The Linear Stability of Liquid Film with Oscillatory Gas Velocity
Abstract
:1. Introduction
2. Program Formulation
3. Results and Discussion
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Nomenclature
undetermined coefficients | |
undetermined coefficient | |
undetermined coefficient | |
undetermined coefficient | |
undetermined coefficient | |
a function of time | |
thickness of gas | |
thickness of liquid | |
wavenumber | |
nondimensional wavenumber | |
pressure disturbance | |
disturbed pressure of gas | |
disturbed pressure of liquid | |
Reynolds number | |
time | |
axial gas velocity in coordinate system moving with liquid jet | |
axial gas velocity in fixed coordinate system | |
axial velocity of liquid | |
axial velocity of gas | |
axial basic velocity of liquid | |
axial basic velocity of gas | |
oscillations amplitude of gas velocity | |
velocity vector | |
wavelength | |
Weber number | |
thickness ratio between gas and liquid | |
perturbation potential of gas | |
perturbation potential of liquid | |
nondimensional oscillations amplitude | |
disturbed amplitude | |
gas viscosity | |
liquid viscosity | |
gas–liquid density ratio | |
gas density | |
liquid density | |
surface tension coefficient | |
heat flux ratio | |
nondimensional complex growth rate | |
nondimensional inherent frequency | |
nondimensional disturbance frequency | |
nondimensional temporal growth rate | |
nondimensional forcing oscillation frequency | |
contribution of viscosity to the instability | |
complex growth rate | |
complex disturbance frequency | |
temporal growth rate | |
forcing oscillations frequency | |
gas–liquid viscosity ratio |
Appendix A
Appendix B
References
- Gater, R.A.; L’Ecuyer, M.R. A Fundament Investigation of the Phenomena That Characterize Liquid-Film Cooling. Int. J. Heat Mass Transf. 1970, 13, 1925–1939. [Google Scholar] [CrossRef] [Green Version]
- Petrarolo, A.; Kobald, M.; Schlechtriem, S. Understanding Kelvin–Helmholtz instability in paraffin-based hybrid rocket fuels. Exp. Fluids 2018, 59, 62. [Google Scholar] [CrossRef] [Green Version]
- Petrarolo, A.; Kobald, M.; Schlechtriem, S. Optical analysis of the liquid layer combustion of paraffin-based hybrid rocket fuels. Acta Astronaut. 2019, 158, 313–322. [Google Scholar] [CrossRef]
- Yang, L.J.; Fu, Q.F.; Qu, Y.Y.; Gu, B.; Zhang, M.Z. Breakup of a power-law liquid sheet formed by an impinging jet injector. Int. J. Multiph. Flow 2012, 39, 37–44. [Google Scholar] [CrossRef]
- Fu, Q.F.; Yang, L.J.; Qu, Y.Y. Measurement of annular liquid film thickness in an open-end swirl injector. Aerosp. Sci. Technol. 2011, 15, 117–124. [Google Scholar] [CrossRef]
- Fu, Q.F.; Yang, L.J.; Qu, Y.Y.; Gu, B. Linear Stability Analysis of a Conical Liquid Sheet. J. Propul. Power 2010, 26, 955–968. [Google Scholar] [CrossRef]
- Fu, Q.F.; Yang, L.J.; Wang, X.D. Theoretical and Experimental Study of the Dynamics of a Liquid Swirl Injector. J. Propul. Power 2010, 26, 94–101. [Google Scholar] [CrossRef]
- Yang, L.J.; Qu, Y.Y.; Fu, Q.F.; Gu, B. Linear Stability Analysis of a Non-Newtonian Liquid Sheet. J. Propul. Power 2010, 26, 1212–1224. [Google Scholar] [CrossRef]
- Yang, L.J.; Fu, Q.F.; Qu, Y.Y.; Zhang, W.; Du, M.L.; Xu, B.R. Spray characteristics of gelled propellants in swirl injectors. Fuel 2012, 97, 253–261. [Google Scholar] [CrossRef]
- Lasheras, J.C.; Hopfinger, E.J. Liquid Jet Instability and Atomization in a Coaxial Gas Stream. Annu. Rev. Fluid Mech. 2000, 32, 275–308. [Google Scholar] [CrossRef]
- Varga, C.M.; Lasheras, J.C.; Hopfinger, E.J. Initial breakup of a small-diameter liquid jet by a high-speed gas stream. J. Fluid Mech. 2003, 497, 405–434. [Google Scholar] [CrossRef] [Green Version]
- Aliseda, A.; Hopfinger, E.J.; Lasheras, J.C.; Kremer, D.M.; Berchielli, A.; Connolly, E.K. Atomization of viscous and non-Newtonian liquids by a coaxial, highspeed gas jet. Experiments and droplet size modeling. Int. J. Multiphase Flow. 2008, 34, 161–175. [Google Scholar] [CrossRef] [Green Version]
- Mayer, E. Theory of liquid atomization in high velocity gas streams. ARS J. 1961, 31, 1783–1785. [Google Scholar]
- Qin, L.Z.; Yi, R.; Yang, L.J. Theoretical breakup model in the planar liquid sheets exposed to high-speed gas and droplet size prediction. Int. J. Multiph. Flow 2018, 98, 158–167. [Google Scholar] [CrossRef]
- Fu, Q.F.; Yao, M.W.; Yang, L.J.; Xie, L. Atomization Model of Liquid Jets Exposed to Subsonic Crossflows. AIAA J. 2020, 58, 2347–2351. [Google Scholar] [CrossRef]
- Yang, L.J.; Gao, Y.P.; Li, J.X.; Fu, Q.F. Theoretical atomization model of a coaxial gas–liquid jet. Phys. Fluids 2020, 32, 124108. [Google Scholar] [CrossRef]
- Yang, L.J.; Fu, Q.F. Stability of Confined Gas–Liquid Shear Flows in Recessed Shear Coaxial Injectors. J. Propul. Power 2012, 28, 1413–1424. [Google Scholar] [CrossRef]
- Rayana, F.B.; Cartellier, A.; Hopfinger, E.J. Assisted Atomization of a Liquid Layer: Investigation of the Parameters Affecting the Mean Drop Size Prediction. In Proceedings of the International Conference on Liquid Atomization and Spray Systems (ICLASS), Kyoto, Japan, 27 September 2006; pp. 1–8. [Google Scholar]
- Anderson, W.E.; Yang, V. Liquid Rocket Engine Combustion Instability. In Progress in Astronautics and Aeronautics; AIAA: Washington, DC, USA, 1995; pp. 1–78. [Google Scholar]
- Baillot, F.; Blaisot, J.B.; Boisdron, G.; Dumouchel, C. Behaviour of an air-assisted jet submitted to a transverse high-frequency acoustic field. J. Fluid Mech. 2003, 640, 305–342. [Google Scholar] [CrossRef]
- Ćosić, B.; Moeck, J.P.; Paschereit, C.O. Nonlinear Instability Analysis for Partially Premixed Swirl Flames. Combust. Sci. Technol. 2014, 186, 713–736. [Google Scholar] [CrossRef]
- Kheirkhaha, S.; Cirtwill, J.D.M.; Saini, P.; Venkatesanb, K.; Steinberga, A.M. Dynamics and mechanisms of pressure, heat release rate, and fuel spray coupling during intermittent thermoacoustic oscillations in a model aeronautical combustor at elevated pressure. Combust. Flame 2017, 185, 319–334. [Google Scholar] [CrossRef]
- Jia, B.Q.; Xie, L.; Cui, X.; Yang, L.J.; Fu, Q.F. Linear Stability of Confined Coaxial Jets in the Presence of Gas Velocity Oscillations with Heat and Mass Transfer. Phys. Fluids 2019, 31, 092101. [Google Scholar] [CrossRef]
- Jia, B.Q.; Xie, L.; Cui, X.; Yang, L.J.; Fu, Q.F. Linear instability of viscoelastic planar liquid sheets in the presence of gas velocity oscillations. J. Non-Newton. Fluid Mech. 2019, 273, 104169. [Google Scholar] [CrossRef]
- Jia, B.Q.; Yang, L.J.; Xie, L.; Fu, Q.F.; Cui, X. Linear stability of confined swirling annular liquid layers in the presence of gas velocity oscillations with heat and mass transfer. Int. J. Heat Mass Transf. 2019, 138, 117–125. [Google Scholar] [CrossRef]
- Guan, X.Y.; Jia, B.Q.; Yang, L.J.; Fu, Q.F. Linear instability of an annular liquid jet with gas velocity oscillations. Phys. Fluids 2021, 33, 054110. [Google Scholar] [CrossRef]
- Deng, X.D.; Jia, B.Q.; Cui, X.; Wang, N.F.; Shi, B.L. Temporal Instability of Liquid Jet in Swirling Gas with Axial Velocity Oscillations. AIAA J. 2022, 60, 3852–3862. [Google Scholar] [CrossRef]
- Deng, X.D.; Wang, H.R.; Cui, X.; Xie, L.; Jia, B.Q. Temporal instability of confined three-dimensional liquid jet with heat and mass transfer under longitudinal acoustic oscillations. Phys. Fluids 2022, 34, 102107. [Google Scholar] [CrossRef]
- Fu, Q.F.; Deng, X.D.; Yang, L.J. Kelvin–Helmholtz Instability of Confined Oldroyd-B Liquid Film with Heat and Mass Transfer. J. Non-Newton. Fluid Mech. 2019, 267, 28–34. [Google Scholar] [CrossRef]
- Fu, Q.F.; Deng, X.D.; Jia, B.Q.; Yang, L.J. Temporal instability of a confined liquid film with heat and mass transfer. AIAA J. 2018, 56, 2615–2622. [Google Scholar] [CrossRef]
- Mohanta, L.; Cheung, F.B.; Bajorek, S.M. Stability of coaxial jets confined in a tube with heat and mass transfer. Phys. A 2016, 443, 333–346. [Google Scholar] [CrossRef]
- Hsieh, D.Y. Effects of heat and mass transfer on Rayleigh-Taylor instability. J. Fluid Eng. T. ASME 1972, 94, 156–160. [Google Scholar] [CrossRef]
- Hsieh, D.Y. Interfacial stability with mass and heat transfer. Phys. Fluids 1978, 21, 745–748. [Google Scholar] [CrossRef] [Green Version]
- Asthana, R.; Agrawal, G.S. Viscous potential flow analysis of Kelvin–Helmholtz instability with mass transfer and vaporization. Phys. A 2007, 382, 389–404. [Google Scholar] [CrossRef]
- Asthana, R.; Agrawal, G.S. Viscous potential flow analysis of electrohydrodynamic Kelvin–Helmholtz instability with heat and mass transfer. Int. J. Eng. Sci. 2010, 48, 1925–1936. [Google Scholar] [CrossRef]
- Awasthi, M.K.; Asthana, R.; Agrawal, G.S. Pressure corrections for the potential flow analysis of Kelvin–Helmholtz instability with heat and mass transfer. Int. J. Heat Mass Transf. 2012, 55, 2345–2352. [Google Scholar] [CrossRef]
- Awasthi, M.K.; Asthana, R.; Agrawal, G.S. Viscous correction for the viscous potential flow analysis of Kelvin–Helmholtz instability of cylindrical flow with heat and mass transfer. Int. J. Heat Mass Transf. 2014, 78, 251–259. [Google Scholar] [CrossRef]
- Awasthi, M.K. Kelvin-Helmholtz instability of viscoelastic liquid-viscous gas interface with heat and mass transfer. Int. J. Therm. Sci. 2021, 161, 106710. [Google Scholar] [CrossRef]
- Moatimid, G.M.; Obied Allah, M.H.; Hassan, M.A. Kelvin-Helmholtz instability for flow in porous media under the influence of oblique magnetic fields: A viscous potential flow analysis. Phys. Plasmas 2013, 20, 102111. [Google Scholar] [CrossRef]
- Awasthi, M.K.; Asthana, R.; Uddin, Z. Nonlinear Study of Kelvin-Helmholtz instability of cylindrical flow with mass and heat transfer. Int. J. Commun. Heat Mass 2016, 71, 216–224. [Google Scholar] [CrossRef]
- Scriven, L.E.; Sternling, V.S. The Marangoni effects. Nature 1960, 187, 186–788. [Google Scholar] [CrossRef]
- Oron, A.; Deissler, R.T.; Duh, J.C. Marangoni instability in a liquid sheet. Adv. Space Res. 1995, 16, 83–86. [Google Scholar] [CrossRef]
- Dávalos-Orozco, L.A. Thermocapillar instability of liquid sheets in motion. Colloids Surf. A 1999, 157, 223–233. [Google Scholar] [CrossRef]
- Funada, T. Marangoni instability of thin liquid sheet. J. Phys. Soc. Jpn. 1986, 55, 2191–2202. [Google Scholar] [CrossRef]
- Tong, M.X.; Yang, L.J.; Fu, Q.F. Thermocapillar instability of a two-dimensional viscoelastic planar liquid sheet in surrounding gas. Phys. Fluid 2014, 26, 033105. [Google Scholar] [CrossRef]
- Zhang, S.; Lan, X.D.; Zhou, M. Thermocapillary instability of a liquid sheet with centrifugal force. J. Braz. Soc. Mech. Sci. 2018, 47, 40–47. [Google Scholar] [CrossRef]
- Hu, K.X.; He, M.; Chen, Q.S. Instability of thermocapillary liquid layers for Oldroyd-B fluid. Phys. Fluid 2016, 28, 033105. [Google Scholar] [CrossRef] [Green Version]
- Hu, K.X.; He, M.; Chen, Q.S.; Liu, R. Linear stability of thermocapillary liquid layers of a shear-thinning fluid. Phys. Fluid 2017, 29, 073101. [Google Scholar] [CrossRef]
- Chandrasekhar, S. Hydrodynamic and Hydro-Magnetic Stability; Dover publications, Inc.: New York, NY, USA, 1961. [Google Scholar]
- Hasegawa, K.; Manzaki, Y. Marangoni fireworks: Atomization dynamics of binary droplets on an oil pool. Phys. Fluids 2021, 33, 034124. [Google Scholar] [CrossRef]
- Moezzi, M.; Sajjadi, M.; Hossein Hejazi, S. Thermally driven Marangoni effects on the spreading dynamics of droplets. Int. J. Multiph. Flow 2023, 159, 104335. [Google Scholar] [CrossRef]
- Cui, X.; Jia, B.Q. Thermal Effect on the Instability of Annular Liquid Jet. Aerospace 2021, 8, 382. [Google Scholar] [CrossRef]
- Funada, T.; Joseph, D.D. Viscous potential flow analysis of Kelvin-Helmholtz instability in a channel. J. Fluid Mech. 2001, 445, 263–283. [Google Scholar] [CrossRef] [Green Version]
- Yang, L.; Zhang, Q.; Zhang, H.Q.; Wang, B. Numerical investigation on the performance of internal flow and atomization in the recessed gas-centered swirl coaxial injectors. Aerosp. Sci. Technol. 2022, 129, 107858. [Google Scholar] [CrossRef]
- Benjamin, T.B.; Ursell, F. The stability of the plane free surface of a liquid in vertical periodic motion. Proc. R. Soc. Lond. Ser. A 1954, 225, 505–515. [Google Scholar] [CrossRef]
- Kumar, K. Linear Theory of Faraday Instability in Viscous Liquids. Proc. R. Soc. A 1996, 452, 1113–1126. [Google Scholar] [CrossRef]
- Funada, T.; Joseph, D.D.; Yamashita, S. Stability of a Liquid Jet into Incompressible Gases and Liquids. Int. J. Multiph. Flow 2004, 30, 1279–1310. [Google Scholar] [CrossRef]
- Zeng, P.; Sarholz, S.; Iwainsky, C.; Binninger, B.; Peters, N.; Herrmann, M. Simulation of Primary Breakup for Diesel Spray with Phase Transition. In Recent Advances in Parallel Virtual Machine and Message Passing Interface: 16th European PVM/MPI Users’ Group Meeting, Espoo, Finland, 7–10 September 2009; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar]
- Chan, S.H.; Wang, Y.S.; Tan, C.C. The effect of mass transfer on Kelvin-Helmholtz instability at the gas-liquid interface of a sonic reacting and non-reacting gas jet submerged in a liquid. Int. J. Heat Mass Transf. 1994, 37, 1123–1132. [Google Scholar] [CrossRef]
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Deng, X.; Shi, B.; Tang, Y.; Wang, N. The Linear Stability of Liquid Film with Oscillatory Gas Velocity. Aerospace 2023, 10, 691. https://doi.org/10.3390/aerospace10080691
Deng X, Shi B, Tang Y, Wang N. The Linear Stability of Liquid Film with Oscillatory Gas Velocity. Aerospace. 2023; 10(8):691. https://doi.org/10.3390/aerospace10080691
Chicago/Turabian StyleDeng, Xiangdong, Baolu Shi, Yong Tang, and Ningfei Wang. 2023. "The Linear Stability of Liquid Film with Oscillatory Gas Velocity" Aerospace 10, no. 8: 691. https://doi.org/10.3390/aerospace10080691