Model for transitional turbulence in a planar shear flow

TL;DR

This work develops a reduced yet physically faithful model for transitional turbulence in planar shear flows by projecting the Reynolds-averaged Navier–Stokes equations onto a minimal vertical-mode basis, yielding six coupled fields that describe large-scale vector flow and turbulent kinetic energy. The model, calibrated to Direct Numerical Simulations of Waleffe flow, reproduces key transition phenomena, including oblique turbulent bands, band formation from localized patches, and large-scale quadrupolar circulation, and it reveals a linear instability of uniform turbulence at $Re_c$ that selects a finite band angle. Through a long-wavelength reduction, the authors derive a tractable stability framework that yields a concrete bound on the onset angle $0<|\theta_c|<45^\circ$, consistent with observed oblique bands. The approach links first-principles Navier–Stokes dynamics to a low-dimensional, analysable model, enabling detailed exploration of pattern formation, selection mechanisms, and potential extensions to fluctuations and other flow geometries.

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.

Figures (10)

• Figure 1: Sketch of the basic transition scenario in many wall-bounded shear flows. Flow states are illustrated for a pipe and shear flow between parallel planes, with the inset depicting the geometry rotated into perspective. Black signifies turbulent regions, where the fluid motion is highly fluctuating, while white signifies quiescent regions of smooth laminar motion. Three regimes are encountered as a function of the Reynolds number. At low $Re$, laminar motion is the only asymptotic state available to the system. At large $Re$, turbulence can be triggered, and its asymptotic state is spatially uniform, meaning that it adopts the full symmetry afforded by the system. Separating these two extremes is an intermittent regime where, once triggered, turbulence develops into a spatio-temporally intermittent form, composed of turbulent puffs or oblique turbulent bands. Two band states are illustrated: sparse bands (at lower $Re$) and dense bands (at higher $Re$). The sketches are not to scale: the length scale $L$ of turbulent structures, whether puffs or bands, is much larger than the wall separation $\ell$, which sets the scale of turbulent fluctuations. Coarse-grained modelling exploits that $\ell/L \ll 1$.
• Figure 2: (a) Waleffe flow geometry. Fluid is confined between two parallel, stress-free boundaries and is driven by a body force of the form ${\bf f} = f \sin(\pi y /2) {\bf e}_x$. Shown are illustrations of the five wall-normal (vertical) modes in the model used to represent the large-scale flow. The mode labelled $u_1$ has the same form as the forcing and the laminar flow: $\sin( \pi y/2) {\bf e}_x$. (b) Turbulent band in Waleffe flow. Visualized is the time- and vertically-averaged turbulent kinetic energy (TKE) (colour) and velocity field (arrows) from a direct numerical simulation at $Re=140$. The averages were taken over 990 advective time units. Along- and across-band directions are shown and are referred to frequently in the text.
• Figure 3: Functional dependence of (a) the first mode amplitude of $-\langle u^\prime v^\prime \rangle$ on $q_0$, and (b) the vertically averaged pseudo-dissipation rate $\varepsilon_0$ on $q_0$. Data is from a DNS of a steady band in Wf at Re = 140, with averaging along the band and over 1000 time units. The black curve tracks the values of $-\langle u^\prime v^\prime \rangle$ or $\varepsilon_0$ with $q_0$ as the across-band direction is traversed. The maximum of $q_0$ occurs at the band centre, while small $q_0$ corresponds to the quasi-laminar regions between turbulent bands. Two lines appear because the averages extracted from DNS are not exactly symmetric about the band centre. See for example figure \ref{['fig:closures']}b for the variation of $\varepsilon_0$ across the band direction. In (a) the dashed line shows $A(q_0)$ for the model and the vertical grey line highlights $q_0 = \eta$, where $\eta=5\times 10^{-3}$. In (b) the dashed line (displaced vertically for clarity) represent best fits of the DNS data.
• Figure 4: Comparison of Reynolds-stress force and pseudo-dissipation from DNS, model closures, and a model solution. Panel (a) shows the Reynolds-stress force in the $u_1$ mode. DNS results are obtained from a steady band in Wf at $Re = 140$, with averaging along the band and over 1000 time units. The dashed curve shows the corresponding force using closure \ref{['eq:A_closure']} with $q_0$ obtained from the DNS. The model curve (green) shows the force for a steady band in the model at $Re = 75$. (See § \ref{['subsec:params']} for a discussion on the shift in $Re$.) Panels (b) and (c) show similar plots, but for the pseudo-dissipation rates $\varepsilon_0$ and $\varepsilon_1$, respectively.
• Figure 5: Across-band profiles of mode amplitudes from (a) DNS of a steady band in Wf at Re = 140, with averaging along the band and over 1000 time units, and (b) a model band at $Re = 75$. Although curves are plotted as a function of the across-band direction, the vectors have not been rotated, so that, for example, $u_0$ and $u_1$ still represent vertical modes of the streamwise velocity.