Stretching of a vortical structure: filaments of vorticity

Abstract

r ro b) The first term represents the temporal evolution of the vorticity, the second is an advection term, the third is the stretching term and the fourth represents the viscous effects. When stretching is parallel and in the same direction as the vorticity, the term (m. V) is positive and amplifies the vorticity (om/ot). The vorticity increases and the viscous term (v~m) becomes large enough to exactly coWlter balance the amplification term (0). V)V.An equilibrium is then reached which imposes the diameter of the vortex r m= vcolT02, hence TO (v/i) 1/2. From the general perspective of view, the stretching of the vorticity in the flow leads to its amplification and its confinement. Note that it is not the vorticity which is conserved, since it is amplified by stretching, but the circulation r=oJcU.dl. Circulation aroWld curve C is the total vorticity inside this curve. Vortex stretching is very important mechanism in fluid dynamics: it usually corresponds to the presence ofvortical structures which are much more intense that those produced by simple shear or rotating flows. In particular, it is now well known that local stretching of vorticity in turbulent flows produces very intense vortices called ofvorticity. A large part of the scientific community working on turbulence believes that these structures are very important in the dynamics of turbulent flows, although their structure, their dynamics or their instabilities are not fully Wlderstood. The study ofthese structures ofintense vorticity is hence extremelyimportant for both fundamental research and applied science (flow control, ... ). Two experiments have been built to produce filaments without turbulent flows. Only the fundamental ingredients have been kept, i.e. initial vorticity and stretching. These set-ups allow the study of the filaments of vorticity with control over various parameters of the experiments. In that way, an isolate om/dt+(V. V) 00 = (00. V) V + v~m. In this example, one observes indeed that the vorticity is concentrated in the vortex core. A flow with Vii 1fT outside the core has been chosen. This gives zero vorticity outside this region. Note that such velocity field depends on r although its axial vorticityis zero: Wz = liT [%r(Ve!r)T-Tdv,:tOO] =O. In similar way, a non-zero vorticity does not mean that vortex is present: A shear flow such as bOWldary layer flow on flat wall can be stable Wltillarge Reynolds number (Re=UL/v). For all vortical flows, vorticity is very important parameter since it locates vortices and gives their intensity. An important mechanism that enhances the vorticity is the stretching. Stretching vortex along its axis will make it rotate faster and decrease its diameter in order to maintain its kinetic momentum constant. This is analogous to the kinetic momentum conservation law in solid mechanics. A well known example is the ice-skater who turns faster as she brings her hands near her body, and vice versa. An example in fluid mechanics is the bath-tub vortex that rotates faster and becomes smaller as it goes from the fluid surface to the exit. More precisely, stretching ris an acceleration of the axial velocity along the vortex (r= VV). The vorticity equation is obtained from the curl of the Navier-Stokes equation: