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Introduction to turbulence/Nature of turbulence

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Introduction to turbulence
Nature of turbulence
Statistical analysis
Reynolds averaged equation
Turbulence kinetic energy
Stationarity and homogeneity
Homogeneous turbulence
Free turbulent shear flows
Wall bounded turbulent flows
Study questions

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The turbulent world around us

The turbulent motion of fluids has captured the fancy of observers of nature for most of recorded history. From howling winds to swollen floodwaters, the omnipresence of turbulence paralyzes continents and challenges our quest for authority over the world around us. But it also delights us with its unending variety of artistic forms. Subconsciously we find ourselves observing exhaust jets on a frosty day; we are willingly hypnotized by licking flames in an open hearth. Babbling brooks and billowing clouds fascinate adult and child alike. From falling leaves to the swirls of cream in steaming coffee, turbulence constantly competes for our attention.

Turbulence by its handiwork immeasurably enriches the lives of even those who cannot comprehend its mysteries. Art museums are filled with artists attempts to depict turbulence in the world around us. The classic sketch of Italian renaissance artist and engineer, Leonardo da Vinci, shown in Figure 1.1 represents both art and early science. And as the tongue-in-cheek poem below by Corrsin (one of the turbulence greats of the past century) shows, even for those who try, the distinction between art and research is often difficult to make.


For Hans Liepmann [2] on the occasion of his 70th birthday, with apologies to Bill S. and Liz B.B.

Shall we compare you to a laminar flow?
You are more lovely and more sinuous.
Rough winter winds shake branches free of snow,
And summer’s plumes churn up in cumulus.
How do we perceive you? Let me count the ways.
A random vortex field with strain entwined.
Fractal? Big and small swirls in the maze
May give us paradigms of flows to find.
Orthonormal forms non-linearly renew
Intricate flows with many free degrees
Or, in the latest fashion, merely few —
As strange attractor. In fact, we need Cray 3’s [3].
Experiment and theory, unforgiving;
For serious searcher, fun ... and it’s a living!

Not uploaded yet: Figure 1.1: Leonardo da Vinci’s observation of turbulent flow: Drawing of a free water jet issuing from a square hole into a pool (courtesy of

The turbulence texts in CFD-Wiki mostly deal with the equations used to describe the mechanics of turbulence. It is only equations which can give us the hope of predicting turbulence. But your study of this subject will be missing a great deal if this is all you learn. The advantage of studying turbulence is that you truly can see it almost everywhere as it mixes and diffuses, disrupts and dissipates the world around us.

So teach yourself to observe the natural and manmade processes around you. Not only will your life become more interesting, but your learning will be enhanced as well. Be vigilant. Whenever possible relate what you are learning to what you see. Especially note what you do not understand, and celebrate when and if you do. Then you will find that the study of turbulence really is fun.

What is turbulence?

Turbulence is that state of fluid motion which is characterized by apparently random and chaotic three-dimensional vorticity. When turbulence is present, it usually dominates all other flow phenomena and results in increased energy dissipation, mixing, heat transfer, and drag. If there is no three-dimensional vorticity, there is no real turbulence. The reasons for this will become clear later; but briefly, it is ability to generate new vorticity from old vorticity that is essential to turbulence. And only in a three-dimensional flow is the necessary stretching and turning of vorticity by the flow itself possible.

For a long time scientists were not really sure in which sense turbulence is “random”, but they were pretty sure it was. Like anyone who is trained in physics, we believe the flows we see around us must be the solution to some set of equations which govern. (This is after all what mechanics is about — writing equations to describe and predict the world around us.) But because of the nature of the turbulence, it wasn’t clear whether the equations themselves had some hidden randomness, or just the solutions. And if the latter, was it something the equations did to them, or a consequence of the initial conditions?

All of this began to come into focus as we learned about the behavior of strongly non-linear dynamical systems in the past few decades. Even simple nonlinear equations with deterministic solutions and prescribed initial conditions were found to exhibit chaotic and apparently random behavior. In fact, the whole new field of chaos was born in the 1980’s [4], complete with its new language of strange attractors, fractals, and Lyapunov exponents. Such studies now play a major role in analyzing dynamical systems and control, and in engineering practice as well.

Not uploaded yet: Figure 1.2: Turbulence in a water jet. Photo from Dimotakis, Miake-Lye and Papantoniou, Phys. Flds., 26 (11), 3185 – 3192.

Turbulence is not really chaos, at least in the sense of the word that the dynamical systems people use, since turbulent flows are not only time-dependent but space dependent as well. But as even the photos of simple turbulent jets and wakes shown in Figures 1.2 and 1.3 make clear, turbulence has many features that closely resemble chaos. Obvious ones include spatial and temporal intermittency, dissipation, coherent structures, sensitive dependence of the instantaneous motions on the initial and upstream conditions, and even the near-fractal distribution of scales. In fact, the flows we see themselves bear an uncanny resemblance to the phase plane plots of strange attractors. No one would ever confuse a jet with a wake, but no two wakes seem to be quite alike either.

Because of the way chaos has changed our world view, most turbulence researchers now believe the solutions of the fluid mechanical equations to be deterministic. Just like the solutions of non-linear dynamical systems, we believe turbulent solutions to be determined (perhaps uniquely) by their boundary and initial conditions [5]. And like non-linear dynamical systems, these deterministic solutions of the non-linear fluid mechanics equations exhibit behavior that appears for all intents and purposes to be random. We call such solutions turbulent, and the phenomenon turbulence. Because of this chaotic-like and apparently random behavior of turbulence, we will need statistical techniques for most of our study of turbulence.

Not uploaded yet: Figure 1.3: Axisymmetric wakes from four different generators. Photo from S.C. Cannon, Ph.D. Dissertation., U.of Ariz, 1991.

The lack of a satisfactory understanding of turbulence presents one of the great remaining fundamental challenges to scientists — and to engineers as well, since most technologically important flows are turbulent. The advances in understanding over the past few decades, together with the advent of large scale computational and experimental capabilities, present the scientist and engineer with the first real capabilities for understanding and managing turbulent flows. As a result, this is a really wonderful time to study this subject.

Why study turbulence?

There really are the TWO reasons for studying turbulence - engineering and physics! And they are not necessarily complementary, at least in the short run.

Certainly a case can be made that we don’t know enough about turbulence to even start to consider engineering problems. To begin with (as we shall see very quickly over the next few lectures), we always have fewer equations than unknowns in any attempt to predict anything other than the instantaneous motions. This is the famous turbulence closure problem.

Of course, closure is not a problem with the so-called direct numerical simulation in which we numerically produce the instantaneous motions in a computer using the exact equations governing the fluid. Unfortunately we won’t be able to perform such simulations for real engineering problems until at least a few hundred generations of computers have come and gone. And this won’t really help us too much, since even when we now perform a DNS simulation of a really simple flow, we are already overwhelmed by the amount of data and its apparently random behavior. This is because without some kind of theory, we have no criteria for selecting from it in a single lifetime what is important.

The engineer’s counter argument to the scientists’ lament above is:

  • airplanes must fly,
  • weather must be forecast,
  • sewage and water management systems must be built,
  • society needs ever more energy-efficient hardware and gadgets.

Thus, the engineer argues, no matter the inadequate state of our knowledge, we have the responsibility as engineers to do the best we can with what we have. Who, considering the needs, could seriously argue with this? Almost incredibly -some physicists do!

The same argument happens in reverse as well. Engineers can become so focused on their immediate problems they too lose the big picture. The famous British aerodynamicist M. Jones captured this well when he said,

"A successful research enables problems which once seemed hopelessly complicated to be expressed so simply that we soon forget that they ever were problems. Thus the more successful a research, the more difficult does it become for those who use the result to appreciate the labour which has been put into it. This perhaps is why the very people who live on the results of past researches are so often the most critical of the labour and effort which, in their time, is being expended to simplify the problems of the future."

It seems evident then that there must be at least two levels of assault on turbulence. At one level, the very nature of turbulence must be explored. At the other level, our current state of knowledge — however inadequate it might be - must be stretched to provide engineering solutions to real problems.

The great danger we face is of being deceived by the successes and good fortune of our “engineering solutions” into thinking we really understand the “physics”. But the real world has a way of shocking us back to reality when our “tried and tested” engineering model fails miserably on a completely new problem for which we have not calibrated it. This is what happens when we really don’t understand the “physics” behind what we are doing. Hopefully this course will get you excited about both the physics and the applications, so you won’t fall into this trap.

The cost of our ignorance

It is difficult to place a price tag on the cost of our limited understanding of turbulence, but it requires no imagination at all to realize that it must be enormous. Try to estimate, for example, the aggregate cost to society of our limited turbulence prediction abilities which result in inadequate weather-forecasts alone. Or try to place a value on the increased cost to the consumer need of the designer of virtually every fluid-thermal system-from heat exchangers to hypersonic planes- to depend on empiricism and experimentation, with the resulting need for abundant safety factors and non-optimal performance by all but the crudest measures.Or consider the frustration to engineers and cost to management of the never-ending need for 'code-validation' experiments every time a new class of flows is encountered or major design change is contemplated. The whole idea of 'codes' in the first place was to be able to evaluate designs without having to do experiments or build prototypes.

Some argue that our quest for knowledge about turbulence should be driven solely by the insatiable scientific curiosity of the researcher, and not by the applications. Whatever the intellectual merits of this argument, it is impossible to consider the vastness and importance of the applications and not recognize a purely financial imperative for fundamental turbulence research. The problem is, of course, that the cost of our ignorance is not confined to a single large need or to one segment of society, but is spread across the entire economic spectrum of human existence. If this were not the case, it would be easy to imagine federal involvement at the scale of America’s successful moon venture or the international space station, or at very least a linear accelerator or a Galileo telescope. Such a commitment of resources would certainly advance more rapidly our understanding.

But the turbulence community - those who study and those who use the results - have failed ourselves to recognize clearly the need and nature of what we really do. Thus in turbulence, we have been forced to settle for far, far less than required to move us forward very fast, or maybe at all. Hopefully you will live to see this change. Or even better, perhaps you will be among the ones who change it.

What do we really know for sure?

Turbulence is a subject on which still studies are going on. We really don't know a whole lot for sure about turbulence. And worse, we even disagree about what we think we know! There are indeed some things some researchers think we understand pretty well - like for example the Kolmogorov similarity theory for the dissipative scales and the Law of the Wall for wall-bounded flows. These are based on assumptions and logical constructions about how we believe turbulence behaves in the limit of infinite Reynolds number. But even these ideas have never been tested in controlled laboratory experiments in the limits of high Reynolds number, because no one has ever had the large scale facilities required to do so.

It seems to be a characteristic of humans(and contrary to popular belief, scientists and engineers are indeed human) that we tend to accept ideas which have been around a while as fact, instead of just working hypotheses that are still waiting to be tested. One can reasonably argue that the acceptance of most ideas in turbulence is perhaps more due to the time lapsed since they were proposed and found to be in reasonable agreement with limited data base, than that they have been subjected to experimental tests over the range of their assumed validity. Thus it might be wise to view most 'established' laws and theories of turbulence as more like religious creeds than matters of fact.

The whole situation is a bit analogous to the old idea that the sun and stars revolved around the earth - it was a fine idea, and even good today for navigational purposes. The only problem was that one day someone (Copernicus, Brahe and Galileo among them) looked up and realized it wasn't true. So it may be with a lot of what we believe today to be true about turbulence - some day you may be the one to look at evidence in a new way and decide that things we thought to be true are wrong.


  1. Stan Corrsin was a famous and much beloved turbulence researcher and professor at the Johns Hopkins University.
  2. Hans Liepmann was another famous turbulence researcher and professor at Cal Tech, who was Corrsin’s Ph.D. dissertation advisor.
  3. At the time this poem was written, the Cray 2 was the world’s most powerful computer.
  4. The delightful book by James Gleik “Chaos: the making of a new science” provides both interesting reading and a mostly factual account.
  5. If it comes as a surprise to you that we don’t even know this for sure, you might be even more surprised to learn that there is a million dollar prize for the person who proves it.


This text was based on "Lectures in Turbulence for the 21st Century" by Professor William K. George, Professor of Turbulence, Chalmers University of Technology, Gothenburg, Sweden.

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