Tag Archives: bernoulli principle

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Fluid Dynamics – Week 8: Bernoulli’s Equation and Principle

Hello again everybody, and welcome back to my blog! This week’s blog post is revisiting a topic that we covered way back at the start of the term, and that is Bernoulli’s Principle. The principle itself is defined from Bernoulli’s equation, which I will derive here as well. With that out of the way, let’s get started! 🙂

Bernoulli’s Equation, and its derivation

We start our derivation by re-introducing Euler’s Equation, a topic that I covered in my last blog post:

\dfrac{D\vec{u}}{Dt} = \dfrac{-1}{\rho}\nabla p + \vec{F}

If we assume that the body force is conservative (i.e. \vec{F} = -\nabla\phi) then we can rewrite Euler’s equation as follows:

\dfrac{D\vec{u}}{Dt} = \dfrac{-1}{\rho}\nabla p -\nabla\phi

With our knowledge of the convected acceleration (as mentioned in my blog post on flow visualisation), we can then expand the L.H.S. as follows:

\dfrac{\partial\vec{u}}{\partial t} + \nabla\left(\dfrac{1}{2}|\vec{u}|^2\right) - \vec{u} \times \text{curl}(\vec{u}) = -\dfrac{1}{\rho}\nabla p - \nabla\phi

If we now make another assumption in that we assume the flow to be steady (i.e. the velocity of the flow does not depend on time), then our first term vanishes, leaving:

\nabla\left(\dfrac{1}{2}|\vec{u}|^2\right) - \vec{u} \times \text{curl}(\vec{u}) = -\dfrac{1}{\rho}\nabla p - \nabla\phi

Just for convenience, I’ll combine all the terms that are affected by a \nabla, so that we get:

\nabla\left(\dfrac{1}{2}|\vec{u}|^2 + \dfrac{p}{\rho} + \phi\right) = \vec{u} \times \text{curl}(\vec{u})

Now this derivation involves us finding Bernoulli’s equation along a streamline, a fact that I have not brought up until now. So now, we need to find the component of our previous equation in the direction of said streamline.

For those of you who did Vector Calculus last year, we know that we can do this by simply defining a tangential vector \vec{s} (which is also defined as a unit vector) to the streamline, and then take the dot product of this with the equation above. This will then give the following:

\vec{s}.\nabla\left(\dfrac{1}{2}|\vec{u}|^2 + \dfrac{p}{\rho} + \phi\right) = \vec{s}.\vec{u} \times \text{curl}(\vec{u})

The R.H.S. cancels to zero, due to the fact (from Vector Calculus) that the resultant dot product of two perpendicular vectors \left(\vec{u} \times \text{curl}(\vec{u})\right) must always be equal to zero. Therefore we can now simplify our above equation to:

\dfrac{d}{ds}\left(\dfrac{1}{2}|\vec{u}|^2 + \dfrac{p}{\rho} + \phi\right) = 0

Since the derivative equals zero, then we know that the contents in the brackets must be equal to a constant along our streamline. Thus we get streamline version of Bernoulli’s equation:

\dfrac{1}{2}|\vec{u}|^2 + \dfrac{p}{\rho} + \phi = \text{const}

Bernoulli’s Principle

If we now consider the case where our body forces are negligible (\phi = 0), then we write Bernoulli’s equation in the following way:

\dfrac{1}{2}|\vec{u}|^2 = \text{const} - \dfrac{p}{\rho}

Thus we have produced a relationship between the speed of the flow, and its pressure, which if you recall, is the basis of the observations that I spoke about in one of my first blog posts.

Therefore, by examining the relationship, we can see that:

High speed \implies low pressure

Low speed \implies high pressure

The various applications of Bernoulli’s Equation

During the lectures, we were shown various applications for Bernoulli’s equation. These included measuring the flow from a reservoir, a Pitot tube, and a Venturi tube, as well as more complex examples, such as forces exerted on a wall due to a vortex, and the flow around a cylinder.

In the future, I would like to revisit these applications, as a means of further research, and to help me solidify my understanding of Bernoulli’s equation for revision purposes.

Example Sheet Questions

Below you will find my scanned answers to (most of) question 2 of Example Sheet 3.

Example Sheet 3 - Question 2

Example Sheet 3 – Question 2

Reflection

Much like the derivation of Euler’s equations, I believe that the derivation of Bernoulli’s equation is relatively straight forward, and the conclusion from the Bernoulli’s Principle is quite a simple, but yet powerful, observation.

However, I did have a few difficulties with understanding it in regards to the second question on Example Sheet 3. The change in sign within \psi (once I had introduced the arctangent), its subsequent derivation, and upon checking the solutions, my inclusion of $\rho$ (that is seemingly missing in the solutions), have all caused me problems with this question. When it comes to revision, its clear that I need to spend some time swatting up on common derivatives of trigonometric functions, and perhaps getting some further clarification about the assumptions made during the calculation.

If I had more time in which to complete these blog posts, I would have done significantly more background research and revision. For this post alone, I should really do more reading into the variety of applications that Bernoulli’s equation has in practice. This is something that I’ll keep in mind and carry forward into my second term, and begin doing as soon as I have the opportunity to do so. I am pleased though with what I have done with them, even though most of my more recent posts are inconsistent in style to my older ones.

Thanks for a great first term, and I’ll see you in the New Year! 🙂

References and Sources:

1) All of the concepts talked about in this blog post are derived from notes taken from Chapter 2 of the Inviscid Flow PowerPoints on Bernoulli’s equation, and my own notes derived from the lectures.

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Introduction to Fluid Dynamics: Bernoulli’s Principle

Hello everybody, and welcome back to my blog on fluid dynamics! 🙂 In this particular blog post, I will be going over the content covered in our first lecture on the subject, which took place on the 23rd of September.

As this was our first lecture on the subject, David was kind enough to not overload us with anything too technical, so please don’t expect anything substantial in the way of equations, derivations and definitions (in this blog post as least! ;)). With that out of the way, let the fun begin!

Experiments and their results

In this particular lecture, David showed us a variety of experiments which demonstrated an important aspect of fluid dynamics, although for our amusement, this was initially veiled through the use of straws, old juice bottles and ping pong balls.

The ball and the straw observation

The experiments were inherently simple in nature, and yet when conducted, seemed to produce results that were naturally counter-intuitive to what we would expect. The first example of this was David blowing through an angled straw, with some force I might add, on to a ping pong ball, that had a green line painted across the centre.

What we would expect, naturally, is for the ball to go flying away if it comes into contact with the stream of air coming out of the straw. The ball would just fall to the ground under gravity otherwise. However, we observed something totally different! Given that if David could provide enough puff, the ball actually stayed suspended in the air while rotating in the direction of the air flow. The actual rotation itself would have been hard to spot, if it were not for the green line. This then lead on to the next ‘experiment’…

The ping pong ball is suspended in the air due to Bernoulli's Principle.

The ping pong ball is suspended in the air due to Bernoulli’s Principle. [1]

The bottle observation

For our next observation, we were asked in what direction the bottles would move if David were to blow through the middle of them. Naturally, most people assumed that the bottles would move away from each other. However, once again, this proved not to be the case! As you can see from the diagram below, the bottles actually ended up being moved towards each other!

By this point, most of us were mildly amused by David running out of breath from attempting to blow on various objects, but a lot of us were also thinking “Well, what’s going on then? Why are these things not acting in the way we expected them to?”. As this was near the end of the lecture, David then mentioned in a few words that this effect was known as Bernoulli’s Principle, and what it actually meant in layman’s terms.

Counter-intuitively, the bottles move towards each other, rather than away.

Counter-intuitively, the bottles move towards each other, rather than away. [2]

The Bernoulli Principle

The Bernoulli Principle, as put forward by Daniel Bernoulli in his book Hydrodynamica in 1738 [3], states that if the pressure of a fluid decreases, then inversely, the velocity must increase. Of course, by extension and observation, the opposite is also true. Indeed, it is this simple statement that defines the seemingly counter-intuitive behaviour that we observed.

By looking back at the experiments, the reasons are clear. In our first experiment, we can now deduce that since the air flow is moving with greater velocity, then we must have an area of low pressure to compensate. In line with this, we must know that at the base of the ball, there must be an area of higher pressure, since the air flow is not as strong here. It is the difference in the pressure on the ping pong ball that causes the phenomenon we observe: the ball is forced to move from the area of high pressure to the area of low pressure, and thus with the constant stream of air, the ball is suspended in the air while spinning rapidly.

In the second experiment, a similar observation is now made. The stream of air flows quickly between the two bottles, and thus causes low pressure. On the outer sides of the bottles, there is little air flow and thus there is an increase in the pressure. Therefore, the areas of high pressure on the outsides of the bottles push the bottles inward, as we know from our observation.

Reflection

The concepts detailed here, although now seemingly quite simple in nature, were initially a bit confusing to observe. However, David’s explanation of Bernoulli’s Principle makes sense upon reflection, and his rather amusing displays will certainly stick in my mind for awhile. Hopefully I’ll be able to understand the more complicated topics, that will be covered, in a similar manner.

I took my own advice from last time, and I’m glad that I managed to get this uploaded faster than I had done my first post. However, I’m still not uploading new blog posts as fast as I should be, so this will have to improve over the coming weeks.

Hopefully this [somewhat] brief but informative post about Bernoulli’s Principle was useful to you. Many more interesting ideas and observations will be discussed in later blog posts, as we delve further into the mathematics of fluid dynamics. Who knows, maybe we will encounter a more strict definition of this observation at a later point in the module? Many thanks to those of you who got this far!

References and Sources:

[1] [2] Many thanks to my friend Darren Mooney who gave me permission to re-use his images. He’s also got a blog on fluid dynamics! Why don’t you check him out? Here’s the address: http://math3402darrenmooney.wordpress.com/

[3] http://www.britannica.com/EBchecked/topic/62591/Daniel-Bernoulli#ref200813