Vorticity vortex stretching

The term (ui V) V, which is called vortex stretching, originates from the acceleration terms (2.3.5) in the Navier-Stokes equations, and not the viscous terms. In two-dimensional flow, the vorticity vector is orthogonal to the velocity vector. Thus, in cartesian coordinates (planar flow), the vortex-stretching term must vanish. In noncartesian or three-dimensional flows, vortex stretching can substantially alter the vorticity field. 

For the two-dimensional problem the body force must be purely in the two-dimensional plane. Therefore Vxf must be purely orthogonal to the plane for example, in the r-6 problem, it must point in the z plane. It can be shown that the vortex-stretching term vanishes under these conditions. As a result the vorticity-transport equation is a relatively straightforward scalar parabolic partial differential equation,... 

This shows how the vorticity of a fluid is affected by the forcing and by the diffusion of vorticity. In addition, the first term on the right-hand-side representing vortex stretching also affects the vorticity field and the intensification of vorticity by the stretching of vortex tubes allows the transfer of vorticity and energy from large to smaller scales. 

It turns out that there are significant differences in some aspects of turbulence between three- and two-dimensional systems. As the vorticity vector points in the direction perpendicular to the plane of the flow u> T v, fluid motion can be fully described by a scalar field u (x,y). A consequence of two-dimensionality is that the vortex stretching term u> Vv vanishes in the vorticity equation (1.11) that becomes... 

This so-called lamellar structure- is thought to be formed by the action of vorticity in the fluid, whereby, as a vortex winds up, alternating layers of A-rich and B-rich fluid are incorporated . Vortices are stretched by larger eddies which cause their layers to become thinner, convecting A and B towards each other and reducing the diffusion path between them. For the single, second-order reaction... 

The left-hand side represents the advection (or convection) of vorticity by the velocity u, and the second term on the right-hand side represents the transport of vorticity by diffusion (with diffusivity = the kinematic viscosity v). These two terms are familiar in the sense that they resemble the convection and diflusion terms appearing in the transport equation for any passive scalar. A counterpart to the second term does not appear in these transport equations, however. Known as the production term, it is associated with the intensification of vorticity that is due to stretching of vortex lines. It is not a true production term, however, because it cannot produce vorticity where none exists. Indeed, because (10-5) contains to linearly in every term, it is clear that vorticity can be neither created nor destroyed in the interior of an isothermal, incompressible fluid It can only be convected, diffused, or changed in magnitude once it is already present.6... 

Such a concept is compatible with a comprehensive model of ejections which has been proposed by PERRY CHONG (1982) who call them-vortices, -vortices are simplified picture of events which are called vortex loops, horseshoe or hairpin vortices, all of which are topologically equivalent but are different stage of stretching. 

The authors postulate that a 1-vortex consists of viscous sublayer material and that it formed from a sheet of such material which rolls up at the edges into rods. This model is close to the wall combined with the model of a viscous tornado. Using such a A-vortex model GYR SCHMID (1984) show that the onset of the drag reducing effect can be explained by events just able to stretch the molecules. The local rheology in these events is changed by the stretched molecules and it is therefore of interest in which way the internal flow in these events is altered. Of main interest is the interaction of the vorticity stretching and the diffusion under these new material conditions. 

Fig. 2.2 Schematic of the/i-vortex formation and definition scotch Initially the vorticity is organized spanwise at it is uniformly distributed in flow direction. By an instability process, the vorticity starts to wrap up into a vortex which by stretching takes the idealized form of a A composed mainly of two side-vortex-rods with an internal flow behaviour as described by eq.(2.2). This process ends when the vorticity cannot be concentrated anymore and viscous diffusion processes start to dominate the flow field in the event, which starts from thereon to decay. Since this model also has to account for the non-slip condition, the wall near flow can be described by a viscous tornado. Therefore the question arises whether by incorporating the model of the viscous tornado into the 1-vortex model it would be possible to describe the flow field completely.

In turbulent flows, large scale eddies with coherent structmes are primarily responsible for the mixing of passive scalars. The large scale eddies embody themselves in the form of identifiable and organized distributions of vorticity. In addition, the mixing process involves all mechanisms typically found in vortex dynamics, such as stretching, breakup, concatenation, and self-induction of... 

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