Multiscale quasi time-periodic coherent structures in shear flows

TL;DR

This work argues that multiscale features of shear-flow turbulence require states as intricate as quasi-time-periodic solutions, not merely travelling waves. It introduces a quasi-linear vortex-wave interaction (QL-VWI) framework aligned with large-$R$ asymptotics, enabling efficient approximation of quasi-time-periodic states by coupling multiple neutral waves to the mean flow. Numerical results validate the approach: plane Couette flow is accurately captured with two waves producing a GPO, while plane Poiseuille flow yields quasi-periodic states through multi-wave superpositions that align with DNS and reveal wall-attached, multiscale vortex structures consistent with Taylor's frozen-flow hypothesis. The method narrows the gap between exact coherent states and full turbulence, offering a computationally tractable, asymptotically justified route to analyze and extend coherent structures toward more turbulence-like dynamics.

Abstract

Attempts to disentangle shear-flow turbulence often focus on identifying relatively simple solutions, such as travelling waves or periodic orbits. We show, however, that capturing multiscale features requires considering states at least as complex as quasi-time-periodic solutions. Approximations of these states can be computed efficiently using a quasi-linear model, consistent with the large-Reynolds-number asymptotic analysis. The quasi-linear structure is key to producing multiscale critical layers that generate vortices obeying Taylor frozen-flow hypothesis.

Figures (4)

• Figure 1: Comparison of Navier–Stokes results (symbols) and QL-VWI results (lines) in plane Couette flow at $R=3000$. Our choice of $\beta=1.667$ ensures that, at $\alpha=1.14$ (indicated by the vertical line), the solutions fit the minimal box used in hamilton1995. The black plots show the drag on the lower wall $D$, i.e. the $x$--$z$--$t$ average of $\partial_y u$ at $y=-1$. Dashed lines and open circles represent the NBW, while solid lines and filled circles represent the GPO. The frequency $\omega$ of the GPO is shown by the blue plots. The left and right colourmaps show $\tilde{u}_{\text{rms}}$ of the NBW ($\alpha=1$) and the GPO ($\alpha=1.5$), respectively, computed using the QL-VWI. The plotted domain spans $y\in [-1,1]$ and $z\in [0,2\pi/\beta]$. The green lines denote the critical levels. The grey contours are the level curves of $\overline{u}$.
• Figure 2: Flow visualisation of the NBW (a) and the GPO (b) at $R=20000$ obtained by the QL-VWI. The colourmaps denote the streamwise velocity of the streak field and the blue/green isosurfaces denote the 50% maximum/minimum value of $\partial_y\tilde{w}-\partial_z \tilde{v}$. The yellow isofurfaces show the 50% maximum value of $|\tilde{u}|$.
• Figure 3: Bifurcation diagram of plane Poiseuille flow solutions. Lines show the QL-VWI results and symbols are the Navier-Stokes results. The colourmaps use the same format as figure 1; all panels show the domain $(y,z) \in [-1,0]\times [0,2\pi/8]$, except for one. (a) TWA with $\alpha/\beta=3/4$. (b) TWB with $\alpha/\beta=1/2$. (c) $N=2$ QL-VWI computation with $\alpha_1/\beta=6/8$ and $\alpha_2/\beta=12/8$. (d) $N=3$ QL-VWI solution with $\alpha_1/\beta=6/8$, $\alpha_2/\beta=12/8$ and $\alpha_3/\beta=28/8$. The colourmap of the solution indicated by the blue diamond shows the domain $(y,z) \in [-1,0]\times [0,2\pi/6]$.
• Figure 4: The flow field of the solution at the blue diamond in figure \ref{['fig:fig4_DNS']}-(d). (a) Visualisation in the box $[0,2\pi/1.5]\times [-1,0.25] \times[0,2\pi/6]$. See figure \ref{['fig:fig2']} caption for definitions of the surfaces and colourmap. (b) Energy of the streak associated with the $l$th spanwise Fourier mode. (c) The $x$--$z$--$t$ mean streamwise velocity $U(y)$. The green vertical lines indicate the projected locations of the critical layers, whose horizontal positions correspond to the associated phase speeds.