On the governing equations for the compressing process and its coupling with other processes

As a continuation of a recent linear analysis by Mao et al.(Acta Mech Sin,2010,26:355),in this paper we propose a general theoretical formulation for the compressing process in complex Newtonian fluid flows,which covers gas dynamics,aeroacoustics,nonlinear thermoviscous acoustics,viscous shock layer,etc.,as its special branches.The principle on which our formulation is based is the maximally natural and dynamic Helmholtz decomposition of the Navier-Stokes equation,along with the kinematic Helmholtz decomposition of the velocity field.The central results are the new dilatation equation and velocity-potential equation,which are the counterparts of vorticity transport equation and vector stream-function equation for the shearing process,respectively.Various couplings of the compressing process with shearing and thermal processes,including its physical sources,are carefully identified.While the possible applications and influences of the new formulation are yet to be explored,our preliminary discussion on the pros and cons of previous formulations pertain to acoustic analogy and that on the process splitting and coupling in highly compressible turbulence indicates that at least the formulation can serve as a new frame of reference by which one may gain some additional insight and thereby develop new approaches to the multi-process complex flow problems.