Study On Nonlinear Stability Of Multi-order Surface Waves Of Annular Liquid Film

In this paper,a non-linear physical model is established,the fragmentation process of a non-viscous annular liquid film ejected into an incompressible axial flow.The continuity control equation and the Bernoulli governing equation of the nonlinear dimension of the three-order wave are derived.The kinematic boundary conditions and dynamic boundary conditions of the first,second and third surface waves are then established at the nonlinear boundariesr （?）and （?）,and then the boundary conditions are expanded at （?） and （?） by Taylor series.The kinematic boundary conditions and dynamic boundary conditions of the first,second and third surface waves at linear boundaries （?）and （?） are obtained.The initial function expression of the surface wave disturbance amplitude proposed by Professor Cao Jianming proposed the solution of the surface waves of the first,second and third levels.The nonlinear space-time mode surface wave disturbance amplitude initial function expression can be well connected with the linear stability theory.Finally,we will use the Fortran language programming program to numerically calculate the solution of the spatiotemporal pattern,and draw the waveforms of the first,second,and third positive opposition surface waves,as well as the waveform overlay.The fragmentation length and time of the surface wave are predicted and compared with the experimental results.the result shows:（1）The surface wave growth rate expressions of the inner and outer rings of the first,second and third waves are different,and the magnitude of the surface wave disturbance amplitude of the inner and outer rings is also different,and there is a phase difference.（2）The annular liquid film,the circular frequency expression of the first-order wave has no source term,and the second-stage positive anti-symmetric waveform has source terms S_ai2、S_aso2 and S_avo2 related to the first-order wave circular frequency.The third-level positive antisymmetric waveform has a source term S_ai31 、S_ai31-2 、S_aso31、S_aso31-2、S_avo31 and S_avo31-2,associated with the first-order wave and the first-order and second-order wave circular frequencies,and these source terms have explicit analytical expressions.（3）The initial thickness of the annular liquid film does not start from the nozzle outlet,but deviates.This is due to the effect of the surface tension of the liquid.（4）Regardless of whether it is a positive symmetrical waveform or an antisymmetric waveform,the wavelength of the surface wave does not change much with time,and the amplitude increases continuously and changes in stages.（5）It can be seen from the waveform overlay that for the positive antisymmetric waveform,the first-order wave has a greater influence on the waveform superposition map,while the second-order and third-order waves have less influence.The waveform overlay has little difference from the waveform of the first-order waveform.It is indicated that the first-order wave is the main factor causing the jet to break.（6）In the following,by comparing the numerical calculation results of the fragmentation length and the fragmentation time of the antisymmetric waveform with the experimental data,it can be seen that the relative error of the fragmentation time is Δ_（?）b=65.1%,and the relative error of the fragmentation length is Δ_（?）b=7.2%;By comparing the numerical calculation results of the fragmentation length and the fragmentation time of the positive symmetrical waveform with the experimental data,it can be seen that the relative error of the fragmentation time is Δ_（?）b=71.1%,and the relative error of the fragmentation length is Δ_（?）b=4.3%.The fit is better.It is worth noting that both the circular liquid film positive symmetry and the antisymmetric nonlinear third-order surface wave have 64 sets of solutions.After comparison,we have chosen the solution that is closest to the experimental data.（7）The annular liquid film is injected into the gas environment.Since the inner ring is wrapped by the liquid film,the inner ring is less disturbed;the outer ring is affected by the air disturbance,which causes the outer ring to fluctuate greatly.The theoretically derived results are consistent with the actual situation.