Model for transitional turbulence in a planar shear flow
arXiv:2309.12879 · doi:10.1098/rspa.2025.0391
Abstract
A central obstacle to understanding the route to turbulence in wall-bounded flows is that the flows are composed of complex, highly fluctuating, and strongly nonlinear states. In the case of pipe flow, models have deepened our understanding of turbulent onset by providing valuable theory to complement experiments and simulations. In planar cases, the large-scale flows associated with transitional turbulence are considerably more complex than for pipes, limiting our ability to develop models and provide theoretical analyses for these cases. We address this challenge here by deriving from the Navier-Stokes equations a simplified model for transitional turbulence in a planar setting. The Reynolds-averaged and turbulent-kinetic-energy equations are projected onto a minimal set of wall-normal modes and justified model closures are used for the Reynolds stresses and turbulent dissipation and transport. The model reproduces phenomena found at the onset of turbulence in planar shear flows, such as turbulent-laminar patterns (turbulent bands) oriented obliquely to the streamwise direction and large-scale flows associated with both stationary patterns and growing turbulent spots. We demonstrate the model's utility by showing that patterns arise with decreasing Reynolds number via a linear instability of uniform turbulence and by deriving a selection criterion for the pattern orientation at onset.
28 pages in the main text; 10 figures
References in corpus (38)
- Dedalus: A Flexible Framework for Numerical Simulations with Spectral Methods
- Computational Study of Turbulent-Laminar Patterns in Couette Flow
- "Barber pole turbulence" in large aspect ratio Taylor-Couette flow
- The rise of fully turbulent flow
- Mean flow of turbulent-laminar patterns in plane Couette flow
- Simplifying the complexity of pipe flow
- Ecological collapse and the emergence of traveling waves at the onset of shear turbulence
- Universal continuous transition to turbulence in a planar shear flow
- Turbulent-laminar patterns in plane Poiseuille flow
- Self-sustaining turbulence in a restricted nonlinear model of plane Couette flow
- Evolution of turbulent spots in a parallel shear flow
- From temporal to spatiotemporal dynamics in transitional plane Couette flow
- Bifurcations to turbulence in transitional channel flow
- Speed and structure of turbulent fronts in pipe flow
- Transition to turbulence in wall-bounded flows: Where do we stand?
- On the transition to turbulence of wall-bounded flows in general, and plane Couette flow in particular
- Invariant Solution underlying Oblique Stripe Patterns in Plane Couette Flow
- Modeling transitional plane Couette flow
- Turbulent-laminar patterns in shear flows without walls
- Velocity Profile in a Two-Layer Kolmogorov-Like Flow
- On modelling transitional turbulent flows using under-resolved direct numerical simulations: The case of plane Couette flow
- The growth mechanism of turbulent bands in channel flow at low Reynolds numbers
- Extreme events in transitional turbulence
- A statistical state dynamics approach to wall-turbulence
- Large scale flow around turbulent spots
- Turbulence in the highly restricted dynamics of a closure at second order: comparison with DNS
- Flow Statistics in the Transitional Regime of Plane Channel Flow
- Laminar-Turbulent Patterning in Transitional Flows
- Turbulent patterns in wall-bounded flows: a Turing instability?
- On the growth of laminar-turbulent patterns in plane Couette flow
- Linear instability of turbulent channel flow
- Routes to turbulence in Taylor-Couette flow
- Collapse of transitional wall turbulence captured using a rare events algorithm
- Patterns in transitional shear turbulence. Part 2: Emergence and optimal wavelength
- Dynamics and proliferation of turbulent stripes in plane-Poiseuille and plane-Couette flows
- Laminar-turbulent patterning in wall-bounded shear flows: a Galerkin model
- Patterns in transitional shear turbulence. Part 1. Energy transfer and mean-flow interaction
- Far field of turbulent spots