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In the injecting process of liquid jet, the disturbance wave on jet interface will grow continually, leading to the spatial development and atomization of liquid jet. Studying the spatial evolution of liquid jet will help to deepen the understanding of the mechanism of jet breakup and atomization. In this paper, based on the linear and nonlinear stability theories, the first-order and second-order dispersion equations describing the stability of liquid jet with cavitation bubbles in a coaxial swirling compressible airstream are built, respectively, and the dispersion equation and its solving method are verified by the data in the literature. On this basis, the developments of first-order and second-order disturbance are analyzed, and the spatial evolutions of liquid jet are compared under linear and nonlinear stability theories. The results show that the wavelength and amplitude of the second-order disturbance are much smaller than those of the first-order disturbance. The disturbance development on jet surface is mainly dominated by the development of the first-order disturbance along the axial direction. With the increasing of axial distance, the second-order disturbance gradually begins to play a role in the developing of disturbance. The role of second-order disturbance is mainly reflected in three aspects, i. e., obviously increasing the disturbance amplitude at wave crest, reducing the disturbance amplitude at wave trough (sometimes ups and downs occur), and changing the waveform to a certain degree. The dominant disturbance mode on jet surface will not change under two kinds of theories. By using the nonlinear stability theory, satellite droplets which are found on jet surface in experiments can be reflected, and the shape of main droplet changes obviously from the ellipsoid to sphere. Also, the change of dimensionless radius of liquid jet is greater by nonlinear stability theory than by linear stability theory, which indicates that the oscillation extent of jet surface increases due to considering the second-order disturbance. Therefore, compared with the linear stability theory, the nonlinear stability theory has the advantage that it considers the effects of high-order disturbance on the spatial evolution of liquid jet in addition to the first-order disturbance on jet surface. The nonlinear stability theory can predict the spatial development of liquid jet in more detail than the linear stability theory.
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Keywords:
- jet /
- spatial evolution /
- linear theory /
- nonlinear theory
[1] Yi S J 1996 Ph. D. Dissertation (Dalian: Dalian University of Technology) (in Chinese) [易世君 1996 博士学位论文(大连: 大连理工大学)]
[2] Zhou Z W, Lin S P 1992 J. Propul. Power 8 736
[3] Ozgen S, Uzol O 2012 J. Fluid Eng. 134 1
[4] Turner M R, Sazhin S S, Healey J J 2012 Fuel 97 288
[5] Liang X, Deng D S, Nave J C 2011 J. Fluid Mech. 683 235
[6] Cao J M 2014 J. Circ. Syst. 2 165 (in Chinese) [曹建明 2014 新能源进展 2 165]
[7] Jazayeri S A, Li X G 2000 J. Fluid Mech. 406 281
[8] Yang L J, Wang C, Fu Q F 2013 J. Fluid Mech. 735 249
[9] Yuen M C 1968 J. Fluid Mech. 33 151
[10] Nayfeh A H 1970 Phys. Fluids 13 841
[11] Lafrance P 1975 Phys. Fluids 18 428
[12] Ibrahim A A, Jog M A 2006 Phys. Fluids 18 114101
[13] Ibrahim A A, Jog M A 2008 Int. J. Multiphase Flow 34 647
[14] Rangel R H, Sirignano W A 1988 Phys. Fluids 31 1845
[15] Lozano A, Olivares A G, Dopazo C 1998 Phys. Fluids 10 2188
[16] Ibrahim E A, Lin S P 1992 J. Appl. Mech. 59 291
[17] Tharakan T J, Ramamurthi K, Balakrishnan M 2002 Acta Mech. 156 29
[18] Ibrahim A A 2006 Ph. D. Dissertation (Cincinnati: University of Cincinnati)
[19] Yan K, Jog M A, Ning Z 2013 Acta Mech. 224 3071
[20] Hadji L, Schreiber W 2007 J. Phys. Nat. Sci. 1 1
[21] Potter M C, Wiggert D C 2009 Mechanics of Fluids (3rd Ed.) (Stamford: Cengage Learning) p213
[22] Lin S P 2003 Breakup of Liquid Sheets and Jets (Cambridge: Cambridge University Press) p109
[23] Zhou H, Zhao G F 2004 Hydrodynamic Stability (Beijing: National Defence Industry Press) p23 (in Chinese) [周恒, 赵耕夫 2004 流动稳定性 (北京: 国防工业出版社) 第23页]
[24] Li Q Y, Wang N C, Yi D Y 2008 Numerical Analysis (Beijing: Tsinghua University Press) p228 (in Chinese) [李庆扬, 王能超, 易大义 2008 数值分析 (第5版) (北京: 清华大学出版社) 第228页]
[25] Lin S P, Lian Z W 1990 AIAA J. 28 120
[26] Sallam K A, Dai Z, Faeth G M 2002 Int. J. Multiphase Flow 28 427
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[1] Yi S J 1996 Ph. D. Dissertation (Dalian: Dalian University of Technology) (in Chinese) [易世君 1996 博士学位论文(大连: 大连理工大学)]
[2] Zhou Z W, Lin S P 1992 J. Propul. Power 8 736
[3] Ozgen S, Uzol O 2012 J. Fluid Eng. 134 1
[4] Turner M R, Sazhin S S, Healey J J 2012 Fuel 97 288
[5] Liang X, Deng D S, Nave J C 2011 J. Fluid Mech. 683 235
[6] Cao J M 2014 J. Circ. Syst. 2 165 (in Chinese) [曹建明 2014 新能源进展 2 165]
[7] Jazayeri S A, Li X G 2000 J. Fluid Mech. 406 281
[8] Yang L J, Wang C, Fu Q F 2013 J. Fluid Mech. 735 249
[9] Yuen M C 1968 J. Fluid Mech. 33 151
[10] Nayfeh A H 1970 Phys. Fluids 13 841
[11] Lafrance P 1975 Phys. Fluids 18 428
[12] Ibrahim A A, Jog M A 2006 Phys. Fluids 18 114101
[13] Ibrahim A A, Jog M A 2008 Int. J. Multiphase Flow 34 647
[14] Rangel R H, Sirignano W A 1988 Phys. Fluids 31 1845
[15] Lozano A, Olivares A G, Dopazo C 1998 Phys. Fluids 10 2188
[16] Ibrahim E A, Lin S P 1992 J. Appl. Mech. 59 291
[17] Tharakan T J, Ramamurthi K, Balakrishnan M 2002 Acta Mech. 156 29
[18] Ibrahim A A 2006 Ph. D. Dissertation (Cincinnati: University of Cincinnati)
[19] Yan K, Jog M A, Ning Z 2013 Acta Mech. 224 3071
[20] Hadji L, Schreiber W 2007 J. Phys. Nat. Sci. 1 1
[21] Potter M C, Wiggert D C 2009 Mechanics of Fluids (3rd Ed.) (Stamford: Cengage Learning) p213
[22] Lin S P 2003 Breakup of Liquid Sheets and Jets (Cambridge: Cambridge University Press) p109
[23] Zhou H, Zhao G F 2004 Hydrodynamic Stability (Beijing: National Defence Industry Press) p23 (in Chinese) [周恒, 赵耕夫 2004 流动稳定性 (北京: 国防工业出版社) 第23页]
[24] Li Q Y, Wang N C, Yi D Y 2008 Numerical Analysis (Beijing: Tsinghua University Press) p228 (in Chinese) [李庆扬, 王能超, 易大义 2008 数值分析 (第5版) (北京: 清华大学出版社) 第228页]
[25] Lin S P, Lian Z W 1990 AIAA J. 28 120
[26] Sallam K A, Dai Z, Faeth G M 2002 Int. J. Multiphase Flow 28 427
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