Stream function
The stream function is defined for two-dimensional flows of various kinds. The stream function can be used to plot streamlines, which represent the trajectories of particles in a steady flow. Streamlines are perpendicular to equipotential lines. In most cases, the stream function is the imaginary part of the complex potential, while the potential function is the real part.
Considering the particular case of fluid dynamics, the difference between the stream function values at any two points gives the volumetric flow rate (or volumetric flux) through a line connecting the two points.
Since streamlines are tangent to the velocity vector of the flow, the value of the stream function must be constant along a streamline. If there were a flux across a line, it would necessarily not be tangent to the flow, hence would not be a streamline.
The usefulness of the stream function lies in the fact that the velocity components in the x- and y- directions at a given point are given by the partial derivatives of the stream function at that point. A stream function may be defined for any flow of dimensions greater than or equal to two, however the two dimensional case is generally the easiest to visualize and derive.
Taken together with the velocity potential, the stream function may be used to derive a complex potential for a potential flow. In other words, the stream function accounts for the solenoidal part of a two-dimensional Helmholtz decomposition, while the velocity potential accounts for the irrotational part.
Contents
Two dimensional stream function
Definitions
The sign of the stream function depends on the definition used.
One way is to define the stream function \(\psi\) for a two dimensional flow such that the flow velocity can be expressed as: \[ \mathbf{u}= \nabla \times \boldsymbol{\psi} \]
Where \(\boldsymbol{\psi} = (0,0,\psi)\) if the velocity vector \(\mathbf{u} = (u,v,0)\).
In Cartesian coordinate system this is equivalent to \[ u= \frac{\partial\psi}{\partial y},\qquad v= -\frac{\partial\psi}{\partial x} \] Where \(u\) and \(v\) are the velocities in the \(x\) and \(y\) coordinate directions, respectively.
Alternative definition (opposite sign)
Another definition (used more widely in meteorology and oceanography than the above) is
\[\mathbf{u}=\mathbf{z}\times\nabla\psi'\equiv(-\psi'_y,\psi'_x,0)\],
where \(\mathbf{z}\) is a unit vector in the \(+z\) direction and the subscripts indicate partial derivatives.
Note that this definition has the opposite sign to that given above (\(\psi'=-\psi\)), so we have \[ u= -\frac{\partial\psi'}{\partial y},\qquad v= \frac{\partial\psi'}{\partial x} \] in Cartesian coordinates.
Both formulations of the stream function constrain the velocity to satisfy the two dimensional continuity equation exactly: \[ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \]
Derivation of the two dimensional stream function
Consider two points A and B in two dimensional plane flow. If the distance between these two points is very small: δn, and a stream of flow passes between these points with an average velocity, q perpendicular to the line AB, the volume flow rate per unit thickness, δΨ is given by: \[\delta \psi = q \delta n\,\]
As δn → 0, rearranging this expression, we get: \[q = \frac{\partial \psi}{\partial n}\,\]
Now consider two dimensional plane flow with reference to a coordinate system. Suppose an observer looks along an arbitrary axis in the direction of increase and sees flow crossing the axis from left to right. A sign convention is adopted such that the velocity of the flow is positive.
Flow in Cartesian coordinates
By observing the flow into an elemental square in an x-y Cartesian coordinate system, we have: \[\delta \psi = u \delta y\,\] \[\delta \psi = -v \delta x\,\]
where u is the velocity parallel to and in the direction of the x-axis, and v is the velocity parallel to and in the direction of the y-axis. Thus, as δn → 0 and by rearranging, we have: \[u = \frac{\partial \psi}{\partial y}\,\] \[v = - \frac{\partial \psi}{\partial x}\,\]
Flow in polar coordinates
By observing the flow into an elemental square in an r–θ polar coordinate system, we have: \[\delta \psi = v_r ( r \delta \theta ),\,\] \[\delta \psi = -v_\theta \delta r,\,\]
where vr is the radial velocity component (parallel to the r-axis), and vθ is the tangential velocity component (parallel to the θ-axis). Thus, as δr → 0 and δθ → 0, by rearranging we have: \[v_r = \frac{1}{r} \frac{\partial \psi}{\partial \theta},\,\] \[v_\theta = -\frac{\partial \psi}{\partial r}.\,\]
Continuity: the derivation
Consider two dimensional plane flow within a Cartesian coordinate system. Continuity states that if we consider incompressible flow into an elemental square, the flow into that small element must equal the flow out of that element.
The total flow into the element is given by: \[\delta \psi_{in} = u \delta y + v \delta x.\,\]
The total flow out of the element is given by: \[\delta \psi_{out} = \left( u + \frac{\partial u}{\partial x}\delta x \right) \delta y + \left( v + \frac{\partial v}{\partial y}\delta y \right) \delta x.\,\]
Thus we have: \[\delta \psi_{in} = \delta \psi_{out}\,\] \[ u \delta y + v \delta x\ = \left( u + \frac{\partial u}{\partial x}\delta x \right) \delta y + \left( v + \frac{\partial v}{\partial y}\delta y \right) \delta x\,\]
and simplifying to: \[\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0.\]
Substituting the expressions of the stream function into this equation, we have:
\[\frac{\partial^2 \psi}{\partial x \partial y} - \frac{\partial^2 \psi}{\partial y \partial x} = 0.\]
Vorticity
In Cartesian coordinates, the stream function can be found from vorticity using the following Poisson's equation: \[\nabla ^2 \psi = -\omega\] or \[\nabla ^2 \psi' = +\omega\]
where \( \vec \omega = ( 0, 0, \omega ) \) and \(\vec \omega = \vec \nabla \times \vec v .\)
See also
References
- B. S. Massey and J. Ward-Smith, Mechanics of Fluids, 7th ed., Nelson Thornes, UK (1998).
- F. M. White, Fluid Mechanics, 5th ed., McGraw-Hill, New York (2003).
- T. W. Gamelin, Complex Analysis, Springer, New York (2001). ISBN 0-387-95093-1.
- AMS Glossary of Meteorology entry: [1]pl:Funkcja prądu
