The potential flow which is symmetric to the axis of a plane flow is known as axisymmetric potential flow. It can be determined using superposition technique of plane flows. Some of the examples are as follows:
Table1: Plane flow and counterpart axisymmetric flow
Spherical Polar Coordinates
Spherical polar coordinates are used to express axisymmetric potential flows. Only two coordinates (r,θ) are used and the flow properties are constant on a circle of radius about the x-axis.
The equation of continuity for incompressible flow in spherical polar coordinates is:
where and are radial and tangential velocities. Therefore a spherical polar stream function exists such that
= = (2)
Similarly a velocity potential exists such that
= = (3)
These formulas help to deduce the and functions for various elementary axisymmetric potential flows.
Uniform stream in the x Direction
The components of a stream in x direction are:
Substituting in eqs. 2 and and 3 and integrating them gives
The arbitrary constants of integration have been neglected.
Point Source or Sink
Consider a volume flux Q emanating from a point source. The flow will spread out radially and at radius r, it will be equal to . Thus
With for convenience. Integrating Eqs. 2 and 3 gives
For a point sink, change m to –m in Eq. 6.
A source can be placed at and an equal sink at . On taking a limit when tends to 'zero' with product being constant
The velocity potential of point doublet can be given by:
Uniform Stream plus a point source
On combination of Eqns. (5) and (7) we get the stream function for a uniform stream and a point source at the origin.
From Eqn. (2), the velocity components can be written after differentiation as:
[[File:Fig 2 Streamlines and potential lines due to a point doublet at the origin, from Eqns. (8) and (9).JPG|thumb|center|800x1200px|upright = 1.5|Fig 2:Streamlines and potential line due to a point doublet at the origin]]
Equating these equations with zero gives a stagnation point at and at , as shown in the Fig. Suppose m = , we can write the stream function as:
The value of stream surface passing through the stagnation point is which forms a half body of revolution enclosing a point source, as shown in Fig. Using this half body, a pitot tube can be simulated. The half body approaches the constant radius about the x-axis far down the stream.
At ,, , there occurs the maximum velocity and minimum pressure along the half body surface. There exists an adverse gradient downstream of this point because Vs slowly decelerates to , but no flow separation is indicated by boundary layer theory. Thus for a real half body flow, Eqn. (12) proves to be a realistic simulation. But if one adds the uniform stream to a sink to form a half body rear surface, the separation will be predictable and inviscid pattern would not be realistic.
Uniform Stream plus Point Doublet
From Eqns. (5) and (8), if we combine a uniform steam and a point doublet at the origin, we get
On examining this relation, the steam surface corresponds to the sphere of radius:
Taking for convenience, we rewrite Eqn. (13) as
Below is the plot of streamlines for this sphere. Differentiating Eqn. (2) , we get the velocity components as
The radial velocity vanishes at the surface of sphere r = a, as expected. A stagnation point exists at the front and the rear of the sphere.
At the shoulder , there is maximum velocity where and . The surface velocity distribution is
Fluid Mechanics - Frank M. White
Fluid Mechanics and Hydraulic mechanics by R.K. Bansal.