Uniform Flow

Satisfaction
Uniform Flow
  • Velocity does not change with y-coordinate
  • There exists only one component of velocity which is in the x direction.
  • Magnitude of the velocity is U0 .
Since   
 
              or,  
 
 
 Thus,    
(20.4)
  
Using stream function ψ for  uniform flow
 
           so 
(20.5)
The constants of integration C1 and K1   are arbitrary.
 The values of ψ and Φ for different streamlines and velocity potential lines may change but flow pattern is unaltered
. The constants of integration may be omitted, without any loss of generality and it is possible to write
(20.6)


Fig 20..2          (a)  Flownet for a Uniform Stream        (b)     Flownet for uniform Stream with an Anglea with x-axis
These are plotted in Fig. 20.2(a) and consist of a rectangular mesh of straight streamlines and orthogonal straight potential-lines (remember streamlines and potential lines are always orthogonal ). It is conventional to put arrows on the streamlines showing the direction of flow.

In terms of polar (r - θ) coordinate, Eq. (20.6) becomes
(20.7)

Flow at an angle
If we consider a uniform stream at an angle α to the x-axis as shown in Fig. 20.2b. we require that
 
             and     
(20.8)
Integrating. we obtain for a uniform velocity U0 at an angle α, the stream function and velocity potential respectively as
(20.9)