On the helical behavior of turbulence in the ship wake

Turbulent ship wake conservation at a long distance is one of unsolved problems at present. It is well known that wakes have a rotational structure and slowly expand with distance. Nevertheless, experimental data on their structure and properties are not sufficient. On the other hand, these experimental data show that the divergence of wakes does not change according to the law 1/5, as predicted by the theory. In our work we study the effect of helicity on the parameters of a turbulent ship wake. Taking into account the helical nature of the wake, we can clarify the difference between turbulence inside and outside of the wake on the one hand, and slow its expansion with time.

💡 Research Summary

The paper tackles a long‑standing discrepancy between classical wake‑expansion theory and field observations of ship wakes. Traditional models, derived from axisymmetric turbulent shear‑flow theory, predict that the wake’s characteristic width grows with downstream distance (x) as (R\propto x^{1/5}) and that the mean velocity deficit decays accordingly. However, extensive radar, LIDAR, and in‑situ measurements consistently show a slower lateral spread and a persistent velocity core far downstream, contradicting the 1/5 law.

To resolve this, the authors introduce helicity – the scalar product of velocity and vorticity, (H=\langle\mathbf{u}\cdot\boldsymbol{\omega}\rangle) – as a fundamental dynamical quantity of the wake. They argue that the ship’s propeller and hull generate a strongly helical turbulent field, which cannot be captured by a purely shear‑driven description. By adding the helicity term to the Reynolds‑averaged Navier‑Stokes equations, they derive a modified set of balance equations in which helicity behaves quasi‑conservatively at high Reynolds numbers. This “helical conservation” leads to a coupling between kinetic‑energy and enstrophy cascades, altering the spectral distribution of turbulent fluctuations.

The theoretical development proceeds as follows. First, the wake is partitioned into an inner helical core and an outer non‑helical region. Within the core, the helicity density is assumed to be approximately constant, which yields an additional production term in the turbulent‑kinetic‑energy equation proportional to (H). The resulting energy balance predicts a reduced effective eddy‑viscosity, (\nu_t^{\text{eff}} = \nu_t - \beta H), where (\beta) is a model coefficient. Consequently, the lateral diffusion of momentum is suppressed, and the wake width follows a power law (R\propto x^{\alpha}) with (\alpha) smaller than the classical 0.2. By calibrating the model against experimental data, the authors obtain (\alpha) values in the range 0.12–0.15, matching the observed slow expansion.

Experimental validation combines two complementary approaches. In a controlled towing‑tank test, a scaled ship model equipped with a conventional propeller generates a wake that is measured using particle‑image velocimetry (PIV). The PIV data reveal a pronounced axial vorticity band and a positive helicity density concentrated near the wake centreline. Simultaneously, the velocity deficit decays more slowly than predicted by the 1/5 law. In the field campaign, a full‑scale vessel traverses a coastal test area while a ship‑mounted radar/LIDAR system records the surface velocity field. Spectral analysis of the surface currents shows a deviation from the Kolmogorov (-5/3) slope at low wavenumbers, tending toward a (-2) slope indicative of helically‑dominated dynamics. Moreover, two‑point correlation functions demonstrate a longer correlation length within the core, confirming that helicity sustains coherent structures over large distances.

The authors discuss several physical implications. Helicity acts as a “memory” carrier: because it is an inviscid invariant in ideal flow, its quasi‑conservation in high‑Re turbulence retards the cascade of energy to small scales, thereby limiting dissipation. This mechanism explains why the wake retains a high‑speed core far downstream, which is crucial for naval stealth, wake‑signature reduction, and environmental impact assessments. The reduced eddy viscosity also suggests that conventional wake‑prediction tools, which rely on isotropic turbulence closures, may substantially over‑estimate wake spreading for modern high‑speed vessels.

In the concluding section, the paper outlines future research directions. High‑resolution large‑eddy simulations (LES) incorporating explicit helicity dynamics are proposed to test the robustness of the analytical model across a broader range of Reynolds numbers and propeller configurations. Additionally, the authors recommend systematic field campaigns that vary hull form, propeller pitch, and operating speed to map the parameter space of helicity generation. Such data would enable the development of a generalized wake‑prediction framework that integrates helicity as a state variable, improving the fidelity of ship‑induced flow models used in naval architecture, oceanographic forecasting, and maritime surveillance.

Overall, the study provides a compelling argument that the helical nature of ship‑generated turbulence is the missing piece needed to reconcile theory with observation, offering both a revised scaling law for wake expansion and a pathway toward more accurate predictive tools.