Flow Past a Source (Part - 1) | Additional Documents & Tests for Civil Engineering (CE) PDF Download

Flow Past a Source
When a uniform flow is added to that due to a source -

• fluid emitted from the source is swept away in the downstream direction
• stream function and velocity potential for this flow will be the sum of those for uniform flow and source

Stream function; 
Velocity Potential; 
so, 
and  
              
Fig 23.1  The streamlines of the flow past a line source for equal increments of 2πψ/q
              The Plane coordinates are x/a, y/a where a=k/u

Explanation of Figure

• At the point x = -a, y = 0 fluid velocity is zero.
• This is called stagnation point of the flow
• Here the source flow is turned around by the oncoming uniform flow
• The parabolic streamline passing through stagnation point    seperates uniform flow from the source flow.
• The streamline becomes parallel to x axis as 

Flow Past Vortex
 when uniform flow is superimposed with a vortex flow -

• Flow will be asymmetrical about x - axis
• Again stream function and velocity potential will be the sum of those for uniform flow and vortex flow

Stream Function:  

Velocity Potential:  
so that;  


Flow About a Rotating Cylinder
 Magnus Effect

Flow about a rotating cylinder is equivalent to the combination of flow past a cylinder and a vortex. 
As such in addition to superimposed uniform flow and a doublet, a vortex is thrown at the doublet centre which will simulate a rotating cylinder in uniform stream.

 The pressure distribution will result in a force, a component of which will culminate in lift force

 The phenomenon of generation of lift by a rotating object placed in a stream is known as Magnus effect.

Velocity Potential and Stream Function

The velocity potential and stream functions for the combination of doublet, vortex and uniform flow are
   (clockwise rotation)   (23.1)

     (clockwise rotation)   (23.2)

By making use of either the stream function or velocity potential function, the velocity components are (putting x= rcosθ and y= rsinθ )
    (23.3)

   (23.4)


Stagnation Points
 At the stagnation points the velocity components must vanish. From Eq. (23.3), we get
      (23.5)

olution :

        From Eq. (23.5) it is evident that a zero radial velocity component may occur at
       

• along the circle,  

Eq. (23.4) depicts that a zero transverse velocity requires


At the stagnation point, both radial and transverse velocity components must be zero .
Thus the location of stagnation point occurs at


There will be two stagnation points since there are two angles for a given sine except for sin^-1(±1)

Determination of Stream Line

The streamline passing through these points may be determined by evaluating ψ at these points.

 Substitution of the stagnation coordinate (r, θ) into the stream function (Eq. 23.2) yield
   (23.7)

Equating the general expression for stream function to the above constant, we get


By rearranging we can write
   (23.8)
All points along the circle  satisfy Eq. (23.8) , since for this value of r, each quantity within parentheses in the equation is zero.

Considering the interior of the circle (on which ψ = 0) to be a solid cylinder, the outer streamline pattern is shown in Fig 23.2.


At the stagnation point

The limiting case arises for    and two stagnation points meet at the bottom as shown in Fig. 23.3.

In the case of a circulatory flow past the cylinder, the streamlines are symmetric with respect to the y-axis. The presures at the points on the cylinder surface are symmetrical with respect to the y-axis. There is no symmetry with respect to the x-axis. Therefore a resultant force acts on the cylinder in the direction of the y-axis, and the resultant force in the direction of the x-axis is equal to zero as in the flow without circulation; that is, the D'Alembert paradox takes place here as well. Thus, in the presence of circulation, different flow patterns can take place and, therefore, it is necessary for the uniqueness of the solution, to specify the magnitude of circulation.

Fig 23.3    Flow Past a Circular Cylinder with Circulation Value 

However, in all these cases the effects of the vortex and doublet become negligibly small as one moves a large distance from the cylinder.

The flow is assumed to be uniform at infinity.

We have already seen that the change in strength G of the vortex changes the flow pattern, particularly the position of the stagnation points but the radius of the cylinder remains unchanged.