vorticity (redirected from vorticities)

vorticity [vȯr′tis·əd·ē]

(fluid mechanics)

For a fluid flow, a vector equal to the curl of the velocity of flow.

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Vorticity

A vector proportional to the local angular velocity of a fluid flow. The vorticity, , is a derived quantity in fluid mechanics, defined, for a flow field with velocity , by Eq. (1).

(1) 

As the curl of the velocity vector, the vorticity is a vector with the dimensions of both a frequency and an angular velocity . The component of vorticity along a particular axis is related to the rate of rotation of the fluid about the axis. For this reason, flows for which are described as irrotational. See Dimensional analysis, Velocity

Circulation

Closely related to vorticity is the fluid circulation, &Ggr;, defined, for any closed contour, C, in a fluid, by Eq. (2).

(2) 

In this definition, indicates the conventional counterclockwise contour integral around the contour C, is a unit vector tangent to the contour, and S is an arbitrary curved surface bounded by the contour C. [The equality of the two integrals in Eq. (2) may be deduced by the application of Stokes' theorem.] The circulation is thus a scalar quantity equal to the integrated component of vorticity normal to the surface around which &Ggr; is taken. Circulation is important because the Kutta-Joukowski law of aerodynamics states that the lift generated by a two-dimensional airfoil is . In this expression, is the fluid density, U is the free-stream velocity, and &Ggr; is the bound circulation of the airfoil, defined conventionally as the negative of the definition above. See Aerodynamics, Airfoil

Vortex line and vortex tube

A vortex line is defined as a line that is everywhere tangent to the local vorticity vector (analogous to a streamline). A series of adjacent vortex lines is referred to as a vortex tube. The first Helmholtz vortex law states that at any instant in time the circulation about all loops taken around the exterior of a vortex tube is the same. Thus, vortex tubes must either form loops entirely within a fluid or terminate at some fluid boundary. See Vortex

Kelvin's theorem

Kelvin's theorem considers how the circulation &Ggr; around a material loop in a fluid (a loop that moves with the fluid) varies in time. Starting with the Navier-Stokes equations, Lord Kelvin showed that if (1) the fluid is inviscid along the loop, (2) the fluid is subject only to potential body forces, and (3) the fluid pressure is a function of density alone, then the rate of change of &Ggr; is 0. In other words, the circulation around a material loop is time-independent. Kelvin's theorem may also be stated slightly differently: subject to the above three constraints, vortex lines are material lines, convected with the local fluid velocity. See Kelvin's circulation theorem

Generation

Kelvin's theorem can tell what happens when vorticity is already present in a flow, but it sheds no light on how vorticity is generated. To answer this question, it is useful to consider situations for which Kelvin's theorem is inapplicable: flow with viscosity, with nonpotential body forces, and for which the pressure is not solely a function of the density.

The action of viscosity has two effects on vorticity. One effect of viscosity is to cause the diffusion of vorticity in a fluid. A second effect of viscosity is the generation of vorticity at a wall where there is a pressure gradient at the wall. See Viscosity

A common example of a nonpotential body force is the Coriolis force, which is present in a rotating frame of reference. This force generates vorticity in a fluid, and is a major cause of the large-scale circulation in the atmosphere and oceans.

There are many flows for which the pressure may not be solely a function of the density (so-called baroclinic flows), such as the flow of gas with heat addition and the flow of water with salinity variations. Pressure gradients in such flows generate vorticity. This source of vorticity is called baroclinic torque, and is important in atmospheric flow, buoyancy-driven flow, and oceanographic flow.