It was shown previously that two-dimensional incompressible, inviscid, and irrotational flow can be described by the
velocity potential, φ, or
stream function, ψ, using the Laplace's equation:
In the following sections, some simple plane potential flows (e.g., uniform flow, source and sink, vortex and doublet) will be introduced. Also, since Laplace's equation is linear, various solutions can be combined to form other solutions. Therefore, some of the real flow problems (e.g., half-body) are obtained by combining these simple plane potential flows using the method of superposition.
Normally, the stream function and/or velocity potential equation (Laplace's Equation) is solved easily using finite difference methods or finite element methods. An example of this is given in the graphic.
Greek Lower Case Letter
Phi can be written two ways
or (each Internet Browser does it differently).
Also, sometimes Upper
Case Phi, , and Upper Case Psi, , is used for
Velocity Potential and Stream Function, respectively.
Uniform flow is the simplest form of potential flow. For flow in
a specific direction, the velocity potential is
φ = U (x cosα + y sinα)
while the stream function is
ψ = U (y cosα - x sinα)
where α represents the angle between the flow direction and the x-axis (as shown in the figure). Recall that the
velocity potential and
stream function are related to the component velocity in the 2 dimensional
flow field as follows:
Note that the lines of the constant velocity potential (equipotential
lines) are orthogonal to the lines of the stream function (streamlines).
When a fluid flows radially outward from a point source, the
vr = m / (2πr) and vθ = 0
where m is the volume flow rate from the
line source per unit length. The velocity potential and stream function
can then be represented as
respectively. When m is
negative, the flow is inward, and it represents a sink. The volume
flow rate per unit depth, m, indicates the strength of the
source or sink. Note that as r approaches zero, the radial velocity goes
to infinity. Hence, the origin represents a singularity. As shown
in the figure, the equipotential lines are given by the concentric circles
while the streamlines are the radial lines.
By combining a source and a sink of equal strength using the method
of superposition, the stream function is given by
ψ = ψsource + ψsink = -(m/2π) (θ1 - θ2)
Through considerable manipulation (i.e., geometric relationships and
trigonometric identities), the above equation can be rewritten as
For small values of a gives,
A doublet is obtained by letting the distance between the source and sink
approach zero (i.e., distance "a" tends to zero) which means r/(r2 - a2) -> 1/r. The stream function for
a doublet then becomes
ψ = -Ksinθ/r
where K is a constand equal to ma/π, and is called the strength of the doublet.
The streamlines of a typical doublet are shown in the figure. The corresponding velocity potential is
φ = Kcosθ/r
For simplicity, the details of the velocity potential derivation are not given here.
Students are encouraged to go through the above derivation process themselves
Flow around a half-body (or commonly referred to as a Rankine shape) can be obtained by the superposition of a uniform
flow with a source. The combined stream function is given by
ψ = ψuniform
flow + ψsource =
U r sinθ + (m/2π) θ
and the corresponding velocity potential is
φ = φuniform
flow + φsource = U r cosθ +
The velocity components are given by
Instead of using the source strength, m, it is common to describe the velocities in terms of b (distance from the source to the stagnation point). The b distance is helpful in graphing and describing the half-body or Rankine shape. The b term can be related to m by considering the stream line going through the stagnation point (θ = π and r = b) where vr and vθ are zero. For vr , this gives,
0 = U cosπ
+ m / (2πb)
b = m / (2πU)
Past a Half-Body
Using b in place of m, the total velocity at any point in the flow field is,
Now back to the interesting half-body or Rankine shape. The stream line that goes through the stagnation point (θ = π and r = b) is also the stream line that defines the surface edge,
U b sinπ + (m/2π) π
= 0 + m/2 = m/2
By replacing this streamline with a solid boundary, one can then clearly see that flow around a half-body can indeed be represented by the superposition of a uniform flow with a source. The equation describing the stream function , ψstag, going through the stagnation point can be determined by,
U r sinθ + (m/2π) θ
m - (m/ π) θ = 2 U r sinθ
m (π - θ) = 2π U r sinθ
Or in terms of b,
b (π - θ) = r sinθ
This equation gives the surface equation in polar coordinates, r and θ.