Localization in a spanwise-extended model of plane Couette flow
arXiv:1410.2035 · doi:10.1103/PhysRevE.91.043005
Abstract
We consider a 9-PDE (1-space and 1-time) model of plane Couette flow in which the degrees of freedom are severely restricted in the streamwise and cross-stream directions to study spanwise localisation in detail. Of the many steady Eckhaus (spanwise modulational) instabilities identified of global steady states, none lead to a localized state. Localized periodic solutions were found instead which arise in saddle node bifurcations in the Reynolds number. These solutions appear global (domain filling) in narrow (small spanwise) domains yet can be smoothly continued out to fully spanwise-localised states in very wide domains. This smooth localisation behaviour, which has also been seen in fully-resolved duct flow (Okino 2011), indicates that an apparently global flow structure needn't have to suffer a modulational instability to localize in wide domains.
14 pages, 18 figures
References in corpus (22)
- Travelling waves in pipe flow
- The Significance of Simple Invariant Solutions in Turbulent Flows
- Visualizing the geometry of state space in plane Couette flow
- Turbulence transition and the edge of chaos in pipe flow
- Low-dimensional dynamics embedded in a plane Poiseuille flow turbulence : Traveling-wave solution is a saddle point ?
- Streamwise-localized solutions at the onset of turbulence in pipe flow
- Transition in pipe flow: the saddle structure on the boundary of turbulence
- Snakes and ladders: localized solutions of plane Couette flow
- Laminar-turbulent boundary in plane Couette flow
- Recurrence of Travelling Waves in Transitional Pipe Flow
- Localised edge states nucleate turbulence in extended plane Couette cells
- Revealing the state space of turbulent pipe flow by symmetry reduction
- Spatially localized solutions of shear flows
- A homoclinic tangle on the edge of shear turbulence
- The genesis of streamwise-localized solutions from globally periodic travelling waves in pipe flow
- Streamwise and doubly-localised periodic orbits in plane Poiseuille flow
- A doubly-localized equilibrium solution of plane Couette flow
- Modeling transitional plane Couette flow
- Increasing lifetimes and the growing saddles of shear flow turbulence
- Periodically bursting edge states in plane Poiseuille flow
- Long wavelength instability of coherent structures in plane Couette flow
- Turbulent transition in a truncated one-dimensional model for shear flow