CFD Online Logo CFD Online URL
Home > Wiki > Stream function

Stream function

From CFD-Wiki

Jump to: navigation, search

The stream function is a scalar field variable which is constant on each streamline. It exists only in two-dimensional and axisymmetric flows.

On a streamline in two-dimensional flow

d\psi = \frac{\partial \psi}{\partial x} dx + \frac{\partial \psi}{\partial y} dy = 0

The equation of a streamline in two-dimensions is

v dx - u dy = 0

Comparing the two equations, we have

u = - \frac{\partial \psi}{\partial y}

v = \frac{\partial \psi}{\partial x}

Conversely, the stream function at any point P can be obtained from the velocity field by a line integral

\psi(P) = \psi(P_o) + \int_{P_o}^P [ v(x,y,t) dx - u(x,y,t) dy ]

where P_o is some reference point and one can assume \psi(P_o) = 0 since the stream function is determined only upto a constant.

If the flow is incompressible, then the continuity equation is identically satisfied

\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = -\frac{\partial^2 \psi}{\partial x \partial y} + \frac{\partial^2 \psi}{\partial y \partial x} = 0
My wiki