Linear-model-based estimation of turbulent fluctuations in compressible turbulent boundary layers

Linear-model-based estimation provides an equation-based route to reconstruct turbulent fluctuations from limited measurements, but its performance in compressible turbulent boundary layers depends critically on both the inter-layer coherence and the modelling of nonlinear forcing. This study systematically extends and assesses linear-model-based estimation for velocity and temperature fluctuations in compressible turbulent boundary layers, using a logarithmic-region measurement plane at Mach numbers upper M equals 2 M = 2 $M=2$ , 4 and 6 under both adiabatic- and cold-wall conditions. With the direct numerical simulation databases, the linear coherence spectra are first analysed. Results show that the recoverable information is mainly confined to large scales especially when the estimation plane is located near the wall, whereas the coherent scale range broadens markedly as the estimation location moves towards the measurement plane. The temperature coherence exhibits stronger sensitivity to Mach number and wall thermal condition than the streamwise velocity coherence, especially in cold-wall cases where the non-monotonic mean temperature profile alters the inter-layer thermal correlation. Based on these coherence characteristics, large-scale velocity and temperature snapshots are reconstructed using three levels of linear-model-based estimation. A controlled hierarchy comprising the baseline linear model with white forcing, an eddy-viscosity-corrected operator and a combined eddy-viscosity and forcing model is used to distinguish the effects of operator correction and forcing closure. Finally, a single-plane-calibrated estimation of wall-normal fluctuation-energy profiles is performed using the response cross-spectral density estimated by the Lyapunov formulation. The principal physical insight is that inter-layer coherence determines which scales are recoverable, the eddy-viscosity correction primarily improves the reconstructed response structure and the forcing model modifies the fluctuation amplitude and wall-normal energy redistribution. These roles are distinct in instantaneous and statistical reconstruction and are particularly important for temperature fluctuations under cold-wall conditions.