Mathematical Model And Numerical Investigation Of Transient Cavitating Flows In Pipelines

The amplitude of the instantaneous pressure produced by the collapse of the vapor cavity in the transient process can reach tens or even hundreds times of the normal working pressure in the pipeline,which could seriously threaten the safe operation of the water systems.Based on the quasi-steady friction model and the discrete cavity model,the traditional one-dimensional transient flow calculation method could not accurately predict the amplitude and phase of the pressure head.In order to guide accurate prediction of the transient cavitating flows in the pipeline,more relatively accurate 1D and quasi two-dimensional(2D)models are established based on theoretical analysis,and the relevant numerical algorithm are also investigated.The rationality and reliability of the above-mentioned models and corresoponding numerical algorithm are verified through a series of experimental data in the literature.The existing commercial water hammer softwares do not consider unsteady friction for calculating transient flow problems and need to be improved in terms of universality,modification and pertinence.Based on the unsteady MIAB model which is more consistent with the flow characteristics,the MIAB-DGCM model is constructed.The comparison of experimental data and simulation results shows that the MIAB-DGCM model can better predict the amplitude and phase of the pressure fluctuation than the quasi-steady friction model.Based on the C#programming language under the.NET framework and modular design ideas,the efficient communication among the interface layer,the method layer and the file storehouse is implemented by the parallel execution of the multi-thread.The transient two-phase flow visualization software with MIAB model is designed and developed,which has an extensible interface,strong pertinence,simple operation and friendly interface.Based on the single component two-phase flow mixture theory and the Rayleigh-Plesset kinetic equation for cavitation bubble,the two phase flow cavitation model based on the vapor phase volume fraction transport equation is construceted.The empirical constants of two different forms of mass transport equation are determined,Ce = 0.02 and Cc = 1 for Singhal model,and Cdeat =1 and Cprod =100,000 for Kunz model.The specific analysis of the transient cavitating flow with different degree of cavitation is carried out.The numerical results show that the pressure fluctuation curves at the monitoring points predicted by the two different forms of two phase flow cavitation model are almost consistent,and they is in good agreement with the experimental data.The numerical results both could reproduce the basic characteristics of transient cavitating flow and predict the timing and duration of the first cavitation and peak pressure caused by the collapse of the vapor cavity.In addition,unlike the assumption in other cavitation models that the wave speed is constant,the proposed models could reflect the actual pressure wave speed in the transient process.For the same space and time discretization,the Vardy-Hwang algorithm,Zhao-Ghidaoui algorithm and Korbar algorithm of the quasi 2D model have the same precision,and the five-region algebraic model,k-? model and k-? model have the same precision.When the ratio of the time scale of radial diffusion of shear to the time scale of wave propagation is small(P? 1),the velocity distribution could be fully developed,and the dissipation rate of kinetic energy and turbulent dissipation rate fluctuates greatly.It shows that the quasi-steady assumption and the frozen assumption would make the calculation results unreliable.When the ratio of the time scale of radial diffusion of shear to the time scale of wave propagation is large(P>>1),the velocity distribution does not have enough time to be fully developed,and the dissipation rate of kinetic energy and turbulent dissipation rate fluctuates slowly.It shows that the quasi-steady assumption and the frozen assumption would make the calculation results accurate.In addition,the quasi 2D DGCM model and quasi 2D two-phase equilibrium cavitation model are constructed based on the instantaneous wall shear stress.The explicit finite-difference scheme is used for 1D continuity equation and the implicit finite-difference scheme for the quasi 2D momentum equation.The pressure head curve predicted by the quasi 2D DGCM model is closer to the experimental results than the 1D DGCM model.The pressure head curve predicted by the quasi 2D equilibrium cavitation model is closer to the experimental results than the 1D two phase equilibrium cavitation model.By introducing the density-pressure equation which reflects the variable wave speed of the compressible fluid,the weak compressible fluid RANS method and its related algorithms are established.The numerical simulations of the transient non-cavitating and cavitating flows caused by the fast closing valve in the reservoir-pipe-valve system are carried out by using the slip technique.The calculation results of the pressure head and the experimental data are correct.Compared with the RANS method,the accuracy and reliability of the weakly compressible fluid are verified.The accuracy and reliability of the RANS method for the weakly compressible fluid are verified by comparing the calculated results with the experimental data.The results indicate that the grid scale has little effect on the calculation results of the pressure head,and the maximum water hammer pressure and pressure fluctuation cycle are almost unaffected by the time step,and the attenuation of the pressure head amplitude increases with the increase of the time step.Furthermore,the simulation results can enrich the flow field information such as velocity,pressure and vapor volume fraction.Through the evolution of the pressure field,the propagation characteristics of pressure waves can be analyzed qualitatively and quantitatively.Through the evolution of the velocity field,it can be seen that the velocity distribution in the wall area changes rapidly and has a high gradient,which mainly depends on the viscosity.The change of the velocity distribution in the core region is related to the velocity distribution of the history of the past time,which mainly depends on the diffusion.The formation,development and collapse of the cavity can be successfully captured,and it can be clearly and visually observed that the uneven distribution of vapor cavity in the direction of pipe length and pipe diameter,and move slowly along the top of the pipe wall.If the rarefaction wave propagates into a zone of decreasing pressure,the vaporous cavitation region will form.Alternatively,the rarefaction wave propagating into a zone of increasing pressure does not result in the formation of the vaporous cavitation zone.