Scaling laws of velocity gradient moments of attached eddies

Townsend's attached-eddy model (AEM) is one of the most widely used models in explaining and predicting the logarithmic region of wall turbulence. Townsend pioneered the postulate that wall-attached eddies exhibit self-similar velocity distributions. This premise has led to the derivation of velocity variance scalings in the logarithmic region. In particular, the attached eddies have been extracted at moderate scales and have been illustrated to contain the most kinetic energies in the logarithmic region. In the present contribution, we derive analytically the scalings of the moments of velocity gradients of attached eddies by using the AEM. The direct numerical simulation data with the moderate-scale extraction of attached eddies show good agreement with the derived scalings. Moreover, the contributions of different-scale structures to the moments of velocity gradients are compared, showing that the wall scalings of all-scale velocity gradients are interestingly half of moderate-scale attached eddies. This also indicates the non-negligible influence of the small-scale eddies on the velocity gradients in the logarithmic region. In addition, there are departures in the moments of velocity Hessian, inspiring future improvement in the extraction method of attached eddies.