Giving the construction of Reynolds stress tensor there some basic constraints which limits how the values for components may assume in a given realisable flow. I currently have no problem understanding that
- the Reynolds stresses are positive semi-definite;
- and have both first and second invariants greater than zero, due to results in functional analysis.
However, I am failing to see why should also the determinant of R be also a positive (or zero) quantity. At the appendix in Schumann's paper there is such a proof, but I didn't quite well follow all his steps in the demonstration.
Has anyone ever demonstrated somewhat differently than Schumann's version? I mean, in a more geometrically way?
Best regards to all and thanks in advance for any advice,
Fernando Soares.
Oh yeah. Now that you've pointed it out I see how trivial it was (since I've already comprehended its positive semi-definiteness and symmetry). Anyways, thank you very much for your answer G. van der Zwan.
Best wishes,
F. Soares
Fernando, I missed the Schumann paper so I am not aware whether I can help or not, but attached you may find a paper which might help indirectly. It is a closed theory based on Kolmogorov 1942 with remarks by Landau, same year. It gives for the first time not only stresses but also the fundamental constants of turbulence like von-Karman as
1/SQRT(2 pi) =0.399. Princeton superpipe gives 0.40 +/- 0.02.
On the principle of independence of the boundary conditions of structure direction of turbulence
Lu Panming
Abstract
This is an introduction to the turbulence modeling theory of the present author, which can take the boundary conditions of turbulence structure into consideration ( the main contents have either not been published in normal journals or only published in chinese papers). It is included that： (I)。Difficulty Problems Encountered in “Second-Moment-Closure” turbulence modelling ，i.e. “Gao-Ge Anomaly” ; (II)。The improvement to the ”Second-Moment-Closure” turbulence modelling in order to allows the boundary conditions of turbulence structure direction could be prescribed; (III)。An explanation to the principle of independence of the boundary conditions of
turbulence structure direction ; (IV)。Six suggestions to the possible future work directions; (V)。 An explanation to the ”Gao-Ge Anomaly”.
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4. The principle of the independence of the boundary conditions is not only to the turbulence modelling, but also suitable to the other subject, because it is frequently meet the equations are not easy to solve, so there is often a need of modelling or predigestion. For example, both the D‘Alembert anomaly (~1752) and the Stokes anomaly (~1856), are due to breach the principle of independence of the boundary conditions, according to today`s point of view .This is because, first, by using the inviscid potential flow`s dynamic equations it is not possible to affiliate the slip-less condition of the real fluid at the wall, and second, using the Stokes equation it is not possible to affiliate the boundary conditions both at the wall of cylinder and at the infinite to the cylinder. The first anomaly lead to the discovery of the Prandlt`s boundary layer theory（1904）, and the second anomaly was solved by Oseen（1910）with introducing a modification. All these reflect the scientific worthiness and broad latency of the present principle of the independence of the boundary conditions.
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above is an example paragraph,for more details or need the full text, please go to the end of this letter, where will be an web address existed.
http://staff.ustc.edu.cn/~pmlu/
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