Reynolds number induced growth of the large-scale rolls in plane Couette flow using resolvent analysis

Abstract

In direct numerical simulations (DNS) of turbulent Couette flow, the observation has been made that the long streamwise rolls increase in length with the Reynolds number (Lee & Moser, J. Fluid Mech., vol. 842, 2018, pp. 128–145). To understand this, we employ both linear stability theory and its extension to resolvent analysis. For this, we emphasise the high Reynolds number ( $Re \rightarrow \infty$ ) and small streamwise wavenumbers ( $\alpha \rightarrow 0$ ) limit, imposing the distinguished limit $Re_{\alpha }=Re \, \alpha = O(1)$ . We find that in case of linear stability theory, $Re_{\alpha }$ acts as a global invariant in the resulting eigenvalue problem, while in case of resolvent analysis, $Re_{\alpha }$ acts as a local invariant in the behaviour of the energy of the system characterised through the first singular value $\sigma _1$ of the resolvent operator within the investigated asymptotic limit. In order to obtain constant streamwise structures for increasing Reynolds numbers, the respective streamwise wavenumber has to decrease, which verifies the observations from DNS studies of an increasing length of the streamwise structures with the Reynolds number. In linear stability theory, a parameter reduction is achieved for the above asymptotic limit, resulting in the modified Orr–Sommerfeld and Squire equations being dependent only on $Re_{\alpha }$ . The behaviour of both the coherent structures obtained from linear stability theory and resolvent analysis are compared with each other and show similar behaviours over $Re_{\alpha }$ .