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This page is about the axisymmetric potential flow. For other flow, see Laminar flow.

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:

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