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Fluid Dynamics – Week 5: The Stream Function and the Principle of Superposition

Hello again everybody, and welcome back for another exciting blog post on your favourite subject; fluid dynamics! 🙂 For once, I don’t really have much to say right here, so let’s just get stuck in, shall we?

Stream functions

In my last post, I introduced the topics of flow visualisation, featuring important aspects such as pathlines and streamlines. In this post, we extend this discussion, by firstly looking at the topic of stream functions.

Stream functions, as you might be able to infer from their name, are related closely to streamlines. In fact, by utilising the fact that streamlines are parallel to the velocity, we can deduce a new streamline by defining a stream function during its derivation. Subsequently, this leads to another feature of the stream function, which is that they are constant along a streamline.

Stream functions also tell us about the flow speed. The volume flow rate between two points joined in a curve can be calculated by simply working out the difference between the values of their respective stream functions, at those points.

So how do we go about defining these stream functions? For a 2-dimensional flow, that is incompressible and has a constant density, we can define the stream function simply as follows:

\dfrac{\partial\psi}{\partial x} = -v, \dfrac{\partial\psi}{\partial y} = u

Similarly, for polar coordinates:

\dfrac{\partial\psi}{\partial r} = -v_{\theta}, \dfrac{1}{r}\dfrac{\partial\psi}{\partial\theta} = v_r

where (u, v) are the components of a Cartesian velocity, and (v_r,v_{\theta}) is the same, but for a polar system. It should be noted that you should also check these derived results, to make sure that they are consistent with the definition of incompressibility.

The Principle of Superposition

The Principle of Superposition, as awesome as it sounds, is really just an expressive way of saying that we can add stream functions together. By combining two velocity fields, which each have respective stream functions, then the combined velocity field has a stream function that is defined by just adding up the respective stream functions together. In layman’s terms, we’ve produced an entirely new flow by just adding the stream functions! Fancy, ain’t it? 😉

During the lectures, this effect was explained to us through a series of diagrams that looked at two sources. I’ll re-interpret this below, as although the concept is quite logical, it can be a little tricky to follow with just words alone, so a visual aid is always beneficial. Not to mention the fact that a question based on drawing a graph for this could appear in an exam, so it’s always good for practice!

The example considered was that of two sources, with the following stream function properties:

\text{(A)} \qquad \psi_1 = m\theta_1, \quad -\pi < \theta_1 \le \pi, \quad \text{with centre at } (a,0)

\text{(B)} \qquad \psi_2 = m\theta_2, \quad -\pi < \theta_2 \le \pi, \quad \text{with centre at } (-a,0)

By using our newly acquired knowledge of superposition, we can now combine these separate stream functions to create a new one, as follows:

\psi = \psi_1 + \psi_2 = m(\theta_1 + \theta_2)

Since we know that stream functions are constant along the streamlines, then we know that:

\theta_1 + \theta_2 = \text{constant}

We can now draw a diagram that represents how the combination of the sources’ stream functions (and their streamlines) turn into a new stream function (with new streamlines). In the diagram, the source A is shown by the blue streamlines, whereas the source B is shown by black streamlines.

Streamlines of two sources based symmetrically around x-axis.

Streamlines of two sources based symmetrically around x-axis.

As you can see, the streamlines drawn are representations of increments of m\alpha, where \alpha = \dfrac{\pi}{12}. This is due to our initial definitions of the sources’ stream functions.

Since we are interested in generating a new stream function and its streamlines, we can now observe from the graph that we can join the components of the sources together. This is done by identifying points on the graph where the sum of the components of the stream functions are the same.

New streamlines have been produced due to superposition.

New streamlines have been produced due to superposition.

For example, on my graph, we can see that at the points P_1, P_2, P_3, P_4, the sum is equal to 13m\alpha, whereas the sum of the points at Q_1, Q_2, Q_3, Q_4 is equal to 11m\alpha.

By extending this idea with the other components, we can gradually draw up a picture of what the new streamlines will look like for our combined stream function.

By extension, we can begin to observe how the streamlines of the new stream function will look.

By extension, we can begin to observe how the streamlines of the new stream function will look.

Example Sheet Questions

Below you will find my scanned answers to question 1 of Example Sheet 2.

Example Sheet 2 - Part 1

Example Sheet 2 – Part 1

Example Sheet 2 - Part 2

Example Sheet 2 – Part 2

Reflection

I feel that I understood the concept of the stream functions quite well, as they seem to be fairly similar concepts to previously looked at material, such as pathlines and streamlines. The overall objective of the principle of superposition makes sense to me, although I must say that it took me awhile to get around the idea of the example that I have re-specified above. I feel that this was mostly due to the number of streamlines coming out of the defined sources, and so it made it a bit difficult to keep track of what streamlines I should have been adding up.

Although I only have the time right now to cover the first question, I feel that I understand the calculations just fine. Like I specified last week, I feel that I will make best use of the questions that I did not cover when it comes to revision for the exam.

I am still learning from the advice that David has given to the class, and I’m trying to be more succinct in my blog posts, while still covering the fundamentals of the lessons. Compared to some of my earlier attempts however, I think the result is much better. Also, the time it has taken me to get around to uploading this has been a little better than the time between my previous two blog posts. I can only hope that any posts that follow will also have more brevity and be more punctual as well!

Thank you to everybody who took the time to read this! Expect more posts soon as the first term begins to wrap up! 🙂

References and Sources:

1) The vast majority of the concepts talked about in this blog post are derived from notes taken from Chapter 1 of the Inviscid Flow PowerPoints on continuity and stream functions, and my own notes derived from the lectures.