Stream function

(Difference between revisions)

Revision as of 11:11, 12 September 2005

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.

@@ Line 22: / Line 22: @@
 v = \frac{\partial \psi}{\partial x}
 </math>
+Conversely, the stream function at any point <math>P</math> can be obtained from the velocity field by a line integral
+<math>
+\psi(P) = \psi(P_o) + \int_{P_o}^P [ v(x,y,t) dx - u(x,y,t) dy ]
+</math>
+where <math>P_o</math> is some reference point and one can assume <math>\psi(P_o) = 0</math> since the stream function is determined only upto a constant.