Derivation On The Linear Stability Theoryof The First Tomization Dispersion Relation In Liquid Jets

The disintegration of circular liquid jets is very important in the analysis of theatomization process in the diesel engines.It is also the hot research project in internationalfluid and combustion fields.At present,the study of the disintegration mostly use the linearstability theory on international,the core and key point is to get the dispersion relation.But thisneeds solid fluid mechanics and mathematical knowledge,and the derivation process isabnormally heavy and complicated.You should do a lot of repeated analysis to make littleprogress.The numerical calculation program of working process in larger engines——KIVA,employs Reitz’s zero-order dispersion relation,was widely used by many famous universitiesand research institutions at home and boroad.The program is one of the important researchtools for developing doctoral candidate of engine’s working process in famous universitiesand research institutions, which had thousands of statements, was writed with Fortranlanguage and developed by the United States.The atomization and breakup of circular liquid jets are of significant importance for thestudy on the science of engine atomization and combustion. The linear stability theory is animportant method for studying on circular liquid jets disintegrated process.To study thedisintegration process of a viscous circular liquid jet moving in a compressible gas medium,this paper employs the linearized Navier-Stokes equations, the linearized kinematic anddynamic boundary conditions, separately applies the stream function method and the dotproduct of the gradient operator with momentum equation method,that means we do thelinear stability analysis for the disintegration process.Results shows that:First, We build the physical mode of the circular liquid jet moving ina compressible gas medium, the linearized Navier-Stokes equations was obtained by thelinearized continuity equation and momentum equations,and the dimensionless Navier-Stokesequations were obtained by dimensional normalization.According to the different medium,wedivide into two parts,liquid phase and gas phase,applies the stream function method and thedot product of the gradient operator with momentum equation method,employs the linearized kinematic and dynamic boundary conditions,get the zero-order dimensionlessdispersion relation and the n-order dimensionless dispersion relation.Second, Throughvalidating,the zero-order dimensionless dispersion relation derived by this paper and thezero-order dispersion relation derived by Reitz and Li are consistent.When the order numberis zero,we can get the the zero-order dimensionless dispersion relation from the n-orderdimensionless dispersion relation,this illustrated that the n-order dimensionless dispersionrelation derived by this paper is correct.Third,These dimensionless dispersion relations ofcircular liquid jets only related with some dimensionless parameters, they are Weber number,We, Reynolds number, Re, Euler number, Eu,Mach number, Ma, gas-liquid velocity ratio,U, and gas-liquid pressure ratio, P.After programming and numerical computation,we canexamine the effect of these dimensionless parameters on the break-up process of circularliquid jets.