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
![](14-3_pressure_dist_fre_vortex_flow_files/image002.gif)
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(14.11) |
![](14-3_pressure_dist_fre_vortex_flow_files/image004.gif) |
(14.12) |
- Integrating Eq. (14.12) with respect
to dr, and considering the flow to be incompressible we
have,
![](14-3_pressure_dist_fre_vortex_flow_files/image006.gif) |
(14.13) |
![](14-3_pressure_dist_fre_vortex_flow_files/image010.gif) |
(14.14) |
- If the pressure at some radius r =
ra, is known to be the atmospheric pressure
patm then equation (14.14) can be written
as
![](14-3_pressure_dist_fre_vortex_flow_files/image012.gif) |
|
![](14-3_pressure_dist_fre_vortex_flow_files/image014.gif) |
(14.15) |
where z and za
are the vertical elevations (measured from any arbitrary
datum) at r and ra.
- Equation (14.15) can also be derived
by a straight forward application of Bernoulli’s equation
between any two points at r = ra 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
![](14-3_pressure_dist_fre_vortex_flow_files/image016.gif) |
(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.
![](images/fig_14.1.gif)
Fig 14.1 Cylindrical Free Vortex
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