Basics of Fluid Mechanics, 2014a

348 CHAPTER 10. POTENTIAL FLOW
Solution
The circulation can be carried by the integration
Γ=
∮ =0 {}}{
U · i rdθds=0
(10.V.a)
Since the velocity is perpendicular to the path at every point on the path, the integral
identically is zero.
End Solution
Thus, there are two kinds of potential functions one where there are single value
and those with multi value. The free vortex is the cases where the circulation add the
value of the potential function every rotation. Hence, it can be concluded that the
potential function of vortex is multi value which increases by the same amount every
time, c 1 2 π. In this case value at θ =0is different because the potential function
did not circulate or encompass a singular point. In the other cases, every additional
enclosing adds to the value of potential function a value.
It was found that the circulation, Γ is zero when
there is no singular point within the region inside the
path.
For the free vortex the integration constant can be found if the circulation is
known as
c 1 = Γ
(10.106)
2 π
In the literature, the term Γ is, some times, referred to as the “strength” of the vortex.
The common form of the stream function and potential function is in the form of
φ = Γ
2 π (θ − θ 0)+φ 0 (10.107a)
Superposition of Flows
ψ = Γ
2 π ln ( r
r 0
)
+ ψ 0 (10.107b)
For incompressible flow and two dimensional the continuity equation reads
∇·U = ∇·∇φ = ∇ 2 φ = ∂2 φ
∂x 2 + ∂2 φ
=0 (10.108)
∂y2 The potential function must satisfy the Laplace’s equation which is a linear partial
differential equation. The velocity perpendicular to a solid boundary must be zero