Free of Irrotational Vortex

irrotational vortex

(free vortex)

Presented by: Fajardo, Maffeo Niccolo A.BSME

What is a vortex???

A vortex is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. These vortices are generally created at a moving boundary due to the shear resulting from the no-slip condition, but can also result from thermal circulation. Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in the wake of a boat, and the winds surrounding a tropical cyclone, tornado or dust devil.

Fig.1.1: An example of a vortex.

What is an irrotational vortex???

If the vorticity at a point in a flow field is nonzero, and the fluid particle that happens to occupy that point in space is rotating; the flow in that region is called rotational. Likewise, if the vorticity in a region of the flow is zero (or negligibly small), and fluid particles there are not rotating; the flow in that region is called irrotational.∇ x V = 0The vorticity of a fluid element cannot change except through the action of viscosity, non-uniform heating (temperature gradients), or other non-uniform phenomena. Thus if a flow originates in an irrotational region, it remains irrotational until some non-uniform process alters it. Irrotational vortices are also called free vortices.

Fig.1.2: An irrotational vortex divided in segments.

irrotational vortex

A free vortex is formed when water flows out of a vessel through a central hole in the base (Figure 1.3). The degree of the rotation depends on the initial disturbance. In a free cylindrical vortex, the velocity varies inversely with the distance from the axis of rotation (Figure 1.4).

Fig. 1.3: The water flows out the bottle through the small opening below.

(1)

The equation governing the surface profile is derived from the Bernoulli’s theorem:

Fig. 1.4: Velocity profile of a free vortex

(2)

irrotational vortex

Substituting the first equation (1) into the second equation (2) will give a new expression:

OR

which is the equation of a hyperbolic curve of nature 𝑦=𝐴/𝑥^2. This curve is asymptotic to the axis of rotation and to the horizontal plane through z=c (Figure 1.5).

Fig.1.5: Surface profile of a free vortex.

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