From aerodynamics towards aeroacoustics: a novel natural velocity decomposition for the Navier-Stokes equations

A novel formulation for the analysis of viscous incompressible and compressible aerodynamics/aeroacoustics fields is presented. The paper is primarily of a theoretical nature, and presents the transition path from aerodynamics towards aeroacoustics. The basis of the paper is a variant of the so-called natural velocity decomposition, as v = ▿φ + w, where w is obtained from its own governing equation and not from the vorticity. With the novel decomposition, the governing equation for w and the generalized Bernoulli theorem for viscous fields assume a very elegant form. Another improvement pertains to the so-called material covariant components of w: For inviscid incompressible flows, they remain constant in time; minor modifications occur when we deal with viscous flows. In addition, interesting simplifications of the formulation are presented for almost-potential flows, namely for flows that are irrotational everywhere except for thin vortex layers, such as boundary layers and wakes. It is shown that, if the thickness is very small, the exact formulation for almost-potential flows is approximately equal to that for quasi-potential flows (zero-thickness wake), as modified by the Lighthill transpiration velocity correction. Simple numerical applications to the flow around a disk and to jet aerodynamics/aeroacoustics (Kelvin-Helmholtz instabilities) are included.