Transition to turbulence from plane Couette flow

Larry K. Forbes

Abstract


Modelling fluid turbulence is perhaps one of the hardest problems in Applied Mathematics. In a recent paper, the author argued that the classical Navier–Stokes equation is not sufficient to describe the transition to turbulence, but that a Reiner–Rivlin type equation is needed instead. This is explored here for the simplest of all viscous fluid flows, the Couette flow, which is a simple shear between two moving plates. It is found that at high wavenumbers, the transition to unstable flow at the critical Reynolds number is characterized by a large number of eigenvalues of the Orr–Sommerfeld equation moving into the unstable zone essentially simultaneously. This would generate high-dimensional chaos almost immediately, and is a suggested mechanism for the transition to turbulence. Stability zones are illustrated for the flow, and a simple asymptotic solution confirms some of the features of these numerical results.

doi:10.1017/S1446181115000176

Keywords


Couette flow; instability; nonNewtonian effects; turbulence



DOI: http://dx.doi.org/10.21914/anziamj.v57i0.8756



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ANZIAM Journal, ISSN 1446-8735, copyright Australian Mathematical Society.