MAE 360 Aerodynamics

Stream Function Tutorial

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, , for which we will define the first derivatives as:

For now, is simply the first derivative of with respect to , and is the negative of the first derivative of with respect to . Notice that, for these definitions of and , the function, , automatically satisfies the steady continuity equation in two dimensions:

Therefore, if we can define a function that satisfies:

1. its -derivative is the -velocity in some flow field, and
2. the negative of its -derivative is the -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:

Which of the following represents the velocity within this flow field?

## Streamlines

The stream function, , is a scalar function of position, i.e., for some value of and some value of the function has some assigned scalar value. In general, can have the same scalar value for different sets of pairs. In the example given above, will equal 1 whenever , so a curve of constant is equivalent to the curve . Consider the case constant . Then the differential of along the curve can be written:

Rearranging the last of these equations leads to

along the curve . This implies that a curve of constant for which and will have slope given by . To envision the meaning of this, refer to the figure, which illustrates the scenario described by equation 2. The velocity vector, , is tangent to the curve at point so that the ratio is the slope at point . Thus, any curve described by