MAE 360 Aerodynamics

Stream Function Tutorial

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This lesson should take you approximately 30 minutes to complete.  Do not forget to SUBMIT your results at the end of the lesson.

The Stream Function

Let us define a function of two variables, $\psi (x,z)$, for which we will define the first derivatives as:


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For now, $u$ is simply the first derivative of $\psi $ with respect to $z$, and $w$ is the negative of the first derivative of $\psi $ with respect to $x$. Notice that, for these definitions of $u$ and $w$, the function, $\psi (x,z)$, automatically satisfies the steady continuity equation in two dimensions:
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Therefore, if we can define a function that satisfies:

  1. its $z$-derivative is the $u$-velocity in some flow field, and
  2. the negative of its $x$-derivative is the $w$-velocity in that same flow field,

that function, called a "stream function," will describe a legitimate steady, two-dimensional flow. It seems like a very tall order to find such a function, but a sizable number of close-to-realistic flows with engineering significance can be described in such a simple manner. Consider the following example:
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Which of the following represents the velocity within this flow field?

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Streamlines

The stream function, $\psi (x,z)$, is a scalar function of position, i.e., for some value of $x$ and some value of $z$ the function $\psi $has some assigned scalar value. In general, $\psi $can have the same scalar value for different sets of $(x,z)$ pairs. In the example given above, $\psi $ will equal 1 whenever $z=1/x$, so a curve of constant $\psi =\psi _{0}$ is equivalent to the curve $z=\psi _{0}/x$. Consider the case $\psi =$ constant $=\psi _{0}$. Then the differential of $\psi $ along the curve $\psi =\psi _{0}$ can be written:
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Rearranging the last of these equations leads to
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along the curve $\psi =\psi _{0}$. This implies that a curve of constant $\psi $ for which MATH and MATH will have slope given by $w/u$. To envision the meaning of this, refer to the figure, which illustrates the scenario described by equation 2. The velocity vector, $\QTR{bf}{V}$, is tangent to the curve at point $P$ so that the ratio $w/u$ is the slope at point $P$. Thus, any curve described by