[Paper Review] On the decay of turbulence in plane Couette flow (long version)

Paul Manneville|arXiv (Cornell University)|Sep 5, 2011

Fluid Dynamics and Turbulent Flows2 references5 citations

TL;DR

This study investigates the decay of turbulence in plane Couette flow using under-resolved direct numerical simulations to model large systems over long timescales. It identifies a two-stage process: initial band rupture due to chaotic dynamics, followed by slow shrinkage of turbulent fragments, offering a spatiotemporal explanation for exponential lifetime distributions in turbulence decay, challenging purely temporal chaotic transient paradigms.

ABSTRACT

Upon decreasing the Reynolds number, plane Couette flow first forms alternately turbulent and laminar oblique bands out of featureless turbulence below some upper threshold R_t. These bands exist down to a global stability threshold R_g below which laminar flow ultimately prevails. We study the fragmentation and decay of these bands in systems that are extended enough for several bands to exist. We use direct numerical simulations appropriately tailored to deal with such large systems during long enough durations. We point out a two-stage process involving the rupture of a band and next its slow shrinking. Previous interpretations of turbulence decay in wall-bounded flows within the chaotic transient or spatiotemporal intermittency paradigms are discussed.

Motivation & Objective

• To understand the mechanisms behind turbulence decay in extended wall-bounded flows, particularly in the presence of laminar-turbulent coexistence.
• To investigate the transition from featureless turbulence to organized oblique bands upon decreasing Reynolds number.
• To clarify the role of band fragmentation and shrinkage in the decay process, especially near the global stability threshold Reg.
• To challenge the conventional chaotic transient paradigm by proposing a spatiotemporal interpretation rooted in nucleation and large-deviation statistics.
• To validate under-resolved simulations as a reliable proxy for fully resolved dynamics in large systems, enabling long-duration studies.

Proposed method

• Conduct under-resolved direct numerical simulations (DNS) of plane Couette flow in large domains to simulate long-time evolution at moderate computational cost.
• Use systems with extended streamwise and spanwise dimensions (Lx, Lz ≫ 2h) to allow for multiple turbulent bands and realistic spatial dynamics.
• Apply numerical techniques tailored to capture the evolution of laminar-turbulent interfaces, including band rupture and fragmentation.
• Model the decay process by analyzing the statistical behavior of band rupture and subsequent shrinkage, focusing on probabilistic dynamics at the interface.
• Incorporate large-scale flow corrections to account for nonlocal couplings, improving hydrodynamic realism beyond minimal flow unit (MFU) approximations.
• Compare simulation results with experimental data on turbulence decay, particularly the exponential distribution of transient lifetimes and intermittent drops in turbulent fraction.

Experimental results

Research questions

• RQ1 What are the dominant physical mechanisms governing the decay of turbulent bands in plane Couette flow at moderate Reynolds numbers?
• RQ2 How does the transition from continuous turbulent bands to fragmented states occur, and what determines the probability of band rupture?
• RQ3 To what extent can the observed exponential distribution of turbulence lifetimes be explained by spatiotemporal dynamics rather than purely temporal chaotic transients?
• RQ4 How do large-scale mean flows influence the growth and breakdown of turbulent structures at band boundaries?
• RQ5 Can under-resolved simulations accurately reproduce the statistical features of turbulence decay observed in experiments and fully resolved simulations?

Key findings

• Turbulence decay in plane Couette flow proceeds via a two-stage process: first, turbulent bands rupture due to localized chaotic dynamics at the laminar-turbulent interface, forming small gaps.
• The rupture process is probabilistic and governed by large-deviation statistics of chaotic fluctuations, consistent with nucleation-like behavior.
• Once a gap forms, the remaining turbulent fragment shrinks slowly and steadily from both ends, consistent with spatiotemporal intermittency (STI) dynamics.
• The overall lifetime distribution of turbulent transients is exponentially decreasing, with mean lifetimes increasing rapidly as Reynolds number approaches the global stability threshold Reg ≈ 325.
• The simulation results closely match experimental observations, including long periods of nearly constant turbulent fraction interrupted by sudden drops due to large-scale band breakdown.
• Under-resolved simulations successfully capture the essential physics of band decay, validating their use as a predictive tool for large-aspect-ratio systems near Reg.