Large-scale structures of scalar and velocity in a turbulent jet flow

We study the relation between large-scale structures in the concentration field with those in the velocity field in a dye-seeded turbulent jet. The scalar concentration in a plane is measured using laser-induced fluorescence. Uniform concentration zones of an advected scalar are indentified using cluster analysis. We simultaneously measure the two-dimensional velocity field using particle image velocimetry. The structures in the velocity field are characterized by finite-time Lyapunov exponents. The measurement of the scalar- and velocity fields moves with the mean flow. In this moving frame, turbulent structures remain in focus long enough to observe well-defined ridges of the finite-time Lyapunov field. This field gauges the rate of point separation along Lagrangian trajectories; it was measured both for future and past times since the instant of observation. The edges of uniform concentration zones are correlated with the ridges of the past-time Lyapunov field, but not with those of the future-time Lyapunov field. We quantify this relation using both conditional averages and the ordinary correlation function.

Figure 1

The experimental setup. (a) Schematic view of the arrangement of cameras, traversing system and jet (not to scale). (b) Schematic view of the turbulent jet and sketch of the comoving observation window (not to scale). In a each run a total of 210 PIV frames and 101 LIF frames is acquired at a rate of 10 Hz. The coordinates within a comoving frame are $(x,y)$, the distance of its right edge to the orifice is $d$. (c) Mean observation time in the comoving frame when integrating the separation of fluid parcels backwards in time. A maximum time of $\left|T\right|=4\mathrm{s}$ corresponds to a traveled distance of $d=0.16\mathrm{m}$ from the jet orifice. In a stationary Eulerian frame fluid structures move with the mean jet velocity. In this case they could only have been observed over $0.08\mathrm{m}$ for fluid parcels starting at the very right border of the frame (and $0.04\mathrm{m}$ for structures starting halfway).

Figure 2

(a) Snapshot of tracer concentration $c(\mathit{x},t)$, taken at $d=0.83\mathrm{m}$ from the jet orifice. (b) Field of panel (a), coarsened into four uniform concentration zones. (c) Finding uniform concentration zones. Blue lines: membership distribution of each of the four zones. Black line: sum of membership distributions, nearly coinciding with the histogram of concentrations in the field of frame (a). Gray line: smoothed histogram used in the kernel density estimate (KDE) of the number of zones ($n_{\mathrm{zones}}=4$).

Figure 3

(a) Degree of clustering Cl [see Eq. (1)] as a function of distance $d$ to the jet orifice at $d=0$. Full line: mean over 20 realizations, the width of the gray band is the rms fluctuation. A single point with error bar represents Cl for the clustering of the fluctuating streamwise velocity in a turbulent boundary layer [data of 4]. (b) Kernel density estimate of the number of zones $n_{\mathrm{zones}}$ as a function of distance $d$. Full line: mean over 20 realizations, gray band: rms fluctuation. (c) Averaged area of zones as a function of $d$. The lines are ordered according to the area of a zone in each individual frame. No significant coarsening of the zones in the jet far field is observed.

Figure 4

Comparison of backward-time Lyapunov field (c), (d) ${\mathrm{\Lambda }}_{-T}$ to the concentration field (a), (b) in two snapshots of a single scan, which also illustrates the temporal evolution of the fields. (a), (c) At $d=0.67\mathrm{m}$, (c), (d) at $d=0.83\mathrm{m}$. The large-scale filamentous structure of ${\mathrm{\Lambda }}_{-T}\left(\mathit{x}\right)$ roughly corresponds to the large-scale structure of the edges of the uniform concentration zones (suggested by arrows).

Figure 5

Evolution with integration time $T$ of the time-backward finite-time Lyapunov field ${\mathrm{\Lambda }}_{-T}$ in the Lagrangian frame. (a) After $T=0.8\mathrm{s}$, (b) after 6.4 s. (c) Evolution of ${\mathrm{\Lambda }}_{-T}$ on the line in (b) which intersects a filament. At increasing integration time, $T=0.1,1.7,3.3,$ and $4.9\mathrm{s}$, structures with increasing detail emerge. The bar indicates the Kolmogorov scale. (d) The evolution of increments $d{\mathrm{\Lambda }}_{-T}$ measured at the 4 intersection points shown in panel (b). The two points with largest $x$ have already left the observation window at $T\approx 2\mathrm{s}$.

Figure 6

Relation between the finite time Lyapunov field ${\mathrm{\Lambda }}_{\pm T}$ and the edges of uniform concentration zones. (a) Illustration of conditional averages on lines perpendicular to zone edges. Three of these lines $\mathit{n}\left(s\right)$ are shown, the corresponding zone edge at $s=0$ is indicated by the circle. In comparison to Fig. 4, small zones with a relative area of ${10}^{-3}$ have been removed. (b) Perpendicular conditional averages of ${\mathrm{\Lambda }}_{\pm T}$. Only the past Lyapunov field ${\mathrm{\Lambda }}_{-T}$ shows a well-defined peak at zone edges $I_e$. (c) Dots conected by lines: height of the peak in ${\mathrm{\Lambda }}_{-T}$ above the background [as illustrated in (b)] as a function of the delay between ${\mathrm{\Lambda }}_{-T}$ and the zone edge field $I_e$, gray line: same for the peak in ${\mathrm{\Lambda }}_T$. (d, e) Normalized correlation $C\left(\mathit{r}\right)$ between edges and the Lyapunov fields ${\mathrm{\Lambda }}_{\pm T}$. It is defined as $C\left(\mathit{r}\right)={\langle {\mathrm{\Lambda }}_{\pm T}(\mathit{x}+\mathit{r})I_e\left(\mathit{x}\right)\rangle }_{\mathit{x},t,r}$, where $I_e\left(\mathit{x}\right)$ is the zone edge field: $I_e\left(\mathit{x}\right)=1$ when $\mathit{x}$ is on an edge, and zero otherwise.