Tracking coherent structures in massively-separated and turbulent flows

Coherent vortex structures are tracked in simulations of massively-separated and turbulent flows. Topological Lagrangian saddle points are found using intersections of the positive and negative finite-time Lyapunov exponent ridges, and these points are then followed in order to track individual coherent structure motion both in a complex interacting three-dimensional flow (turbulent channel) and during vortex formation (two-dimensional bluff body shedding). For a simulation of wall-bounded turbulence in a channel flow, tracking Lagrangian saddles shows that the average structure convection speed exhibits a similar trend as a previously published result based on velocity and pressure correlations, giving validity to the method. When this tracking method is applied in a study of a circular cylinder in cross-flow it shows that Lagrangian saddles rapidly accelerate away from the cylinder surface as the vortex sheds. This saddle behavior is compared with the time-resolved static pressure distribution on the circular cylinder, yielding locations on a cylinder surface where common sensors could detect this phenomenon, which is not available from force measurements or vortex circulation calculations. The current method of tracking coherent structures yields insight into the behavior of the coherent structures in both of the diverse flows presented, highlighting the breadth of its potential application.

Figure 1

Lagrangian particle evolution around a Lagrangian saddle: (a) $t/T=0.75$, (b) $t/T=0.82$, (c) $t/T=0.89$, and (d) $t/T=0.96$. Particles with values above $0.75{\mathrm{\Lambda }}_{\max }$ are black; other particles are colored by their initial location. Repulsion from the pFTLE ridge is indicated by the thickening of the black ridge between red and blue, and between green and magenta. Attraction to the nFTLE ridge is indicated by the narrowing of the black ridge between red and green, and between blue and magenta.

Figure 2

Negative-time FTLE fields in a plane located at $y^{+}=49.6$ for five different integration times: (a) ${\tau }^{+}=9$, (b) ${\tau }^{+}=18$, (c) ${\tau }^{+}=27$, (d) ${\tau }^{+}=36$, and (e) ${\tau }^{+}=45$. All five panels use the same color axis as shown.

Figure 3

Instantaneous snapshots of pFTLE ridges (blue) and nFTLE ridges (red) (values above $0.65{\mathrm{\Lambda }}_{\max }$) at (a) $y^{+}=10.5$ and (b) $y^{+}=49.6$ in the turbulent channel simulation. Lagrangian saddles are highlighted by cyan circles.

Figure 4

Plane-averaged velocity of Lagrangian saddles in the turbulent channel simulation, plotted against wall-normal distance (red). These data are compared with the simulation mean streamwise velocity profile (blue). The error bars in the figure are one standard deviation of the measured Lagrangian saddle convection velocity in the plane.

Figure 5

Instantaneous snapshots of positive- and negative-time FTLE ridges (blue and red, respectively, with values above $0.67{\mathrm{\Lambda }}_{\max }$) in the flow around a circular cylinder (represented by a green circle) with vortex cores visualized by $Q$ (grey contours): (a) $t/T=0.22$ and (b) $t/T=0.82$. The Lagrangian saddle is highlighted by a black arrow.

Figure 6

pFTLE (blue), nFTLE (red), and $Q$ criterion (grey contours) behind a circular cylinder (green) at $t/T=0.55$. The Lagrangian saddle is highlighted by a red X and the vortex center is highlighted by a red O.

Figure 7

Distance from cylinder to vortex center (blue squares) and Lagrangian saddle (red diamonds).

Figure 8

(a) Coefficient of lift per unit span, $C_L\left(t\right)$, for the cylinder and (b) the normalized circulation for a vortex forming and shedding from the cylinder at $\text{Re}=150$ at the time of Lagrangian saddle lift-off (black dash-dotted line).

Figure 9

Circulation area during (a) vortex formation ($t/T=0.29$) and (b, c) shedding ($t/T=0.58$ and $t/T=0.78$, respectively).

Figure 10

$C_p^{\prime }\left(t\right)$ (blue curves) at (a) ${60}^{\circ }$, (b) ${80}^{\circ }$, (c) ${100}^{\circ }$, and (d) ${120}^{\circ }$ with saddle locations, and Lagrangian saddle distance to cylinder (red diamonds).

Figure 11

pFTLE (blue), nFTLE (red), and $Q$ criterion (grey contours) behind a circular cylinder (green) in left-hand column. $C_p$ (blue), saddle location (red X in left-hand and center columns and red diamond in right column), and vortex center location (red O) at different times. FTLE and $Q$ in cylinder wake at (a) $t/T=0.15$, (d) $t/T=0.29$, (g) $t/T=0.44$, and (j) $t/T=0.58$; $C_p$ distribution around cylinder at (b) $t/T=0.15$, (e) $t/T=0.29$, (h) $t/T=0.44$, and (k) $t/T=0.58$; and (c, f, i, l) $C_p^{\prime }\left(t\right)$ at ${100}^{\circ }$ with saddle locations.