Ghosh - 550Page 109/23/2018

Stream Function

Now we discuss an application of the incompressible continuity equation to introduce an important function in fluid mechanics, widely used for flow visualization and analytical techniques.

The concept of the stream function was introduced as an attempt to solve the incompressible fluid flow problems. In old times when computers were not available, computational fluid dynamics (CFD) solvers were non-existent. Thus the attempt was to define the transformations, which satisfy certain flow equations. One such way was to define a function (x,y) which satisfied the two dimensional mass-conversion law:

Suppose, and

we then get and

Since (x,y) is a continuousfunction of x and y,

Therefore, we see that the above definition of  function satisfies

Physical Interpretation

Now we have discovered the existence of a function (x,y), called the stream function, which satisfies the continuity equation for incompressible fluid flows. If  exists, we know that the flow must be incompressible. But what can we do with such a function? It turns out that this function has some handy properties which we may be able to use for learning more about fluid flows. First of all, the name “stream function” is chosen because this function staysconstant along a particular streamline in a flow. Recall the definition of streamlines, which satisfied:

This expression means , which is a vector parallel to the tangent vector of a streamline, is in the same direction (or, parallel) as the velocity vector, .

[Note: ,  = angle in between the vectors.

 = 0, i.e., ]

In Cartesian coordinates,

and

Therefore, on a streamline in 2-dimensional plane (x,y)

With the definition of (x,y) defined above,

This nice property gives us an opportunity to visualize the streamlines readily. If we can determine a (x,y) in the flow, we can sketch the streamlines by setting a different constant each time for  and plotting the  as shown below.

Here 1, 2, 3 and 4 are different numerical constants (e.g. 1, 2, 3, 4 etc…)

(Remember  = constant plot only yields the sketch of a streamline. One still needs to evaluate the velocity vector to determine the flow direction)

As shown in the sketch,  = 2 at points, A, B, C, D, etc… on one streamline. Similarly,  = 4 at all points A’, B’, C’, D’, etc on the other streamline and so on.

Does the difference of  values between two streamlines have any significance? For example what does  (= 2 - 4 ) mean? It is the volumetric flow rate per unit depth between the two streamlines shown on the sketch.

This means that if we can somehow determine (x,y) in a 2-D, incompressible flow, we can:

(i)Visualize the flow (by sketching streamlines)

(ii)Determine the volumetric flow rate (without computing )

Examples Continue