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

	(14.11)
	(14.12)

• Integrating Eq. (14.12) with respect to dr, and considering the flow to be incompressible we have,

(14.13)

• For a free vortex flow,

• Hence Eq. 14.13 becomes

(14.14)

• If the pressure at some radius r = r_a, is known to be the atmospheric pressure p_atm then equation (14.14) can be written as

	(14.15)

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

• Equation (14.15) 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 (Fig. 14.1), where at each horizontal cross-section, there exists a planar free vortex motion with tangential velocity given by Eq. (14.10).
  
  
• The total energy at any point remains constant and can be written as

(14.16)

• The pressure distribution along the radius can be found from Eq. (14.16) 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. (14.16).
  
  
• If p is measured in gauge pressure, then the value of z, where p = 0 determines the free surface (Fig. 14.1), if one exists.

Fig 14.1 Cylindrical Free Vortex