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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 ${\displaystyle rsin\theta }$ about the x-axis.

The equation of continuity for incompressible flow in spherical polar coordinates is:

${\displaystyle {\frac {\partial }{\partial x}}(r^{2}v_{r}sin\theta )}$ +${\displaystyle {\frac {\partial }{\partial \theta }}(rv_{\theta }sin\theta )=0}$                                (1)

where ${\displaystyle v_{r}}$ and ${\displaystyle v_{\theta }}$ are radial and tangential velocities. Therefore a spherical polar stream function exists such that

${\displaystyle v_{r}}$ = ${\displaystyle {\frac {-1}{r^{2}sin\theta }}{\frac {\partial \psi }{\partial \theta }}}$                     ${\displaystyle v_{\theta }}$ = ${\displaystyle {\frac {1}{rsin\theta }}{\frac {\partial \psi }{\partial r}}}$                    (2)

Similarly a velocity potential ${\displaystyle \phi (r,\theta )}$ exists such that

${\displaystyle v_{r}}$= ${\displaystyle {\frac {\partial \phi }{\partial r}}}$                    ${\displaystyle v_{\theta }}$ = ${\displaystyle {\frac {1}{r}}{\frac {\partial \phi }{\partial x}}}$                                               (3)

Spherical polar coordinates for axisymmetric flow

These formulas help to deduce the ${\displaystyle \psi }$ and ${\displaystyle \phi }$ functions for various elementary axisymmetric potential flows.

Uniform stream in the x Direction

The components of a stream ${\displaystyle U_{\infty }}$ in x direction are:

${\displaystyle v_{r}=U_{\infty }cos\theta }$                     ${\displaystyle v_{\theta }=-U_{\infty }sin\theta }$                            (4)

Substituting in eqs. 2 and and 3 and integrating them gives
${\displaystyle \psi ={\frac {-1}{2}}U_{\infty }r^{2}sin^{2}\theta }$                     ${\displaystyle \phi =U_{\infty }rcos\theta }$                   (5)

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 ${\displaystyle {\frac {Q}{4\pi r^{2}}}}$.  Thus

${\displaystyle v_{r}={\frac {Q}{4\pi r^{2}}}={\frac {m}{r^{2}}}}$                     ${\displaystyle v_{\theta }=0}$                     (6)

With ${\displaystyle m={\frac {Q}{4\pi }}}$ for convenience. Integrating Eqs. 2 and 3 gives

${\displaystyle \psi =mcos\theta }$                     ${\displaystyle \phi =-{\frac {m}{r}}}$                         (7)

For a point sink, change m to –m in Eq. 6.

Point Doublet

A source can be placed at ${\displaystyle (x,y)=(-a,0)}$ and an equal sink at ${\displaystyle (+a,0)}$. On taking a limit when ${\displaystyle a}$ tends to 'zero' with product ${\displaystyle 2am=\lambda }$ being constant

${\displaystyle \psi _{doublet}=\lim _{a\to 0\ 2am=\lambda }(mcos\theta _{source}-mcos\theta _{sink})={\frac {\lambda sin^{2}\theta }{r}}}$                     (8)

The velocity potential of point doublet can be given by:

${\displaystyle \phi _{doublet}=\lim _{a\to 0\ 2am=\lambda }}$ ${\displaystyle ({\frac {-m}{r_{source}}}+{\frac {m}{r_{sink}}})={\frac {\lambda cos\theta }{r^{2}}}}$                                           (9)

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.

${\displaystyle \psi ={\frac {-1}{2}}U_{\infty }r^{2}sin^{2}\theta +mcos\theta }$                                               (10)

From Eqn. (2), the velocity components can be written after differentiation as:

${\displaystyle v_{r}}$= ${\displaystyle U_{\infty }cos\theta +{\frac {m}{r^{2}}}}$                     ${\displaystyle v_{\theta }=-U_{\infty }sin\theta }$                    (11)

[[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 ${\displaystyle \theta =180^{o}}$ and at ${\displaystyle r=a=({\frac {m}{U_{\infty }}})^{1/2}}$, as shown in the Fig. Suppose m = ${\displaystyle U_{\infty }a^{2}}$, we can write the stream function as:

${\displaystyle {\frac {\psi }{U_{\infty }a^{2}}}=cos\theta -{\frac {1}{2}}({\frac {r}{a^{2}}})^{2}sin^{2}\theta }$                                              (12)

The value of stream surface passing through the stagnation point ${\displaystyle (r,\theta )=(a,\pi )}$ is ${\displaystyle \psi =-U_{\infty }a^{2}}$ 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 ${\displaystyle R=2a}$ about the x-axis far down the stream.

Fig 3:Streamlines for Rankine half-body of revolution.

At ${\displaystyle \theta =70.5^{o}}$,${\displaystyle r=a{\sqrt {3}}}$, ${\displaystyle V_{s}=1.155U_{\infty }}$, 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 ${\displaystyle U_{\infty }}$, 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

${\displaystyle \psi ={\frac {-1}{2}}U_{\infty }r^{2}sin^{2}\theta +{\frac {\lambda }{r}}sin^{2}\theta }$                              (13)

On examining this relation, the steam surface ${\displaystyle \psi =0}$ corresponds to the sphere of radius:

${\displaystyle r=a=({\frac {2\lambda }{U_{\infty }}})^{\frac {1}{3}}}$                                                          (14)

Taking${\displaystyle \lambda ={\frac {1}{2}}U_{\infty }a^{3}}$ for convenience, we rewrite Eqn. (13) as

${\displaystyle {\frac {\psi }{{\frac {1}{2}}U_{\infty }a^{2}}}}$= ${\displaystyle -sin^{2}\theta ({\frac {r^{2}}{a^{2}}}-{\frac {a}{r}})}$                                       (15)

Below is the plot of streamlines for this sphere. Differentiating Eqn. (2) , we get the velocity components as

${\displaystyle v_{r}=U_{\infty }cos\theta (1-{\frac {a^{3}}{r^{3}}})}$                                             (16)

${\displaystyle v_{\theta }=-{\frac {1}{2}}U_{\infty }sin\theta (2+{\frac {a^{3}}{r^{3}}})}$                                      (17)

The radial velocity vanishes at the surface of sphere r = a, as expected. A stagnation point exists at the front ${\displaystyle (a,\pi )}$ and the rear ${\displaystyle (a,0)}$ of the sphere.

Fig 4:Streamlines and potential lines for inviscid flow past a sphere

At the shoulder ${\displaystyle (a,\pm {\frac {1}{2\pi }})}$ , there is maximum velocity where ${\displaystyle v_{r}=0}$ and ${\displaystyle v_{\theta }-{\frac {3}{2}}U_{\infty }}$. The surface velocity distribution is

${\displaystyle V_{s}=-v_{\theta |r=a}={\frac {3}{2}}U_{\infty }sin\theta }$                (18)

References

Fluid Mechanics - Frank M. White
Fluid Mechanics and Hydraulic mechanics by R.K. Bansal.

Category:Mechanical engineering