Modeling The Divergence Of The Two-Point, Third-Order Velocity Correlation

Knowledge of $\Gamma_{ij}(\mathbf{x},\mathbf{r})=\partial \langle u'_i(\mathbf{x})u'_k(\mathbf{x})u'_j(\mathbf{x}+\mathbf{r})\rangle/\partial r_k$ is crucial for modeling the subgrid force using the Optimal LES approach (Langford and Moser, (1999)). Here, a method is developed to obtain an approximation to $\Gamma_{ij}$ in statistically stationary turbulence, given a finite-dimensional representation for $R_{ij}(\mathbf{x},\mathbf{r})=\langle u'_i(\mathbf{x}) u'_j(\mathbf{x}+\mathbf{r})\rangle$. The rotationally invariant representation of $R_{ij}$ is in terms of Structure Tensors (Kassinos, 2001), and accounts for componental and directional anisotropy. Our method is based on the fact that the evolution equation for $R_{ij}$ is $D R_{ij}/D t=F_{ij}(\Gamma_{ij},\Pi_{ij},R_{ij},G_{ij})$, where $\Pi_{ij}(\mathbf{x},\mathbf{r})$ is the two-point pressure-strain correlation and $G_{ij}$ is the mean velocity gradient. For $D R_{ij}/D t=0$, a symmetrized form of $\Gamma_{ij}$ can be obtained in terms of a production term involving $R_{ij}$ and $G_{ij}$ (after projecting out $\Pi_{ij}$ using continuity). The resulting model and it's underlying assumptions are validated by comparison with DNS.