Streamwise inclination angle of wall-attached eddies in turbulent channel flows

We develop a new methodology to assess the streamwise inclination angles (SIAs) of the wall-attached eddies populating the logarithmic region with a given wall-normal height. To remove the influences originating from other scales on the SIA estimated via two-point correlation, the footprints of the targeted eddies in the vicinity of the wall and the corresponding streamwise velocity fluctuations carried by them are isolated simultaneously, by coupling the spectral stochastic estimation with the attached-eddy hypothesis. Datasets produced with direct numerical simulations spanning $Re_{\tau} \sim O(10^2)-O(10^3)$ are dissected to study the Reynolds-number effect. The present results show, for the first time, that the SIAs of attached eddies are Reynolds-number dependent in low and medium Reynolds numbers and tend to saturate at $45^{\circ}$ as the Reynolds number increases. The mean SIA reported by vast previous experimental studies are demonstrated to be the outcomes of the additive effect contributed by multi-scale attached eddies. These findings clarify the long-term debate and perfect the picture of the attached-eddy model.