Path lengths in turbulence

By tracking tracer particles at high speeds and for long times, we study the geometric statistics of Lagrangian trajectories in an intensely turbulent laboratory flow. In particular, we consider the distinction between the displacement of particles from their initial positions and the total distance they travel. The difference of these two quantities shows power-law scaling in the inertial range. By comparing them with simulations of a chaotic but non-turbulent flow and a Lagrangian Stochastic model, we suggest that our results are a signature of turbulence.

💡 Research Summary

In this paper the authors investigate a previously unexplored geometric aspect of Lagrangian particle motion in high‑Reynolds‑number turbulence. Using a von Kármán swirling water flow driven by counter‑rotating disks, they seed the fluid with neutrally buoyant polystyrene tracer particles (diameter ≈ 25 µm, density ≈ 1.06 g cm⁻³) and record three‑dimensional trajectories with three high‑speed cameras at up to 27 kHz. The measurement volume is 5 × 5 × 5 cm³, the integral length scale L ≈ 7.1 cm, and the Taylor‑scale Reynolds numbers examined are Rλ = 690 and 815.

From each trajectory they compute two scalar quantities: (i) the displacement vector R(t) = x(t) − x(0) and its mean‑square ⟨R²⟩, and (ii) the arc length S(t) = ∫₀ᵗ|v(τ)| dτ, i.e. the total distance travelled, together with its mean‑square ⟨S²⟩. Both ⟨R²⟩ and ⟨S²⟩ display the classic ballistic growth (∝ t²) at short times and a crossover to diffusive growth (∝ t) at times larger than the integral time TL. This confirms that, for short intervals, the path length and the net displacement are essentially identical.

The novelty lies in examining the differences between these two measures. The authors analyse (a) the difference of the second moments, Δ₁(t)=⟨S²−R²⟩, and (b) the second moment of the difference, Δ₂(t)=⟨(S−R)²⟩. In the inertial range (τ_η ≪ t ≪ TL) both Δ₁ and Δ₂ follow a power law close to t³·⁷ (the fitted exponent is 3.7 ± 0.2). Dimensional arguments based on Kolmogorov’s 1941 theory suggest a t³ scaling because Δ₁ and Δ₂ have dimensions of length squared, but the measured exponent is systematically larger than three. Local‑slope analysis (d log Δ/d log t) shows extended plateaus at ≈ 3.7, and the width of the plateau grows with Reynolds number, indicating that the scaling is a genuine inertial‑range feature rather than a finite‑size artifact.

To test whether this behavior is specific to turbulence, the authors compare the experimental data with two synthetic flows. The first is the steady Arnold–Beltrami–Childress (ABC) flow, a classic chaotic but non‑turbulent velocity field. Particle trajectories in the ABC flow do not exhibit any clear t³·⁷ scaling; instead, Δ₁ and Δ₂ either saturate to t² at long times or show only short, ambiguous power‑law segments with exponents far from 3.7. The second comparison uses Sawford’s second‑order Lagrangian stochastic model, which incorporates two time scales (the integral time TL and the Kolmogorov time τη) but assumes Gaussian statistics for acceleration and velocity increments (i.e., no intermittency). Simulations of this model at Rλ ≈ 815 reproduce a t³·⁷‑like scaling for Δ₁ and Δ₂, albeit over a narrower time window than the laboratory data. This suggests that the presence of a well‑separated inertial range, rather than intermittency, is sufficient to generate the observed scaling.

The authors also examine the covariance ⟨SR⟩, which is non‑negligible because S and R are strongly correlated. The covariance itself follows the same ballistic‑to‑diffusive crossover as the individual moments, confirming that the cross term contributes significantly to Δ₁ and Δ₂.

In summary, the paper demonstrates that the difference between path length and net displacement provides a new Lagrangian observable that scales as t³·⁷ in the inertial range of high‑Reynolds‑number turbulence. This scaling is absent in a purely chaotic but non‑turbulent flow and is reproduced by a stochastic model that contains only the two characteristic time scales of turbulence. The authors argue that the scaling reflects the separation of time scales inherent to turbulent cascades and may be linked to coherent vortical structures (e.g., vortex tubes) that cause particles to travel longer, spiralling paths while their net displacement remains modest. Because the statistics are second order, intermittency corrections are expected to be small, making the result a robust signature of turbulence. The work opens a new avenue for theoretical modeling of Lagrangian geometry in turbulence and suggests that incorporating path‑length statistics could improve our understanding of particle transport, mixing, and dispersion in complex flows.