Stream function

The volume flux through the curve between the points ${\displaystyle A}$ and ${\displaystyle P.}$

Lamb and Batchelor define the stream function ${\displaystyle \psi (x,y,t)}$ for an incompressible flow velocity field ${\displaystyle (u(t),v(t))}$ as follows.[3] Given a point ${\displaystyle P}$ and a point ${\displaystyle A}$,

${\displaystyle \psi =\int _{A}^{P}\left(u\,{\text{d}}y-v\,{\text{d}}x\right)}$

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

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

${\displaystyle \delta \psi =u\,\delta y-v\,\delta x}$.

From the exact differential

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

the flow velocity components in relation to the stream function ${\displaystyle \psi }$ have to be

${\displaystyle u={\frac {\partial \psi }{\partial y}},\qquad v=-{\frac {\partial \psi }{\partial x}},}$

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

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

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

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

${\displaystyle \mathbf {u} =\nabla \times {\boldsymbol {\psi }}}$

Where ${\displaystyle {\boldsymbol {\psi }}=(0,0,\psi )}$ if the flow velocity vector ${\displaystyle \mathbf {u} =(u,v,0)}$.

In Cartesian coordinate system this is equivalent to

${\displaystyle u={\frac {\partial \psi }{\partial y}},\qquad v=-{\frac {\partial \psi }{\partial x}}}$

Where ${\displaystyle u}$ and ${\displaystyle v}$ are the flow velocity components in the cartesian ${\displaystyle x}$ and ${\displaystyle y}$ coordinate directions, respectively.

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

${\displaystyle \mathbf {u} =\mathbf {z} \times \nabla \psi '\equiv \left(-\psi '_{y},\psi '_{x},0\right)}$,

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

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

${\displaystyle u=-{\frac {\partial \psi '}{\partial y}},\qquad v={\frac {\partial \psi '}{\partial x}}}$

in Cartesian coordinates.

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

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

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

${\displaystyle \nabla \times \left(\psi \mathbf {z} \right)=\psi \nabla \times \mathbf {z} +\nabla \psi \times \mathbf {z} =\nabla \psi \times \mathbf {z} =\mathbf {z} \times \nabla \psi '.}$

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

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:

${\displaystyle \delta \psi =q\delta n\,}$

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

${\displaystyle q={\frac {\partial \psi }{\partial n}}\,}$

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 flow velocity is positive.

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

${\displaystyle {\begin{aligned}\delta \psi &=u\delta y\,\\\delta \psi &=-v\delta x\,\end{aligned}}}$

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:

${\displaystyle {\begin{aligned}u&={\frac {\partial \psi }{\partial y}}\,\\v&=-{\frac {\partial \psi }{\partial x}}\,\end{aligned}}}$

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:

${\displaystyle \delta \psi _{\text{in}}=u\delta y+v\delta x.\,}$

The total flow out of the element is given by:

${\displaystyle \delta \psi _{\text{out}}=\left(u+{\frac {\partial u}{\partial x}}\delta x\right)\delta y+\left(v+{\frac {\partial v}{\partial y}}\delta y\right)\delta x.\,}$

Thus we have:

${\displaystyle {\begin{aligned}\delta \psi _{\text{in}}&=\delta \psi _{\text{out}}\,\\u\delta y+v\delta x&=\left(u+{\frac {\partial u}{\partial x}}\delta x\right)\delta y+\left(v+{\frac {\partial v}{\partial y}}\delta y\right)\delta x\,\end{aligned}}}$

and simplifying to:

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

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

${\displaystyle {\frac {\partial ^{2}\psi }{\partial x\partial y}}-{\frac {\partial ^{2}\psi }{\partial y\partial x}}=0.}$

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

${\displaystyle \nabla ^{2}\psi =-\omega }$

or

${\displaystyle \nabla ^{2}\psi '=+\omega }$

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

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

${\displaystyle {\begin{aligned}&\psi (x+dx,y+dy)-\psi (x,y)\\={}&{\partial \psi \over \partial x}dx+{\partial \psi \over \partial y}dy\\={}&\nabla \psi \cdot d{\boldsymbol {r}}\end{aligned}}}$

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

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

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

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

1. The stream function ${\displaystyle \psi }$ is constant along any streamline.
2. For a continuous flow (no sources or sinks), the volume flow rate across any closed path is equal to zero.
3. For two incompressible flow patterns, the algebraic sum of the stream functions is equal to another stream function obtained if the two flow patterns are super-imposed.
4. The rate of change of stream function with distance is directly proportional to the velocity component perpendicular to the direction of change.