243. Turbulence: brief introduction into phenomenon
APPLIED — By Dmitry Podolsky on February 9, 2009 at 6:11 pm
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About a week ago I have listed top ten open problems in physics. If you say A, you are naturally supposed to say B (especially, if readers ask you 
), so, I guess, I will have to discuss each problem from the list in more details. I would like to start with the problem N3 most compelling to me at the present time – the physics of developed turbulence. In order not to make the post too long, I am not going to discuss various models and theories of turbulence today – and will focus instead just on nature of the phenomenon.
A particular reason why I am so fond of turbulence is the way the simplicity and complexity of the problem are interwoven into each other – while the form of the Navier-Stockes equation describing a turbulent fluid is extremely simple, a solution of this equation that the Mother Nature chooses is so complicated that so far all attempts to analytically describe it has pretty much failed.
1. What is turbulence. Weak vs. strong turbulence
Turbulence is a complex behaviour of a dissipative medium or a field, disordered, stochastic in time and space. What does it mean – stochastic? Even if you set up initial conditions for the medium (say, initial density and velocity distributions for a turbulent fluid) with arbitrary precision, memory about these initial conditions will be inevitably lost as time passes. The same holds for boundary conditions: if you set them with arbitrary precision, correlations of the fluid velocity in the volume of the flow with velocity at the boundary will (exponentially) decay. As we say, the flow has finite correlation length and time scale.
So what, you say, why should I get excited about that? In Nature, there multiple examples of systems where time and space correlations decay but still the overall behavior of the system is trivial: one well known simple example is a system in (or near) the thermal equilibrium.
The reason why turbulence is interesting is that the loss of temporal and spatial correlations in the turbulent flow is a consequence of the complex dynamic behavior of the fluid itself, instabilities in individual pulsations of the flow. It has nothing to do with incomplete description of the system (like in the case of a system at thermal equilibrium) or effects of external noise. In other words, even dynamics of a closed system can be turbulent.
2. How does turbulence affect our lifes
Actually, turbulent, chaotic physical phenomena related to turbulence are more typical in Nature than phenomena characterized by regular behavior of observables. We find turbulent behavior in micro- and macrophysics – starting from cosmological scales (turbulence together with gravitation determines the large scale structure of the Universe; interstellar gas is turbulent) and ending by the scales of particle physics.
As for our everyday life, one can recall that water flows in rivers, seas, oceans are turbulent as well as blood flow in your veins; terrestrial atmospheric circulation is turbulent, and this turbulence directly affects the weather (or better say – our inability to predict it).
In hydro- and aerodynamics which we will focus on, probably the most important effect of turbulence is the resistance, reactance, that a moving body experiences in a turbulent flow.
Actually, reactance can increase or decrease depending on the form of the body that moves in the turbulent flow. For example, the main contribution into this resistance for prolonged bodies of aerodynamic form is given by the total tangential stress. The latter increases after transition to the regime of developed turbulence.
On the other hand, for bodies like a cylinder the main contribution into resistance is given by the face pressure decreasing after transition to the turbulent regime.
Let us now discuss different kinds of turbulent flow.
Weak turbulence is a turbulence of wave fields, when phases of separate waves can be considered random. For example, the distributions of small amplitude waves on the ocean’s surface are described by the theory of weak turbulence (they can scatter on each other, and the overall phase in each wave gets randomized in the process). Actually, it turns out that almost any system in the regime of dynamical chaos is described by the theory of weak turbulence.
Generally, turbulence can be characterized by the dimension of the phase space of the system (more or less, the number of independent modes that contribute into the motion of the medium). In the case of weak turbulence, dimension of the phase space of the system is typically less than 10.
On the other hand, in the regime of strong or developed turbulence random phase approximation ceases to be valid – different modes strongly interact with each other and there exists large coherent contribution. Turbulence of shock waves in media with weak dispersion and soliton turbulence are two examples of the developed turbulence. The phase space of the system in the regime of strong turbulence is huge, its dimension is typically much greater than 100.
3. Transition to turbulence
Usually, a single parameter decides whether a given flow is ordered or turbulent. In dissipative flows this parameter can be constructed out of the velovity of the flow 
, viscosity 
and the characteristic scale of the flow 
.
This parameter is called the Reynolds number, nd typically, if the value of this parameter is large (say, more than 100), then the flow is turbulent.
If we increase the value of the Reynolds number (say, by increasing the velocity of the flow), the transion from regularity to turbulence can happen suddenly or in several steps, with the complexity in the flow motion grows subsequently.
Whether the transition is sudden or happens in steps seems to depend on the geometry on the flow.
For example, suppose that we placed two parallel cylinders the water and started to rotate them with different angular velocities (frequences) 
and 
. Due to viscosity effects, the water gets carried by rotating cylinders (the resulting flow is called the Taylor-Quette flow).
While we increase the Reynolds number of the flow, the following sequence of events happens.
- Of course, First, at very small Reynolds numbers, the water gets involved into the circular motion, with two centers of circles coinciding with the axis of the cylinders.
- Toroidal vortices (Taylor vortices) appear.
- Small excitations (sound waves) appear along Taylor vortices.
- Sound waves along Taylor vortices get modulated with the second (smaller) frequency.
- Further, the motion becomes completely chaotic since sound waves start to strongly interact with each other and carrying vortices themselves.
Taylor vortices
Another example is the flow between two parallel planes (so called Puaseil flow). Here, transition to turbulence happens suddenly – when the Reynolds number gets larger than 5772, the motion of the fluid becomes chaotic (5772 is a funny magic number, isn’t it? but that’s what experiment gives us ).
4. Intermediate conclusion
That’s pretty much all we see on experiment (so far I am leaving apart questions about Kolmogorov’s spectra of turbulence, intermittency etc. – they are beyond the scope of this basic introduction into the subject).
On one hand, behavior of turbulent flows is largely universal (after the regime of developed turbulence is established, all turbulent phenomena – in fluids, in plasmas, in interstellar gas – look remarkably similar to each other). On the other hand, if we study transition to turbulence, universality is lost: for different geometries and setups transition happens at different critical values of the Reynolds number, and the way how the transition happens may be also very different.
Next time I’ll either discuss theories and models or the most interesting open questions in the physics of turbulence – I have not decided yet.
7 Comments
What do you mean by turbulent behavior? It seems you just explained the transition to turbulance.
Hi Daniel,
Kindly see the first paragraph after the subtititle “1. What is turbulence?”
Or does this definition make you unhappy anyhow?
Cheers,
Dmitry.
There are a few things that made me unhappy with the definition.
In the first paragraph, you write “Turbulence is a complex behaviour of a dissipative medium or a field, disordered, stochastic in time and space.”. Now, when you define strong turbulance you write “strong or developed turbulence random phase approximation ceases to be valid – different modes strongly interact with each other and there exists large coherent contribution.”. So, there is no random here, but instead, it indicates something completely, different, that is, something of almost of infinite complexity.
Other thing that makes me sad because stochastic inherent to a continuous system, but I keep thinking of the brownian motion and vapor formation inside the liquid. For example, what makes it sure that the transition of chaos is not merely that the flow is cloged, or disturbed, or cause by an amplification of brownian movement. Another thing it is if the irregular distribution of energy in the system could lead to state transtions in some parts, like, formation of microbubbles, and if they wouldn’t contribute significantly to the motion of the fluid.
There are a few things that made me unhappy. For example, I am confused if turbulance is indeed stochastic, as you pointed in the first part, because when you define strong turbulance you write: ?strong or developed turbulence random phase approximation ceases to be valid – different modes strongly interact with each other and there exists large coherent contribution.?. But I see that the case of interest is the strong one, not the weak, or at least this is what looks like from seeing the transitional steps. Besides, in the strong case, it seems that chaotic is not actualy random.What is it then?
Also, what makes it sure that a fluid can be considered a fluid? You have brownian motion and, perhaps, the formation of bubles due the energy released by shear-stress along the boundary of layers of different reynold numbers. It seems they should be taken seriously when going to chaos.
Dear Daniel,
Regarding the randomness and random phase approximation, I thought it was rather clear what I’ve meant.
In weak turbulence, the only degrees of freedom that you excite are sound waves. In developed turbulence, there is also vorticity. If you consider a single vortex and decompose its velocity field into Fourier modes, you will clearly see that phases of different modes are correlated with each other. This is pretty much all – you have vortices interacting with each other and sound waves interacting with each other as well as with vortices. Random phase approximation for separate Fourier modes is lost but stochasticity – is not, because both velocity and vorticity fields are random.
Regarding he Brownian motion – thermal effects are clearly subdominant (suppose I take nearly zero temperature and study turbulence in liquid Helium: I’ll still have all these patterns with Kolmogorov spectra etc.) Regarding shear stress and bubbles – also, I think, not that relevant for the problem – these effects are included into the definition of viscosity which is macroscopic. Also, consider a turbulence in convection systems for example, where energy of the flow is low (for turbulence, what is important is not only the energy of the flow per unit volume but also the dissipation of energy).
Cheers,
Dmitry.
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