Elliptic approximation model revisited

The elliptic approximation (EA) – rooted in Taylor’s frozen flow hypothesis, Kolmogorov’s theory of small-scale turbulence, and the Kraichnan–Tennekes random sweeping hypothesis – remains a foundational framework for modelling spatiotemporal velocity correlations in incompressible wall-bounded turbulence. This study revisits the model’s theoretical basis, and extends its applicability to velocity and temperature fluctuations in supersonic channel flows. First, we identify non-elliptic distortions in the viscous sublayer, and introduce a shear-induced acceleration that captures the observed deviation from the assumed constant convection velocity at large time separations. Next, we show that the inertial-range scalings underpinning the EA are not valid in regions where the model remains accurate; instead, its validity is supported by extended self-similarity between spatial and temporal structure functions. Finally, we conduct high-fidelity direct numerical simulations of compressible channel flows with fluctuating Mach numbers up to 0.8; our data confirm the robustness of the EA under supersonic conditions, and its effectiveness in characterising both velocity and temperature correlations. Together, these findings provide new theoretical insights into the spatiotemporal structure of wall-bounded turbulence, and broaden the operational envelope of the EA.