Flows where streamlines are concentric circles and the tangential velocity is directly proportional to the radius of curvature are known as plane circular forced vortex flows.
The flow field is described in a polar coordinate system as,
(14.17a)
and
(14.17b)
All fluid particles rotate with the same angular velocity ω like a solid body. Hence a forced vortex flow is termed as a solid body rotation.
Therefore, a forced vortex motion is not irrotational; rather it is a rotational flow with a constant vorticity 2ω. Equation (14.8) is used to determine the distribution of mechanical energy across the radius as
Integrating the equation between the two radii on the same horizontal plane, we have,
(14.18)
Thus, we see from Eq. (14.18) that the total head (total energy per unit weight) increases with an increase in radius. The total mechanical energy at any point is the sum of kinetic energy, flow work or pressure energy, and the potential energy.
Therefore the difference in total head between any two points in the same horizontal plane can be written as,
Substituting this expression of H2-H1 in Eq. (14.18), we get
The same equation can also be obtained by integrating the equation of motion in a radial direction as
To maintain a forced vortex flow, mechanical energy has to be spent from outside and thus an external torque is always necessary to be applied continuously.
Forced vortex can be generated by rotating a vessel containing a fluid so that the angular velocity is the same at all points.
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