Stream function

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For three-dimensional flows with axisymmetry, see Stokes stream function.
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 then 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.[1] The Stokes stream function is for axisymmetrical three-dimensional flow, and is named after George Gabriel Stokes.[2]

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. 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.

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[edit]

Definitions[edit]

The volume flux through the curve between the points A and P.

Lamb and Batchelor define the stream function \psi(x,y,t) – in the point P with two-dimensional coordinates (x,y) and as a function of time t – for an incompressible flow by:[3]

\psi = \int_A^P \left( u\, \text{d}y - v\, \text{d}x \right).

So the stream function \psi is the volume flux through the curve AP, that is: the integral of the dot product of the flow velocity vector (u,v) and the normal (+\text{d}y,-\text{d}x) to the curve element (\text{d}x,\text{d}y). The point A is a reference point defining where the stream function is zero: a shift of A results in adding a constant to the stream function \psi.

An infinitesimal shift \delta P=(\delta x,\delta y) of the position P results in a stream function shift:

\delta \psi = u\, \delta y - v\, \delta x,

which is an exact differential provided

\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0.

This is the condition of zero divergence resulting from flow incompressibility. Since

\delta\psi = \frac{\partial\psi}{\partial x}\, \delta x + \frac{\partial\psi}{\partial y}\, \delta y,

the velocity components have to be

u= +\frac{\partial\psi}{\partial y}   and   v = -\frac{\partial\psi}{\partial x}

in relation to the stream function \psi.

Definition by use of a vector potential[edit]

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 through the vector potential \boldsymbol{\psi} :


\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)[edit]

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}=(0,0,1) 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.

All 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

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

\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'.

Note that \boldsymbol{\psi}=\psi\mathbf{z} in this two-dimensional flow.

Derivation of the two-dimensional stream function[edit]

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[edit]

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[edit]

By observing the flow into an elemental square in an rθ 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[edit]

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[edit]

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

\nabla ^2 \psi = -\omega

or

\nabla ^2 \psi' = +\omega

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

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

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


\psi (x+dx,y+dy) - \psi (x,y) = {\partial \psi \over \partial x} dx + {\partial \psi \over \partial y} dy

\qquad \qquad = \nabla \psi \cdot d \boldsymbol{r}

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


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

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


\boldsymbol{u} \cdot \nabla \psi = {\partial \psi \over \partial y} {\partial \psi \over \partial x} + \Big( -{\partial \psi \over \partial x} \Big) {\partial \psi \over \partial y} = 0.

References[edit]

Inline[edit]

  1. ^ Lagrange, J.-L. (1868), "Mémoire sur la théorie du mouvement des fluides (in: Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin, année 1781)", Oevres de Lagrange, Tome IV, pp. 695–748 
  2. ^ Stokes, G.G. (1842), "On the steady motion of incompressible fluids", Transactions of the Cambridge Philosophical Society 7: 439–453, Bibcode:1848TCaPS...7..439S 
    Reprinted in: Stokes, G.G. (1880), Mathematical and Physical Papers, Volume I, Cambridge University Press, pp. 1–16 
  3. ^ Lamb (1932, pp. 62–63) and Batchelor (1967, pp. 75–79)

Other[edit]