Shunichi Tsuge
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MOTIVE AND PERSPECTIVE
Fluid turbulence as the unsolved problem in the world of physics, aeronautics, mechanical engineering, combustion, meteorology, etc. is something like Fermat's problem in mathematics. It is the last problem in classical mechanics that has been left unsolved from first principles over a century since the days of Osborn Reynolds.
Latest approach to turbulence along this line was the direct numerical simulation (DNS) which attempts 'precise look' to the phenomena in the extreme of capacity of contemporary supercomputers. Its success is the finding that the dissipation region in the sense of Kolmogorov has the shape like a 'worm' with the radius of the Kolmogorov length aligned randomly streamwise.
Its failure, however, is the impracticability for engineering problems with high Reynolds numbers R = 107 -- 108 because the computer memory required increases with R9/4 .
Here, instinct of physicists to 'look into more precisely' works as if monkeys keep peeling off the onion skin. Thus, DNS in the engineering field of turbulence is concluded as not rewarding compared with transonics where it provided with the most powerful tool for the jumbo jet design.
In most cases turbulence results eventually from the chaos, or more accurately, the deterministic chaos.DNS looks at
turbulence as such chaos, in other words, as 'chaos in order'. The approach employed in the proposed paper, in contrast,
pursues turbulence as something having 'order in chaos' . We seek a deterministic equation to govern stochastic variables
of turbulent fluctuation like Schroedinger's equation governing wave function subject to the uncertainty principle. The
equation thus derived does not have the high Reynolds number difficulty mentioned above, but has another difficulty of
increased degree of freedom in independent variables. It has the form of the Navier-Stokes equation in 6D space (x,s),
namely, the physical space (x) plus the eddy space (s), in contrast with the classical one in 3D space x.
In this paper the classical Kolmogorov turbulence theory is reconsidered in the light of the 6D Navier-Stokes equation. The dissipation region is mapped onto an energy black hole near the origin of the eddy space (s=0)around which the turbulence wave function forms a sharp ridgeline of O(R7/8). It is one of the new findings that have been hidden behind the classical dimensional analysis, yet is totally consistent with the Kolmogorov theory, leading, as an example, to the -5/3 law for the power spectrum.
LEAD TO 6-D NAVIER-STOKES EQ. FOR TURBULENCE
DISSIPATION REGION: |s| ~ O(Kolmogorov scale)
COMPARISON WITH KOLMOGOROV THEORY |
