Mean topological evolution in a turbulent boundary layer

Lagrangian mean evolution of a developing zero pressure gradient turbulent boundary layer (Re_θ = 730 to 1954) is investigated using data from a direct numerical simulation performed by Wu & Moin (2010). Conditional mean trajectories (CMTs) for the evolution of the invariants of the velocity gradient tensor (VGT) are calculated based on the mean rate of change of the invariants, conditioned on their location in the (R_A,Q_A) invariants plane. Following Chong et al. (1990) the location in this plane distinguish the focal or non-focal nature of flow at that point, such that CMTs represent the mean topological evolution of points in the flow. In the present case CMTs for strong gradients in all regions of the boundary layer pass around a focus at the origin and asymptote towards the right-hand side of a saddle point located near the of the line dividing unstable focal and unstable nodal structures. Closer to the origin weaker gradients follow an almost periodic clockwise spiralling evolution from stable-focus stretching to unstable-focus contraction, unstable-node saddle/saddle and stable-node saddle/saddle topology. Increasing time-scales are observed for both the strong and weak gradient trajectories further above the wall. Mean timescales associated with the spiralling evolution in terms of inner scales are 67.9 ν/u² _τ in the viscous layer, 151 ν/u² _τ in the buffer layer and 658 ν/u² _τ in the log and wake region or 1.25 estimated eddy turnover times.