  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.
