Well, the other day I saw a solution of a DNS (Direct Numeric Simulation) of a turbulent flow. My questions are a bit phylosophical, so that if I do not obtain some opinion in some day in this forum, it will be not a strange thing.

-The turbulence is inherently three-dimensional and unsteady. A flow over a flat plate has to have a third dimension (the width of the plate). It seemed a mesh filled with a lot of very small eddies. So that, symmetry seems to be broken up. If there is no symmetry possible in spite of having a symmetric geometry and a symmetric initial condition, then you will see a different event depending on your angle of view. Don't you think it violates the common sense? In spite of having initial symmetry, the flow does not behave in a symmetric form. It is not usual to see this phenomena in other physics problems.

-My principal question is about unsteadyness and the apparent alleatory behaviour. I'm going to quote the book "Computational Fluid Dynamics and Heat Transfer" of Anderson:

The question is: if such certain random behaviour is real, Would I obtain two different solutions for the same problem? If I would solve the turbulent equations twice, would I obtain two distinct solutions each time?.

Thanks for pay attention to this auto-quiz-puzzle.

If one took a random signal generator (noise) and sample say 100 times, then 200 times, one might find a slightly different mean, but the values would probably be very close. If one samples 10,000 and then 20,000 times, the amplitude about the mean would probably be the same within a small deviation, say about 0.1% or less.

In the real world, initial conditions are never the same, and the boundary conditions are noisy, but perhaps bounded.

I see where you are getting at. But I'm talking to you about mathematical models, equations, PDE's that are closed and self-contained. Imagine a problem of the Heat Equation. Would you think of obtaining distinct solutions for each time you try to resolve it. The reality seems to be closed at these types of problems. So far, I haven't got the confirmation of that, (i.e random solutions or bounded random solutions for each simulation for turbulent flow).

But boundness doesn't seem to fade away my surprising. Navier Stokes equations are not written on terms of mean values.

On the other hand, you are right about the randomness at experimental events, and about their boundness. But I'm talking about well posed mathematical problems.

It would be nice to have an analytical solution for turbulence, but that's impossible!

The PDE's are special cases of the relationships within physical systems. Navier Stokes is an idealized model.

One can appreciate this when developing Finite Element or Finite Difference Models to predict a physical phenomenon, as in Computational Fluid Dynamics. One can develop an approximation that works quite well, i.e. within an acceptable margin of 'error' or 'uncertainty'.

In CFD, trying to model complicated phenomenon such as eddies and vortex shedding with an analytical model cannot be done. And extending this into fluid-structure interactions again cannot be done analytically.

Heat conduction problems are much more well behaved, as long as properties are relatively uniform, i.e. compositions, phases, etc.

Elastic mechanics can be solved using analytical solutions, but when non-linearities enter, its a different ball game.

I think here you're wrong. N-S equations predict that. Sure it doesn't exist any analitycal solution, but there are numerical solutions. The problem with turbulent flows was to model small and large eddies at the same time (disparity of scales). So that people used averaged methods (RANS equations) and small eddies aproximations (LES methods).

Nowadays we have enough computer power to deal directly with the N-S equations. They are a model, sure, but they have more amount of reality contained than we have observed yet from RANS equations. I'm talking to you about DNS (simulating directly the N-S equations). N-S equations are a powerful analytical model for turbulent phenomena, but it is only actually when people is enabled to simulate them numerically.

Based on first hand experience with cross flow and axial flow in heat exchanger tube banks, and axial flows in nuclear fuel assemblies, where in all cases, flow-induced vibration may lead to fretting wear at the contact points of the mechanical restraints - there is now way that one can get a precise solution with N-S - especially when considering the fluid structure interactions, where the damping and structure stiffness is not so precisely defined, and is also, changing non-linearly with time, is spatially asymmetric and temporally aperiodic.

My experience is that state of the art CFD codes still give an approximate solution - as can be demonstrated by comparing with laser doppler anemometry measurements.

But I am interested in DNS now that you brought it to my attention.

The paper of Patankar is a bit advanced for me. Patankar is a genius of Fluid Dynamics. I'm only a begineer. But I have seen that caculating events at such a small scales is now possible due to DNS. You've said that N-S equations are a simple mathematical model. That's right, but we actually know a 1% of that model. The paper of Patankar likely was impossible of thinking of it some years ago. The CFD codes give an approximate solution. That's right again. But think of it and realize yourself that the approximation is really good and accurate. Computer power is the threshold at CFD, not the proper mathematical model or algorithms.

Turning to my first question, does anybody know the answer?. Is the turbulent flow so random? I mean, bounded random at sighting the mean values but random sighting the fluctuations. If the fluctuations are random, and DNS is able to calculate them, then will we obtain two different solutions for the same flow at the same time interval in each simulation?

BTW: I see you have some idea about CFD, as a nuclear engineer. Take into account my experience is relatively short. I'm only studying the last courses of Mech. Engineering.

I have had the explanation in my last fluid mech class by the professor. Imagine I have two identical and experimental flows. They have the same boundary conditions (I'm including rugosity and noise). That experiment is imaginary of course. But it is enough for me to understand it. The flow is fully turbulent. Well, as N-S equations are determinist, it will be hoped that all that randomness (unsteady variations of velocity) is yielded by some theoretical and well defined path= the N-S equations. As you said it is bounded chaos. But it is not chaos after all. The variations are described by the N-S equations which are determinist and not alleatory. Such numerical computation (including all the experimental factors) will be the hardest one. But what I was trying to discover is if the N-S equations had a "alleatory generator" inside them. On the contrary, the model is absolutely determinist. If I repeat these two experiments, I will obtain the same result each time and the same small variations.

And I have learn something about DNS. From the Kolmogorov Scales of Turbulence one can derive the approximate number of computational spatial nodes that are necessary for the computation of the complete turbulent flow field, including the small eddy scales and the large eddy scales. That number, for a three-dimensional flow would be:

[tex] Number_{nodes}=Re_L^{9/4}[/tex] where L is the characteristic lenght of the flow and Re is the Reynolds Number.

Surprising isn't it? A fully turbulent flow may have [tex]Re\approx 10^5[/tex] so that
[tex] N=1.77\cdot 10^{11}[/tex]

:surprised :surprised

Now I realize how necessary are the supercomputers, but I doubt there is someone able to do it !!!

Unified Framework for Development of Pressure-Strain Correlation Turbulent Transport and Sub-Grid Stress Closure Models for Turbulence Simulation

Authors: Sharath S. Girimaji; TEXAS A AND M UNIV COLLEGE STATION DEPT OF AEROSPACE ENGINEERING

Abstract: Study of turbulence is of vital scientific, military and economic interest. Advances in several areas of external aerodynamics and internal combustor flows of interest to Air Force hinge on our ability to clearly understand and adequately predict the effects of turbulence. At the current time, however, there exists a substantial gap between our knowledge of the physics of turbulence phenomenon and the physics that is incorporated into turbulence models, especially subgrid closures. The disciplines of turbulence theory/analysis (e.g., rapid distortion theory, spectral closure models), high- order turbulence modeling (e.g., second-moment closures, structure-based models and realizability constraints) and turbulence simulation (DNS- direct numerical simulations, and LES- large eddy simulations) are evolving independently with very little cross fertilization of ideas. For example, the currently popular LES subgrid closures (e.g., Smagorinsky, dynamic Smagorinsky) are algebraic in nature; completely insensitive to extra rates of strain such as rotation, curvature, and buoyancy and, further, may not even be realizable. These major deficiencies in the LES-SGS modeling are tolerated despite the fact that, in higher order closures, these physical effects and mathematical constraints have long been represented adequately. Further, we would like to point out that the very premise of detached-eddy simulation (DES) approach - that is seen as the practical computational tool for turbulence - is erroneous. This is due to the fact that inhomogeneous spatial filtering is inevitable in this method, and yet the governing equations ignore the effects that necessarily arise with inhomogeneous filtering of the velocity field.

Heeemm.. now I am remebering this article unfortunately I didn't save it in my HD. Have you got it over there?. It seems the url doesn't work now I need it.