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A generalised quasilinear (GQL) approximation (Marston \emph{et al.},\emph{Phys. Rev. Lett.}, vol. 116, 104502, 2016) is applied to turbulentchannel flow at $Re_\tau \simeq 1700$ ($Re_\tau$ is the friction Reynoldsnumber), with emphasis on the energy transfer in the streamwise wavenumberspace. The flow is decomposed into low and high streamwise wavenumber groups,the former of which is solved by considering the full nonlinear equationswhereas the latter is obtained from the linearised equations around the former.The performance of the GQL approximation is subsequently compared with that ofa QL model (Thomas \emph{et al.}, \emph{Phys. Fluids.}, vol. 26, no. 10,105112, 2014), in which the low-wavenumber group only contains zero streamwisewavenumber. It is found that the QL model exhibits a considerably reducedmulti-scale behaviour at the given moderately high Reynolds number. This isimproved significantly by the GQL approximation which incorporates only a fewmore streamwise Fourier modes into the low-wavenumber group, and it reasonablywell recovers the distance-from-the-wall scaling in the turbulence statisticsand spectra. Finally, it is proposed that the energy transfer from the low tothe high-wavenumber group in the GQL approximation, referred to as the`scattering' mechanism, depends on the neutrally stable leading Lyapunovspectrum of the linearised equations for the high wavenumber group. Inparticular, it is shown that if the threshold wavenumber distinguishing the twogroups is sufficiently high, the scattering mechanism can completely be absentdue to the linear nature of the equations for the high-wavenumber group.

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