Theory of Turbulent Flow Between Parallel Plates 



While the same result might have been suspected from the core equations just 

 by assuming a mean velocity in the core, it now appears intrinsically excited. 

 Beyond this, the mean velocity characteristics near the wall determine the 

 dispersion of the wave system. 



[The following comments may help to "explain" the process of satisfying 

 the boundary conditions that lead to the final secular equation. Corresponding 

 to the primitive eJ<:°y^^^*"^) ( ^^ negative), expressing a temporarily coherent 

 traveling wave system traveling downstream, there is really a dual set, ap- 

 proaching and reflecting from the walls. This dual set must satisfy the bound- 

 ary conditi ons. The viscous and thermal diffusive modes have leading terms 

 g+v' jojx^ g±/jcrwx ^ over the entire cross section. The outgoing system of waves 

 grows large in the face of the local pressure gradient, while the incoming wave 

 is highly damped. This suggests that the boundary condition need be satisfied 

 by only one of the two wave systems, namely, the outgoing one which has re- 

 sulted from the excited pressure mode moving also in the downstream direction. 

 Near the wall, the mean gradient is svifficient, by perturbation, to split the two 

 viscous diffusive modes into two with slightly different propagative velocities. 

 (In the laminar — small velocity amplitude ~ case one clearly can see the source 

 of the two viscous diffusive modes, which Kovasznay [12] refers to as vorticity 

 modes. They are eigenvalues for two components of the vector velocity 

 potential — the solenoidal components that give rise to vorticity. The two un- 

 split repeated roots are seen clearly in the core solutions, associated with 

 X = 0, and X ^ 0.) It is their interaction with the pressure gradient in the 

 boundary layer that turns the waves over into an eddy, and thus provides a 

 source of radiated acoustic eddies that emerge from the wall region. Further, 

 only one of the two propagation constants — say, c^ j— can satisfy the boundary 

 conditions. What emerges is that neither an upstream propagated system ( s 

 positive), nor c^ ^j the second possible "elastic" mode can satisfy the bound- 

 ary conditions, it is rapidly attenuated or absorbed. 



Actually, it is the inability of the second mode c^ 2 to provide a trapped 

 self-generated vortical filament that is crucial. (It could very well have been 

 that the first mode might not have been able to, also, or that some other mode — 

 given other boundary conditions — might have been the source.) There then 

 emerges linear combinations of the other diffusive modes which provide trapped 

 limit cycle structures. Here the outward radiated "acoustic" components are 

 exhibited. 



While some added nonlinear distortion may visually deform the local field 

 even further, the intrinsic modal interaction should essentially persist as shown 

 in this elementary derivation.] 



More globally, these results may be interpreted as follows: There may be 

 many waves that can be excited. For any e'^s, say, there will be a possible 

 e^^. The nondenumerable class of all such waves forms a complex stochastic 

 system that fits Kraichnan's allusions to the inordinately complex picture of a 

 turbulent field. (This was a salient point in Kraichnan's keynote address at the 

 10th Annual Meeting of the Fluid Mechanics Division of the American Physical 

 Society, Lehigh U., November, 1967.) With the existence of such a complex 

 picture, we would concur. However, practically all of the waves are dissipated. 



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