I've recently come across the paper "Micromorphic Description of Turbulent Channel Flows" of A. C. Eringen (attached) and, among all the theory and the striking (analytical) results contained therein, I was really curious about the following excerpt (right in the introduction):
"[...] (the) equations are closed and require no further a priori conjectures on the nature of the fluid or modifications for the viscosity coefficients. Becauseof the extra degrees of freedom present in the theory, many interesting phenomena can be described mathematically."
Could anyone please comment a little bit deeper on this statement? It seems to me the author avoids the closure issue of classical turbulence approach by extending the theory for a generalised continuum. I simply find it to good to be true and no one be talking about that when dealing with turbulent flows.
I went on in investigating his work and found his treatises on micro continuum theories, viz. "Microcontinuum Field Theories", vol. I and II which shed some light on some aspects of his work.
Any contribution would be sincerely appreciated.
Best regards,
F. Soares.
Fernando, there is a notorious comment by Wolfgang Pauli to this sort of papers: "Not even wrong". There are thousands of similar papers without any real result, but full of assumptions and equations etc. They all have in common to use at the end of the day empirical/phenomenological closure relations. They are not able to give us meaningful values for the universal constants of turbulence (van-Karman's, Kolmogorov's, free decay etc. ) at asymptotically high Reynolds numbers. Typically they do not consider the vortex dipole as the elementary particle of turbulence.
My own view you may find attached. I put the vortex dipole into the center of attention. But it is dangerous to read it because it is like Copernicus' case who was justified by the Church only in 1992 ...
Article Universal equations and constants of turbulent motion
With a first, brief look, the paper studies steady, laminar flow in a channel-not the case of turbulent channel flow. It seems (and I admit that I did not work through the details) to give a 20 equation model that replicates what is known from the already-closed 4 equation model of conservation of mass and linear momentum (the Navier-Stokes equations). This is no feat-with enough parameters one can fit an elephant and make it wiggle its trunk as von Neumann once said. It may be a valuable model if it could be used to give a first principle development of viscoelastic fluids for example but that is not done in this paper.
Fernando,
do not forget that we accept (so far) that turbulent flows are governed by the Navier-Stokes equations that are closed... ok, they have a model in the Newtonian stress, Fourier heat flux, etc. but once accepted these assumption we now can solve numerically the equations without any arbitrary assumption for turbulence. That is the Direct Numerical Simulation.
The paper you posted is dated 1972, I don't understand if it considers microfluidic (where many physical effects are more relevant and turbulence can be dumped by the small-scale problem). It considers a steady 2D case that can be approached only in statistical sense for the statistically averaged NS equations (and I do not understand the Reynolds stress defined in the paper by means of a local volume average...). This was the hystorical approach for studyng turbulence and introduces the closure problem for the statistically averaged NS. But that happens as the original equations are transformed.
I would like to thank all the kind answers given above. I sincerely appreciate your time.
I just want to state that I am just getting acquainted to the field of generalised continua. I realize Eringen didn't dedicate his career strictly to studying turbulence, but rather develop many branches of continuum mechanics, specially his theories for sophisticated material systems.
However he was not the first one to consider the importance of angular momentum balance in the description of turbulent flows. Some others, like the pioneering work of Mattioli, emphasise the incompleteness of not considering inner spin of fluid turbulent motion, and how it is balanced, to properly describe the motion.
I still have a long path ahead in order to understand this theory really and I would like to thank you all for the contributions.
Best regards,
F. Soares.
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