Physics:Stream function

Short description: Function for incompressible divergence-free flows in two dimensions

Streamlines – lines with a constant value of the stream function – for the incompressible potential flow around a circular cylinder in a uniform onflow.

The stream function is defined for incompressible (divergence-free) flows in two dimensions – as well as in three dimensions with axisymmetry. The flow velocity components can be expressed as the derivatives of the scalar stream function. The stream function can be used to plot streamlines, which represent the trajectories of particles in a steady flow. The two-dimensional Lagrange stream function was introduced by Joseph Louis Lagrange in 1781. The Stokes stream function is for axisymmetrical three-dimensional flow, and is named after George Gabriel Stokes.

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 flow velocity vector of the flow, the value of the stream function must be constant along a streamline. The usefulness of the stream function lies in the fact that the flow 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.

For two-dimensional potential flow, streamlines are perpendicular to equipotential lines. Taken together with the velocity potential, the stream function may be used to derive a complex potential. 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.

Two-dimensional stream function

Definitions

The volume flux through the curve between the points [math]\displaystyle{ A }[/math] and [math]\displaystyle{ P. }[/math]

Lamb and Batchelor define the stream function [math]\displaystyle{ \psi(x,y,t) }[/math] for an incompressible flow velocity field [math]\displaystyle{ (u(t),v(t)) }[/math] as follows. Given a point [math]\displaystyle{ P }[/math] and a point [math]\displaystyle{ A }[/math],

[math]\displaystyle{ \psi = \int_A^P \left( u\, \text{d}y - v\, \text{d}x \right) }[/math]

is the integral of the dot product of the flow velocity vector [math]\displaystyle{ (u,v) }[/math] and the normal [math]\displaystyle{ (+\text{d}y,-\text{d}x) }[/math] to the curve element [math]\displaystyle{ (\text{d}x,\text{d}y). }[/math] In other words, the stream function [math]\displaystyle{ \psi }[/math] is the volume flux through the curve [math]\displaystyle{ AP }[/math]. The point [math]\displaystyle{ A }[/math] is simply a reference point that defines where the stream function is identically zero. A shift in [math]\displaystyle{ A }[/math] results in adding a constant to the stream function [math]\displaystyle{ \psi }[/math] at [math]\displaystyle{ P }[/math].

An infinitesimal shift [math]\displaystyle{ \delta P=(\delta x,\delta y) }[/math] of the position [math]\displaystyle{ P }[/math] results in a change of the stream function:

[math]\displaystyle{ \delta \psi = u\, \delta y - v\, \delta x }[/math].

From the exact differential

[math]\displaystyle{ \delta\psi = \frac{\partial\psi}{\partial x}\, \delta x + \frac{\partial\psi}{\partial y}\, \delta y, }[/math]

the flow velocity components in relation to the stream function [math]\displaystyle{ \psi }[/math] have to be

[math]\displaystyle{ u= \frac{\partial\psi}{\partial y}, \qquad v = -\frac{\partial\psi}{\partial x}, }[/math]

in which case they indeed satisfy the condition of zero divergence resulting from flow incompressibility, i.e.

[math]\displaystyle{ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0. }[/math]

Definition by use of a vector potential

The sign of the stream function depends on the definition used.

One way is to define the stream function [math]\displaystyle{ \psi }[/math] for a two-dimensional flow such that the flow velocity can be expressed through the vector potential [math]\displaystyle{ \boldsymbol{\psi} : }[/math]

[math]\displaystyle{ \mathbf{u}= \nabla \times \boldsymbol{\psi} }[/math]

where [math]\displaystyle{ \boldsymbol{\psi} = (0,0,\psi) }[/math] if the flow velocity vector [math]\displaystyle{ \mathbf{u} = (u,v,0) }[/math].

In Cartesian coordinate system this is equivalent to

[math]\displaystyle{ u= \frac{\partial\psi}{\partial y},\qquad v= -\frac{\partial\psi}{\partial x} }[/math]

where [math]\displaystyle{ u }[/math] and [math]\displaystyle{ v }[/math] are the flow velocity components in the cartesian [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] coordinate directions, respectively.

Alternative definition (opposite sign)

Another definition (used more widely in meteorology and oceanography than the above) is

[math]\displaystyle{ \mathbf{u} = \mathbf{z}\times\nabla\psi' \equiv \left(-\psi'_y, \psi'_x, 0\right) }[/math],

where [math]\displaystyle{ \mathbf{z} = (0, 0, 1) }[/math] is a unit vector in the [math]\displaystyle{ +z }[/math] direction and the subscripts indicate partial derivatives.

Note that this definition has the opposite sign to that given above ([math]\displaystyle{ \psi' = -\psi }[/math]), so we have

[math]\displaystyle{ u = -\frac{\partial\psi'}{\partial y},\qquad v = \frac{\partial\psi'}{\partial x} }[/math]

in Cartesian coordinates.

All formulations of the stream function constrain the velocity to satisfy the two-dimensional continuity equation exactly:

[math]\displaystyle{ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 }[/math]

The last two definitions of stream function are related through the vector calculus identity

[math]\displaystyle{ \nabla\times\left(\psi\mathbf{z}\right) = \psi\nabla \times \mathbf{z} + \nabla\psi\times\mathbf{z} = \nabla\psi \times \mathbf{z} = \mathbf{z} \times \nabla\psi'. }[/math]

Note that [math]\displaystyle{ \boldsymbol{\psi} = \psi\mathbf{z} }[/math] in this two-dimensional flow.

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:

[math]\displaystyle{ \delta \psi = q \, \delta n }[/math]

As δn → 0, rearranging this expression, we get:

[math]\displaystyle{ q = \frac{\partial \psi}{\partial n} }[/math]

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 flow velocity is positive.

Flow in Cartesian coordinates

By observing the flow into an elemental square in an x-y Cartesian coordinate system, we have:

[math]\displaystyle{ \begin{align} \delta\psi &= u\, \delta y\, \\ \delta\psi &= -v\, \delta x\, \end{align} }[/math]

where u is the flow velocity parallel to and in the direction of the x-axis, and v is the flow velocity parallel to and in the direction of the y-axis. Thus, as δn → 0 and by rearranging, we have:

[math]\displaystyle{ \begin{align} u &= \frac{\partial\psi}{\partial y}\, \\ v &= -\frac{\partial\psi}{\partial x}\, \end{align} }[/math]

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:

[math]\displaystyle{ \delta\psi_\text{in} = u\, \delta y + v\, \delta x.\, }[/math]

The total flow out of the element is given by:

[math]\displaystyle{ \delta\psi_\text{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.\, }[/math]

Thus we have:

[math]\displaystyle{ \begin{align} \delta\psi_\text{in} &= \delta\psi_\text{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\, \end{align} }[/math]

and simplifying to:

[math]\displaystyle{ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0. }[/math]

Substituting the expressions of the stream function into this equation, we have:

[math]\displaystyle{ \frac{\partial^2\psi}{\partial x \partial y} - \frac{\partial^2\psi}{\partial y \partial x} = 0. }[/math]

Vorticity

The stream function can be found from vorticity using the following Poisson's equation:

[math]\displaystyle{ \nabla ^2 \psi = -\omega }[/math]

or

[math]\displaystyle{ \nabla ^2 \psi' = +\omega }[/math]

where the vorticity vector [math]\displaystyle{ \boldsymbol{\omega} = \nabla \times \mathbf{u} }[/math] – defined as the curl of the flow velocity vector [math]\displaystyle{ \mathbf{u} }[/math] – for this two-dimensional flow has [math]\displaystyle{ \boldsymbol{\omega} = ( 0, 0, \omega ), }[/math] i.e. only the [math]\displaystyle{ z }[/math]-component [math]\displaystyle{ \omega }[/math] can be non-zero.

Proof that a constant value for the stream function corresponds to a streamline

Consider two-dimensional plane flow within a Cartesian coordinate system. Consider two infinitesimally close points [math]\displaystyle{ P = (x,y) }[/math] and [math]\displaystyle{ Q = (x+dx,y+dy) }[/math]. From calculus we have that

[math]\displaystyle{ \begin{align} \psi (x + dx, y + dy) - \psi(x, y) &= {\partial\psi \over \partial x} dx + {\partial\psi \over \partial y} dy \\ &= \nabla\psi \cdot d\boldsymbol{r} \end{align} }[/math]

Say [math]\displaystyle{ \psi }[/math] takes the same value, say [math]\displaystyle{ C }[/math], at the two points [math]\displaystyle{ P }[/math] and [math]\displaystyle{ Q }[/math], then [math]\displaystyle{ d \boldsymbol{r} }[/math] is tangent to the curve [math]\displaystyle{ \psi = C }[/math] at [math]\displaystyle{ P }[/math] and

[math]\displaystyle{ 0 = \psi(x + dx, y + dy) - \psi(x, y) = \nabla \psi \cdot d \boldsymbol{r} }[/math]

implying that the vector [math]\displaystyle{ \nabla \psi }[/math] is normal to the curve [math]\displaystyle{ \psi = C }[/math]. If we can show that everywhere [math]\displaystyle{ \boldsymbol{u} \cdot \nabla \psi = 0 }[/math], using the formula for [math]\displaystyle{ \boldsymbol{u} }[/math] in terms of [math]\displaystyle{ \psi }[/math], then we will have proved the result. This easily follows,

[math]\displaystyle{ \boldsymbol{u} \cdot \nabla\psi = {\partial\psi \over \partial y}{\partial\psi \over \partial x} + \left(-{\partial\psi \over \partial x}\right) {\partial\psi \over \partial y} = 0. }[/math]

Existence of the stream function

For this discussion we'll assume we're working in a three-dimensional space domain with a right-handed Cartesian coordinate system. Suppose we have two-dimensional plane flow with velocity vector

[math]\displaystyle{ \mathbf{u} = \begin{bmatrix} u \\ v \\ 0 \end{bmatrix}. }[/math]

Then [math]\displaystyle{ \mathbf{u} }[/math] satisfies the curl-divergence equation

[math]\displaystyle{ (\nabla \cdot \mathbf{u})\, \mathbf{k} = - \nabla \times (R\, \mathbf{u}) }[/math]

where [math]\displaystyle{ \mathbf{k} }[/math] is a unit vector pointing in the positive [math]\displaystyle{ z }[/math] direction

[math]\displaystyle{ \mathbf{k} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} }[/math]

and [math]\displaystyle{ R }[/math] is the [math]\displaystyle{ 3 \times 3 }[/math] rotation matrix corresponding to a [math]\displaystyle{ 90^\circ }[/math] anticlockwise rotation about the positive [math]\displaystyle{ z }[/math] axis:

[math]\displaystyle{ R = R_z(90^\circ) = \begin{bmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}. }[/math]

If the flow is incompressible (i.e., [math]\displaystyle{ \nabla \cdot \mathbf{u} = 0 }[/math]) then the curl-divergence equation gives

[math]\displaystyle{ \mathbf{0} = \nabla \times (R\, \mathbf{u}) }[/math].

Then by Stokes' theorem the line integral of [math]\displaystyle{ R\, \mathbf{u} }[/math] over every closed loop vanishes

[math]\displaystyle{ \oint_{\partial\Sigma} (R\, \mathbf{u}) \cdot \mathrm{d}\mathbf{\Gamma}= 0. }[/math]

Hence, the line integral of [math]\displaystyle{ R\, \mathbf{u} }[/math] is path-independent. Finally, by the converse of the gradient theorem, a scalar function [math]\displaystyle{ \psi (x,y,z) }[/math] exists such that

[math]\displaystyle{ R\, \mathbf{u} = - \nabla \psi }[/math].

Here [math]\displaystyle{ \psi }[/math] represents the stream function.

Conversely, if the stream function exists, then [math]\displaystyle{ R\, \mathbf{u} = - \nabla \psi }[/math]. Substituting this result into the curl-divergence equation yields [math]\displaystyle{ \nabla \cdot \mathbf{u} = 0 }[/math] (i.e., the flow is incompressible).

In summary, the stream function for two-dimensional plane flow exists if and only if the flow is incompressible.

Properties of the stream function

1. The stream function [math]\displaystyle{ \psi }[/math] is constant along any streamline.
2. The rate of change of stream function with distance is equal in absolute value to the velocity component perpendicular to the direction of change.
3. If two incompressible flow fields are superimposed, then the stream function of the resulting flow field is the algebraic sum of the two original stream functions.

See also

• Elementary flow

References

Sources

• Batchelor, G. K. (1967), An Introduction to Fluid Dynamics, Cambridge University Press, ISBN 0-521-09817-3 
• Lamb, H. (1932), Hydrodynamics (6th ed.), Cambridge University Press, republished by Dover Publications, ISBN 0-486-60256-7 
• Massey, B. S.; Ward-Smith, J. (1998), Mechanics of Fluids (7th ed.), UK: Nelson Thornes 
• White, F. M. (2003), Fluid Mechanics (5th ed.), New York: McGraw-Hill 
• Gamelin, T. W. (2001), Complex Analysis, New York: Springer, ISBN 0-387-95093-1 
• "Streamfunction", AMS Glossary of Meteorology (American Meteorological Society), http://glossary.ametsoc.org/wiki/Streamfunction, retrieved 2014-01-30 

External links

• Joukowsky Transform Interactive WebApp