Role of Vorticity in Generation of Pressure Sources and its Implication for Maintenance of Turbulence

B Mikic

doi:10.1007/s10494-012-9397-9

Abstract

The Poisson equation for pressure, together with the evolutions equations for the velocity gradients, reveals the role of vorticity in generation of pressure sources. Specifically, it was shown how a pressure field created by a local source, acting on nearby vorticity, would create new pressure sources. It was further es- tablished that a moving pressure field, which moves with the velocity of its source, but extends well beyond the source location, could lead to generation of fast and slow streaks as wells as contribute to formation of flow structures in the wall region. These processes, which are part of central mechanisms of maintenance of turbulence, suggest that turbulence could be self-sustaining only if the perturbation pressure force could overcome the diffusion effects; the value of friction Reynolds number reflects the balance between the two.

Highlights

In the following, the symbols used are standard for the field, and the notation of Cartesian tensor assumes its common conventions
Using for the first time the evolution equations for the velocity gradients, it was explicitly shown how a perturbation pressure field created by a local pressure source could generate fresh pressure sources, and it revealed the role of vorticity in those processes
These findings, together with the processes involving relative motion between pressure fields and the fluid in the wall regions that could lead to formations of flow structures, point to contributing mechanisms for spreading and maintaining of turbulent motions

Summary

Key Properties of Poisson Equation for Pressure

The symbols used are standard for the field, and the notation of Cartesian tensor assumes its common conventions. Divergence of the momentum equation yields the Poisson equation for pressure, ∇2 p = −S, where the source term S is given as: S. The following important features of the Poisson equations could be readily identified: i) The length scale of a pressure field is always larger that the length scale of the source that created it; ii) for S > 0, the source would increase nearby pressure, otherwise, for S < 0, it would lower the pressure; iii) each of the three cross-product terms. ∂uj ∂ xi terms were negative would imply a rollup of vorticity, follows a second corollary: the square root of the sum of squares of negative pressure source terms can be used to identify instantaneous locations of vortices

Evolution Equations for Velocity Gradients

On Generation of Flow Structures in the Wall Region

Concluding Remarks