Vortex stretching

As noted above, the process of vortex stretching leads to successively smaller eddies being more nearly isotropic. The microscale eddies are statistically isotropic and, as a result... 

The remaining two terms in (2.125) are the vortex-stretching term V, defined by... 

For example, the vortex-stretching term is a triple-correlation term that corresponds to the rate at which dissipation is created by spectral energy passing from the inertial range to the dissipative range of the energy spectrum (see (2.75), p. 43). Letting /cdi 0.1 denote... 

Likewise, because the gradient-dissipation term must balance the vortex-stretching term at spectral equilibrium, it also must scale with the Reynolds number as... 

Figure 3.6. Sketch of vortex stretching at small scales.

For scalar eddies smaller than the Kolmogorov scale, the physics of scalar mixing changes. As illustrated in Fig. 3.6, vortex stretching causes the scalar field to become one-dimensional at a constant rate (Batchelor 1959). Thus, for 1 < rj, the mixing rate can be approximated by /e i/2... 

When the scalar spectrum is not fully developed, the vortex-stretching term Vf will depend on the scalar spectral energy transfer rate evaluated at the scalar-dissipation wavenumber 7 (kd, O-32 Like the vortex-stretching term Vf appearing in the transport... 

As for the scalar dissipation rate, the remaining terms in (3.151) will be large in high-Reynolds-number turbulent flows. The vortex-stretching term Vf is defined by... 

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,... 

Gyr A (1968) Analogy between vortex stretching by drag-reducing additives and vortex stretching by fine suspensions Nature 219 928... 

Other proposed DR mechanisms for both types of additives include high extensional viscosity providing additional resistance to vortex stretching and turbulent eddy growth and viscoelasticity, which dampens small turbulent eddies and stores turbulent fluctuation energy that would otherwise be dissipated. 

---