Parameterization of travelling waves in plane Poiseuille flow

• Published in: IMA journal of applied mathematics Vol. 79; no. 1; pp. 22 - 32
• Main Authors: Smith, Warren R, Wissink, Jan G
• Format: Journal Article
• Language: English
• Published: 01-02-2014
• Subjects: Laminar flow; Mathematical analysis; Mathematical models; Navier-Stokes equations; Parametrization; Phase velocity; Planes; Traveling waves
• ISSN: 0272-4960; 1464-3634
• DOI: 10.1093/imamat/hxs037

The first finite-dimensional parameterization of a subset of the phase space of the Navier-Stokes equations is presented. Travelling waves in two-dimensional plane Poiseuille flow are numerically shown to approximate maximum-entropy configurations. In a coordinate system moving with the phase velocity, the enclosed body of the flow exhibits a hyperbolic sinusoidal relationship between the vorticity and stream function. The phase velocity and two-amplitude parameters describe the stable manifold on the slow viscous time scale. This original parameterization provides a valuable visualization of this subset of the phase space of the Navier-Stokes equations. These new results provide physical insight into an important intermediate stage in the instability process of plane Poiseuille flow.