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) |
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) |