Generalised quasilinear approximations of turbulent channel flow. Part 1. Streamwise nonlinear energy transfer

Abstract

A generalised quasilinear (GQL) approximation (Marston et al., Phys. Rev. Lett., vol. 116, 2016, 104502) is applied to turbulent channel flow at $Re_\tau \simeq 1700$ ( $Re_\tau$ is the friction Reynolds number), with emphasis on the energy transfer in the streamwise wavenumber space. The flow is decomposed into low- and high-streamwise-wavenumber groups, the former of which is solved by considering the full nonlinear equations whereas the latter is obtained from the linearised equations around the former. The performance of the GQL approximation is subsequently compared with that of a QL model (Thomas et al., Phys. Fluids, vol. 26, 2014, 105112), in which the low-wavenumber group only contains zero streamwise wavenumber. It is found that the QL model exhibits a considerably reduced multi-scale behaviour at the given moderately high Reynolds number. This is improved significantly by the GQL approximation which incorporates only a few more streamwise Fourier modes into the low-wavenumber group, and it reasonably well recovers the distance-from-the-wall scaling in the turbulence statistics and spectra. Finally, it is proposed that the energy transfer from the low- to the high-wavenumber group in the GQL approximation, referred to as the ‘scattering’ mechanism, depends on the neutrally stable leading Lyapunov spectrum of the linearised equations for the high-wavenumber group. In particular, it is shown that if the threshold wavenumber distinguishing the two groups is sufficiently high, the scattering mechanism can be completely absent due to the linear nature of the equations for the high-wavenumber group.

Generalised quasilinear approximations of turbulent channel flow. Part 2. Spanwise triadic scale interactions

“…Continuing from Part 1 (Hernández, Yang & Hwang 2022), the starting point of the present study is the RNL model, which we have referred to as the QL model. The important advantage of this model is that it can capture the so-called ‘self-sustaining process’ (Hamilton, Kim & Waleffe 1995; Waleffe 1997) – i.e.…”

“…In Part 1 (Hernández et al. 2022), the QL model was extended by applying the GQL approximation to the streamwise direction in turbulent channel flow at to explore a route to improvement of the QL model. The GQL approximation employs the flow decomposition in the QL model in a more flexible manner: the nonlinear equations for the streamwise mean flow are replaced with nonlinear equations for a set of low-wavenumber streamwise Fourier modes (i.e.…”

“…In Part 1 (Hernández et al. 2022), the GQL approximation was found to improve the QL model effectively. By incorporation of only a few more streamwise Fourier modes into the low-wavenumber group, the GQL model was able to recover the self-similar scaling with the distance from the wall of the turbulence statistics and spectra.…”

Recent Developments in Theories of Inhomogeneous and Anisotropic Turbulence

“…Furthermore as Λx, Λy → ∞ the GQL system consists solely of fully interacting low modes and returns to fully NL Direct Numerical Simulation, albeit not necessarily monotonically. In particular, if the cutoff Λ is large (but not infinite) it is possible that high modes will be stable and (unphysically) not have any energy (Hernández et al 2022a). GQL like QL is a conservative approximation but one that systematically interpolates between QL and NL.…”

“…The utility of the GQL approximation compared with that of QL has been tested on a number of paradigm turbulent fluid problems and MHD. These include the stochastic driving of jets on a spherical surface and β-plane (Marston et al 2016), three dimensional plane Poiseuille and rotating Couette flow (Kellam 2019, Hernández et al 2022a,b, Tobias & Marston 2017, convectively driven zonal flows in a rotating annulus (Tobias et al 2018) and the helical magnetorotational instability that is crucial to angular momentum transport in disks (Child et al 2016). Depending on the nature of the problem -in particular the degree of non-normality of the linear operator (see below) -the QL and GQL approximations may perform well or poorly in describing the statistics of the full system.…”