Tag Archives: navier-stokes

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Fluid Dynamics – Week 7: Euler’s Equations

Welcome everybody, to yet another blog post on fluid dynamics! 😀 This blog post will try and follow a slightly different format to my previous posts. Hopefully this one will be more succinct than past attempts, which should mean less sore eyes for you! 😛 Without further ado, let us begin.

Euler’s Equations, and their derivation

As you will see soon, Euler’s equations are very closely linked to the Navier-Stokes equations, which I mentioned in a previous blog post. However, we must still be aware that these equations are related to inviscid flow (as has most of the previous content I’ve covered), instead of the viscous flow that the Navier-Stokes equations cover.

Let me now describe the scenario of the derivation (but don’t worry, I’ll include a picture below from my lecture notes for clarity).

Visual overview of the derivation of Euler's Equations, from the lecture notes.

Visual overview of the derivation of Euler’s Equations, from the lecture notes.

Once again, we are examining but a small fraction of a moving fluid, which is composed of: a small volume V, that has a small surface \delta S (with corresponding normal vector \vec{n}), all the while moving with a velocity \vec{u}.

From this, we can deduce the other following properties:

  • (Convected) Acceleration = \dfrac{D\vec{u}}{Dt}
  • Mass = \rho V
  • Body force = \vec{F}
  • Pressure (due to force of moving fluid) = -p\vec{n} \delta S

By substituting these appropriately into an integral, we get:

\dfrac{D\vec{u}}{Dt}\int_V \rho dV = \int_V \rho\vec{F} dV - \int_S p\vec{n} dS

where the first integral equates to a total mass multiplied by the acceleration (a force), which is equivalent to the total body force minus the total surface force.

For convenience, we can now apply Gauss’ theorem to the integral over the surface, to convert it into an integral over the volume (like the other integrals), and thus produce:

\dfrac{D\vec{u}}{Dt}\int_V \rho dV = \int_V (\rho\vec{F} - \nabla p) dV

As stated above, we have already assumed that the volume is small, so if we now take V \to 0, we can conclude that:

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

The above is known as Euler’s equation (in its whole form), and is also obtained when we allow the viscosity to equal zero in the Navier-Stokes equations.

To be more explicit, here is the Cartesian version of Euler’s equation in summation form:

\rho\left(\dfrac{\partial u_i}{\partial t} + u_j \dfrac{\partial u_i}{\partial x_j}\right) = \dfrac{\partial\rho}{\partial x_i} + \rho(F_B)_i

A polar form is also shown similarly, but I will not state it here.

Example Sheet Questions

Below you will find my scanned answers to question 1 of Example Sheet 3.

Example Sheet 3 - Part 1

Example Sheet 3 – Part 1

Reflection

In my honest opinion, I think that the derivation of Euler’s equations was not too bad (as I had mentioned, the process to deduce them is essentially a simplified Navier-Stokes method), although I still feel a bit apprehensive about certain details of it. In particular, I don’t think I quite understand the logic in deducing the force due to the pressure. Perhaps this is some anxiety I still hold from my experiences with similar properties in Vector Calculus last year? Either way, when it comes to revision, I will make sure that I fully understand the derivation.

Even though I’ve only attempted the first question of Example Sheet 3 here, I feel a bit more confident about using Euler’s equation now. The only problem I have with my uploaded solution is that, upon checking, it seems I have a factor of \rho in front of the equation, whereas the uploaded solutions do not. If I have made a mistake, then perhaps I will clarify this with one of my peers, or through David directly.

On the whole, I am especially pleased with how this new ‘format’ went, as it seems to be much more succinct, and cuts to the chase. Of course, I still felt inclined to show a derivation, but that felt necessary in order to explain where the equations came from.

Thanks to anyone who has taken the time to read this blog post! See you soon! 🙂

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 Euler’s equation, and my own notes derived from the lectures.

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

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 1

Example Sheet 0 - Part 2

Example Sheet 0 – Part 2

Example Sheet 0 - Part 3

Example Sheet 0 – Part 3

Example Sheet 1 - Part 1

Example Sheet 1 – Part 1

Example Sheet 1 - Part 2

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.

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Fluid Dynamics – Week 3: Stress, Pressure and the Navier-Stokes equations

Hello everybody, and welcome back for the fourth instalment on my blog about fluid dynamics! 🙂 Following on from my previous blog post introducing viscous flow, we will continue our look at the properties of viscous fluids (yes, including that momentum conservation equation!), and we will finally derive the famous Navier-Stokes equations, which we will use extensively in the second term of the module for solving a variety of viscous flow systems.

Oh, and please don’t be misled by the title of this particular blog post. It most certainly isn’t going to be a gushing recount of my hardships at university, since I try to avoid drama whenever possible! 😉 With that out of the way, let the “fun” begin!

Definition of the Momentum Conservation Equation

As I mentioned previously, the deduced momentum conservation equation is vital in the derivation of the Navier-Stokes equations. As such, I feel that it is important that I introduce it now, especially since that I did not produce the result last time. Without further ado, here it is:

\dfrac{d}{dt}\int_V \rho\vec{u} dV = -\int_S \rho\vec{u}(\vec{u}.\vec{n}) dS +\int_V \rho\vec{F}_B dV +\int_S \vec{\sigma} dS

I did warn you last time that it wasn’t pretty, so I apologise now if you have nightmarish visions of this engraved in your conscience. To make it somewhat less painful to read, let me try and explain what some of the individual terms mean, especially in regards to our expression of the principle in layman’s terms.

The first term (the component of the equation before the equals sign) relates to ‘the rate of change of momentum inside the volume’. Since momentum is defined as being the mass times the velocity, then thus the rate of change of momentum will have units matching those of mass times by acceleration (the same units as a force). From checking this first term, we can verify the units are correct.

As for the second term, the integrand represents ‘the net rate of inflow of momentum into the volume’. The term is also negative because we have to consider that the normal is acting outwards of the surface of the volume, when we are considering what is occurring inside the volume.

The final two components of the equation represent ‘the total force acting on the fluid inside the volume’, the first of which refers to \vec{F}_B, which is the body force (per unit mass) acting on the fluid. For most applications we will look at, the body force is normally gravity. The latter term of the pair refers to stress (the force per unit area) acting upon the surface of the volume (represented by \vec{\sigma}), and it arises due to the motion of the surrounding fluid.

Thus that summarises the equation for momentum conservation. See, perhaps it’s not so bad after all? Well OK, maybe it is a bit. Regardless, the final term in this equation neatly leads on to the next aspect of the derivation of the Navier-Stokes equations, and that is by taking a closer look at what we mean by the stress.

Specifying stress and the stress tensor

As specified above, the stress given off by the fluid is the force that the fluid is exerting upon a particular surface. We can define the stress as a continous function with components of x, the position in space, t, the time, and n, which is the normal vector to the surface element in question, i.e.

\vec{\sigma} = \vec{\sigma}(\vec{n},\vec{x},t)

The stress is similarly defined for an inviscid fluid, where \vec{\sigma} has a non-zero component of n only (in other words, \vec{\sigma} and n are parallel), and therefore

\vec{\sigma}(\vec{n},\vec{x},t) = -p(\vec{x},t)\vec{n}

where the scalar quantity p(\vec{x},t) is the mechanical pressure of the fluid. This pressure is considered to be a positive quantity when acting inwards on the surface of the fluid, hence why there is a minus sign present in the equation.

An important observation to take away from the definition of the stress of an inviscid fluid above is that, for a viscous fluid, \vec{\sigma} will not in general, be defined in the same direction as the normal vector to the surface, n. This is simply because the force acting upon on the surface is unlikely to be perpendicular to that particular element.

Due to this observation that \vec{\sigma} and n will generally be in different directions, we have to consider a concept known as the stress tensor to further clarify this point.

The stress tensor is initially defined as

\sigma_{ij}(\vec{x},t) = \sigma_i(\vec{N}_j,\vec{x},t)

where \vec{N}_j is defined as being the unit vectors in the directions of the coordinate axes. An explanation of the above is that \sigma_{ij} is the i-th component of the stress vector \vec{\sigma} at x and t, on an element whose normal is in the j direction. As there are three axes, both i and j are summed from one to three respectively.

This initial derivation is found by considering properties of a tetrahedron (a triangular-based pyramid, if you will), but I will not cover it here. Instead, I will consider the main points of this derivation only, and how important factors that are deduced from it, will further our understanding of what we mean by stress.

From defining the stress tensor as we have, we now need to consider two directions to help us define the overall stress imposed by the fluid:

1) The direction in which the stress acts, and

2) The direction of the normal of the surface element on which the stress is acting.

From here, when taking geometrical considerations into account, we were able to determine that having knowledge of the stress tensor was sufficient enough to write down the stress vector on a surface of arbitrary orientation (the full derivation is in the notes for those interested.)

Also of note is that the stress tensor is often displayed as

\sigma_{ij} = \begin{pmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33}\end{pmatrix}

although it should be made clear that the stress tensor is not a matrix!

Using the same derivation, we can equally determine the stress tensor for an inviscid fluid to be

\sigma_{ij} = -p\delta_{ij}

where \delta_{ij}, as you all know by now, is the Kronecker delta, where

\delta_{ij} = \begin{cases} 0 & \text{if i} \neq \text{j} \\ 1 & \text{if i = j}\end{cases}

For the sake of completeness, we can then express the stress tensor of the inviscid fluid like so

\sigma_{ij} = \begin{pmatrix} -p & 0 & 0 \\ 0 & -p & 0 \\ 0 & 0 & -p\end{pmatrix}

Just quickly, it should also be noted that the conservation of angular momentum does come into play, as it shows that the stress tensor is symmetric, i.e.

\sigma_{ij} = \sigma_{ji}

however, we are mostly just interested in this as a result. As such, no rigorous derivation was given for this result.

The Constitutive Equation for an Incompressible Newtonian Fluid

The individual components of the stress tensor depend on what is known as the rate of deformation of the fluid. In fact, what distinguishes one fluid from another is the precise form of said dependence. As this particular subtitle states, we are looking at Newtonian fluids, which satisfy what is known as Newton’s Law of Viscosity. For an incompressible fluid, this law is represented mathematically as

\sigma_{ij} = -p\delta_{ij} + \mu\left(\dfrac{\partial u_i}{\partial x_j} + \dfrac{\partial u_j}{\partial x_i}\right)

where \mu is known as the dynamic viscosity and is constant, provided that the temperature and pressure are fixed, for any given Newtonian fluid. You should also clearly see that the inviscid form for the stress tensor is found if you allow for \mu = 0.

No attempt was given within the notes to justify the above form for the stress tensor, but a simple analysis was used to hint towards the argument from which the above equation is derived.

To round off this particular subsection of the blog, I will include details of two important properties of any fluid, that is the aforementioned dynamic viscosity, and the kinematic viscosity.

A change in the pressure will do little to vary dynamic viscosity, but changing the temperature can have quite a dramatic effect on its value. As I mentioned way back in my first post, viscosity (I should have specified dynamic viscosity at the time) has S.I. units of kilogram per metre-seconds (\text{kg m}^{-1}\text{s}^{-1}), which is equivalent to Pascal-seconds (Pa.s). This translates, as you can see, to dimensions of \dfrac{\text{mass}}{\text{length} \times \text{time}}.

The kinematic viscosity is related to the dynamic viscosity by this equation

\nu = \dfrac{\mu}{\rho}

Since the density, \rho, is defined as \dfrac{\text{mass}}{\text{volume}}, then it follows simply that it must have dimensions of \dfrac{\text{mass}}{\text{length}^{3}}. From this, it is pretty easy to deduce that the kinematic viscosity, \nu, has dimensions of \dfrac{\text{length}^{2}}{\text{time}}. In S.I. units, this is equivalent to metres-squared per second (\text{m}^{2}s^{-1}).

The Navier-Stokes Equations

At last, we’ve finally got here! It took us long enough, didn’t it? We’ve now defined just about everything we need in order to deduce the equations we need to finally solve those pesky viscous flow systems! 😀

Annoyingly though, the derivation is still quite long, so I’ll try and summarise the main points below:

1) Do you remember that momentum conservation equation? Yep, the one at the start of this blog. Sorry to bring that up again, but we need it! Thankfully we’ve already defined that, so I won’t post it again here, but just keep it in mind.

2) Since the volume of our fluid is fixed (i.e. it does not vary with time), and only the density and velocity vector do change with respect to time, then we can switch the order of the first term of the momentum conservation equation, so that the derivative with respect to time is brought inside the integrand. We also know that the stress vector can be represented as a product of the stress tensor and the normal vector to the surface element, so the final term is replaced with this product.

3) Now using Gauss’ Divergence Theorem, and some clever manipulation, we can change all of our surface integrals into integrals over the volume. Now every single integral is being integrated over the volume, we can now combine all these integrands into one large integral, and move them all onto the left hand side of the equation. Thus the right hand side of the equation now equals zero. Since we are considering our volume to be arbitrary, then it must mean that our integrand must be identically equal to zero.

4) Now by re-arranging our equation so that the stress tensor and the body force terms are on the right hand side, we have now successfully expressed the momentum conservation principle for a general fluid! Congratulations! Well OK, we aren’t quite done yet.

5) By inserting the definition of the stress tensor in terms of Newton’s Law of Viscosity, we will eventually arrive at the Navier-Stokes equations for an incompressible Newtonian fluid. However, there is some algebraic manipulation involved for this. Just for convenience though, I’ll state the results below.

Here are the Navier-Stokes equations for an incompressible, constant viscosity fluid using the summation notation (this form of the equation is particularly useful for re-defining it in terms of Cartesian co-ordinates.)

\rho\left(\dfrac{\partial u_i}{\partial t} + u_j\dfrac{\partial u_i}{\partial x_j}\right) = -\dfrac{\partial\rho}{\partial x_i} + \mu\dfrac{\partial^2 u_i}{\partial (x_j)^2} + \rho(F_B)_i

The above can be expressed in a more condensed form in terms of vectors

\rho\dfrac{D\vec{u}}{Dt} = -\nabla p + \mu\nabla^2\vec{u} + \rho\vec{F}_B

That’s it! Hallelujah! 🙂 With the above defined equations, and the continuity equation that I defined in my last post, we can now solve systems of viscous fluids (subject to appropriate boundary and initial conditions of course).

Just a final note though on boundary and initial conditions. It is strongly advised to consider applying the no-slip boundary condition, which requires that at a solid surface, the fluid velocity should be equal to the surface velocity. If the flow is unsteady (i.e. the flow is time-dependent), then initial conditions are also required.

Reflection

Although it took a long time to get there, it was fairly clear why we were doing what we were doing; to get to the Navier-Stokes equations, which we will use extensively throughout the second term of the module. However, there are a lot of components to consider for its derivation, with lots of subtleties, especially in relation to the stress vector and tensor, and the momentum conservation equation. I feel more comfortable with the aspects of the derivation that were covered in last week’s lectures.

Everyone knows that the best way to practice mathematics is by doing it, so I’ll have to wait until next term when we cover this material again, before I can say with any degree of certainty as to whether I understand it or not.  Although it should be stated that, while we need to have a grasp of the derivation, the vast majority of the second term will be to do with applying the appropriate Navier-Stokes equations and continuity equation to multiple scenarios.

While from a revision point of view, I’m glad that I’ve covered a significant amount of material in this post, I feel that I’ve ultimately sacrificed brevity for additional detail and clarification. Just for future reference to any readers, I won’t be writing to this level of depth in my future blog posts. The time investment was far more significant that was expected of me, and simply put, a blog post of such length can get a little tiring for most people. At least I’m uploading on a more regular schedule now, so progress has been made there! 🙂

Now that we are moving back onto the inviscid flow part of the module, maybe, just maybe, my blog posts might be slightly shorter? Well, we can always hope! 😉 Thanks for reading all of this! If you didn’t…well…I hope it was good enough to send you into a peaceful night’s sleep. See you next time!

References and Sources:

1) The vast majority of the concepts talked about in this blog post are derived from notes taken from Chapter 1 and 2 of the MATH3402 Fluid Dynamics – Viscous Flow lecture notes.

2) I have found and used this particularly useful book on \LaTeX, in order to help me write appropriate matrices and piecewise functions as part of this blog post. Perhaps you’ll find something useful too? Here’s the link: http://en.wikibooks.org/wiki/LaTeX/Mathematics