velocity vector | math3402liamsnow

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.

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.

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.

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 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.

Hello again everybody, and welcome back to another addition in my blog! 😀 Before we start, I just need to clarify a few things:

1) No, I wasn’t on vacation, as delightful as that would have been for me! I’m afraid that the workload of uni caught up with me, and as such, I didn’t put as much time into my blogs as I should’ve done. Don’t worry though, you’ll hopefully be seeing a lot more of my blog posts in the next few days, so stay tuned!

2) As I mentioned in my last post, the rest of the content that is to be featured in my blog will be all about inviscid flow, as opposed to the viscous flow that we have been looking at previously.

With that out of the way, let us begin!

Flow Visualisation

It makes sense for us to be able to describe the properties of fluids as they flow, since we can see this natural phenomena around us all the time. The wind swaying the branches in the trees, and the raging torrents of a river, are all naturally occurring examples of fluid flows. The ability for us to mathematically deduce the properties of these types of flow, and how we can visualise them, can have a significant impact on our understanding of complex problems, such as modelling the weather.

Two such ways that we can represent the flows of fluids is through pathlines and streamlines (technically, there is another way through streaklines, but we won’t go into more depth with them here.)

Pathlines

We can think of pathlines as representing the trajectory of particles through the flow. For a 3-dimensional fluid flow, we can define the pathlines in the form of the ODE:

$\dfrac{d\vec{x}}{dt} = \vec{u} = (u,v,w)$

In this equation, the position vector $\vec{x}$ is defined as $(x(t),y(t),z(t))$ , while the time derivative of the position vector is simply the velocity vector $\vec{u}$ .

Pathlines are similarly defined in polar coordinates, but with the position vector defined as $(r(t),\theta(t),z(t))$ and the velocity vector as $(u_{r},u_{\theta},u_{z})$ .

Streamlines

Streamlines can be thought of as integral curves of the velocity field, which crucially means that a streamline is simply everywhere that is parallel to the velocity. From a physical perspective, a streamline is a snapshot of the flow of particles within the fluid at a fixed time, $t$ .

From some basic manipulation of the definition of the pathlines above, we can define the streamlines as the following:

$\dfrac{dx}{u} = \dfrac{dy}{v} = \dfrac{dz}{w}$

which can be rewritten as

$\dfrac{dy}{dx} = \dfrac{v}{u}, \dfrac{dz}{dx} = \dfrac{w}{u}, \dfrac{dz}{dy} = \dfrac{w}{v}$

Like the pathlines, streamlines can also be written in the form of polar coordinates. Once again, the time aspects of the equation are eliminated, to reveal the definitions that we want, i.e. (for 2-dimensions):

$\dfrac{1}{r}\dfrac{dr}{d\theta} = \dfrac{u_{r}}{u_{\theta}} \text{ or } r\dfrac{d\theta}{dr} = \dfrac{u_{\theta}}{u_{r}}$

Convected Derivative

In my introduction to viscous flow, I defined what it meant for a fluid to be incompressible, which relied upon the definition of a new form of derivative; the convected derivative. Using the definition of the convected derivative, we can now consider the acceleration of the fluid flow, since we can translate it as a convected time derivative of the velocity vector, like so:

$\dfrac{D\vec{u}}{Dt} = \dfrac{\partial\vec{u}}{\partial t} + \vec{u}.\nabla\vec{u}$

Alternatively, the convected acceleration can also be written as follows:

$\dfrac{D\vec{u}}{Dt} = \dfrac{\partial\vec{u}}{\partial t}$ $+ \dfrac{1}{2} \nabla (|\vec{u}|^2) - \vec{u} \text{ x curl(}\vec{u}\text{)}$

As another extension from viscous flow, it can be quite easily observed that the left hand side of the Navier-Stokes equations is simply the density of the fluid ( $\rho$ ), multiplied by our time derivative and convected acceleration. The same can also be said in polar coordinate form.

Example Sheet Questions

Below you will find my scanned answers to questions 1 and 2 of Example Sheet 0 and question 1 of Example Sheet 1.

Example Sheet 0 – Part 1

Example Sheet 0 – Part 2

Example Sheet 0 – Part 3

Example Sheet 1 – Part 1

Example Sheet 1 – Part 2

Reflection

The general principles highlighted during this particular week’s worth of lectures were fairly intuitive to understand, although I do need to make sure that I understand the physical interpretations of the flow visualisations, such as the pathlines and streamlines. I feel that the equations for these visualisations are straight-forward, so it should be a case of memorisation when it comes to revision.

In terms of the exercises, all of the processes were fairly repetitive, and as such, there weren’t many hidden surprises. In fact, the only aspect of the questions that I struggled with, was the interpretation of the first part of question 1, of Example Sheet 0. I was not confident with the graph, and I only convinced myself to put down what I did for it after experimentation with the equation in Maple.

Ideally, I would have liked to have covered more of the questions to further my understanding of the topic. However, with the current levels of workload, I will instead use these additional questions as part of my revision for the exam at the end of the year. I am glad though that I was able to significantly shorten my blog post (particularly after my last post) without leaving out the crucial details, which will be extremely useful when it comes to revision.

As I also mentioned at the start of this post, I do apologise sincerely for not uploading this sooner, but the workload has been getting to me recently. If I have to take away anything from this term so far, it is that I have to make significant improvements to my overall time management, and not get too focused on a few activities! Oh, and that I really need to start on work as soon as I receive it, if it can be helped! 😛

Thank you to anybody who took the time to read this post! I hope you’ll be seeing more of this sooner rather than later! 🙂

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 flow visualisation, continuity and stream functions, and my own notes derived from the lectures.

math3402liamsnow

Nothing but a torrent of fluid dynamics.

Tag Archives: velocity vector

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

Fluid Dynamics – Week 4: Flow Visualisation, Streamlines and Pathlines, and the Convective Derivative