A data-driven quasi-linear approximation for turbulent channel flow

A data-driven implementation of a quasi-linear approximation is presented, extending a minimal quasi-linear approximation (MQLA) (Hwang&Ekchardt, J. Fluid Mech., 2020, 894:A23) to incorporate non-zero streamwise Fourier modes. A data-based approach is proposed, matching the two-dimensional wavenumber spectra for a fixed spanwise wavenumber between a direct numerical simulation (DNS) (Lee&Moser, J. Fluid Mech., 2015, 774:395-415) and that generated by the eddy viscosity-enhanced linearised Navier-Stokes equations at $Re{\tau} \simeq 5200$. Leveraging the self-similar nature of the energy-containing part in the DNS velocity spectra, a universal self-similar streamwise wavenumber weight is determined for the linearised fluctuation equations at $Re_{\tau} \simeq 5200$. This data-driven quasi-linear approximation (DQLA) offers qualitatively similar findings to the MQLA, with quantitative improvements in the turbulence intensities and additional insights from the streamwise wavenumber spectra. By comparing the one-dimensional streamwise wavenumber spectra and two-dimensional spectra to DNS results, the limitations of the presented framework are discussed, mainly pertaining to the lack of the streak instability (or transient growth) mechanism and energy cascade from the linearised model. The DQLA is subsequently employed over a range of Reynolds numbers up to $Re_{\tau} = 10^5$. Overall, the turbulence statistics and spectra produced by the DQLA scale consistently with the available DNS and experimental data, with the Townsend-Perry constants displaying a mild Reynolds dependence (Hwang, Hutchins&Marusic, J. Fluid Mech., 2022, 933:A8). The scaling behaviour of the turbulence intensity profiles deviates away from the classic $\ln(Re_{\tau})$ scaling, following the inverse centreline velocity scaling for the higher Reynolds numbers.