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.

Uniform Flow
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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:

Cartesian coordinates:

       and 

Cylindrical coordinates:

       and  

Note that the lines of the constant velocity potential (equipotential lines) are orthogonal to the lines of the stream function (streamlines).

Source

Sink

When a fluid flows radially outward from a point source, the velocities are

     v_r = m / (2πr)   and   v_θ = 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.

Vortex

Vortex Example: Tornado

A vortex can be obtained by reversing the velocity potential and stream functions for a point source such that

     φ = Kθ   and     ψ = -K ln(r)

where K is a constant indicating the strength of the vortex. Now, the equipotential lines are radial lines while the streamlines are given by the concentric circles. The velocities of a vortex are,

     v_r = 0   and   v_θ = K/r

The strength of a vortex can be described using the concept of circulation (Γ), which is defined as,

     

where the integral is taken around a closed arbitrary curve C. For irrotational flows (), the circulation becomes

     

However, when the closed curve consists of a singularity point (such as the case of a vortex), the circulation is non-zero,

Superposition of a Source and a Sink

Streamlines for a Doublet

By combining a source and a sink of equal strength using the method of superposition, the stream function is given by

     ψ = ψ_source + ψ_sink = -(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 for practice.

Superposition of a Uniform Flow
and a Source

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 = ψ_{uniform flow} + ψ_source = U r sinθ + (m/2π) θ

and the corresponding velocity potential is

     φ = φ_{uniform flow} + φ_source = U r cosθ + (m/2π) ln(r)

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 v_r and v_θ are zero. For v_r , this gives,

     0 = U cosπ + m / (2πb)

     b = m / (2πU)

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,

     ψ_stag = 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 , ψ_stag, going through the stagnation point can be determined by,

     m/2 = 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.

Combination of Uniform Flow and Point Source