Identification of Lagrangian Coherent Structures in a Turbulent Boundary Layer

In this study, we identify Lagrangian coherent structures (LCS) in a flat plate turbulent boundary layer at Re$_{\theta}$ 0f $19\:100$. To detect the LCS, we compute direct Lyapunov exponents (DLE) (Haller, G., Physica D, vol 149, pp 248-277, 2001). Specifically we use the velocity field obtained from stereo PIV measurements to compute trajectories, $\mathbf{x} (t,t_{0},\mathbf{x} _{0})$, from initial positions, $\mathbf{x}_{0}$, at time $t_{0}$. For fixed integration times, $\left| t-t_{0} \right|$, we numerically differentiate the flow map, given by $F_{t_{0}}^{t}(\mathbf{x}_{0}) = \mathbf{x}(t, t_{0}, \mathbf{x}_{0})$, and then compute the deformation gradient tensor field $\Delta ^{t}_{t_{0}}(\mathbf{x}_{0}) = \left[ \nabla F_{t_{0}}^{t}(\mathbf{x}_{0}) \right]^{T} \left[ \nabla F_{t_{0}}^{t}(\mathbf{x}_{0}) \right]$. The DLE field is then found as $\mathrm{DLE}_{t_{0}}^{t}(\mathbf{x_{0}}) = \ln \left( \lambda _{\mathrm{max}} \left( \Delta _{t_{0}}^{t}(\mathbf{x}_{0}) \right) \right)/ \left (2\left| t-t_{0} \right| \right)$. Two dimensional gradient climbing is then used to find points on the locally maximizing, LCS surfaces of the field, $\mathrm{DLE}_{t_{0}}^{t}(\mathbf{x}_{0})$. To determine whether these surfaces truly repel (attract) near by fluid particles, the \emph{hyperbolicity criterion} is applied (Mathur et al., Phys. Rev. Lett., vol 98, pp 144502, 2007). In particular we compute normal strain rates, $\langle \mathbf{n},\mathbf{Sn}\rangle$, to locate repelling surfaces $\left( t>>t_{0} \mathrm{\:and\: } \langle \mathbf{n},\mathbf{Sn}\rangle >0\right)$ and attracting surfaces $\left( t<