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In fluid dynamics, vortex stretching is the lengthening of
In fluid dynamics, vortex stretching is the lengthening of vortices
In fluid dynamics, a vortex ( : vortices or vortexes) is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in th ...
in three-dimensional fluid flow, associated with a corresponding increase of the component of vorticity
In continuum mechanics, vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along wi ...
in the stretching direction—due to the conservation of angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syste ...
.
Vortex stretching is associated with a particular term in the vorticity equation. For example, vorticity transport in an incompressible inviscid flow is governed by
:
where ''D/Dt'' is the material derivative
In continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field. The material der ...
. The source term on the right hand side is the vortex stretching term. It amplifies the vorticity when the velocity is diverging in the direction parallel to .
A simple example of vortex stretching in a viscous flow is provided by the Burgers vortex In fluid dynamics, the Burgers vortex or Burgers–Rott vortex is an exact solution to the Navier–Stokes equations governing viscous flow, named after Jan Burgers and Nicholas Rott. The Burgers vortex describes a stationary, self-similarity, self- ...
.
Vortex stretching is at the core of the description of the turbulence energy cascade
In continuum mechanics, an energy cascade involves the transfer of energy from large scales of motion to the small scales (called a direct energy cascade) or a transfer of energy from the small scales to the large scales (called an inverse energy ...
from the large scales to the small scales in turbulence
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
. In general, in turbulence fluid elements are more lengthened than squeezed, on average. In the end, this results in more vortex stretching than vortex squeezing. For incompressible flow
In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. A ...
—due to volume conservation of fluid elements—the lengthening implies thinning of the fluid elements in the directions perpendicular to the stretching direction. This reduces the radial length scale of the associated vorticity. Finally, at the small scales of the order of the Kolmogorov microscales
In fluid dynamics, Kolmogorov microscales are the smallest scales in the turbulent flow of fluids. At the Kolmogorov scale, viscosity dominates and the turbulence kinetic energy is dissipated into thermal energy. They are defined by
where
* is ...
, the turbulence kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acc ...
is dissipated into heat through the action of molecular viscosity.Tennekes & Lumley (1972) pp. 75–92.
Notes
References
* * Fluid dynamics Turbulence {{fluiddynamics-stub