A Contribution to the Theory 

 of Turbulent Flow Between 



Parallel Plates 



A. S. Iberall 

 General Technical Services, Inc. 

 Upper Darby, Pennsylvania 



INTRODUCTION 



The study of the stability of motion of a viscous fluid was begun by Reynolds 

 to determine those conditions under which laminar flow might no longer persist. 

 Theory and background are presented by Lin [l]. In this report, an alternative 

 study is undertaken of conditions under which stable nonlinear limit cycles might 

 persist at Reynolds number well beyond the laminar flow limit. The case of tur- 

 bulent, low-Mach-number flow between parallel plates is discussed. Since the 

 turbulent field beyond the critical Reynolds number appears to be stable and 

 marked by a stationary though stochastic spectrum of fluctuations, there is con- 

 siderable reason for attempting to identify the suggestive nonlinear behavior 

 with limit cycles. Since sustained oscillations in a distributed field are associ- 

 ated with propagation, one type of which is concerned with compressible waves, 

 compressibility is retained. While an apparent added complexity, if it is not 

 needed in any particular hydrodynamic problem, it should drop out naturally as 

 negligible. Actually, it will be shown to be needed to establish limit cycles in 

 parallel-plate flow. Turbulence in that problem is thereby traced to a coupling 

 of acoustic waves with the hydrodynamic field. 



EQUATIONS FOR TURBULENCE 



In Cartesian tensor form, the continuum equations of hydrodynamics for a 

 fluid, which is not concerned with mass diffusion, body forces, or radiation, are: 



Momentum equation 



DV: 



P 



Dt 



-P,i + 



a(V ■ . + V. .) 



3^- M^i.3 



Energy equation 

 DS 



/oT 



Dt 



kT: 



+ M 



V. . + V. . 

 > . J J . 1 



(1) 



705 



