The energetics and colour for linearised models of wall turbulence

By comparing the budget of a data-driven quasi-linear approximation (DQLA) (Holford, Lee & Hwang, J. Fluid Mech. , vol. 980, 2024, A12) and direct numerical simulation (DNS) (Lee & Moser, J. Fluid Mech. , vol. 860, 2019, pp. 886–938), the energetics of linear models for wall-bounded turbulence are assessed. The DQLA is implemented with the linearised Navier–Stokes equations with a stochastic forcing term and an eddy viscosity diffusion model. The self-consistent nature of the DQLA allows for a global comparison across all wavenumbers to assess the role of the various terms in the linear model in replicating the features present in DNS. Starting from the steady-state second-order statistics of a Fourier mode, a spectral budget equation is derived, connecting Lyapunov-like equations to the transport budget equations obtained from DNS. It is found that the DQLA and DNS are in good qualitative agreement for the streamwise-elongated structures present in DNS, comparing well for production, viscous transport and wall-normal turbulent transport. However, the DQLA does not have an energy-conservative nonlinear term. This results in no dissipation under molecular viscosity, with energy instead being dissipated locally through the eddy viscosity model, which models the energy removal by the nonlinear term at integral length scales. Comparison of the pressure–strain statistics also highlights the absence of the streak instability, with production and forcing mainly being retained in the streamwise and wall-normal components or shifted to the spanwise component. It is demonstrated that the eddy viscosity diffusion term locally enforces a self-similar budget, making the model for the nonlinear term self-consistent with a logarithmic mean profile. Implications and recommendations to improve the current eddy viscosity enhanced linear models are also discussed concerning the comparison with DNS, as well as considerations with regard to pressure statistics to mimic the role of the streak instability through colour of turbulence models.