Review on Parabolized Navier-Stokes (PNS) equation and Gao's PNS theory with inferences and applications

PNS equations are in use very widespread, it forms the basis of industry-standard aerodynamic computations. There are some PNS with different slightly viscous terms and eight different names, in which PNS and thin-layer (TL) NS equations are often used. But they have same mathematical property i.e., when Mach number in streamwise direction is larger than unity, they are parabolic-type equations and can be solved by space-marching algorithm (SMA). PNS-SMA computation reduces space dimension number and can save greatly computer inner-storage and CPU-time compared with time marching computation of NS and Reynolds average (RA) NS equations. PNS-SMA method is greatly developed. However, PNS is rather vague in the aspect of fluid mechanics in the early days of PNS study, which is improved by the viscous/inviscid interacting shear flow (ISF) theory presented by Z. Gao in the 1990s. This theory illustrates basic flow field described by PNS equations and extracts motion law and mathematic definitions and deduces ISF equations, which is a kind of PNS equations. For the sake of not adding new name we call it Gao's PNS equations and relevantly call ISF theory Gao's PNS equations. Especially, this theory also has some important applications in computations of NS and RANS equations. Such as PNS's optimal coordinates used to decrease false diffusion in computations of ISF's viscous layer, the length scaling laws used to generate adaptive mesh in order to capture heat flux etc. abrupt changes in hypersonic flows over a body, the wall-surface relations deduced by PNS theory used to new pressure boundary conditions and to verify numerical results given by NS and RANS equations solving near wall flows, and so on. Some initial applications are presented, further applications and integration of PNS-SMA and RANS-SMA and PSE-SMA computations are worth expecting, here PSE is parabolized stability equations.