Representation of Two-Point Velocity Correlations in Terms of Structure Tensors

A general representation of the homogeneous, anisotropic two-point second-order correlation of turbulent velocity fluctuation of the form $R_{ij}=\langle u'_i(\mathbf{x}) u'_j(\mathbf{x+r}) \rangle=\Sigma_n f^{(n)}(r) T^{(n)}_{ij}$ is constructed, where $12$ basis tensors $T^{(n)}_{ij}$ are expressed in terms of the separation vector $\mathbf{r}$ and structure tensors introduced by Kassinos and Reynolds (1995). The structure tensors are one-point correlations of the derivatives of fluctuating streamfunctions and are given by componentality $b_{ij}$, dimensionality $y_{ij}$ and stropholysis $Q_{ijk}$. These tensors are shown to contain information about the anisotropy of $R_{ij}$ (thus motivating such a representation). Using continuity and an additional constraint, only four scalar functions $f^{(n)}$ are shown to remain linearly independent. A comparison of the representation with two-point correlation data from DNS of channel flow turbulence is made in order to assess the suitability of this representation.