A stream function defines the geometry of a streamline
in a given flow. NOTE: This explanation holds for incompressible flow
only. For compressible flow
, the changes in density must be taken into account.
Usually denoted by ψ, a stream function is useful in characterizing the flow over an arbitrary body. This is done by combining fundamental flow components to achieve the desired effect. Flow components are explained as follows, in polar coordinates
Uniform - ψ = V*r*sin(θ), where V is the relative wind
Source - ψ = (Λ⁄2π)*θ, where Λ is the source strength. A source flow can be considered a flow where the velocity is inversely proportional to distance (r) from the center.
Sink - ψ = (-Λ⁄2π)*θ. A sink flow is effectively the opposite of a source flow.
Vortex - ψ = (Γ⁄2π)*ln(r), where Γ is the vortex strength. A vortex flow is rotationally opposite to a source flow, in that the tangential
component of a vortex is equivalent to the radial
component of a source, and vice versa.
An important fact of the stream function is its relationship to the velocity field
, explained as follows:
= 1⁄r ∂ψ⁄∂θ