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.%分别基于线性和非线性稳定性理论，建立了描述同轴旋转可压缩气体中含空泡液体射流稳定性的一阶与二阶色散方程，并对色散方程进行验证分析；在此基础上，进行了射流表面一阶与二阶扰动及其发展的分析，线性与非线性稳定性理论下射流空间发展的对比研究。研究结果表明，二阶扰动波的波长和振幅明显小于一阶扰动波；沿射流方向，射流表面的扰动发展主要由一阶扰动波的发展所主导；随着轴向距离的增大，二阶扰动波才开始逐渐对扰动的发展起一定的作用。两种稳定性理论下射流表面的占优扰动模式不会发生改变；采用非线性稳定性理论时，可以反映一些实验中发现的射流表面出现“卫星液滴”的现象，由于考虑了射流表面的二阶扰动，射流界面振荡程度加剧。
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