A. S. Iberall 



However, another possibility exists. The first family has produced a stationary 

 fluctuating set marked by a, 8, co. These wavelets — radiating from each wall in 

 a certain "regular" manner (actually stochastic in phase, which really makes 

 each subsequent chunk stochastic) — then can act as coupled sources for the 

 second set. However, for this set a, &, ^ are not unchanged. 



We may assume that a and h are unchanged (the size of the radiating ele- 

 ment has been fixed by a scale) but that a complex co^ 



CO = CO ± K] 



is developed. Thus, the second family becomes e^(°'y+^^'^'^^)-*^^, with the previ- 

 ous amplitudes (via reflection). This new instantaneously nonstationary field 

 (since it has attenuating and growing components) represents a stochastic field 

 that "fluctuates" around the mean fluctuations. Namely, this is a clue to the 

 theory of the fluctuation band width associated with each "stationary" limit cycle 

 system that crosses through the field. This family instantaneously can create 

 the horribly complex picture of the turbulent field. 



Note this field cannot come into existence except as it is created by the 

 stable limit cycle field. Thus our inquiry is "justified." By this verbal "picture," 

 however, we have shown how the three components of the turbulent field can be 

 arrived at by a decomposition: namely a mean field; stationary limit cycles; and 

 a fluctuating band width for each spectral line. I believe that many other non- 

 linear field quantizations arise by related mechanisms. 



In conclusion: Thus we have shown 



(1) Frequency limits to the spectrum of turbulence. 



(2) A better than order of magnitude estimate of the fluctuating amplitude. 



(3) Rough estimable form for the mean velocity distribution. 



(4) Estimate of the critical Reynolds number. 



All consistent with this self-generated standing and wave system, I have sug- 

 gested a "reason" for the apparent random nature for the fluctuating field, rather 

 than this estimated "stationary" field (albeit with the "same" spectral 

 characteristics) . 



In this crude but suggestive fashion, a deterministic nonlinear theory for 

 turbulence has thereby been proposed. 



SUMMARY 



The compressible equations of hydrodynamics are investigated for condi- 

 tions under which self- sustained propagative primitives would persist for the 

 particular boundary value problem of turbulent flow between parallel plates 

 under a constant pressure gradient. This requires assuming a form for the 

 mean velocity distribution that satisfies the boundary condition. It is shown 

 that an extra condition other than the equality of pressure gradient and viscous 

 shear (proportional to the velocity gradient) at the wall is required. As in 



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