Basics of Fluid Mechanics, 2014a

342 CHAPTER 10. POTENTIAL FLOW
Example 10.4:
Is there a potential based on the following stream function
ψ =3x 5 − 2 y
(10.IV.a)
Solution
Equation (10.74) dictates what are the requirements on the stream function. According
to this equation the following must be zero
∂ 2 ψ
∂y 2 + ∂2 ψ
∂x 2 ?
=0 (10.IV.b)
In this case it is
0 ? =0+60x 3 (10.IV.c)
Since x 3 is only zero at x =0the requirement is fulfilled and therefor this function
cannot be appropriate stream function.
End Solution
10.3 Potential Flow Functions Inventory
This section describes several simple scenarios of the flow field. These flow fields will
be described and exhibits utilizition of the potential and stream functions. These flow
fields can be combined by utilizing superimposing principle.
Uniform Flow
The trivial flow is the uniform flow in which the fluid field moves directly and
uniformly from one side to another side. This flow is further simplified, that is the
coordinates system aligned with to flow so the x–coordinate in the direction of the flow.
In this case the velocity is given by
U x = U 0
U y =0
(10.75)
and according to definitions in this chapter
Hence, it can be obtained that
U x = ∂φ
∂x = ∂ψ
∂y = U 0 (10.76)
φ = U 0 x + f y (y)
ψ = U 0 x + f x (x)
(10.77)