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Characteristics of liquid upwash formed on a splash plate
International Journal of Multiphase Flow 99 (2018) 446–453
Contents lists available at ScienceDirect
International Journal of Multiphase Flow journal homepage: www.elsevier.com/locate/ijmulflow
Characteristics of liquid upwash formed on a splash plate Takao Inamura∗, Sosuke Endo, Takahiro Okabe, Koji Fumoto Graduate School of Science and Technology, Hirosaki University, 3 Bunkyo-cho, Hirosaki, Aomori 036-8561, Japan
a r t i c l e
i n f o
Article history: Received 13 June 2017 Revised 10 November 2017 Accepted 14 November 2017 Available online 15 November 2017 Keywords: Upwash, Impinging jet Sheet impingement Sheet breakup Gas turbine injector Splash plate atomization
a b s t r a c t The present study attempts to clarify the characteristics, i.e., the sheet thickness and sheet velocity, of a liquid upwash formed by the oblique impingement of two equal liquid jets on a splash plate. Moreover, the results are compared with those obtained by the theoretical analysis derived in the previous paper. First, the equations of sheet thickness and sheet velocity of the upwash were derived using the theoretical analysis developed in the previous paper. Second, the upwash thickness and velocity were measured by the electric resistance method and the PIV method, respectively. Finally, the predictions of the upwash thickness and velocity obtained by the theoretical analysis were compared to the measurement results, and the validity of the theoretical analysis was verified. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction The phenomenon whereby a liquid jet impinges on a solid wall is observed in terms of liquid jet impingement cooling and in the combustor of an internal combustion engine. The splash plate atomization that issues the liquid sheet on a solid wall into the atmosphere and generates fine droplets after free liquid sheet fragmentation has been used in a small combustor, because this requires no auxiliary machine, such as a compressor, and easily generates fine droplets. In the cases of liquid jet impingement cooling or preparation of a fuel spray, multiple liquid jets are generally impinged on the solid wall in order to broaden the cooling area or increase the fuel flow rate. In these cases, adjacent liquid sheets on a solid wall impinge upon each other, and the liquid sheet normal to the solid wall generates an upwash. In the case of liquid jet impingement cooling, the liquid lumps and drops scattered around have an adverse influence on cooling equipment. Since the upwash is generally thicker than the liquid sheet on a solid wall, coarse droplets are generated by the disintegration of an upwash. Therefore, for the case in which splash plate atomization is used for the fuel injector, the upwash formation is anticipated to have an adverse effect on the fuel spray characteristics. Thus, the upwash formation and its characteristics have a significant influence on the performance of cooling equipment or a fuel injector. Splash plate atomization, which generates fine droplets by issuing a free liquid sheet into the atmosphere, where the sheet is generated by the impingement of a liquid jet on a splash plate,
∗
Corresponding author. E-mail address: [email protected] (T. Inamura).
https://doi.org/10.1016/j.ijmultiphaseflow.2017.11.011 0301-9322/© 2017 Elsevier Ltd. All rights reserved.
has been widely investigated (Ashgriz, 2011). Inamura and Tomoda (2004) developed a new fuel injector that applies splash plate atomization and measured the spray characteristics of the new fuel injector. Inamura et al. (2004) theoretically analyzed the liquid sheet flow on a solid wall due to the impingement of a liquid jet on a splash plate and experimentally measured the sheet thickness distribution. Finally, they compared the sheet thickness distribution calculated by the theoretical analysis and the measurements and verified the theoretical analysis. Ren and Marshall (2014) experimentally investigated splash plate atomization under a largeWeber-number condition. They deduced a semi-theoretical equation for mean droplet size and verified this equation through measurements. The phenomenon whereby two gas jets impinge on the ground and generate a gas upwash is observed during takeoff and landing of a VTOL (Vertical Take-Off and Landing). Since the gas upwash greatly influences the attitude stability of a fuselage, a number of studies have theoretically and experimentally investigated the generation of an upwash and its effect on attitude instability in the field of aeronautics (Jenkins and Hill Jr., 1977; Siclari et al., 1981; Rizk and Menon, 1989). However, despite its importance, as mentioned above, few studies have investigated liquid upwash. Kate et al. investigated the normal impingement of two equal liquid jets on a solid wall and phenomenologically discussed the formation of a liquid upwash (Kate et al., 2007). They also investigated the normal impingement of two unequal jets. Inamura theoretically analyzed the formation of a liquid upwash by oblique impingement of two equal liquid jets on a solid wall (Inamura, 2016). Moreover, he compared the upwash formation conditions and upwash shape
T. Inamura et al. / International Journal of Multiphase Flow 99 (2018) 446–453
447
Fig. 2. Momentum of the upwash sheet.
Fig. 1. Symbols and coordinate system.
predicted by the theoretical analysis to the measurements and verified the validity of the theoretical analysis. Although the characteristics of an upwash formed by the liquid jet impingement on a solid wall are very important from the point of view of liquid atomization, they remain unclear at present. In the present study, the characteristics of a liquid upwash formed by the oblique impingement of two equal liquid jets on a splash plate were theoretically and experimentally investigated. First, the equations of the sheet thickness and velocity of an upwash were theoretically deduced. Second, the sheet thickness and velocity of an upwash were measured experimentally. Finally, the upwash thickness and its velocity distributions predicted by the theoretical analysis were compared with the measurements, and the validity of the theoretical analysis was verified. 2. Theoretical analysis of upwash formation According to the theoretical analysis reported in the previous paper (Inamura, 2016), the sheet thickness and velocity of a liquid upwash generated by the oblique impingement of two equal liquid jets on a splash plate were deduced. Prior to the theoretical analysis, the following assumptions of the previous paper are established: 1) The liquid flows in an impinging jet and liquid sheet on the splash plate are two-dimensional and laminar. 2) The velocity distribution across the liquid jet is uniform. 3) The liquid sheet flows radially from the stagnation point on the splash plate, and the liquid flow in the circumferential direction can be ignored. 4) On the splash plate, the laminar boundary layer develops in the liquid sheet from the stagnation point (see Fig. 1). 5) The effects of the airflow and gravity on a liquid sheet flow and upwash flow can be ignored. 6) The velocity distribution across an upwash is uniform. 7) The x-component of the momentum of both liquid sheets is transferred to the z-component in an upwash after the impingement (see Fig. 2). The y-component of the momentum of both liquid sheets is transferred to the y-component in an upwash. At the impingement of two liquid sheets on the splash plate, momentum loss due to impingement occurs in only the z-direction, and the y-component of the liquid sheet momentum is maintained after impingement. The momentum loss co-
efficient Cimp in the z-direction was assumed to be 0.85 based on the comparisons between the calculations and measurements of the upwash shape. 8) The effect of a hydraulic jump generated at the periphery of a liquid sheet on the upwash can be neglected. 9) The liquid velocity in an upwash is constant along the stream line. In assumption 7), the coefficient Cimp was assumed to be 0.9 in the previous paper. However, in the present study, Cimp is taken as 0.85 based on careful examinations. After a liquid jet impinges on a splash plate, the laminar boundary layer develops along the stream line from stagnation point P (see Fig. 1) and reaches the liquid surface at rf = rf0 . According to the previous paper, rf0 is expressed by the following equation (Inamura et al., 2004):
rφ 0 ∗ =
0.564
1/3
( 4π )
A · B2
2 / 3
(1)
where A and B are constant and are defined as
A=
sin θ
φ + cos2 φ · sin2 θ 2 sin θ ε 2 tan2 φ B = ±ε + 1 − 2 2 tan2 φ + sin θ tan2 φ + sin θ sin
2
(2)
(3)
in which θ , φ , and ε indicate the impingement angle, the azimuthal angle, and the coefficient of the gap between the center line and the stagnation line of a liquid jet, respectively (see Fig. 1). Here, ε is given as follows (Hasson and Peck, 1964):
ε = cos θ
(4)
The sign of the first term in the right-hand side of Eq. (3) depends on the azimuth-position in the cross section of an impinging jet (Inamura et al., 2004). The dimensionless radial distance from the stagnation point, rφ ∗ , and the thickness of a liquid sheet on the splash plate, hφ ∗ , at the azimuthal angle, φ , and the jet Reynolds number, Re, are defined as follows:
rφ ∗ = ∗
hφ = Re =
rφ 1 · a Re1/3
(5)
hφ 1/3 Re a
(6)
Q a νl
(7)
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T. Inamura et al. / International Journal of Multiphase Flow 99 (2018) 446–453
where a, Q, and ν l indicate the liquid jet radius, the volume flow rate of a liquid jet, and the liquid kinetic viscosity, respectively. Next, the upwash velocity Vφ at point C in Fig. 2 is expressed by the following equation (Inamura, 2016):
Vφ =
2mφ
2
1 − 1 − Cimp 2 sin
φ
(8)
ρl Q φ
where point C is located on the y-axis at the azimuthal angle, φ , and mφ and Qφ indicate the liquid momentum and the volumetric flow rate of a liquid sheet over the infinitesimal azimuthal angle, dφ , respectively. Moreover, Mψ in Fig. 2 indicates the upwash momentum over the infinitesimal azimuthal angle, dψ , and mφ and Qφ are classified by the magnitude correlation between rφ of point C and rφ 0 , as follows (Inamura, 2016): (1) For rφ < rφ 0 ,
mφ = ρl rφ dφ U0
Re
a
Qφ = rφ dφ U0
Re
a
2
1/3
1/3
A · B2 − 0.702 rφ ∗ 2rφ ∗
A · B2 rφ ∗
(9)
(10)
where ρ l and U0 indicate the liquid density and the impingement velocity of a liquid jet on the splash plate, respectively. (2) For rφ > rφ 0 ,
mφ = 0.583 ρl rφ dφ Uφ 2 hφ Qφ =
(11)
7 r dφ Uφ hφ 5 φ
(12)
where Uφ and hφ indicate the surface velocity and the thickness of a liquid sheet on a splash plate at the azimuthal angle, φ , respectively, and are expressed by the following equations: ∗
hφ =
5.03rφ ∗2 0.642 2 A · B + rφ ∗ A · B2
Uφ 1 = U0 0.899 + 7.04
(13)
(14)
rφ ∗ 3
(A·B2 )
2
for which the relationship between hφ and hφ ∗ is given by Eq. (6). According to assumption 7), the relationship between the azimuthal angle, φ , on the splash plate and the azimuthal angle, ψ , on the upwash plane is given by the following equation:
tan ψ = Cimp tan φ ∴ dψ =
(15)
Cimp
2
1 − 1 − Cimp 2 sin
φ
dφ
(16)
According to assumption 9), since the upwash velocity is constant along the stream line, the upwash velocity, Vm , at point D (z = zm ) in Fig. 3 is expressed by the following equation:
Vm = Vφ =
2mφ
2
2
1 − 1 − Cimp sin
ρl Q φ
φ
(17)
Since the liquid flow rate of a liquid sheet on the splash plate over the infinitesimal azimuthal angle, dφ , is equal to that of an upwash over the infinitesimal azimuthal angle, dψ , the upwash thickness, hm , at point D is expressed by the following equation:
hm =
Qφ Vm
1 p dφ 2 sin φ
+
zm d ψ sin ψ
(18)
Fig. 3. Variables of the upwash sheet.
where the relationship between φ and ψ and that between dφ and dψ are expressed by Eqs. (15) and (16), respectively, and p indicates the distance between the impingement points of two liquid jets. The y-coordinate of point C, ym , is expressed as follows:
ym =
p zm + 2 tan φ tan ψ
(19)
3. Experimental setup and method 3.1. Experimental setup Fig. 4 shows a schematic diagram of the experimental setup used in the present study. This setup is the same as that used in the previous paper (Inamura, 2016). Tap water was used as a test fluid. Pressurized water in a water tank was transferred to two equal liquid nozzles. The liquid nozzle is made of stainless steel with an inner diameter of 1.4 mm and a length of 150 mm. The liquid jet issued from a liquid nozzle impinges on a splash plate 5 mm downstream. In the previous study, the distance between the liquid nozzle exit and the impingement point on the splash plate was maintained at 25 mm. However, in the present study, this distance was maintained at 5 mm in order to prevent the influence of turbulent waves of the liquid jet surface on the liquid sheet flow on the splash plate. The impingement plate is made of acrylic and is placed horizontally in order to allow the liquid sheet flow to be observed from below. As a preliminary experiment, we examined the impingement of two liquid jets onto a vertical solid wall and the consequent formation of horizontal upwash. Through this preliminary experiment, we confirmed that the effect of gravity on the thickness and velocity of an upwash is small under the present experimental conditions. The upwash flow was observed using a still camera (D800E, Nikon Co.) with a stroboscope (MS-220, SUGAWARA Laboratories Inc.) as a light source. Image analysis of still photographs was performed using MATLAB (The MathWorks, Inc.). 3.2. Measurement of upwash thickness The sheet thickness was measured by the electric resistance method. The measurement probe is shown in Fig. 5. The measurement probe is configured by two parallel copper wires. The space between the two wires is 1 mm, and the wire diameter is 100 μm. A DC voltage of 10 V is impressed between the two copper wires, and the electric current passing through the liquid sheet is output after being converted to voltage. Fig. 6 shows one side of a slit nozzle used in the preliminary experiment to calibrate the thickness measurement probe. The slit nozzle is constructed by placing one processed plate, as shown in Fig. 6, on another processed
T. Inamura et al. / International Journal of Multiphase Flow 99 (2018) 446–453
Fig. 4. Experimental setup.
Fig. 5. Sheet thickness measurement probe.
plate. The gap of the slit nozzle exit is changed by placing spacers of various thicknesses between the plates. During the preliminary experiment, the measurement probe was positioned 1 mm downstream from the slit nozzle exit in the spanwise direction. Fig. 7 shows the relationship between the liquid sheet thickness and the output voltage from the thickness measurement probe. The output voltage is proportional to the sheet thickness, and its correlation coefficient is approximately equal to 1.00.
Fig. 6. Slit nozzle.
3.3. Measurement of upwash velocity The upwash velocity was measured using the PIV method. In the PIV method, a high-speed video camera (HX-3, NAC Image Technology Inc.) was used, and Koncert (Seika Co.) was used for image processing. The frame rate used for high-speed photography was 20,0 0 0 fps. In the present study, since the upwash is thin, with a thickness of approximately 100 μm at maximum, and is a free liquid sheet, the velocity of a turbulent wave on an upwash surface was assumed to be equal to that of the upwash itself, and the
Fig. 7. Calibration of the sheet thickness probe.
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cide with the measurements, the theoretical analysis overestimates the upwash thickness as a whole. 4.2. Upwash velocity
Fig. 8. PIV measurement of sheet velocity.
turbulent wave on the upwash surface was used as the tracer of the PIV method. Based on this assumption, we hereinafter refer to the measured velocity as the upwash velocity. In order to confirm this assumption, a preliminary experiment was carried out. The slit nozzle used in the preliminary experiment was same as that used in the calibration of the thickness measurement probe shown in Fig. 6. Fig. 8 shows the photograph used in the calibration of upwash velocity measurements. In the center of the liquid sheet, the PIV measurements reveal velocities in the downstream direction, and the velocity of a disturbance in the sheet surface is approximately constant in the downstream direction in the measurement area. In the periphery of the liquid sheet, the disturbance velocity cannot be measured exactly due to the thick rim. Fig. 9 shows the photograph used in the actual upwash velocity measurements. The photograph shows that the upwash velocity decreases as the measurement point leaves the impingement point and that the upwash velocity becomes parallel to the impingement wall in the vicinity of the wall. Fig. 10 shows the relationship between the sheet velocity calculated using the volumetric flow rate and the sectional area of the slit nozzle exit and the surface wave velocity measured using the PIV method. The surface wave velocity was measured 2.5 mm downstream from the slit nozzle exit and at the center in the span-wise direction. The measurements were approximately coincident with the calculated velocities within a measurement error of 10%. The correlation coefficient is 0.989. 4. Results and discussion 4.1. Upwash thickness Fig. 11 shows the measurement results for the upwash thickness along with the theoretical analysis calculations at zm = 3 mm. The theoretical analysis calculations are indicated by the broken line (case of β = 0.0). The impingement angle, θ , the distance between impingement points, and the jet impingement velocity are 60 deg., p = 10 mm, and U0 = 3.5 m/s, respectively. The upwash thickness increases with increasing y-coordinate, except at y = 20 mm. The calculated upwash thickness qualitatively agrees with the measured upwash thickness. However, the theoretical analysis overestimates the upwash thickness as a whole. Figs. 12 and 13 show the upwash thickness at U0 = 6.5 m/s and 8.0 m/s, respectively, for θ = 60 deg. and p = 20 mm. The upwash thickness increases with increasing y-coordinate. In both cases, although the calculations of the upwash thickness qualitatively coin-
Fig. 14 shows the measurements for the upwash velocity along with the theoretical analysis calculations at zm = 3 mm. The theoretical analysis calculations are indicated by the broken line (case of β = 0.0). The impingement angle, θ , the distance between impingement points, and the jet impingement velocity are 60 deg., p = 10 mm, and U0 = 3.5 m/s, respectively. The upwash velocity has a peak at around y = 10 mm and decreases on both sides. The theoretical analysis calculations have a peak at approximately y = 8 mm and coincide qualitatively with the measurements. However, the theoretical analysis quantitatively underestimates the upwash velocity. At low jet impingement velocities, the turbulent waves of the upwash surface are considerably small. Since the tracers used in the PIV method become invisible at low impingement velocities, the measurement error of the upwash velocity becomes high. Furthermore, as mentioned above, the velocity measured by the PIV method is not the upwash velocity itself, but rather the velocity of a turbulent wave on the upwash surface. Therefore, the measured velocity may differ from the actual upwash velocity. At present, it is difficult to directly measure the actual velocity of the upwash because the upwash is thin and fluctuates randomly with time. As such, further research is needed in order to clarify whether the assumption that the upwash velocity is equal to the disturbance velocity of the upwash surface is correct. This is a subject for future research. Fig. 15 shows the upwash velocity at θ = 60 deg., p = 20 mm, and U0 = 6.5 m/s. Under these conditions, the calculations approximately coincide both qualitatively and quantitatively with the measurements. However, the theoretical analysis slightly underestimates the upwash velocity. Fig. 16 shows the upwash velocity at θ = 60 deg., p = 20 mm, and U0 = 8.0 m/s. Although the calculations approximately coincide qualitatively and quantitatively with the measurements, the theoretical analysis slightly underestimates the upwash velocity. 4.3. Influence of vortex formation on upwash characteristics Thus, in every case, although the calculations of the upwash thickness qualitatively coincide with the measurements, there are differences between the calculations using the theoretical analysis and the measurements. Kate et al. observed the upwash generated by the normal impingement of two equal liquid jets on a solid wall (Kate et al., 2007). They assumed that the vortices are generated in the base of the upwash generated by the impingement of two adjacent liquid sheets on a solid wall, as shown in Fig. 17. They reported that the liquid taken into a vortex flows outward along the stagnation line and generates an interacting jump segment on the extended line of the stagnation line, as shown in Fig. 18. Fig. 19 shows the liquid sheet flow on a solid wall. The interacting jump segment, as reported by Kate et al., is indicated by the white ellipse. Based on the above observation, the same liquid flows in the base of an upwash as reported by Kate et al. are assumed to occur in the oblique impingement of the present study. Thus, liquid with a thickness of β hφ was assumed to have been taken into the vortices and to have flowed outward along a stagnation line, as shown in Fig. 20. The liquid taken into a vortex does not participate in the formation of an upwash. Adopting this assumption, the liquid momentum, mφ , and the volumetric flow rate of a liquid sheet, Qφ , over the infinitesimal azimuthal angle, dφ , are given by the following equations (Inamura et al., 2004):
T. Inamura et al. / International Journal of Multiphase Flow 99 (2018) 446–453
451
Fig. 9. PIV measurement of upwash velocity.
Fig. 10. Calibration of the sheet velocity measurement.
Fig. 12. Upwash thickness (p = 20 mm, U0 = 6.5 m/s).
Fig. 11. Upwash thickness (p = 10 mm, U0 = 3.5 m/s).
Fig. 13. Upwash thickness (p = 20 mm, U0 = 8.0 m/s).
(1) For rφ < rφ 0 ,
mφ = ρl rφ dφ U0 2
Qφ = rφ dφ U0
a Re1/3
a Re
1/3
A · B2 − 0.702 rφ ∗ − 5.97 rφ ∗ Aβ ∗ 2rφ
A · B2 − 5.97 rφ ∗ Bβ ∗ rφ
Aβ = β 3
3
−
(2) For rφ > rφ 0 ,
(23)
mφ = ρl rφ dφ Uφ 2 hφ 0.583 − Aβ
(21)
Qφ = 2 rφ dφ Uφ hφ
8 2 2 3 4 4 1 5 1 6 β + β + β − β + β 5 3 7 2 9
1 2 1 3 β + β 2 5
(20)
where Aβ and Bβ are constant and are defined as follows:
4
Bβ = β 2 1 −
(22)
7 10
− Bβ
(24)
(25)
where Aβ and Bβ are expressed by Eqs. (22) and (23), respectively. Fig. 11 shows the upwash thickness calculated using Eq. (18) under the above assumption. The theoretical analysis
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T. Inamura et al. / International Journal of Multiphase Flow 99 (2018) 446–453
Fig.17. Liquid flow at the upwash base.
Fig. 14. Upwash velocity (p = 10 mm, U0 = 3.5 m/s).
Fig. 18. Liquid flow along the stagnation line. Fig. 15. Upwash velocity (p = 20 mm, U0 = 6.5 m/s).
Fig. 19. Formation of interacting jump segments. Fig. 16. Upwash velocity (p = 20 mm, U0 = 8.0 m/s).
calculations are indicated by the solid line (case of β = 0.40). The calculations using β = 0.0 indicate the upwash thickness without considering the liquid taken into a vortex. In the present study, β is assumed to be 0.40 based on comparisons of the calculations and measurements. This figure shows the upwash thickness at θ = 60 deg., p = 10 mm, and U0 = 3.5 m/s. The results of the upwash thickness calculations that consider the liquid taken into a vortex are considerably smaller than those obtained without considering
Fig. 20. Liquid taken into a vortex.
T. Inamura et al. / International Journal of Multiphase Flow 99 (2018) 446–453
the liquid taken into a vortex and coincide approximately with the measurements. Fig. 12 and 13 show the upwash thickness at U0 = 6.5 m/s and 8.0 m/s, respectively. In both cases, the calculations that consider the liquid taken into a vortex coincide both qualitatively and quantitatively with the measurements. Fig. 14 shows the upwash velocity calculated by Eq. (17) at θ = 60 deg., p = 10 mm, and U0 = 3.5 m/s. The theoretical analysis calculations are indicated by the solid line (case of β = 0.40). Considering the liquid taken into a vortex, the calculated upwash velocities are higher than those that do not consider the liquid taken into the vortex and approach the measurement results. However, there are still large differences between the calculations and measurements. This is because the tracers used in the PIV method become invisible at low impingement velocities, and the measurement error becomes large, as mentioned above. Figs. 15 and 16 show the upwash velocity at U0 = 6.5 m/s and 8.0 m/s, respectively. Although, at high impingement velocities, the differences between the calculations that consider the liquid taken into a vortex and those that do not consider the liquid taken into a vortex are small, the calculations that consider the liquid taken into a vortex more closely approach the measurements, as compared to those that do not consider the liquid taken into a vortex. Thus, since part of the liquid film near the solid wall is taken into the vortex formed in the base of an upwash and flows outward along a stagnation line, the calculations of upwash thickness that consider the liquid taken into the vortex more closely approach the measurements. In the present study, β was assumed to be equal to 0.40. However, β is anticipated to change according to the impingement conditions, which include the impingement angle, the impingement velocity, and the distance between impingement points. In order to elucidate the relationship between β and the impingement conditions, further investigations are needed. 5. Conclusions In the present study, equations of the upwash thickness and its velocity were deduced using the theoretical analysis developed in the previous paper. The upwash thickness and its velocity were experimentally measured, and the measurement results were compared with the results of the theoretical analysis. Consequently, the following results were obtained: (1) The upwash thickness increases with increasing y-coordinate of a measurement point. (2) Theoretical analysis overestimates the upwash thickness for all impingement conditions.
453
(3) The theoretical calculations predict well the upwash velocity, except at low impingement velocities. (4) It is thought that part of the liquid film near the splash plate is taken into a vortex formed at the base of an upwash and flows outward along a stagnation line. The liquid taken into the vortex does not participate in the upwash formation. (5) Assuming that part of the liquid film is taken into a vortex and does not participate in the upwash formation, the calculations of upwash thickness and its velocity by the present theoretical analysis are approximately coincident with the measurement results. (6) Although, in the present study, β was assumed to be 0.40, its value is anticipated to change according to the impingement conditions. Further research is necessary in order to elucidate the relationship between β and the impingement conditions. (7) In the present research, the upwash velocities were measured under the assumption that the velocity of the turbulent wave on the upwash surface is equal to the actual velocity of the upwash. This assumption should be verified through further research. Acknowledgement The authors would like to thank Ms. Kobayashi for her help in conducting the experiments. References Ashgriz, N., 2011. Handbook of Atomization and Sprays. Springer. Hasson, D., Peck, R.E., 1964. Thickness distribution in a sheet formed by impinging jets. AIChE J. 10-5, 752–754. Inamura, T., Tomoda, T., 2004. Characteristics of sprays through a wall impingement injector. Atomization Sprays 14, 375–395. Inamura, T., Yanaoka, H., Tomoda, T., 2004. Prediction of mean droplet size of sprays issued from wall impingement injector. AIAA J. 42-3, 614–621. Inamura, T., 2016. Upwash formation on splash plate atomization. Int. J. Multiphase Flow. 85, 67–75. Jenkins, R. C., Hill, Jr., W. G., 1977. Investigation of VTOL upwash flows by two impinging jets. ADA047805. Kate, R.P., Das, P.K., Chakraborty, S., 2007. An experimental investigation on the interaction of hydraulic jumps formed by two normal impinging circular liquid jets. J. Fluid Mech. 590, 355–380. Ren, N., Marshall, A.W., 2014. Characterizing the initial spray flow large Weber number impinging jets. Int. J. Multiphase Flow 58, 205–213. Rizk, M.H., Menon, S., 1989. Large-eddy simulation of excitation effects on a VTOL upwash fountain. Phys. Fluids A. 1-4, 732–740. Siclari, M.J., Hill Jr, W.G., Jenkins, R.C., 1981. Stagnation line and upwash formation of two impinging jets. AIAA J. 19-10, 1286–1293.
Contents lists available at ScienceDirect
International Journal of Multiphase Flow journal homepage: www.elsevier.com/locate/ijmulflow
Characteristics of liquid upwash formed on a splash plate Takao Inamura∗, Sosuke Endo, Takahiro Okabe, Koji Fumoto Graduate School of Science and Technology, Hirosaki University, 3 Bunkyo-cho, Hirosaki, Aomori 036-8561, Japan
a r t i c l e
i n f o
Article history: Received 13 June 2017 Revised 10 November 2017 Accepted 14 November 2017 Available online 15 November 2017 Keywords: Upwash, Impinging jet Sheet impingement Sheet breakup Gas turbine injector Splash plate atomization
a b s t r a c t The present study attempts to clarify the characteristics, i.e., the sheet thickness and sheet velocity, of a liquid upwash formed by the oblique impingement of two equal liquid jets on a splash plate. Moreover, the results are compared with those obtained by the theoretical analysis derived in the previous paper. First, the equations of sheet thickness and sheet velocity of the upwash were derived using the theoretical analysis developed in the previous paper. Second, the upwash thickness and velocity were measured by the electric resistance method and the PIV method, respectively. Finally, the predictions of the upwash thickness and velocity obtained by the theoretical analysis were compared to the measurement results, and the validity of the theoretical analysis was verified. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction The phenomenon whereby a liquid jet impinges on a solid wall is observed in terms of liquid jet impingement cooling and in the combustor of an internal combustion engine. The splash plate atomization that issues the liquid sheet on a solid wall into the atmosphere and generates fine droplets after free liquid sheet fragmentation has been used in a small combustor, because this requires no auxiliary machine, such as a compressor, and easily generates fine droplets. In the cases of liquid jet impingement cooling or preparation of a fuel spray, multiple liquid jets are generally impinged on the solid wall in order to broaden the cooling area or increase the fuel flow rate. In these cases, adjacent liquid sheets on a solid wall impinge upon each other, and the liquid sheet normal to the solid wall generates an upwash. In the case of liquid jet impingement cooling, the liquid lumps and drops scattered around have an adverse influence on cooling equipment. Since the upwash is generally thicker than the liquid sheet on a solid wall, coarse droplets are generated by the disintegration of an upwash. Therefore, for the case in which splash plate atomization is used for the fuel injector, the upwash formation is anticipated to have an adverse effect on the fuel spray characteristics. Thus, the upwash formation and its characteristics have a significant influence on the performance of cooling equipment or a fuel injector. Splash plate atomization, which generates fine droplets by issuing a free liquid sheet into the atmosphere, where the sheet is generated by the impingement of a liquid jet on a splash plate,
∗
Corresponding author. E-mail address: [email protected] (T. Inamura).
https://doi.org/10.1016/j.ijmultiphaseflow.2017.11.011 0301-9322/© 2017 Elsevier Ltd. All rights reserved.
has been widely investigated (Ashgriz, 2011). Inamura and Tomoda (2004) developed a new fuel injector that applies splash plate atomization and measured the spray characteristics of the new fuel injector. Inamura et al. (2004) theoretically analyzed the liquid sheet flow on a solid wall due to the impingement of a liquid jet on a splash plate and experimentally measured the sheet thickness distribution. Finally, they compared the sheet thickness distribution calculated by the theoretical analysis and the measurements and verified the theoretical analysis. Ren and Marshall (2014) experimentally investigated splash plate atomization under a largeWeber-number condition. They deduced a semi-theoretical equation for mean droplet size and verified this equation through measurements. The phenomenon whereby two gas jets impinge on the ground and generate a gas upwash is observed during takeoff and landing of a VTOL (Vertical Take-Off and Landing). Since the gas upwash greatly influences the attitude stability of a fuselage, a number of studies have theoretically and experimentally investigated the generation of an upwash and its effect on attitude instability in the field of aeronautics (Jenkins and Hill Jr., 1977; Siclari et al., 1981; Rizk and Menon, 1989). However, despite its importance, as mentioned above, few studies have investigated liquid upwash. Kate et al. investigated the normal impingement of two equal liquid jets on a solid wall and phenomenologically discussed the formation of a liquid upwash (Kate et al., 2007). They also investigated the normal impingement of two unequal jets. Inamura theoretically analyzed the formation of a liquid upwash by oblique impingement of two equal liquid jets on a solid wall (Inamura, 2016). Moreover, he compared the upwash formation conditions and upwash shape
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447
Fig. 2. Momentum of the upwash sheet.
Fig. 1. Symbols and coordinate system.
predicted by the theoretical analysis to the measurements and verified the validity of the theoretical analysis. Although the characteristics of an upwash formed by the liquid jet impingement on a solid wall are very important from the point of view of liquid atomization, they remain unclear at present. In the present study, the characteristics of a liquid upwash formed by the oblique impingement of two equal liquid jets on a splash plate were theoretically and experimentally investigated. First, the equations of the sheet thickness and velocity of an upwash were theoretically deduced. Second, the sheet thickness and velocity of an upwash were measured experimentally. Finally, the upwash thickness and its velocity distributions predicted by the theoretical analysis were compared with the measurements, and the validity of the theoretical analysis was verified. 2. Theoretical analysis of upwash formation According to the theoretical analysis reported in the previous paper (Inamura, 2016), the sheet thickness and velocity of a liquid upwash generated by the oblique impingement of two equal liquid jets on a splash plate were deduced. Prior to the theoretical analysis, the following assumptions of the previous paper are established: 1) The liquid flows in an impinging jet and liquid sheet on the splash plate are two-dimensional and laminar. 2) The velocity distribution across the liquid jet is uniform. 3) The liquid sheet flows radially from the stagnation point on the splash plate, and the liquid flow in the circumferential direction can be ignored. 4) On the splash plate, the laminar boundary layer develops in the liquid sheet from the stagnation point (see Fig. 1). 5) The effects of the airflow and gravity on a liquid sheet flow and upwash flow can be ignored. 6) The velocity distribution across an upwash is uniform. 7) The x-component of the momentum of both liquid sheets is transferred to the z-component in an upwash after the impingement (see Fig. 2). The y-component of the momentum of both liquid sheets is transferred to the y-component in an upwash. At the impingement of two liquid sheets on the splash plate, momentum loss due to impingement occurs in only the z-direction, and the y-component of the liquid sheet momentum is maintained after impingement. The momentum loss co-
efficient Cimp in the z-direction was assumed to be 0.85 based on the comparisons between the calculations and measurements of the upwash shape. 8) The effect of a hydraulic jump generated at the periphery of a liquid sheet on the upwash can be neglected. 9) The liquid velocity in an upwash is constant along the stream line. In assumption 7), the coefficient Cimp was assumed to be 0.9 in the previous paper. However, in the present study, Cimp is taken as 0.85 based on careful examinations. After a liquid jet impinges on a splash plate, the laminar boundary layer develops along the stream line from stagnation point P (see Fig. 1) and reaches the liquid surface at rf = rf0 . According to the previous paper, rf0 is expressed by the following equation (Inamura et al., 2004):
rφ 0 ∗ =
0.564
1/3
( 4π )
A · B2
2 / 3
(1)
where A and B are constant and are defined as
A=
sin θ
φ + cos2 φ · sin2 θ 2 sin θ ε 2 tan2 φ B = ±ε + 1 − 2 2 tan2 φ + sin θ tan2 φ + sin θ sin
2
(2)
(3)
in which θ , φ , and ε indicate the impingement angle, the azimuthal angle, and the coefficient of the gap between the center line and the stagnation line of a liquid jet, respectively (see Fig. 1). Here, ε is given as follows (Hasson and Peck, 1964):
ε = cos θ
(4)
The sign of the first term in the right-hand side of Eq. (3) depends on the azimuth-position in the cross section of an impinging jet (Inamura et al., 2004). The dimensionless radial distance from the stagnation point, rφ ∗ , and the thickness of a liquid sheet on the splash plate, hφ ∗ , at the azimuthal angle, φ , and the jet Reynolds number, Re, are defined as follows:
rφ ∗ = ∗
hφ = Re =
rφ 1 · a Re1/3
(5)
hφ 1/3 Re a
(6)
Q a νl
(7)
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where a, Q, and ν l indicate the liquid jet radius, the volume flow rate of a liquid jet, and the liquid kinetic viscosity, respectively. Next, the upwash velocity Vφ at point C in Fig. 2 is expressed by the following equation (Inamura, 2016):
Vφ =
2mφ
2
1 − 1 − Cimp 2 sin
φ
(8)
ρl Q φ
where point C is located on the y-axis at the azimuthal angle, φ , and mφ and Qφ indicate the liquid momentum and the volumetric flow rate of a liquid sheet over the infinitesimal azimuthal angle, dφ , respectively. Moreover, Mψ in Fig. 2 indicates the upwash momentum over the infinitesimal azimuthal angle, dψ , and mφ and Qφ are classified by the magnitude correlation between rφ of point C and rφ 0 , as follows (Inamura, 2016): (1) For rφ < rφ 0 ,
mφ = ρl rφ dφ U0
Re
a
Qφ = rφ dφ U0
Re
a
2
1/3
1/3
A · B2 − 0.702 rφ ∗ 2rφ ∗
A · B2 rφ ∗
(9)
(10)
where ρ l and U0 indicate the liquid density and the impingement velocity of a liquid jet on the splash plate, respectively. (2) For rφ > rφ 0 ,
mφ = 0.583 ρl rφ dφ Uφ 2 hφ Qφ =
(11)
7 r dφ Uφ hφ 5 φ
(12)
where Uφ and hφ indicate the surface velocity and the thickness of a liquid sheet on a splash plate at the azimuthal angle, φ , respectively, and are expressed by the following equations: ∗
hφ =
5.03rφ ∗2 0.642 2 A · B + rφ ∗ A · B2
Uφ 1 = U0 0.899 + 7.04
(13)
(14)
rφ ∗ 3
(A·B2 )
2
for which the relationship between hφ and hφ ∗ is given by Eq. (6). According to assumption 7), the relationship between the azimuthal angle, φ , on the splash plate and the azimuthal angle, ψ , on the upwash plane is given by the following equation:
tan ψ = Cimp tan φ ∴ dψ =
(15)
Cimp
2
1 − 1 − Cimp 2 sin
φ
dφ
(16)
According to assumption 9), since the upwash velocity is constant along the stream line, the upwash velocity, Vm , at point D (z = zm ) in Fig. 3 is expressed by the following equation:
Vm = Vφ =
2mφ
2
2
1 − 1 − Cimp sin
ρl Q φ
φ
(17)
Since the liquid flow rate of a liquid sheet on the splash plate over the infinitesimal azimuthal angle, dφ , is equal to that of an upwash over the infinitesimal azimuthal angle, dψ , the upwash thickness, hm , at point D is expressed by the following equation:
hm =
Qφ Vm
1 p dφ 2 sin φ
+
zm d ψ sin ψ
(18)
Fig. 3. Variables of the upwash sheet.
where the relationship between φ and ψ and that between dφ and dψ are expressed by Eqs. (15) and (16), respectively, and p indicates the distance between the impingement points of two liquid jets. The y-coordinate of point C, ym , is expressed as follows:
ym =
p zm + 2 tan φ tan ψ
(19)
3. Experimental setup and method 3.1. Experimental setup Fig. 4 shows a schematic diagram of the experimental setup used in the present study. This setup is the same as that used in the previous paper (Inamura, 2016). Tap water was used as a test fluid. Pressurized water in a water tank was transferred to two equal liquid nozzles. The liquid nozzle is made of stainless steel with an inner diameter of 1.4 mm and a length of 150 mm. The liquid jet issued from a liquid nozzle impinges on a splash plate 5 mm downstream. In the previous study, the distance between the liquid nozzle exit and the impingement point on the splash plate was maintained at 25 mm. However, in the present study, this distance was maintained at 5 mm in order to prevent the influence of turbulent waves of the liquid jet surface on the liquid sheet flow on the splash plate. The impingement plate is made of acrylic and is placed horizontally in order to allow the liquid sheet flow to be observed from below. As a preliminary experiment, we examined the impingement of two liquid jets onto a vertical solid wall and the consequent formation of horizontal upwash. Through this preliminary experiment, we confirmed that the effect of gravity on the thickness and velocity of an upwash is small under the present experimental conditions. The upwash flow was observed using a still camera (D800E, Nikon Co.) with a stroboscope (MS-220, SUGAWARA Laboratories Inc.) as a light source. Image analysis of still photographs was performed using MATLAB (The MathWorks, Inc.). 3.2. Measurement of upwash thickness The sheet thickness was measured by the electric resistance method. The measurement probe is shown in Fig. 5. The measurement probe is configured by two parallel copper wires. The space between the two wires is 1 mm, and the wire diameter is 100 μm. A DC voltage of 10 V is impressed between the two copper wires, and the electric current passing through the liquid sheet is output after being converted to voltage. Fig. 6 shows one side of a slit nozzle used in the preliminary experiment to calibrate the thickness measurement probe. The slit nozzle is constructed by placing one processed plate, as shown in Fig. 6, on another processed
T. Inamura et al. / International Journal of Multiphase Flow 99 (2018) 446–453
Fig. 4. Experimental setup.
Fig. 5. Sheet thickness measurement probe.
plate. The gap of the slit nozzle exit is changed by placing spacers of various thicknesses between the plates. During the preliminary experiment, the measurement probe was positioned 1 mm downstream from the slit nozzle exit in the spanwise direction. Fig. 7 shows the relationship between the liquid sheet thickness and the output voltage from the thickness measurement probe. The output voltage is proportional to the sheet thickness, and its correlation coefficient is approximately equal to 1.00.
Fig. 6. Slit nozzle.
3.3. Measurement of upwash velocity The upwash velocity was measured using the PIV method. In the PIV method, a high-speed video camera (HX-3, NAC Image Technology Inc.) was used, and Koncert (Seika Co.) was used for image processing. The frame rate used for high-speed photography was 20,0 0 0 fps. In the present study, since the upwash is thin, with a thickness of approximately 100 μm at maximum, and is a free liquid sheet, the velocity of a turbulent wave on an upwash surface was assumed to be equal to that of the upwash itself, and the
Fig. 7. Calibration of the sheet thickness probe.
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cide with the measurements, the theoretical analysis overestimates the upwash thickness as a whole. 4.2. Upwash velocity
Fig. 8. PIV measurement of sheet velocity.
turbulent wave on the upwash surface was used as the tracer of the PIV method. Based on this assumption, we hereinafter refer to the measured velocity as the upwash velocity. In order to confirm this assumption, a preliminary experiment was carried out. The slit nozzle used in the preliminary experiment was same as that used in the calibration of the thickness measurement probe shown in Fig. 6. Fig. 8 shows the photograph used in the calibration of upwash velocity measurements. In the center of the liquid sheet, the PIV measurements reveal velocities in the downstream direction, and the velocity of a disturbance in the sheet surface is approximately constant in the downstream direction in the measurement area. In the periphery of the liquid sheet, the disturbance velocity cannot be measured exactly due to the thick rim. Fig. 9 shows the photograph used in the actual upwash velocity measurements. The photograph shows that the upwash velocity decreases as the measurement point leaves the impingement point and that the upwash velocity becomes parallel to the impingement wall in the vicinity of the wall. Fig. 10 shows the relationship between the sheet velocity calculated using the volumetric flow rate and the sectional area of the slit nozzle exit and the surface wave velocity measured using the PIV method. The surface wave velocity was measured 2.5 mm downstream from the slit nozzle exit and at the center in the span-wise direction. The measurements were approximately coincident with the calculated velocities within a measurement error of 10%. The correlation coefficient is 0.989. 4. Results and discussion 4.1. Upwash thickness Fig. 11 shows the measurement results for the upwash thickness along with the theoretical analysis calculations at zm = 3 mm. The theoretical analysis calculations are indicated by the broken line (case of β = 0.0). The impingement angle, θ , the distance between impingement points, and the jet impingement velocity are 60 deg., p = 10 mm, and U0 = 3.5 m/s, respectively. The upwash thickness increases with increasing y-coordinate, except at y = 20 mm. The calculated upwash thickness qualitatively agrees with the measured upwash thickness. However, the theoretical analysis overestimates the upwash thickness as a whole. Figs. 12 and 13 show the upwash thickness at U0 = 6.5 m/s and 8.0 m/s, respectively, for θ = 60 deg. and p = 20 mm. The upwash thickness increases with increasing y-coordinate. In both cases, although the calculations of the upwash thickness qualitatively coin-
Fig. 14 shows the measurements for the upwash velocity along with the theoretical analysis calculations at zm = 3 mm. The theoretical analysis calculations are indicated by the broken line (case of β = 0.0). The impingement angle, θ , the distance between impingement points, and the jet impingement velocity are 60 deg., p = 10 mm, and U0 = 3.5 m/s, respectively. The upwash velocity has a peak at around y = 10 mm and decreases on both sides. The theoretical analysis calculations have a peak at approximately y = 8 mm and coincide qualitatively with the measurements. However, the theoretical analysis quantitatively underestimates the upwash velocity. At low jet impingement velocities, the turbulent waves of the upwash surface are considerably small. Since the tracers used in the PIV method become invisible at low impingement velocities, the measurement error of the upwash velocity becomes high. Furthermore, as mentioned above, the velocity measured by the PIV method is not the upwash velocity itself, but rather the velocity of a turbulent wave on the upwash surface. Therefore, the measured velocity may differ from the actual upwash velocity. At present, it is difficult to directly measure the actual velocity of the upwash because the upwash is thin and fluctuates randomly with time. As such, further research is needed in order to clarify whether the assumption that the upwash velocity is equal to the disturbance velocity of the upwash surface is correct. This is a subject for future research. Fig. 15 shows the upwash velocity at θ = 60 deg., p = 20 mm, and U0 = 6.5 m/s. Under these conditions, the calculations approximately coincide both qualitatively and quantitatively with the measurements. However, the theoretical analysis slightly underestimates the upwash velocity. Fig. 16 shows the upwash velocity at θ = 60 deg., p = 20 mm, and U0 = 8.0 m/s. Although the calculations approximately coincide qualitatively and quantitatively with the measurements, the theoretical analysis slightly underestimates the upwash velocity. 4.3. Influence of vortex formation on upwash characteristics Thus, in every case, although the calculations of the upwash thickness qualitatively coincide with the measurements, there are differences between the calculations using the theoretical analysis and the measurements. Kate et al. observed the upwash generated by the normal impingement of two equal liquid jets on a solid wall (Kate et al., 2007). They assumed that the vortices are generated in the base of the upwash generated by the impingement of two adjacent liquid sheets on a solid wall, as shown in Fig. 17. They reported that the liquid taken into a vortex flows outward along the stagnation line and generates an interacting jump segment on the extended line of the stagnation line, as shown in Fig. 18. Fig. 19 shows the liquid sheet flow on a solid wall. The interacting jump segment, as reported by Kate et al., is indicated by the white ellipse. Based on the above observation, the same liquid flows in the base of an upwash as reported by Kate et al. are assumed to occur in the oblique impingement of the present study. Thus, liquid with a thickness of β hφ was assumed to have been taken into the vortices and to have flowed outward along a stagnation line, as shown in Fig. 20. The liquid taken into a vortex does not participate in the formation of an upwash. Adopting this assumption, the liquid momentum, mφ , and the volumetric flow rate of a liquid sheet, Qφ , over the infinitesimal azimuthal angle, dφ , are given by the following equations (Inamura et al., 2004):
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451
Fig. 9. PIV measurement of upwash velocity.
Fig. 10. Calibration of the sheet velocity measurement.
Fig. 12. Upwash thickness (p = 20 mm, U0 = 6.5 m/s).
Fig. 11. Upwash thickness (p = 10 mm, U0 = 3.5 m/s).
Fig. 13. Upwash thickness (p = 20 mm, U0 = 8.0 m/s).
(1) For rφ < rφ 0 ,
mφ = ρl rφ dφ U0 2
Qφ = rφ dφ U0
a Re1/3
a Re
1/3
A · B2 − 0.702 rφ ∗ − 5.97 rφ ∗ Aβ ∗ 2rφ
A · B2 − 5.97 rφ ∗ Bβ ∗ rφ
Aβ = β 3
3
−
(2) For rφ > rφ 0 ,
(23)
mφ = ρl rφ dφ Uφ 2 hφ 0.583 − Aβ
(21)
Qφ = 2 rφ dφ Uφ hφ
8 2 2 3 4 4 1 5 1 6 β + β + β − β + β 5 3 7 2 9
1 2 1 3 β + β 2 5
(20)
where Aβ and Bβ are constant and are defined as follows:
4
Bβ = β 2 1 −
(22)
7 10
− Bβ
(24)
(25)
where Aβ and Bβ are expressed by Eqs. (22) and (23), respectively. Fig. 11 shows the upwash thickness calculated using Eq. (18) under the above assumption. The theoretical analysis
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Fig.17. Liquid flow at the upwash base.
Fig. 14. Upwash velocity (p = 10 mm, U0 = 3.5 m/s).
Fig. 18. Liquid flow along the stagnation line. Fig. 15. Upwash velocity (p = 20 mm, U0 = 6.5 m/s).
Fig. 19. Formation of interacting jump segments. Fig. 16. Upwash velocity (p = 20 mm, U0 = 8.0 m/s).
calculations are indicated by the solid line (case of β = 0.40). The calculations using β = 0.0 indicate the upwash thickness without considering the liquid taken into a vortex. In the present study, β is assumed to be 0.40 based on comparisons of the calculations and measurements. This figure shows the upwash thickness at θ = 60 deg., p = 10 mm, and U0 = 3.5 m/s. The results of the upwash thickness calculations that consider the liquid taken into a vortex are considerably smaller than those obtained without considering
Fig. 20. Liquid taken into a vortex.
T. Inamura et al. / International Journal of Multiphase Flow 99 (2018) 446–453
the liquid taken into a vortex and coincide approximately with the measurements. Fig. 12 and 13 show the upwash thickness at U0 = 6.5 m/s and 8.0 m/s, respectively. In both cases, the calculations that consider the liquid taken into a vortex coincide both qualitatively and quantitatively with the measurements. Fig. 14 shows the upwash velocity calculated by Eq. (17) at θ = 60 deg., p = 10 mm, and U0 = 3.5 m/s. The theoretical analysis calculations are indicated by the solid line (case of β = 0.40). Considering the liquid taken into a vortex, the calculated upwash velocities are higher than those that do not consider the liquid taken into the vortex and approach the measurement results. However, there are still large differences between the calculations and measurements. This is because the tracers used in the PIV method become invisible at low impingement velocities, and the measurement error becomes large, as mentioned above. Figs. 15 and 16 show the upwash velocity at U0 = 6.5 m/s and 8.0 m/s, respectively. Although, at high impingement velocities, the differences between the calculations that consider the liquid taken into a vortex and those that do not consider the liquid taken into a vortex are small, the calculations that consider the liquid taken into a vortex more closely approach the measurements, as compared to those that do not consider the liquid taken into a vortex. Thus, since part of the liquid film near the solid wall is taken into the vortex formed in the base of an upwash and flows outward along a stagnation line, the calculations of upwash thickness that consider the liquid taken into the vortex more closely approach the measurements. In the present study, β was assumed to be equal to 0.40. However, β is anticipated to change according to the impingement conditions, which include the impingement angle, the impingement velocity, and the distance between impingement points. In order to elucidate the relationship between β and the impingement conditions, further investigations are needed. 5. Conclusions In the present study, equations of the upwash thickness and its velocity were deduced using the theoretical analysis developed in the previous paper. The upwash thickness and its velocity were experimentally measured, and the measurement results were compared with the results of the theoretical analysis. Consequently, the following results were obtained: (1) The upwash thickness increases with increasing y-coordinate of a measurement point. (2) Theoretical analysis overestimates the upwash thickness for all impingement conditions.
453
(3) The theoretical calculations predict well the upwash velocity, except at low impingement velocities. (4) It is thought that part of the liquid film near the splash plate is taken into a vortex formed at the base of an upwash and flows outward along a stagnation line. The liquid taken into the vortex does not participate in the upwash formation. (5) Assuming that part of the liquid film is taken into a vortex and does not participate in the upwash formation, the calculations of upwash thickness and its velocity by the present theoretical analysis are approximately coincident with the measurement results. (6) Although, in the present study, β was assumed to be 0.40, its value is anticipated to change according to the impingement conditions. Further research is necessary in order to elucidate the relationship between β and the impingement conditions. (7) In the present research, the upwash velocities were measured under the assumption that the velocity of the turbulent wave on the upwash surface is equal to the actual velocity of the upwash. This assumption should be verified through further research. Acknowledgement The authors would like to thank Ms. Kobayashi for her help in conducting the experiments. References Ashgriz, N., 2011. Handbook of Atomization and Sprays. Springer. Hasson, D., Peck, R.E., 1964. Thickness distribution in a sheet formed by impinging jets. AIChE J. 10-5, 752–754. Inamura, T., Tomoda, T., 2004. Characteristics of sprays through a wall impingement injector. Atomization Sprays 14, 375–395. Inamura, T., Yanaoka, H., Tomoda, T., 2004. Prediction of mean droplet size of sprays issued from wall impingement injector. AIAA J. 42-3, 614–621. Inamura, T., 2016. Upwash formation on splash plate atomization. Int. J. Multiphase Flow. 85, 67–75. Jenkins, R. C., Hill, Jr., W. G., 1977. Investigation of VTOL upwash flows by two impinging jets. ADA047805. Kate, R.P., Das, P.K., Chakraborty, S., 2007. An experimental investigation on the interaction of hydraulic jumps formed by two normal impinging circular liquid jets. J. Fluid Mech. 590, 355–380. Ren, N., Marshall, A.W., 2014. Characterizing the initial spray flow large Weber number impinging jets. Int. J. Multiphase Flow 58, 205–213. Rizk, M.H., Menon, S., 1989. Large-eddy simulation of excitation effects on a VTOL upwash fountain. Phys. Fluids A. 1-4, 732–740. Siclari, M.J., Hill Jr, W.G., Jenkins, R.C., 1981. Stagnation line and upwash formation of two impinging jets. AIAA J. 19-10, 1286–1293.