Structure Functions in Wall-bounded Flows at High Reynolds Number

The scaling of the structure function $D_{\mathrm{ij}}=$\textless ($u_{\mathrm{i}}$(\textbf{x}$+$\textbf{r})-$u_{\mathrm{i}}$(x))($u_{\mathrm{j}}$(\textbf{x}$+$\textbf{r})-$u_{\mathrm{j}}$(\textbf{x}))\textgreater (where $i=$1,2,3 and \textbf{r }is the two-point displacement, $u_{i}$ is the velocity fluctuation in the $x_{i}$ direction), is studied in wall-bounded flows at high Reynolds number within the framework of the Townsend attached eddy model. While the scaling of $D_{\mathrm{ij}}$ has been the subject of several studies, previous work focused on the scaling of $D_{\mathrm{11}}$ for \textbf{r}$=(\Delta x$,0,0) (for streamwise velocity component and displacements only in the streamwise direction). Using the Hierarchical-Random-Additive formalism, a recently developed attached-eddy formalism, we propose closed-form formulae for the structure function$ D_{\mathrm{ij}}$ with two-point displacements in arbitrary directions, focusing on the log region$.$ The work highlights new scalings that have received little attention, e.g. the scaling of $D_{\mathrm{ij}}$ for \textbf{r}$=$(0, $\Delta $y, $\Delta $z) and for $i\ne j$. As the knowledge on D$_{ij\thinspace }$leads directly to that of the Reynolds stress, statistics of the filtered flow field, etc., an analytical formula of D$_{ij}$ for arbitrary \textbf{r} can be quite useful for developing physics-based models for wall-bounded flows and validating existing LES and reduced order models.