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Free Vortex Forced Vortex

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, we have,

• Integration of Eq 1 gives

• 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. (2), Ω becomes zero. Therefore we conclude that a free vortex flow is irrotational, and hence, it is also referred to as irrotational 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. (2)]. 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.

Pressure Distribution in a Free Vortex Flow

• Pressure distribution in a vortex flow is usually found out by integrating the equation of motion in the r direction. The equation of motion in the radial direction for a vortex flow can be written as

• Integrating Eq. (4) with respect to dr, and considering the flow to be incompressible we have,

• For a free vortex flow,

• Hence Eq. 5 becomes

• If the pressure at some radius r = ra, is known to be the atmospheric pressure patm then equation (6) can be written as



where z and za are the vertical elevations (measured from any arbitrary datum) at r and ra.

• Equation (7) can also be derived by a straight forward application of Bernoulli’s equation between any two points at r = r_a and r = r.

• In a free vortex flow total mechanical energy remains constant. There is neither any energy interaction between an outside source and the flow, nor is there any dissipation of mechanical energy within the flow. The fluid rotates by virtue of some rotation previously imparted to it or because of some internal action.

• Some examples are a whirlpool in a river, the rotatory flow that often arises in a shallow vessel when liquid flows out through a hole in the bottom (as is often seen when water flows out from a bathtub or a wash basin), and flow in a centrifugal pump case just outside the impeller.

Cylindrical Free Vortex

• A cylindrical free vortex motion is conceived in a cylindrical coordinate system with axis z directing vertically upwards, where at each horizontal cross-section, there exists a planar free vortex motion with tangential velocity given by Eq. (2).

• The total energy at any point remains constant and can be written as

• The pressure distribution along the radius can be found from Eq. (8) by considering z as constant; again, for any constant pressure p, values of z, determining a surface of equal pressure, can also be found from Eq. (8).

• If p is measured in gauge pressure, then the value of z, where p = 0 determines the free surface, if one exists.

Forced Vortex Flows

• 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,


• 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.

• The vorticity Ω for the flow field can be calculated as

• Therefore, a forced vortex motion is not irrotational; rather it is a rotational flow with a constant vorticity 2ω. Equation (1) 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,

• Thus, we see from Eq. (10) 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. (10), 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.