Two-scale solution for tripped turbulent boundary layer

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Recent findings on the Reynolds-number-dependent behaviour of near-wall turbulence in terms of the ‘foot-printing’ of outer large-scale structures call for a new modelling development. A two-scale framework was proposed to couple a local fine-mesh solution with a global coarse-mesh solution by He (Intl J. Numer. Meth. Fluids, vol. 86, 2018, pp. 655–677). The methodology was implemented and demonstrated by Chen & He (J. Fluid Mech, vol. 933, 2022, p. A47) for a canonical turbulent channel flow, where the mesh-count scaling with Reynolds number is potentially reduced from $O(R{e^2})$ for a conventional wall-resolved large-eddy simulation (WRLES) to $O(R{e^1})$ . The present work extends the two-scale method to turbulent boundary layers. A two-dimensional roughness element is used to trip a turbulent boundary layer. It is observed that large-scale disturbances originating at the trip have a much shorter lifetime and weaker foot-printing signatures on near-wall flow compared to those long streaky coherent structures in well-developed wall-bounded turbulent flows. Modal analyses show that the impact of trip-induced large scales can be adequately captured by a locally embedded fine-mesh block. For the tripped turbulent boundary layer, a Chebyshev block-spectral mapping is adopted to propagate source terms from the local fine-mesh blocks to the global coarse-mesh domain, driving to a target solution for the upscaled equations. The computed mean statistics and energy spectra are in good agreement with corresponding experimental data, WRLES and direct numerical simulation (DNS) results. The overall mesh count– $Re$ scaling is estimated to reduce from $O(R{e^{1.8}})$ for the full wall-resolved LES to $O(R{e^{0.9}})$ for the present two-scale solution.