Orbital instability for moderately developed Couette turbulence

Abstract

We perform a long-time Lyapunov analysis of plane Couette turbulence up to friction Reynolds number h Superscript plus Baseline equals 269 h + = 269 $h^+=269$ . The energy of the leading Lyapunov vector is predominantly concentrated in the buffer layer, with characteristic streamwise and spanwise wavelengths lamda 1 Superscript plus Baseline asymptotically equals 100 λ 1 + ≃ 100 $\lambda _1^+\simeq 100$ and lamda 3 Superscript plus Baseline asymptotically equals 50 λ 3 + ≃ 50 $\lambda _3^+\simeq 50$ , respectively. These wavelengths are smaller than the corresponding streamwise and spanwise periods of a minimal flow unit. Through Schmidt–Lyapunov analysis, we find that the first and second Lyapunov exponents are close in value, and that the energy spectra of the first Lyapunov vector and the second Schmidt vector have peaks at similar characteristic wavelengths in the buffer layer. These imply that the fastest orbital instability is not organised by a single minimal-flow-unit-scale perturbation, but by a nearly degenerate set of small-scale buffer-layer perturbation directions, which may be interpreted as a superposition of such perturbations. Coarse graining with a Gaussian low-pass filter further shows that the vortical structures of the local Lyapunov vector tend to appear around near-wall large-scale quasi-streamwise vortices. These results suggest that the orbital instability in wall-bounded turbulence is associated with small-scale buffer-layer perturbations influenced by the strain field around near-wall large-scale structures.