Turbulence and Lorenz Attractor

[Clausius2]

When one tries to have an insight into what the turbulence regime really is, one comes across the question of: is the turbulence a deterministic process or not? Even though we are able to simulate accurately turbulence with Direct Numerical Simulation (DNS) of the Navier-Stokes Equations, it is very difficult to test the results unless done assumming an statistically steady state. What I mean is that even though the DNS and its results are showing the major turbulent coherent structures and it comes from a deterministic model (N-S equations), it is very difficult to test its results with experiments unless the numerical results are averaged and compared with those flow experimentally averaged quantities.

Nowadays fluid dynamicists are prone to think that a turbulent process is governed by a set of deterministic equations such as N-S equations. That is, the process itself seems to be deterministic (how the coherent structures evolutes in time in space seems to have a physical explanation), but it is not quite the case of how the turbulence is triggered. The onset of turbulence seems to me to be influenciated by lot of random variables. I mean, even though we are able to use DNS, the whole key stays in the Initial and Boundary Conditions of integration. Those are kind of random and extraordinarily difficult to account for. The IC and BC in a laboratory facility may be impossible to simulate because they could belong to the randomness of some environmental conditions. As the Reynolds number becomes large, the sensivity of the flow to these small perturbations grows enormously, triggering the global instability.

What about the Lorenz Attractor http://mathworld.wolfram.com/LorenzAttractor.html?? Lorenz worked out this model thinking of the instabilty of a pure convective regime (large Rayleigh Number). It seems to me that this system is a generator of randomness even it is "deterministically determined" by a set of differential equations. My question is, what is triggering this apparently chaotic behavior?? i) Is it because the proper nature of the equations? ii) Or is it because a numerical instability impossible to overcome with nowadays methods??.

If i) is true, the lorenz attractor seems to me contradictory, because then a mathematical model is not as deterministic as I thought. But if ii) is true, that is the instability is caused by the numerical integration, I think it is telling us that despites we are unable of integrate the equations properly, the process is deterministic at the bottom. The Lorenz attractor seems a great comparison with the N-S equations at high Reynolds Numbers. Are they instable per se, generators of randomness, or on the contrary are we unable of integrating the properly propagating numerical errors which trigger finally the instability??.

 

[desA]

Interesting topic indeed... It is the current focus of my PhD study program... :-)

The turbulence behaviour could appear to be chaotic, when observed correctly - rather than using the typical statistical perspective favoured by many Fluid Dynamicists. There is a further physical mechanism however which seems to link back quite nicely with the 'chaotic' theme in the sense of determinism within turbulence, & in fact overlays a sense of order over this apparent chaos. Most Fluid Dynamicists have unfortunately been looking too hard in the wrong places... :-)

I have discovered this mechanism contained within the Navier-Stokes equations themselves & am presently writing a paper series describing this phenomenon. All of a sudden, much of the mystery surrounding these 'magic equations' has been exposed. Not all Fluids folks will, however, agree.

I'll have to contain my input at this point in order not to pre-empt my papers, but suffice it to say that I am able to substantiate these findings in numerous high-resolution numerical simulations.

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I'll look more deeply into the Lorenz Attractor & see if it could link back into my current findings. Why not? :-)

 

[charng]

I find this question about the nature of turbulence and its deterministic or chaotic behavior to be a very interesting and complex topic. The concept of turbulence has been studied for many years and yet, there is still much debate and uncertainty about its true nature.

One of the main challenges in studying turbulence is the difficulty in obtaining accurate and reliable data. As mentioned in the content, even with advanced simulations using DNS, the results can only be validated by comparing them with experimentally averaged quantities. This highlights the fact that turbulence is a highly complex and dynamic phenomenon that is difficult to fully understand and predict.

In terms of the question about whether turbulence is a deterministic process or not, it is important to note that the Navier-Stokes equations, which are the fundamental equations used to describe fluid flow, are deterministic in nature. This means that given a set of initial conditions, the equations can be solved to determine the future state of the flow. However, turbulence is a highly nonlinear process and even small changes in the initial conditions can lead to significantly different outcomes. This sensitivity to initial conditions is known as the "butterfly effect" and is a key characteristic of chaotic systems.

The Lorenz Attractor is a mathematical model that was developed to study the behavior of a convective flow system. It is a good comparison to the Navier-Stokes equations at high Reynolds numbers, as both systems exhibit chaotic behavior. However, it is important to note that the Lorenz Attractor is a simplified model and does not fully capture the complexity of real-world turbulence.

In terms of what triggers the chaotic behavior in the Lorenz Attractor, it is a combination of both the nature of the equations and the numerical instability caused by the integration process. The equations themselves are highly nonlinear and sensitive to initial conditions, which can lead to chaotic behavior. However, numerical errors and limitations in the integration process can also contribute to this chaotic behavior.

In conclusion, the nature of turbulence is a complex and ongoing topic of research. While the underlying equations may be deterministic, the behavior of turbulence is highly nonlinear and chaotic, making it difficult to fully understand and predict. The Lorenz Attractor serves as a useful comparison to the Navier-Stokes equations at high Reynolds numbers, but it is important to keep in mind the limitations and simplifications of this mathematical model. Further research and advancements in technology will continue to shed light on the true nature of turbulence.