A conceptual generalization of Taylor microscales with applications to isotropic and wall turbulence

Two new concepts of the Taylor microscale matrix (TMM) and Taylor microscale tensor (TMT) are proposed to extend the classical concept of Taylor microscale of isotropic turbulence to the general scenario of anisotropic turbulence. The first concept TMM is derived from the classical definition of two-point auto-correlation function, which has shortcomings for not being a tensor. The second concept TMT is proposed based on a new correlation function that rigorously maintains tensor properties. The three eigenvalues of TMM or TMT represent the general Taylor microscales (GTMs). When these three new general concepts, TMM, TMT, and GTM, are applied to homogeneous and isotropic turbulence, the classical Taylor microscale is automatically recovered. The proposed concepts can also be used for characterizing anisotropic turbulence. To demonstrate, direct numerical simulations (DNSs) of turbulent plane-channel flows of three Reynolds numbers are performed. The asymptotic near-wall behaviors of GTM are derived analytically and validated using DNS data. In the current literature, it is popular to study the orientation of turbulent flow structures qualitatively based on the isopleths of a two-point auto-correlation function. As a new advancement, a rigorous explanation of this popular approach is obtained based on the concept of GTM such that the characteristic inclination angle of flow structures (such as the mean angle of hairpin packets) can be precisely defined and computed. It is interesting to observe that in the outer layer, the mean streamwise spacing between two successive hairpin vortices in a hairpin packet is approximately twice the proposed GTM λT1(1).