The attached-eddy model (AEM) is one of the most successful coherent-structure-based phenomenological models in wall turbulence. In the classical AEM, the probability density of eddy $p_e$ is assumed to follow an inverse law with the eddy size $h_e$, i.e., $p_e\propto h_e^{-1}$, to satisfy a constant Reynolds-shear-stress distribution in the inertial layer of canonical wall-bounded turbulent flows. In this paper, we first extend the AEM to general attached eddies with $p_e\propto h_e^{-\alpha }$, where $\alpha$ is an arbitrary positive real number. Scaling laws for velocity covariance (Reynolds stress) by general attached eddies are derived. Preliminary evidence for the validity of the model is provided from adverse-pressure-gradient turbulent boundary layer and turbulent wing boundary layer flows. Second, considering that the eddy cascade self-similarity is manifested by generalized power laws for probability density $p_e$, population density $M_e$, area coverage $C_e$, and volume fraction $V_e$ of eddies, i.e., $p_e\propto h_e^{-\alpha }, M_e\propto h_e^{-\beta }, C_e\propto h_e^{-\gamma }$, and $V_e\propto h_e^{-\zeta }$, we directly connect the exponents with the fractal dimension $D_e$ of the general attached eddies in a simple and clear way. The present paper highlights that the scaling laws of velocity covariance in the inertial layer of wall-bounded turbulent flows can be directly linked to the characteristics of the cascade self-similarity of the general attached eddies. We believe that the scaling laws derived here and the generalized power-law relationships are useful for a deeper understanding of the connection between coherent structures and turbulence statistics.

• Received 9 December 2022
• Accepted 5 April 2023

DOI:https://doi.org/10.1103/PhysRevFluids.8.044603