Plane Circular Vortex Flows

Plane Circular Vortex Flows

• Plane circular vortex flows are defined as flows where streamlines are concentric circles. Therefore, with respect to a polar coordinate system with the centre of the circles as the origin or pole, the velocity field can be described as

where V_θ and V_r are the tangential and radial component of velocity respectively.

• The equation of continuity for a two dimensional incompressible flow in a polar coordinate system is

which for a plane circular vortex flow gives  i.e. V_θ is not a function of θ. Hence, V_θ is a function of  r only.

• We can write for the variation of total mechanical energy with radius as

(14.8)

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Free Vortex Flows

• Free vortex flows are the plane circular vortex flows where the total mechanical energy remains constant in the entire flow field. There is neither any addition nor any destruction of energy in the flow field.
• Therefore, the total mechanical energy does not vary from streamline to streamline. Hence from Eq. (14.8), we have,

or,	(14.9)

• Integration of Eq 14.9 gives

(14.10)

• The Eq. (14.10) describes the velocity field in a free vortex flow, where C is a constant in the entire flow field. The vorticity in a polar coordinate system is defined by -

• In case of vortex flows, it can be written as

• For a free vortex flow, described by Eq. (14.10),Ω becomes zero. Therefore we conclude that a free vortex flow is irrotational, and hence, it is also referred to asirrotational vortex.
• It has been shown before that the total mechanical energy remains same throughout in an irrotational flow field. Therefore, irrotationality is a direct consequence of the constancy of total mechanical energy in the entire flow field and vice versa.
• The interesting feature in a free vortex flow is that as  [Eq. (14.10)]. It mathematically signifies a point of singularity at r = 0 which, in practice, is impossible. In fact, the definition of a free vortex flow cannot be extended as r = 0 is approached.
• In a real fluid, friction becomes dominant as r→0 and so a fluid in this central region tends to rotate as a solid body. Therefore, the singularity at r = 0 does not render the theory of irrotational vortex useless, since, in practical problems, our concern is with conditions away from the central core.