation coefficient
F
surface wave generation frequency (Hz)
Rn
distance between jetting axial and neck of gas–liquid interface (mm)
Fexp
experimental value of generation
Rw
frequency (Hz) Fcal
calculation value of generation frequency
distance between jetting axial and waist of gas–liquid interface (mm)
Re
Reynolds number
(Hz) g
gravity acceleration (m/s2)
Ma
Mach number
H
liquid sheet thickness (mm)
Fr
Froude number
Hc
critical liquid sheet thickness (mm)
ρG
density of gas (kg/m3)
h
jet height (mm)
ρL
density of liquid (kg/m3)
L1
fitting coefficient of the natural
??
kinetic viscosity of air (m2/s)
γ
gas–liquid surface tension (N/m)
θs
static contact angle of the solid substrate
exponential function (Hz) L2
fitting coefficient of the power function (Hz)
M1
fitting coefficient of the natural exponential function
M2
fitting coefficient of the power function
N1
fitting coefficient of the natural exponential function (Hz)
(°) λ
characteristic wavelength (mm)
2. Experimental setup The experimental setup (illustrated in Fig. 1) was developed to study the surface wave generation via a gas-jet penetration into a liquid sheet under various penetration parameters, including nozzle diameters, jet heights, gas flow rates and liquid sheet thicknesses. In the penetration process, the air (density ρG=1.2 kg/m3) provided by the compressor passed through a mass-flow controller and jetted out of a flat tipped needle, finally penetrated vertically into a liquid sheet of purified water (density ρL=998 kg/m3) in a rectangular glass container (length of 250 mm, width of 250mm and height of 50 mm). The gas-jet with suitable penetration parameters completely ruptured the liquid sheet, leaving a dry spot on the solid substrate. It was then deflected horizontally, forming a wall-jet that produced a steady air cavity bounded by the vertical gas–liquid interface and the solid substrate [50].The vertical gas–liquid interface impinged by the normal wall-jet would exhibit dynamic oscillations accompanied surface wave generation and propagation.
Fig. 1. Schematic of experimental setup. Table 1 Experiment parameters. Nozzle diameter, d(mm)
Jet height, h(mm)
Flow rate, Q(l/min)
Re
Ma
0.40
13.2-18.0 (33d-45d)
0.59-0.75
2115-2688
0.23-0.29
0.51
9.18-16.83 (18d-33d)
0.92-1.20
2586-3373
0.22-0.28
0.67
10.05-18.09 (15d-27d)
1.30-1.80
2782-3852
0.18-0.25
0.84
10.08-17.64 (12d-21d)
1.40-2.60
2390-4438
0.12-0.23
Standard flat-tipped needles with various d (see Table 1)were mounted on a movable platform such that jet height h was adjusted vertically from the nozzle to the solid substrate of the container over 20 mm range with an accuracy of 0.01 mm. If the jet height set to a lower value in experiments, the vertical gas–liquid interface cannot be stably existed as well as accompanied by a few droplets splattering at the interface [49]. Thus, a serial of suitable jet heights were qualified to generate the stable vertical gas–liquid interface shown in Table 1. The mass flow controller (Alicat-10-KM1178) regulated flow rate with a reading accuracy of 0.8% and 0.2% full scale accuracy. The interfacial dynamics were captured at a speed of 1000 frame/s with a high-speed video camera (Phantom M320S) fitted with a 50-mm Nikon lens. A high-intensity lamp with a thin paper diffuser was used for non-flickering backlighting. The liquid sheet in the big container was deemed infinite because the container width was more than 200 times d. Thus, reflected waves from the container walls were minimized [51]. To judge surface waves generating at the stable vertical gas–liquid interface obviously, the experimental range of the flow rate was designated and accordingly the ranges of two dimensionless parameters Reynolds number Re V0 d and Mach number Ma V0 Vs were determined to estimate the status of jetting flow (see Table 1). Here, V0 is the mean jet velocity at outlet of the nozzle which can be yield from V0 4Q d . The kinetic viscosity of air ??=1.48×10-5 m2/s and the speed of 2
sound Vs=344 m/s. To produce stable interfacial motion, the thickness of the liquid sheet was greater than the critical thickness [52]: H c 2 L g sin s 2 . Gravity was the main factor in the penetration process, 12
and the ruptured liquid sheet could be spontaneously healed and restored to a smooth sheet as soon as the gas-jet was withdrawn. Thus, Hc was about 1.1 mm for a gas–liquid surface tension γ=0.073 N/m, g=9.8 m/s2 , and a static contact angle θs=24º of the quartz glass substrate, which was measured with a theta optical tension meter (T200-Auto1B). The range of the liquid sheet thickness was 3.0–5.0 mm, such that the thinnest value was greater than Hc, and the thickness could be ruptured on the substrate by the incompressible gas-jet. Each experimental case was under the suitable penetration parameters, including one gas flow rate, one nozzle diameter, one jet height and one liquid sheet thickness. The mass flow controller was calibrated before and after all cases for different nozzle diameter to ensure its accuracy. The jet height was confirmed by repeating the measurement after each case. The liquid sheet thickness was measured using the nozzle to locate the horizontal liquid surface and the solid substrate, respectively [53]. The measurement of this thickness was repeated for three times before every experiment. Experiments were carried out at the temperature of 20±1℃ and the environment pressure of 101.3kPa. Each experimental case was completed in a short time (in one minute), thus the liquid sheet thickness was deemed unchanged due to the evaporation of liquid sheet can be neglected. Uncertainties of penetration parameters were displayed in Table 2. Table 2 Uncertainties of penetration parameters.
Parameter
Uncertainty
Nozzle diameter, d(mm)
±0.01mm
Flow rate, Q(l/min)
0.2% full scale and 0.8% reading value
Jet height, h(mm)
±0.01mm
Liquid sheet thickness, H(mm)
±0.02mm
To study the behavior of surface waves by capturing the dynamic oscillation of the vertical gas–liquid interface, each case was repeated for six times to ensure about the repeatability and reproducibility of experimental data. In order obtain reliable experimental results (especially the generation frequency which discussed below), every case the camera started to capture until the vertical gas–liquid interface oscillated steadily at a fixed flow rate. After every experiment, all penetration parameters would be checked before the next one to confirm the repeatability of the experiments.
3. Results and discussion The gas-jet with suitable penetration parameters completely ruptured the liquid sheet, leaving a dry spot on the substrate. It was then deflected horizontally, forming a wall-jet that pushed the gas–liquid interface a specific radial distance. Meanwhile, a steady air cavity was produced by the surrounding vertical gas–liquid interface and the substrate [50]. Above the substrate, the vertical gas–liquid interface would exhibit dynamic oscillatory behavior that accompanied surface wave generation and propagation. 3.1 Mechanism of the surface wave generation
Fig. 2. Series of images of the dynamic vertical gas–liquid interface for nozzle diameter d=0.51 mm, flow rate Q=0.92 l/min, jet height h=12.24 mm, and liquid sheet thickness H=3.0 mm.
In Fig. 2, a series of high-speed video images are shown of vertical gas–liquid interface induced by the normal wall-jet impingement. The clipped images are of half the vertical gas–liquid interface that depicted interfacial motion. In Figs. 2(a-e), the half cross-section of the interface was S-shaped and exhibited continuous dynamic oscillations along the radial direction under wall-jet impingement. Accompanied by an interfacial oscillation, a surface wave was generated on the upper side of the vertical gas–liquid interface [yellow ellipse in Fig. 2(b)]; then it radially propagated away [yellow ellipse in Fig. 2(c)]. Another surface wave was then generated and propagated away for another interface oscillation [see Figs. 2(d,e)]. The wave generation and propagation would be endless for the continuous interface oscillations. In the generation process, this S-shaped interface radially oscillated like a spring. Wave generation always appeared when the S-shaped interface had a slim profile [Figs. 2(b,d)] in every oscillation. That is, the waist and neck of the S-shaped interface [indicated respectively by the red and blue dots in Fig. 2(a)] approached the nearest radial position here. This could be explained from the aspect of energy transfer. When the waist expanded and the neck shrunk [Fig. 2(b) to Fig. 2(c)], the S-shape became fatter and the deformation of the S-shape interface became aggravated, resulting in intensified interfacial elastic energy transferred from the wall-jet to the maximum at the farthest radial location of
the waist and the neck. Conversely, when the waist shrunk and the neck expanded [Fig. 2(c) to Fig. 2(d)], the interfacial elastic energy was released by smoothing the interface profile towards the minimum at the nearest radial location of the waist and the neck. Accordingly, the released interfacial elastic energy increased and reached a maximum at the nearest site where surface wave generated. Hence, the surface wave was the product when the maximum interfacial elastic energy was released. Generally, in the oscillatory vertical gas–liquid interface, surface waves could be generated by releasing the maximum interfacial elastic energy when the waist was shrunk to the minimum and the neck expanded to the maximum radial location.
Fig. 3. Characteristic amplitude A* during several interfacial oscillations with d=0.51 mm, h=12.24 mm, H=3.0 mm, and various Q in (a); Q= 0.99 l/min, H=3.0 mm, and various h in (b); Q=1.06 l/min, h=15.3 mm, and various H in (c).
The waist and neck of the S-shaped interface can be deemed as the trough and crest of the wavy interface, respectively. Thus, the radial distance between them [indicated in Fig. 2(a)] was defined as the amplitude A:
A Rw Rn .
(1)
Where Rw was the radial distance between the waist and jetting axial, Rn was the radial distance between the neck and jetting axial. Besides, the characteristic amplitude A* was adopted to evaluate the deformation of the vertical gas–liquid interface during dynamic oscillations:
A* A / Amax .
(2)
Where, Amax was the maximum amplitude during dynamic oscillations. The time dependence of the characteristic amplitude A* during several interface oscillations was plotted in Fig. 3. It exhibited a periodic variation for a fixed Q in Fig. 3(a), a fixed h in Fig. 3(b) or a fixed H in Fig. 3(c). In addition, the amplitude variation △A between the minimum and the maximum of the amplitude was used to evaluate the interfacial oscillatory intensity produced by the wall-jet impingement:
A Amax Amin
.
(3)
Where, Amin was the minimum amplitude during dynamic oscillations. In Figs. 4(a-c), △A for several interfacial oscillations was plotted as a function of Q, h, and H, for a 0.51-mm nozzle diameter. The average △A decreased with increased Q [Fig. 4(a)], h [Fig. 4(b)], and H [Fig. 4(c)]. This implied that the vertical gas–liquid interface underwent significant oscillations at a lower flow rate, a lower jet height, and a thinner liquid sheet because of the large interfacial elastic energy transferred from the wall-jet.
Fig. 4. Characteristic amplitude variation △A as a function of Q, h, and H, with a 0.51-mm nozzle diameter. h=12.24 mm, H=3.0 mm in (a); Q=0.99 l/min, H=3.0 mm in (b); Q=1.06 l/min, h=15.3 mm in (c).
The characteristic wavelength was defined as the vertical distance between trough and crest (waist and neck) of the wavy interface [indicated in Fig. 2(a)]. It was normalized by H in Fig. 5 as a function of the penetration parameters. The normalized characteristic wavelength was independent of the flow rate, jet height, and liquid sheet thickness, and remained a constant 0.336 (indicated by the red dashed lines). That indicated that the positions of the trough and crest at the vertical gas–liquid interface induced by the normal wall-jet impingement could be determined, and that the distance between them occupied about one third of the liquid sheet thickness.
Fig. 5. Normalized characteristic wavelength as a function of Q, h, and H, with a 0.51-mm nozzle diameter. h=12.24 mm, H=3.0 mm in (a); Q=1.06 l/min, H=3.0 mm in (b); Q=1.06 l/min, h=15.3 mm in (c).
3.2 Frequency of surface wave generation Along with the periodic oscillations, surface waves would be periodically generated at the vertical gas–liquid interface and then propagate away. Fig. 6 displays images of surface wave generation and propagation for various Q, with d=0.51 mm, h=12.24 mm, and H=3.0 mm. In the first row, the surface wave generated at the upper side of the vertical gas–liquid interface [yellow ellipses in Figs. 6 (a2, a4)] propagated away [yellow ellipses in Figs. 6(a3, a5)]. The surface wave generating period was determined by the time difference when adjacent waves moved through the same position, as in Figs. 6(a2,a4). Similar observations were recorded in the second and third rows. The generating period under different flow rates (different rows in Fig. 6), revealed that a larger Q produced a longer period. According to Ref. [49], the surface wave generation frequency F, defined by the number of surface waves generated per second at the upper side of the vertical gas–liquid interface, can be used to characterize the interfacial oscillation caused by the wall-jet impingement. More importantly, it was associated with the Rayleigh instability that can be triggered at high frequencies by observing droplets splattering at the gas–liquid interface. The position of the generated surface wave was at the upper side of the vertical gas–liquid interface, very close to the corner of the vertical and horizontal gas–liquid interfaces [Figs. 2(b,d)]. This was where the Rayleigh instability would easily occur with liquid finger shooting and then finally evolve to droplet splattering (see Fig. 7 in Ref. [49]). The liquid finger that readily appeared at high generation frequencies may be attributed to selective amplification of the
generated surface wave. Generation frequency F was determined by the reciprocal of the generating period and was plotted in Fig. 7 as a function of Q for a 0.51-mm nozzle diameter. In Fig. 7(b), a higher generation frequency resulted from a lower Q. According to Ref. [50], with a lower Q and other penetration parameters fixed, the closest radial location of the vertical gas–liquid interface from the jetting axis could be determined by the wall-jet. Thus, a shorter distance of the wall-jet would maintain a stronger impingement on the vertical gas–liquid interface. With the stronger wall-jet, the gas–liquid interface would be more violent, producing a higher surface wave generation frequency. Inversely, farther radial position of the vertical gas–liquid interface would have milder interfacial oscillations with a lower generation frequency. In Fig. 7, analogous events were detected for other nozzle diameters.
Fig. 6. Images of surface wave generation and propagation for d= 0.51 mm, h=12.24 mm, H=3.0 mm under various Q (the first row Q=0.92 l/min, the second row Q=1.06 l/min and the third row Q=1.20 l/min).
Fig. 7. Surface wave generation frequency F as a function of Q.
In Fig. 8, a more chaotic gas–liquid interface with a shorter surface wave generating period was observed in the first row, with a lower h=9.18 mm. Thus, a higher generation frequency would be produced by a wall-jet with a lower jet height. The generation frequency was plotted as a function of normalized jet height in Fig. 9. The frequency gradually increased with decreasing h for each nozzle diameter. A similar explanation could be given to the effect of h on the surface wave generation frequency. The closer vertical gas–liquid interface was formed by a wall-jet with a lower h, leading to more chaotic oscillations at a higher surface wave generation frequency.
Fig. 8. Images of surface wave generation and propagation with d= 0.51 mm, Q=0.92 l/min, and H=3.0 mm for various h (first row h=9.18 mm, second row h=12.24 mm, third row h=15.3 mm).
Fig. 9. Surface wave generation frequency F as a function of the normalized jet height h/d.
In Fig. 10, a relative closer vertical gas–liquid interface could be found in the third row with the liquid film thickness of H=5.0mm. However, the S-shaped interface was not evidently exhibited in the first row with the liquid film thickness of H=3.0mm due to the radial position of the neck was sometimes farther than the waist. This also confirmed the negative value of the characteristic amplitude A* in Fig. 2(c) with H=3.0mm. As illustrated in Fig. 10, a shorter generating period was observed when H was increased to 5.0 mm (the third row). The dependence of the generation frequency as a function of H was plotted in Fig. 11. Higher frequencies occurred in thicker liquid sheets. With the other penetration parameters fixed, a closer vertical gas–liquid interface was obtained for a thicker liquid sheet, resulting in stronger wall-jet impingement on the gas–liquid interface. Thus, a higher generation frequency induced by a stronger impingement can be observed in a thicker liquid sheet.
Fig.10. Images of surface wave generation and propagation for d= 0.51 mm, Q=1.20 l/min, h=15.3 mm, and various H (first row H=3.0 mm, second row H=4.0 mm, third row H=5.0 mm)
Fig. 11. Surface wave generation frequency F for various H.
3.3 Fitting analysis of generation frequency The surface wave generation frequency F was affected by various penetration parameters, and, given a normalized correlation between the frequency and various penetration parameters was significant for scientific research and industrial applications. As we known, the surface tension was an important factor in gas–liquid interfacial behavior. However, in our penetration process, the gas-jet, i.e. wall-jet was served as the driving role to generate the surface waves at the vertical gas–liquid interface.
In addition, the gravity and surface tension were the resilience to restore the wavy interface to the smooth one. Comparing with the surface tension, the gravity of the liquid sheet, which presented as the penetration parameter of liquid sheet thickness, played a dominant role to affect the wall-jet development and impingement on the vertical gas–liquid interface, and accordingly the surface wave generation frequency. The dimensionless Froude number Fr was adopted to normalize penetration parameters, in which the important effects of the gas-jet impingement and gravity of liquid sheet on generation frequency F were considered. It is defined as:
Fr
GV02 L gH
.
(4)
Furthermore, the normalized jet height h/d was used for considering the effect of jet height h on the generation frequency F. A combined dimension parameter of Fr multiplying h/d was introduced to normalize the relationship between of F and all penetration parameters:
V2 h h F f G 0 f Fr . d L gH d
(5)
Abundant of experiments were conducted to get the surface wave generation frequency F under various penetrating parameters. The dependence of F on the penetration parameters and the results of F with (Fr·h/d) were arranged and plotted in Fig. 12. As shown in Fig. 12, F generally decreased by increasing (Fr·h/d). To depict the decrease in F with (Fr·h/d), two familiar curves were used for data fitting (red and blue curves in Fig. 12). The curves were the natural exponential function Eq. (6a) and the power function Eq. (6b), respectively: Fr
F L1 e
h d
M1
N1 ,
h F L2 Fr d
M2
(6a)
N2 .
(6b)
The fitting coefficients were L1=5284.7 Hz, L2=-178.7 Hz, M1=-18.48, M2=0.35, N1=71.5 Hz, and N2=970.4 Hz. The correlation coefficient R=0.64 for the natural exponential curve and R=0.60 for the power curve; they were compared to get the best fitting curve. The natural exponential curve with a larger R was thus better to present the relationship between F and (Fr·h/d).
Fig. 12. Surface wave generation frequency F plotted as a function of the combined dimensionless parameter of the Froude number Fr and the ratio of jet height to nozzle diameter h/d, for d=0.51 mm (fitted by red and blue curves).
Every absolute advantage deviation (A.A.D%) of each experimental case was calculated according to the formula:
A. A.D%
Fexp Fcal 1 100 . n Fexp
(7)
Where n is the times of repeating each experimental case. Each experimental case was repeated six times to ensure about the repeatability and reproducibility of experimental data (i.e. n=6). Fexp is the experimental value of generation frequency and Fcal is the calculation value of generation frequency through the fitting function (Eq. (6a)). As can be seen in Fig. 12, two red dashed curves with absolute advantage deviation of 20 percent (A.A.D%=20) departing from the natural exponential curve were used to evaluate the experimental results. It can be found that nearly 85% experimental data collapsed into the region between two dashed curves. Thus, experimental data was reliable and also was shown in agreement with the natural exponential curve.
Fig. 13. Surface wave generation frequency for different nozzle diameters (fitted by the natural exponential curves).
The decreasing natural exponential relationship between F and (Fr·h/d) was also fit for other nozzle diameters, as plotted in Fig. 13. It seemed that better fits were for larger 0.84-mm nozzle diameters [Fig. 13(d)], relative to those for smaller 0.41-mm diameters [Fig. 13(a)]. Also, two dashed curves with absolute advantage deviation of 20 percent (A.A.D%=20) departing from the natural exponential curve were depicted to evaluate the experimental results. The percent of experimental data dropped into the region between two dashed curves were about 76% for d=0.40mm, 85% for d=0.51mm, 94% for d=0.67mm and 92% for d=0.84mm, respectively. Although the ranges of the penetration parameters were different, a relatively close radial position of the vertical gas–liquid interface frequently occurred for small nozzle diameters. This was sensitive to observational precision of F for the intense wall-jet impingement on a close interface, resulting in more scattered data that deviated significantly from the curve fit in Fig. 13(a). 3.4 Experimental uncertainty Due to the turbulent gas jetting out the nozzle, the turbulence of the wall-jet was unavoidable to affect the wavy behavior of the vertical gas–liquid interface. In addition, the compressibility of gas in the cavity bounded by the vertical gas–liquid interface and the solid substrate would also be considered. Besides, in the experimental result extraction, the surface wave generation frequency depended on the artificial observation was sensitive to the images of wavy interface especially in the situation of the violent interfacial oscillation. These were the main uncertainties in experiments affecting the experimental results. And for the future study, these effects should be considered and improved to obtain the good results.
4. Conclusions The generating mechanism and frequency characteristics of surface waves at a vertical gas–liquid interface were experimentally examined for gas-jet penetration into a liquid sheet. Surface waves were generated when the waist of the vertical gas–liquid interface shrunk to its minimum and its neck expanded to its maximum radial location. In addition, an analysis on energy transfer revealed that surface wave generation was a means of releasing the maximum interfacial elastic energy. The time dependence of the characteristic amplitude revealed the periodic deformation of the vertical gas–liquid interface during dynamic oscillations, while the characteristic wavelength was about one third of the liquid sheet thickness and was independent of the penetration parameters. Because of the stronger wall-jet impingement on the closer interface, more intense interfacial oscillation accompanying higher surface wave generation frequency was produced at a lower flow rate, a lower jet height, or a thicker liquid sheet, when fixing other penetration parameters. Finally, from the fitting analysis, the frequency of surface waves generation could be expressed by a natural exponential function of the combined dimensionless normalization parameter of the Froude number Fr and the ratio of jet height to nozzle diameter h/d. In conclusion, this work should be helpful for understanding interfacial flow in industrial applications.
Acknowledgements We are grateful to the Natural Science Foundation of Zhejiang Province (No. LQ17A020002), the Joint Project from National Natural Science Foundation of China and Liaoning Province (No. U1608258), the Young Researchers Foundation of Zhejiang Provincial Top Key Academic Discipline of Mechanical Engineering of Zhejiang Sci-Tech University (No. ZSTUME02B03) and the Science Foundation of Zhejiang Sci-Tech University (No. 16022090-Y). We also thank Alan Burns, PhD, for improving the English writing.
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Highlights Surface wave generated by normal wall-jet was experimentally characterized. Surface wave was a means to release maximum interfacial elastic energy. A combined dimensionless parameter was used to normalize all penetration parameters. Generation frequency exponentially decreased with the normalization parameter.
S0894-1777(18)30073-6 https://doi.org/10.1016/j.expthermflusci.2018.03.035 ETF 9432
To appear in:
Experimental Thermal and Fluid Science
Received Date: Revised Date: Accepted Date:
26 January 2018 27 March 2018 28 March 2018
Please cite this article as: Q. Liu, Z. Lin, X-p. Chen, Z-c. Zhu, B-l. Cui, Surface wave generation via a gas-jet penetration into a liquid sheet, Experimental Thermal and Fluid Science (2018), doi: https://doi.org/10.1016/ j.expthermflusci.2018.03.035
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Surface wave generation via a gas-jet penetration into a liquid sheet
Qi Liu, Zhe Lin*, Xiao-ping Chen, Zu-chao Zhu, Bao-ling Cui Key Laboratory of Fluid Transmission Technology of Zhejiang Province, Zhejiang Sci-Tech University, Hangzhou, 310018, China
Abstract The mechanism and frequency of surface wave generated by normal wall-jet impingement at a vertical gas–liquid interface were experimentally characterized for a gas-jet penetrating into a liquid sheet. The oscillatory gas–liquid interface accompanied by surface wave generation and propagation were captured with a high-speed camera. Periodic oscillations of the vertical gas–liquid interface were revealed in the context of energy transfer from the wall-jet to surface waves. In addition, wave generation was considered as a means of releasing the maximum interfacial elastic energy. The radial distance between the waist and neck of the wavy interface was defined as the amplitude and the characteristic amplitude was adopted to reveal periodic deformation of the wavy interface during dynamic oscillations. Furthermore, the characteristic wavelength, i.e. vertical distance between waist and neck, was about one third of the liquid sheet thickness, irrespective of penetration parameters, such as gas flow rate, jet height, and liquid sheet thickness. The effects of those parameters on the surface wave generation frequency were analyzed. A combined dimensionless parameter of Froude number and the ratio of jet height to nozzle diameter was used to normalize the surface wave generation frequency with all the penetration parameters. It was found that the frequency could be expressed by a natural exponential function with the combined dimensionless parameter for normalization via fitting analysis. This work should be helpful to characterize interfacial motion in actual industrial applications. Keywords: Surface wave, Wall-jet, Liquid sheet, Oscillation, Generation frequency
1. Introduction The behavior of gas–liquid interface in two-phase flow is an important issue in many scientific fields. One well-known is the bubble formation by desorbing gas in solution when increasing the heat flux. Due to bubble formation, the interaction between gas and liquid became greatly stronger and accordingly intensified the convection effect in solution, which served as the major mechanism of enhancing heat transfer [1-3]. And, the physical characteristics of nucleate and rate of bubble formation were often focused on to reveal its relationship with the heat flux in diverse liquids [4-6]. When increasing the heat flux during the flow boiling in nanofluids, thermal performances of that were also significantly intensified due to the fast bubble formation and agitation in nucleate boiling region [7-9]. In addition, the performance of bubble formation can be improved by the deposition of the nano-particles in nanofiluds [10]. However, the behavior of bubbles formed in nanofluids can negatively affect the stability of nanofluids [11]. The effect of mixed convection of nanofluid on heat
*
Corresponding author. Tel.: +86 0571-86843348 E-mail address: [email protected]
transfer was explained by the Lattice Boltzmann method [12, 13], and physical and thermal characteristics were experimentally studied under the effect of solid volume fraction of various nanofluids [14-17]. With the good thermal performance, nanofluids were investigated as the potential coolant for actual applications [18, 19]. Besides, another classical behavior of gas–liquid interface is surface waves. As an interfacial flow, the wavy motion was also related to the heat transfer in fluids as well as the stability of the gas–liquid interface [20-22]. Surface waves generated by a gas at a gas–liquid interface, especially shearing effects, have been widely studied by many researchers in fluid mechanics. Phillips [23] proposed a resonance mechanism for surface wave generation in which pressure fluctuations initiated waves by selective amplification. Miles [24, 25] used the Orr-Sommerfeld equation to study surface wave generation in terms of momentum transfer from the gas phase to the liquid phase via shearing. Benjamin [26] extended the work of Miles by theoretically analyzing normal and tangential stress variations over the wavy interface during wavy flow generation. Many have attributed interfacial stability to surface wave generation induced by shearing gas flow parallel to the liquid surface. The Kelvin-Helmholtz instability modelled by Miles [27] was adequate to interpret the formation of surface waves in relative motional fluids with less viscosity. Katsis and Akylas [28] revealed that wavy generation sheared by high turbulent speeds was related to the Kelvin-Helmholtz instability mechanism, which predicted a minimum gas speed necessary to trigger wavy growth against the stabilizing effects of viscosity. Shtemler et al. [29] compared the wavy generation mechanisms of the Kelvin-Helmholtz instability and Miles theory by using an asymptotic model and found that wavy generation flow speed differed quantitatively in the two regimes. Young and Wolfe [30] considered linear stability in the generation of surface waves and reported that two unstable modes existed in the interaction between the waves and the air, or in the water for inviscid air parallel-shearing over water with gravity and capillarity. Furthermore, the characteristics of a wavy interface, like the evolution of wave amplitude, wave shape, and wave speed were also fascinating. Sutherland [31] found that the spectral growth of a wave train driven by parallel shear gas flow, characterized by power spectral density measurements, was a function of wave frequency. Wilson et al. [32] experimentally determined the relationship between surface wave amplitude attenuation or growth rate with air-flow shearing velocity. Jurman and McCready [33] derived a nonlinear wave equation using boundary-layer approximations to quantitatively describe surface waves induced by a turbulent gas flow shearing a horizontal liquid film. In addition, a reduced wave equation was derived for predicting periodic wave amplitudes and shapes. Ottens et al. [34] analyzed video images for accurate measurements of wave speed and length during gas–liquid flows. Frank [35, 36] used Navier-Stokes equations to analyze the evolution of the shape, amplitude, and speed of long nonlinear waves traveling on a liquid film that was sheared by a laminar gas flow. Lin et al. [37] developed an air-water couple model to initiate a direct simulation on wind-wave generation processes which consisted of linear and exponential growth stages at low wind speed. Moisy et al. [38] characterized an optical method of free-surface synthetic schlieren for the potential measurement of instantaneous wavy topography. Trifonov [39] theoretically analyzed the flooding of the wavy film in the context of counter-current gas shearing the liquid film between vertical plates. Tseluiko et al. [40] modified the Kuramoto-Sivashinsky equation by adding an additional term due to the presence turbulent gas in order to investigate the dynamics of nonlinear waves in counter-current gas-liquid film flowing under an inclined channel. Grare et al. [41] investigated the growth and dissipation of wing-forced, deep-water waves by obtaining the tangential stresses in the
water and air adjacent to the interface through the microphysical measurement techniques, in which the input of energy by wind shearing over water based on the measured tangential stresses was analyzed comparatively with the previous theory. Tian and Choi [42] numerically investigated on the evolution of deep-water waves under wind shearing and the breaking effects by introducing a wave prediction model based on pseudo-spectral method, and the numerical results were verified by the detailed experiments. Vellingiri et al. [43] theoretically found that waves propagating in a liquid film due to co-flowing turbulent gas shearing had increased speed by enhancing the gas and liquid flow rates. Zonta et al. [44] used direct numerical simulation to analyze the dynamics of interface waves, including initial generation process, amplitude and wavenumber spectra in the context of air shearing countercurrent water. Paquier et al. [45] used the free surface synthetic schlieren to accurately determine the spatio-temporal structure of the surface deformation that was for two regimes of wave generation induced by a turbulent wind shearing over a viscous liquid surface. They also revealed that different types of surface deformation, such as disorganized wrinkles and regular gravity-capillary waves, were dependent on the wind speed, which also determined the growth of deformations. Finally, Paquier et al. [46] examined the effects of liquid viscosity on wave thresholds and the corresponding characteristic amplitudes in different generation regimes. Buckley and Veron [47] presented the novel experimental techniques of PIV and LIF to detect the movement of surface waves and measure the airflow velocity field on the side of air-water interface. Kofman et al. [48] carried out a comprehensive investigation on the characteristics including shape, amplitude and velocity of waves on a falling liquid film sheared by a turbulent counter-current gas flow. However, Very few studies have examined wavy flow driven by normal impingement instead of shearing. An experimental setup was developed in our laboratory to study the mechanism and frequency of surface waves induced by the normal wall-jet impingement on a vertical gas-liquid interface via a gas-jet penetration into a liquid sheet. We hope to have a good understand for behavior of the liquid film lateral surface under the sealing effect of a gas wall-jet in immersion lithography through this experimental setup. It was found in our experiments that the surface wave generation was the notable behavior of the vertical gas-liquid interface. Due to the normal wall-jet impingement, the generating mechanism was considered to be different from the traditional surface wave caused by the shearing effect. And the generation frequency of surface waves was the dominant factor of reflecting the stability of the vertical gas-liquid interface, which was confirmed as a key role to trigger the Rayleigh instability of gas-liquid interface [49]. Thus, these were attracted us to study in depth the generating mechanism and frequency characteristics of surface wave at a vertical gas-liquid interface under the normal impingement of wall-jet via a gas-jet penetration into a liquid sheet. In this paper, the generating mechanism and frequency of surface waves experimentally generated at a vertical gas–liquid interface by normal wall-jet impingement were characterized for a gas-jet penetration into a liquid sheet. Interfacial oscillations accompanied by surface wave generation and propagation were captured with a high-speed camera. Periodic oscillations of the vertical gas–liquid interface were revealed and regarded as external energy transferred from the wall-jet to surface waves. In addition, wave generation was considered as a way of releasing the maximum interfacial elastic energy. The characteristic amplitude was defined by normalizing the amplitude of the radial distance between the waist and neck of the wavy interface, and was used to reveal the periodic deformation of the wavy interface during dynamic oscillations. The characteristic wavelength was defined as the vertical distance between the waist and neck of the wavy interface. It was independent of the penetration parameters and was always about one third of the liquid sheet thickness. The effects on the
frequency of surface wave generation by penetration parameters such as nozzle diameter, gas flow rate, jet height, and liquid sheet thickness were analyzed. To normalize that frequency with all those parameters, a combined dimensionless parameter of the Froude number and the ratio jet-height/nozzle-diameter was used. The frequency could be expressed by a natural exponential function with the dimensionless normalization parameter via fitting. Overall, this work will be helpful concerning interfacial motion in industrial applications.
Nomenclature A
amplitude (mm)
N2
fitting coefficient of the power function (Hz)
A
*
characteristic amplitude
n
times of repeating each experimental case
△A
amplitude variation (mm)
Q
flow rate (l/min)
Amax
maximum amplitude (mm)
V0
mean jet velocity at outlet of the nozzle (m/s)
Anin
minimum amplitude (mm)
Vs
sound speed (m/s)
d
nozzle diameter (mm)
R
correlation coefficient
F
surface wave generation frequency (Hz)
Rn
distance between jetting axial and neck of gas–liquid interface (mm)
Fexp
experimental value of generation
Rw
frequency (Hz) Fcal
calculation value of generation frequency
distance between jetting axial and waist of gas–liquid interface (mm)
Re
Reynolds number
(Hz) g
gravity acceleration (m/s2)
Ma
Mach number
H
liquid sheet thickness (mm)
Fr
Froude number
Hc
critical liquid sheet thickness (mm)
ρG
density of gas (kg/m3)
h
jet height (mm)
ρL
density of liquid (kg/m3)
L1
fitting coefficient of the natural
??
kinetic viscosity of air (m2/s)
γ
gas–liquid surface tension (N/m)
θs
static contact angle of the solid substrate
exponential function (Hz) L2
fitting coefficient of the power function (Hz)
M1
fitting coefficient of the natural exponential function
M2
fitting coefficient of the power function
N1
fitting coefficient of the natural exponential function (Hz)
(°) λ
characteristic wavelength (mm)
2. Experimental setup The experimental setup (illustrated in Fig. 1) was developed to study the surface wave generation via a gas-jet penetration into a liquid sheet under various penetration parameters, including nozzle diameters, jet heights, gas flow rates and liquid sheet thicknesses. In the penetration process, the air (density ρG=1.2 kg/m3) provided by the compressor passed through a mass-flow controller and jetted out of a flat tipped needle, finally penetrated vertically into a liquid sheet of purified water (density ρL=998 kg/m3) in a rectangular glass container (length of 250 mm, width of 250mm and height of 50 mm). The gas-jet with suitable penetration parameters completely ruptured the liquid sheet, leaving a dry spot on the solid substrate. It was then deflected horizontally, forming a wall-jet that produced a steady air cavity bounded by the vertical gas–liquid interface and the solid substrate [50].The vertical gas–liquid interface impinged by the normal wall-jet would exhibit dynamic oscillations accompanied surface wave generation and propagation.
Fig. 1. Schematic of experimental setup. Table 1 Experiment parameters. Nozzle diameter, d(mm)
Jet height, h(mm)
Flow rate, Q(l/min)
Re
Ma
0.40
13.2-18.0 (33d-45d)
0.59-0.75
2115-2688
0.23-0.29
0.51
9.18-16.83 (18d-33d)
0.92-1.20
2586-3373
0.22-0.28
0.67
10.05-18.09 (15d-27d)
1.30-1.80
2782-3852
0.18-0.25
0.84
10.08-17.64 (12d-21d)
1.40-2.60
2390-4438
0.12-0.23
Standard flat-tipped needles with various d (see Table 1)were mounted on a movable platform such that jet height h was adjusted vertically from the nozzle to the solid substrate of the container over 20 mm range with an accuracy of 0.01 mm. If the jet height set to a lower value in experiments, the vertical gas–liquid interface cannot be stably existed as well as accompanied by a few droplets splattering at the interface [49]. Thus, a serial of suitable jet heights were qualified to generate the stable vertical gas–liquid interface shown in Table 1. The mass flow controller (Alicat-10-KM1178) regulated flow rate with a reading accuracy of 0.8% and 0.2% full scale accuracy. The interfacial dynamics were captured at a speed of 1000 frame/s with a high-speed video camera (Phantom M320S) fitted with a 50-mm Nikon lens. A high-intensity lamp with a thin paper diffuser was used for non-flickering backlighting. The liquid sheet in the big container was deemed infinite because the container width was more than 200 times d. Thus, reflected waves from the container walls were minimized [51]. To judge surface waves generating at the stable vertical gas–liquid interface obviously, the experimental range of the flow rate was designated and accordingly the ranges of two dimensionless parameters Reynolds number Re V0 d and Mach number Ma V0 Vs were determined to estimate the status of jetting flow (see Table 1). Here, V0 is the mean jet velocity at outlet of the nozzle which can be yield from V0 4Q d . The kinetic viscosity of air ??=1.48×10-5 m2/s and the speed of 2
sound Vs=344 m/s. To produce stable interfacial motion, the thickness of the liquid sheet was greater than the critical thickness [52]: H c 2 L g sin s 2 . Gravity was the main factor in the penetration process, 12
and the ruptured liquid sheet could be spontaneously healed and restored to a smooth sheet as soon as the gas-jet was withdrawn. Thus, Hc was about 1.1 mm for a gas–liquid surface tension γ=0.073 N/m, g=9.8 m/s2 , and a static contact angle θs=24º of the quartz glass substrate, which was measured with a theta optical tension meter (T200-Auto1B). The range of the liquid sheet thickness was 3.0–5.0 mm, such that the thinnest value was greater than Hc, and the thickness could be ruptured on the substrate by the incompressible gas-jet. Each experimental case was under the suitable penetration parameters, including one gas flow rate, one nozzle diameter, one jet height and one liquid sheet thickness. The mass flow controller was calibrated before and after all cases for different nozzle diameter to ensure its accuracy. The jet height was confirmed by repeating the measurement after each case. The liquid sheet thickness was measured using the nozzle to locate the horizontal liquid surface and the solid substrate, respectively [53]. The measurement of this thickness was repeated for three times before every experiment. Experiments were carried out at the temperature of 20±1℃ and the environment pressure of 101.3kPa. Each experimental case was completed in a short time (in one minute), thus the liquid sheet thickness was deemed unchanged due to the evaporation of liquid sheet can be neglected. Uncertainties of penetration parameters were displayed in Table 2. Table 2 Uncertainties of penetration parameters.
Parameter
Uncertainty
Nozzle diameter, d(mm)
±0.01mm
Flow rate, Q(l/min)
0.2% full scale and 0.8% reading value
Jet height, h(mm)
±0.01mm
Liquid sheet thickness, H(mm)
±0.02mm
To study the behavior of surface waves by capturing the dynamic oscillation of the vertical gas–liquid interface, each case was repeated for six times to ensure about the repeatability and reproducibility of experimental data. In order obtain reliable experimental results (especially the generation frequency which discussed below), every case the camera started to capture until the vertical gas–liquid interface oscillated steadily at a fixed flow rate. After every experiment, all penetration parameters would be checked before the next one to confirm the repeatability of the experiments.
3. Results and discussion The gas-jet with suitable penetration parameters completely ruptured the liquid sheet, leaving a dry spot on the substrate. It was then deflected horizontally, forming a wall-jet that pushed the gas–liquid interface a specific radial distance. Meanwhile, a steady air cavity was produced by the surrounding vertical gas–liquid interface and the substrate [50]. Above the substrate, the vertical gas–liquid interface would exhibit dynamic oscillatory behavior that accompanied surface wave generation and propagation. 3.1 Mechanism of the surface wave generation
Fig. 2. Series of images of the dynamic vertical gas–liquid interface for nozzle diameter d=0.51 mm, flow rate Q=0.92 l/min, jet height h=12.24 mm, and liquid sheet thickness H=3.0 mm.
In Fig. 2, a series of high-speed video images are shown of vertical gas–liquid interface induced by the normal wall-jet impingement. The clipped images are of half the vertical gas–liquid interface that depicted interfacial motion. In Figs. 2(a-e), the half cross-section of the interface was S-shaped and exhibited continuous dynamic oscillations along the radial direction under wall-jet impingement. Accompanied by an interfacial oscillation, a surface wave was generated on the upper side of the vertical gas–liquid interface [yellow ellipse in Fig. 2(b)]; then it radially propagated away [yellow ellipse in Fig. 2(c)]. Another surface wave was then generated and propagated away for another interface oscillation [see Figs. 2(d,e)]. The wave generation and propagation would be endless for the continuous interface oscillations. In the generation process, this S-shaped interface radially oscillated like a spring. Wave generation always appeared when the S-shaped interface had a slim profile [Figs. 2(b,d)] in every oscillation. That is, the waist and neck of the S-shaped interface [indicated respectively by the red and blue dots in Fig. 2(a)] approached the nearest radial position here. This could be explained from the aspect of energy transfer. When the waist expanded and the neck shrunk [Fig. 2(b) to Fig. 2(c)], the S-shape became fatter and the deformation of the S-shape interface became aggravated, resulting in intensified interfacial elastic energy transferred from the wall-jet to the maximum at the farthest radial location of
the waist and the neck. Conversely, when the waist shrunk and the neck expanded [Fig. 2(c) to Fig. 2(d)], the interfacial elastic energy was released by smoothing the interface profile towards the minimum at the nearest radial location of the waist and the neck. Accordingly, the released interfacial elastic energy increased and reached a maximum at the nearest site where surface wave generated. Hence, the surface wave was the product when the maximum interfacial elastic energy was released. Generally, in the oscillatory vertical gas–liquid interface, surface waves could be generated by releasing the maximum interfacial elastic energy when the waist was shrunk to the minimum and the neck expanded to the maximum radial location.
Fig. 3. Characteristic amplitude A* during several interfacial oscillations with d=0.51 mm, h=12.24 mm, H=3.0 mm, and various Q in (a); Q= 0.99 l/min, H=3.0 mm, and various h in (b); Q=1.06 l/min, h=15.3 mm, and various H in (c).
The waist and neck of the S-shaped interface can be deemed as the trough and crest of the wavy interface, respectively. Thus, the radial distance between them [indicated in Fig. 2(a)] was defined as the amplitude A:
A Rw Rn .
(1)
Where Rw was the radial distance between the waist and jetting axial, Rn was the radial distance between the neck and jetting axial. Besides, the characteristic amplitude A* was adopted to evaluate the deformation of the vertical gas–liquid interface during dynamic oscillations:
A* A / Amax .
(2)
Where, Amax was the maximum amplitude during dynamic oscillations. The time dependence of the characteristic amplitude A* during several interface oscillations was plotted in Fig. 3. It exhibited a periodic variation for a fixed Q in Fig. 3(a), a fixed h in Fig. 3(b) or a fixed H in Fig. 3(c). In addition, the amplitude variation △A between the minimum and the maximum of the amplitude was used to evaluate the interfacial oscillatory intensity produced by the wall-jet impingement:
A Amax Amin
.
(3)
Where, Amin was the minimum amplitude during dynamic oscillations. In Figs. 4(a-c), △A for several interfacial oscillations was plotted as a function of Q, h, and H, for a 0.51-mm nozzle diameter. The average △A decreased with increased Q [Fig. 4(a)], h [Fig. 4(b)], and H [Fig. 4(c)]. This implied that the vertical gas–liquid interface underwent significant oscillations at a lower flow rate, a lower jet height, and a thinner liquid sheet because of the large interfacial elastic energy transferred from the wall-jet.
Fig. 4. Characteristic amplitude variation △A as a function of Q, h, and H, with a 0.51-mm nozzle diameter. h=12.24 mm, H=3.0 mm in (a); Q=0.99 l/min, H=3.0 mm in (b); Q=1.06 l/min, h=15.3 mm in (c).
The characteristic wavelength was defined as the vertical distance between trough and crest (waist and neck) of the wavy interface [indicated in Fig. 2(a)]. It was normalized by H in Fig. 5 as a function of the penetration parameters. The normalized characteristic wavelength was independent of the flow rate, jet height, and liquid sheet thickness, and remained a constant 0.336 (indicated by the red dashed lines). That indicated that the positions of the trough and crest at the vertical gas–liquid interface induced by the normal wall-jet impingement could be determined, and that the distance between them occupied about one third of the liquid sheet thickness.
Fig. 5. Normalized characteristic wavelength as a function of Q, h, and H, with a 0.51-mm nozzle diameter. h=12.24 mm, H=3.0 mm in (a); Q=1.06 l/min, H=3.0 mm in (b); Q=1.06 l/min, h=15.3 mm in (c).
3.2 Frequency of surface wave generation Along with the periodic oscillations, surface waves would be periodically generated at the vertical gas–liquid interface and then propagate away. Fig. 6 displays images of surface wave generation and propagation for various Q, with d=0.51 mm, h=12.24 mm, and H=3.0 mm. In the first row, the surface wave generated at the upper side of the vertical gas–liquid interface [yellow ellipses in Figs. 6 (a2, a4)] propagated away [yellow ellipses in Figs. 6(a3, a5)]. The surface wave generating period was determined by the time difference when adjacent waves moved through the same position, as in Figs. 6(a2,a4). Similar observations were recorded in the second and third rows. The generating period under different flow rates (different rows in Fig. 6), revealed that a larger Q produced a longer period. According to Ref. [49], the surface wave generation frequency F, defined by the number of surface waves generated per second at the upper side of the vertical gas–liquid interface, can be used to characterize the interfacial oscillation caused by the wall-jet impingement. More importantly, it was associated with the Rayleigh instability that can be triggered at high frequencies by observing droplets splattering at the gas–liquid interface. The position of the generated surface wave was at the upper side of the vertical gas–liquid interface, very close to the corner of the vertical and horizontal gas–liquid interfaces [Figs. 2(b,d)]. This was where the Rayleigh instability would easily occur with liquid finger shooting and then finally evolve to droplet splattering (see Fig. 7 in Ref. [49]). The liquid finger that readily appeared at high generation frequencies may be attributed to selective amplification of the
generated surface wave. Generation frequency F was determined by the reciprocal of the generating period and was plotted in Fig. 7 as a function of Q for a 0.51-mm nozzle diameter. In Fig. 7(b), a higher generation frequency resulted from a lower Q. According to Ref. [50], with a lower Q and other penetration parameters fixed, the closest radial location of the vertical gas–liquid interface from the jetting axis could be determined by the wall-jet. Thus, a shorter distance of the wall-jet would maintain a stronger impingement on the vertical gas–liquid interface. With the stronger wall-jet, the gas–liquid interface would be more violent, producing a higher surface wave generation frequency. Inversely, farther radial position of the vertical gas–liquid interface would have milder interfacial oscillations with a lower generation frequency. In Fig. 7, analogous events were detected for other nozzle diameters.
Fig. 6. Images of surface wave generation and propagation for d= 0.51 mm, h=12.24 mm, H=3.0 mm under various Q (the first row Q=0.92 l/min, the second row Q=1.06 l/min and the third row Q=1.20 l/min).
Fig. 7. Surface wave generation frequency F as a function of Q.
In Fig. 8, a more chaotic gas–liquid interface with a shorter surface wave generating period was observed in the first row, with a lower h=9.18 mm. Thus, a higher generation frequency would be produced by a wall-jet with a lower jet height. The generation frequency was plotted as a function of normalized jet height in Fig. 9. The frequency gradually increased with decreasing h for each nozzle diameter. A similar explanation could be given to the effect of h on the surface wave generation frequency. The closer vertical gas–liquid interface was formed by a wall-jet with a lower h, leading to more chaotic oscillations at a higher surface wave generation frequency.
Fig. 8. Images of surface wave generation and propagation with d= 0.51 mm, Q=0.92 l/min, and H=3.0 mm for various h (first row h=9.18 mm, second row h=12.24 mm, third row h=15.3 mm).
Fig. 9. Surface wave generation frequency F as a function of the normalized jet height h/d.
In Fig. 10, a relative closer vertical gas–liquid interface could be found in the third row with the liquid film thickness of H=5.0mm. However, the S-shaped interface was not evidently exhibited in the first row with the liquid film thickness of H=3.0mm due to the radial position of the neck was sometimes farther than the waist. This also confirmed the negative value of the characteristic amplitude A* in Fig. 2(c) with H=3.0mm. As illustrated in Fig. 10, a shorter generating period was observed when H was increased to 5.0 mm (the third row). The dependence of the generation frequency as a function of H was plotted in Fig. 11. Higher frequencies occurred in thicker liquid sheets. With the other penetration parameters fixed, a closer vertical gas–liquid interface was obtained for a thicker liquid sheet, resulting in stronger wall-jet impingement on the gas–liquid interface. Thus, a higher generation frequency induced by a stronger impingement can be observed in a thicker liquid sheet.
Fig.10. Images of surface wave generation and propagation for d= 0.51 mm, Q=1.20 l/min, h=15.3 mm, and various H (first row H=3.0 mm, second row H=4.0 mm, third row H=5.0 mm)
Fig. 11. Surface wave generation frequency F for various H.
3.3 Fitting analysis of generation frequency The surface wave generation frequency F was affected by various penetration parameters, and, given a normalized correlation between the frequency and various penetration parameters was significant for scientific research and industrial applications. As we known, the surface tension was an important factor in gas–liquid interfacial behavior. However, in our penetration process, the gas-jet, i.e. wall-jet was served as the driving role to generate the surface waves at the vertical gas–liquid interface.
In addition, the gravity and surface tension were the resilience to restore the wavy interface to the smooth one. Comparing with the surface tension, the gravity of the liquid sheet, which presented as the penetration parameter of liquid sheet thickness, played a dominant role to affect the wall-jet development and impingement on the vertical gas–liquid interface, and accordingly the surface wave generation frequency. The dimensionless Froude number Fr was adopted to normalize penetration parameters, in which the important effects of the gas-jet impingement and gravity of liquid sheet on generation frequency F were considered. It is defined as:
Fr
GV02 L gH
.
(4)
Furthermore, the normalized jet height h/d was used for considering the effect of jet height h on the generation frequency F. A combined dimension parameter of Fr multiplying h/d was introduced to normalize the relationship between of F and all penetration parameters:
V2 h h F f G 0 f Fr . d L gH d
(5)
Abundant of experiments were conducted to get the surface wave generation frequency F under various penetrating parameters. The dependence of F on the penetration parameters and the results of F with (Fr·h/d) were arranged and plotted in Fig. 12. As shown in Fig. 12, F generally decreased by increasing (Fr·h/d). To depict the decrease in F with (Fr·h/d), two familiar curves were used for data fitting (red and blue curves in Fig. 12). The curves were the natural exponential function Eq. (6a) and the power function Eq. (6b), respectively: Fr
F L1 e
h d
M1
N1 ,
h F L2 Fr d
M2
(6a)
N2 .
(6b)
The fitting coefficients were L1=5284.7 Hz, L2=-178.7 Hz, M1=-18.48, M2=0.35, N1=71.5 Hz, and N2=970.4 Hz. The correlation coefficient R=0.64 for the natural exponential curve and R=0.60 for the power curve; they were compared to get the best fitting curve. The natural exponential curve with a larger R was thus better to present the relationship between F and (Fr·h/d).
Fig. 12. Surface wave generation frequency F plotted as a function of the combined dimensionless parameter of the Froude number Fr and the ratio of jet height to nozzle diameter h/d, for d=0.51 mm (fitted by red and blue curves).
Every absolute advantage deviation (A.A.D%) of each experimental case was calculated according to the formula:
A. A.D%
Fexp Fcal 1 100 . n Fexp
(7)
Where n is the times of repeating each experimental case. Each experimental case was repeated six times to ensure about the repeatability and reproducibility of experimental data (i.e. n=6). Fexp is the experimental value of generation frequency and Fcal is the calculation value of generation frequency through the fitting function (Eq. (6a)). As can be seen in Fig. 12, two red dashed curves with absolute advantage deviation of 20 percent (A.A.D%=20) departing from the natural exponential curve were used to evaluate the experimental results. It can be found that nearly 85% experimental data collapsed into the region between two dashed curves. Thus, experimental data was reliable and also was shown in agreement with the natural exponential curve.
Fig. 13. Surface wave generation frequency for different nozzle diameters (fitted by the natural exponential curves).
The decreasing natural exponential relationship between F and (Fr·h/d) was also fit for other nozzle diameters, as plotted in Fig. 13. It seemed that better fits were for larger 0.84-mm nozzle diameters [Fig. 13(d)], relative to those for smaller 0.41-mm diameters [Fig. 13(a)]. Also, two dashed curves with absolute advantage deviation of 20 percent (A.A.D%=20) departing from the natural exponential curve were depicted to evaluate the experimental results. The percent of experimental data dropped into the region between two dashed curves were about 76% for d=0.40mm, 85% for d=0.51mm, 94% for d=0.67mm and 92% for d=0.84mm, respectively. Although the ranges of the penetration parameters were different, a relatively close radial position of the vertical gas–liquid interface frequently occurred for small nozzle diameters. This was sensitive to observational precision of F for the intense wall-jet impingement on a close interface, resulting in more scattered data that deviated significantly from the curve fit in Fig. 13(a). 3.4 Experimental uncertainty Due to the turbulent gas jetting out the nozzle, the turbulence of the wall-jet was unavoidable to affect the wavy behavior of the vertical gas–liquid interface. In addition, the compressibility of gas in the cavity bounded by the vertical gas–liquid interface and the solid substrate would also be considered. Besides, in the experimental result extraction, the surface wave generation frequency depended on the artificial observation was sensitive to the images of wavy interface especially in the situation of the violent interfacial oscillation. These were the main uncertainties in experiments affecting the experimental results. And for the future study, these effects should be considered and improved to obtain the good results.
4. Conclusions The generating mechanism and frequency characteristics of surface waves at a vertical gas–liquid interface were experimentally examined for gas-jet penetration into a liquid sheet. Surface waves were generated when the waist of the vertical gas–liquid interface shrunk to its minimum and its neck expanded to its maximum radial location. In addition, an analysis on energy transfer revealed that surface wave generation was a means of releasing the maximum interfacial elastic energy. The time dependence of the characteristic amplitude revealed the periodic deformation of the vertical gas–liquid interface during dynamic oscillations, while the characteristic wavelength was about one third of the liquid sheet thickness and was independent of the penetration parameters. Because of the stronger wall-jet impingement on the closer interface, more intense interfacial oscillation accompanying higher surface wave generation frequency was produced at a lower flow rate, a lower jet height, or a thicker liquid sheet, when fixing other penetration parameters. Finally, from the fitting analysis, the frequency of surface waves generation could be expressed by a natural exponential function of the combined dimensionless normalization parameter of the Froude number Fr and the ratio of jet height to nozzle diameter h/d. In conclusion, this work should be helpful for understanding interfacial flow in industrial applications.
Acknowledgements We are grateful to the Natural Science Foundation of Zhejiang Province (No. LQ17A020002), the Joint Project from National Natural Science Foundation of China and Liaoning Province (No. U1608258), the Young Researchers Foundation of Zhejiang Provincial Top Key Academic Discipline of Mechanical Engineering of Zhejiang Sci-Tech University (No. ZSTUME02B03) and the Science Foundation of Zhejiang Sci-Tech University (No. 16022090-Y). We also thank Alan Burns, PhD, for improving the English writing.
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Highlights Surface wave generated by normal wall-jet was experimentally characterized. Surface wave was a means to release maximum interfacial elastic energy. A combined dimensionless parameter was used to normalize all penetration parameters. Generation frequency exponentially decreased with the normalization parameter.