The effect of a small initial distortion of the basic flow on the subcritical transition in plane Poiseuille flow

Author:
Pu Sun

Journal:
Quart. Appl. Math. **59** (2001), 667-699

MSC:
Primary 76E30; Secondary 35Q30, 76F06

DOI:
https://doi.org/10.1090/qam/1866553

MathSciNet review:
MR1866553

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Abstract: This paper presents an instability theory in which a mean flow and multiple wave interactions in the Poiseuille flow transition process are studied. It is shown that not only can this mean flow term come as the result of the Fourier decomposition of a general disturbance, it can also come as an exact solution to the unsteady Navier-Stokes equations. The presence of this term, though small, can produce totally different linear and nonlinear stability behavior for the flow at subcritical Reynolds numbers. In the linear stability case, with the presence of this mean flow perturbation term, the instabilities are obtained well below the critical value of 5772. When this mean flow term is introduced into the interactions with other harmonic perturbation waves, for the plane Poiseuille flow case with Reynolds numbers around 1200, the nonlinear interactions rapidly modify the total mean flow profile toward the mean flow profile observed in turbulence while the other two- and three-dimensional waves remain small. The initial energies needed to trigger the instabilities are much smaller than those reported by previous investigators. The intermittent character of the disturbance observed in transition experiments is also captured.

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DOI:
https://doi.org/10.1090/qam/1866553

Article copyright:
© Copyright 2001
American Mathematical Society