Vorticity

The vorticity ( Wirbelhaftigkeit, vorticity, and Engl. Vorticity ) is a central quantity of fluid mechanics and meteorology and can be vividly described as the tendency of a fluid element to the intrinsic rotation around an axis.

The vorticity is defined as the rotation speed

And is thus a pseudo- vector field, since the rotation of a vector field is always a pseudo- vector field. In general, any vector field a vorticity be assigned. The SI unit of vorticity is.

The vorticity is equal to the area-related circulation rate:

Especially in the field of meteorology and the symbols used for the vorticity. Except for regular vortices such as tornadoes here are often faced with two-dimensional velocity fields. The corresponding vorticity is for speed fields in the xy plane

And is in the Z - direction.

Hydrodynamics

In hydrodynamics, the vorticity of the fluid, the rotation speed which is oriented toward the rotation axis, respectively, for two-dimensional flow perpendicular to the flow plane. For fluids with a fixed rotation about an axis ( for example, a rotating cylinder ) is the vorticity ω equal to twice the angular velocity ω0 of the fluid element:

Fluids without vorticity hot rotationally or irrotational. However, fluid elements of an irrotational fluid can have an angular velocity equal to 0, that is, move along curved paths - the vorticity refers to forced vortex.

Graphically, can the vorticity represented as follows:

Consider an infinitesimally small, square area of a liquid. If this field rotates, the vorticity of the flow equal to zero.

The idea of the vorticity is a suitable means for fluids with small viscosity. Then the vorticity can be connected to almost all places are the flow considered equal to zero ( except for a small region). This is obvious to a two-dimensional flows in which the flow on the complex plane can be displayed. Such problems can usually be solved analytically.

For each flow, the governing equations can be related to the vorticity rather than the velocity by simply replacing. This leads to the so-called vortex density equation is for incompressible, inviscid fluids as follows:

Also for real flows ( three-dimensional, finite Reynolds number ) is the consideration of the flow over the vortex strength with limitations usable if one assumes that the Vortizitätsfeld can be represented as an array of discrete vortices.

When a viscous fluid is present, the vortices diffuse through the flow. This is described by the vorticity transport equation.

Where the Laplace operator represents. Here, the vortex density equation was supplemented by the diffusion term. For highly viscous flows, such as Couette flow, it may therefore be more appropriate to consider directly the velocity field of the fluid instead of the vorticity, because the high viscosity leads to a very strong diffusion of the vortices.

The concept of the vortex line is closely related to the vorticity: vortex lines are always tangent to the vorticity. The totality of passing through a surface element vortex lines is called a vortex filament. The Helmholtz vortex theorems state that the vortex flow is both temporally and spatially constant.

Meteorology

In meteorology, the rotation of air will be mainly described about an axis with the vorticity. The absolute vorticity of a volume element or of a body in meteorology is composed of two terms, the planetary and relative vorticity:

Due to the Earth's rotation every body learns near the Earth rotation around the Earth's axis and thus has a fixed vorticity. This is determined by the latitude -dependent Coriolisfaktor

Determined and referred to as the planetary vorticity. The relative vorticity is related to the natural rotation of the body size. Adding up the results in the absolute vorticity:

Since most two-dimensional flow fields occur in meteorology, the relative vorticity is often expressed by the rotation in two dimensions:

Thus results for the absolute vorticity

The direction of the vorticity vector can be determined with the corkscrew rule: Business is the fluid counter-clockwise, so shows the vorticity upwards. is positive in this case.

In the northern hemisphere rotation is called counter-clockwise, so positive, as cyclonic rotation and rotation in a clockwise direction, ie negative, as anticyclonic rotation. In the southern hemisphere this is reversed accordingly.

Converted to natural coordinates gives:

In this case,

Here, the curvature of the flow lines as n and s are the components of the coordinate system.

The Helmholtz's law of conservation of the vortex flow lead to the concept of potential vorticity: By combining the vortex density equation with the continuity equation, one can show that

Is preserved in time. PV is called the potential vorticity.

Comments

Alternative definition

In the literature is also the definition of

To find.

Nomenclature

The terms vorticity, vortex density, Wirbelhaftigkeit, vorticity, turbulence, vorticity, vortex filament and the naming of the vortex density and vorticity transport equation are not clearly defined and therefore hard against each delineated. In the literature there are some conflicting data and definitions.