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TRANSITION TO TURBULENCE FROM PLANE COUETTE FLOW

Part of: Numerical linear algebra Turbulence

Published online by Cambridge University Press:  22 September 2015

L. K. FORBES
L. K. FORBES*
Affiliation:
School of Mathematics and Physics, University of Tasmania, Hobart, Tasmania 7004, Australia email [email protected]
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Abstract

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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.

Keywords

Couette flowinstabilitynonNewtonian effectsturbulence

MSC classification

Primary: 76F20: Dynamical systems approach to turbulence
Secondary: 65F15: Eigenvalues, eigenvectors
Type
Research Article
Information
The ANZIAM Journal , Volume 57 , Issue 2 , October 2015 , pp. 89 - 113
Copyright
© 2015 Australian Mathematical Society 

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