A new novel fundamental formula for viscous parallel shear flows

A vorticity-based identity that is contrastingly different to the traditional energy-type identities, such as the renowned Reynolds-Orr equation, has been recently derived. The identity is shown to be more general than the one derived by Synge (1938). Based on this identity, we were able to provide a new set of physical interpretations to different phases of stability in classical 2D parallel shear flows. Specifically, we studied the mechanisms of stability and instability in planar Couette and Poiseuille flows- the two most important representatives of classical shear flow models.

As one direct implementation of the new identity, we demonstrated an example on vorticity boundary control for the planar Poiseuille flow, and it turned out that the flow becomes stable with the addition of a small amount of vorticity control. We foresee the prospect of the applicability of such control scheme in the field of CFD.

The study and application of the identity is only at a preliminary stage, but it has promising potential to be further exploited for unveiling the exact mechanism that governs the onset of turbulence in a wide range of shear flows- a grand task that has remained incomplete to generations of fluid dynamists for over 130 years.