Vortex

A vortex (plural: vortices) is a spinning, often turbulent, flow of fluid. Any spiral motion with closed streamlines is vortex flow. The motion of the fluid swirling rapidly around a center is called a vortex. The speed and rate of rotation of the fluid in a free (irrotational) vortex are greatest at the center, and decrease progressively with distance from the center, whereas the speed of a forced (rotational) vortex is zero at the center and increases proportional to the distance from the center. Both types of vortices exhibit a pressure minimum at the center, though the pressure minimum in a free vortex is much lower.

Contents

Properties

Vortices display some special properties:

Dynamics

A vortex can be any circular or rotary flow. Perhaps unexpectedly, not all vortices possess vorticity. Vorticity is a mathematical concept used in fluid dynamics. It can be related to the amount of "circulation" or "rotation" in a fluid. In fluid dynamics, vorticity is the circulation per unit area at a point in the flow field. It is a vector quantity, whose direction is (roughly speaking) along the axis of the swirl. The vorticity of a free vortex is zero everywhere except at the center, whereas the vorticity of a forced vortex is non-zero. Vorticity is an approximately conserved quantity, meaning that it is not readily created or destroyed in a flow. Therefore, flows that start with minimal vorticity, such as water in a basin, create vortices with minimal vorticity, such as the characteristic swirling and approximately free vortex structure when it drains. By contrast, fluids that initially have vorticity, such as water in a rotating bowl, form vortices with vorticity, exhibited by the much less pronounced low pressure region at the center of this flow. Also in fluid dynamics, the movement of a fluid can be said to be vortical if the fluid moves around in a circle, or in a helix, or if it tends to spin around some axis. Such motion can also be called solenoidal. In the atmospheric sciences, vorticity is a property that characterizes large-scale rotation of air masses. Since the atmospheric circulation is nearly horizontal, the (3 dimensional) vorticity is nearly vertical, and it is common to use the vertical component as a scalar vorticity. Mathematically, vorticity \vec\omega is defined as the curl of the fluid velocity \vec\mathit{u}:

\vec \omega = \nabla \times \vec \mathit{u}.

Two types of vortex

In fluid mechanics, a distinction is often made between two limiting vortex cases. One is called the free (irrotational) vortex, and the other is the forced (rotational) vortex. These are considered below, using the following example:

Types of vortex illustrated by the movement of two autumn leaves
Reference position in a counter-clockwise vortex. In an irrotational vortex, the leaves preserve their original orientation while moving counter-clockwise. In a rotational vortex, the leaves rotate with the counter-clockwise flow.

Free (irrotational) vortex

When fluid is drawn down a plug-hole, one can observe the phenomenon of a free vortex or line vortex. The tangential velocity v varies inversely as the distance r from the center of rotation, so the angular momentum rv is uniform everywhere throughout the flow; the vorticity is zero everywhere (except for a singularity at the center-line) and the circulation about a contour containing r = 0 has the same value everywhere.[1] The free surface (if present) dips sharply (as r −2 ) as the center line is approached.

The tangential velocity is given by:

v_{\theta} = \frac{\Gamma}{2 \pi r}\,

where Γ is the circulation and r is the radial distance from the center of the vortex.

In non-technical terms, the fluid near the center of the vortex circulates faster than the fluid far from the center. The speed along the circular path of flow decreases as you move out from the center. At the same time the inner streamlines have a shorter distance to travel to complete a ring. If you were running a race on a circular track would you rather be on the inside or outside, assuming the goal was to complete a circle? Imagine a leaf floating in a free vortex. The leaf's tip points to the center and the blade straddles multiple streamlines. The outer flow is slow in terms of angle traversed and it exerts a backwards tug on the base of the leaf while the faster inner flow pulls the tip forwards. The drag force opposes rotation of the leaf as it moves around the circle.

Forced (rotational) vortex

In a forced vortex the fluid rotates as a solid body (there is no shear). The motion can be realized by placing a dish of fluid on a turntable rotating at ω radian/s; the fluid has vorticity of 2ω everywhere, and the free surface (if present) is a paraboloid.

The tangential velocity is given by:[1]

v_{\theta} = \omega r\,

where ω is the angular velocity and r is the radial distance from the center of the vortex.

Vortices in magnets

Different classes of vortex waves also exist in magnets. There are exact solutions to classical nonlinear magnetic equations e.g. Landau-Lifshitz equation, continuum Heisenberg model, Ishimori equation, nonlinear Schrödinger equation and so on.

Observations

A vortex can be seen in the spiraling motion of air or liquid around a center of rotation. The circular current of water of conflicting tides often form vortex shapes. Turbulent flow makes many vortices. A good example of a vortex is the atmospheric phenomenon of a whirlwind or a tornado or dust devil. This whirling air mass mostly takes the form of a helix, column, or spiral. Tornadoes develop from severe thunderstorms, usually spawned from squall lines and supercell thunderstorms, though they sometimes happen as a result of a hurricane.

In atmospheric physics, a mesovortex is on the scale of a few miles (smaller than a hurricane but larger than a tornado). [2] On a much smaller scale, a vortex is usually formed as water goes down a drain, as in a sink or a toilet. This occurs in water as the revolving mass forms a whirlpool. This whirlpool is caused by water flowing out of a small opening in the bottom of a basin or reservoir. This swirling flow structure within a region of fluid flow opens downward from the water surface.

Instances

See also

Notes

  1. ^ a b Clancy, L.J., Aerodynamics, sub-section 7.5

References and further reading

External links