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Localization in a spanwise-extended model of plane Couette flow
1School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom.
Summary
Localized, time-periodic solutions in plane Couette flow were discovered. These solutions transition smoothly from domain-filling to spanwise-localized states, challenging traditional instability theories for fluid localization.
Area of Science:
- Fluid Dynamics
- Nonlinear Dynamics
- Computational Physics
Background:
- Plane Couette flow is a fundamental fluid dynamics model.
- Understanding flow localization is crucial for predicting complex fluid behavior.
- Previous studies focused on modulational instabilities for localization.
Purpose of the Study:
- To investigate spanwise localization phenomena in plane Couette flow.
- To explore the nature of localized states beyond traditional instabilities.
- To analyze the transition from global to localized flow structures.
Main Methods:
- A reduced nine-partial-differential-equation model of plane Couette flow was employed.
- Analysis focused on restricted degrees of freedom in streamwise and cross-stream directions.
- Numerical investigation of steady states and bifurcations in Reynolds number was performed.
Main Results:
- No steady Eckhaus instabilities led to localized states.
- Spatially localized, time-periodic solutions were identified via saddle node bifurcations.
- These solutions exhibited smooth transitions from global to spanwise-localized states with increasing domain width.
Conclusions:
- Localized, time-periodic solutions are a key feature of plane Couette flow.
- Flow localization can occur without preceding modulational instabilities.
- The smooth localization behavior suggests a different mechanism for pattern formation in wide domains.
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