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Blog - eddy who?

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Or: A very short introduction to turbulence

Have a look at this picture:

And then look at this one:

Do they look similar?

They should! They are both examples of a Kelvin-Helmoltz instability.

The first graphic is a picture of billow clouds (the fancier name is altostratus undulatus clouds):

The picture is taken from:

• C.Donald Ahrens: Meteorology Today, 9th edition, Books/Cole 2009.

The second graphic:

shows a lab experiment and is taken from:

• G.L. Brown and A. Roshko, online available Density effects and large structure in turbulent mixing layers, Journal of Fluid Mechanics 64 (4), pp. 775-816 (1974), ISSN 0022-1120.

Isn’t it strange that clouds in the sky would show the same pattern as some gases in a small laboratory experiment? The reason for this is not quite understood today. In this post, I would like to talk a little bit about what is known.

Matter that tries to get out of its own way

Fluids like water and air can be well described by Newton’s laws of classical mechanics. When you start learning about classical mechanics, you consider discrete masses, most of the time. Billiard balls, for example. But it is possible to formulate Newton’s laws of motion for fluids by treating them as ‘infinitely many infinitesimally small’ billiard balls, all pushing and rubbing against each other and therefore trying to get out of the way of each other.

If we do this, we get some equations describing fluid flow: the Navier-Stokes equations.

The Navier-Stokes equations are a complicated set of nonlinear partial differential equations. A lot of mathematical questions about them are still unanswered, like: under what conditions is there a smooth solution to these equations? If you can answer that question, you will win one of the a million dollars from the Clay Mathematics Institute.

If you completed the standard curriculum of physics as I did, chances are that you never attended a class on fluid dynamics. At least I never did. When you take a first look at the field, you will notice: the literature about the Navier-Stokes equations alone is huge! Not to mention all the special aspects of numerical simulations, special aspects of the climate and so on.

So it is nice to find a pedagogical introduction to the subject for people who have some background knowledge in partial differential equations, for the mathematical theory:

  • C. Foias, R. Rosa, O. Manley and R. Temam: Navier-Stokes Equations and Turbulence, Cambridge University Press, Cambridge, 2001.

So, there is a lot of fun to be had for the mathematically inclined. But today I would like to talk about an aspect of fluid flows that also has a tremendous practical importance, especially for the climate of the Earth: turbulence!

There is no precise definition of turbulence, but people know it when they see it. A fluid can flow in layers, with one layer above the other, maybe slightly slower or faster. Material of one layer does hardly mix with material of another layer. These flows are called laminar flows. When a laminar flow gets faster and faster, it turns into a turbulent flow at some point:

This is a fluid flow inside a circular pipe, with a layer of some darker fluid in the middle.

As a first guess we could say that a characteristic property of turbulent flow is the presence of circular flows, commonly called eddies.

Tempests in Teapots

A funny aspect of the Navier-Stokes equations is that they don’t come with any recommended length scale. Properties of the fluid flow like velocity and pressure are modelled as smooth functions of continuous time and space. Of course we know that this model does not work on a atomic length scale, where we have to consider individual atoms. Pressure and velocity of a fluid flow don’t make any sense on a length scale that is smaller than the average distance between electrons and the atom nucleus.

We know this, but this is a fact that is not present in the model comprised by the Navier-Stokes equations!

But let us look at bigger length scales. An interesting feature of the solutions of the Navier-Stokes equations is that there are fluid flows that stretch over hundreds of meters that look like fluid flows that stretch over centimeters only. And it is really astonishing that this phenomenon can be observed in nature. This is another example of the unreasonable effectiveness of mathematical models.

You have seen an example of this in the introduction already. That was a boundary layer instability. Here is a full blown turbulent example:

The last two pictures are from the book:

  • Arkady Tsinober: An Informal Conceptual Introduction to Turbulence, 2nd edition, Springer, Fluid Mechanics and Its Applications Volume 92, Berlin, 2009.

This is a nice introduction to the subject, especially if you are more interested in phenomenology than mathematical details.

Maybe you noticed the “Reynolds number” in the label text of the last picture. What is that?

People in business administration like management ratios; they throw all the confusing information they have about a company into a big mixer and extract one or two numbers that tell them where they stand, like business volume and earnings. People in hydrodynamics are somewhat similar; they define all kinds of “numbers” that condense a lot of information about fluid flows.

A CEO would want to know if the earnings of his company are positive or negative. We would like to know a number that tells us if a fluid flow is laminar or turbulent. Luckily, such a number already exists. It is the Reynolds_number! A low number indicates a laminar flow, a high number a turbulent flow. Like the calculation of the revenue of a company, the calculation of the Reynolds number of a given fluid flow is not an exact science. Instead there is some measure of estimation necessary. The definition involves, for example, a “characteristic length scale”. This is a fuzzy concept that usually involves some object that interacts with - in our case - the fluid flow. The characteristic length scale in this case is the physical dimension of the object. While there is usually no objectively correct way to assing a “characteristic length” to a three dimensional object, this concept allows us nevertheless to distinguish the scattering of water waves at a ocean liner (length scale 10 3meter\approx 10^3 meter) from that at a peanut (length scale 10 2meter\approx 10^{-2} meter).

The following graphic shows laminar and turbulent flows and their characteristic Reynolds numbers:

schematic display of laminar and turbulent flow for different Reynolds numbers

This graphic is from the book

  • Thomas Bohr, Mogens H. Jensen, Giovanni Paladin, Angelo Vulpiani: Dynamical systems approach to turbulence., Cambridge Univ. Press, Cambridge, 1998

But let us leave the Reynolds number for now and turn to one of its ingredients: viscosity. Understanding viscosity is important for understanding how eddies in a fluid flow are connected to energy dissipation.

Eddies dissipate kinetic energy

"Eddies," said Ford, "in the space-time continuum."

"Ah," nodded Arthur, "is he? Is he?" He pushed his hands into the pocket of his dressing gown and looked knowledgeably into the distance.

"What?" said Ford.

"Er, who," said Arthur, "is Eddy, then, exactly, then?"
From Douglas Adams: "Life, the Universe and Everything"

A fluid flow can be pictured as consisting of a lot of small fluid packages that move alongside each other. In many situations, there will be some friction between these packages. In the case of fluids, this friction is called viscosity.

It is an empirical fact that at small velocities fluid flows are laminar: There are layers upon layers, with one layer moving at a constant speed, and almost no mixing. At the boundaries, the fluid will attach to the surrounding material, and the relative fluid flow will be zero. If you picture such a flow between a plate that is at rest, and a plate that is moving forward, you will see that due to friction between the layers a force needs to be exerted to keep the moving plate moving:

In the simplest approximation, you will have to exert some force FF per unit area AA, in order to sustain a linear increase of the velocity of the upper plate along the y-axis, u/y\partial u / \partial y. The constant of proportionality is called the viscosity μ\mu:

FA=μuy \frac{F}{A} = \mu \frac{\partial u}{\partial y}

More friction means a bigger viscosity: Honey has a bigger viscosity than water.

If you stir honey, the fluid flow will come to a halt rather fast. The energy that you put in to start the fluid flow is turned to heat by dissipation. This mechanism is of course related to friction and therefore to viscosity.

It is possible to formulate an exact formula for this dissipation process using the Navier-Stokes equations. It is not hard to prove it, but I will only explain the involved gadgets.

A fluid flow in three dimensions can be described by stating the velocity of the fluid flow at a certain time tt and x 3\vec{x} \in \mathbb{R}^3 (I don’t specify the region of the fluid flow or any boundary or initial conditions). Let’s call the velocity u(t,x)\vec{u}(t, \vec{x}).

Let’s assume that the fluid has a constant density ρ\rho. Such a fluid is called incompressible. For convenience we assume that the density is 1, ρ=1\rho = 1. Then the kinetic energy E(u)E(\vec{u}) of a fluid flow at a fixed time tt is given by

E(u)=u(t,x) 2dx E(\vec{u}) = \int \| \vec{u}(t, \vec{x}) \|^2 d \vec{x}

Let’s just assume that this integral is finite for the moment. This is the first gadget we need.

The second gadget is called enstrophy ϵ\epsilon of the fluid flow. This is a measure of how much eddies there are. It is the integral

ϵ=×u 2dx \epsilon = \int \| \nabla \times \vec{u} \|^2 d \vec{x}

where ×\nabla \times denotes the curl of the fluid velocity. The faster the fluid rotates, the bigger the curl is.

(The math geeks will notice that the vector fields u\vec{u} that have a finite kinetic energy and a finite enstrophy are precisely the elements of the Sobolev space H 1( 3)H^1(\mathbb{R}^3))

Here is the relationship of the decay of the kinetic energy and the enstropy, which is a consequence of the Navier-Stokes equations (and suitable boundary conditions):

ddtE=μϵ \frac{d}{d t} E = - \mu \epsilon

This equation says that the energy decays with time; and it decays faster if there is a higher viscosity, and if there are more and stronger eddies.

If you are interested in the mathematically precise derivation of this equation, you can look it up in the book I already mentioned:

  • C. Foias, R. Rosa, O. Manley, R. Temam: Navier-Stokes equations and turbulence.

This connection of eddies and dissipation could indicate that there is also a connection of eddies and some maximum entropy principle. Since eddies maximize dissipation, natural fluid flows should somehow tend towards the production of eddies. It would be interesting to know more about this!

In this post we have seen eddies at different length scales. There are buzzwords in meteorology for this:

You have seen eddies at the microscale (left) and at the mesoscale (middle). A blog post about eddies should of course mention the most famous eddy of the last decade, which formed at the synoptic scale:

Do you recognize it? That was hurricane Katrina.

It is obviously important to understand disasters like this one on the synoptic scale. This is an active topic of ongoing research, both in meteorology and in climate science.

category: blog