[Paper Review] Stokes drift for inertial particles transported by water waves

G. Boffetta, Marco Martins Afonso+3 more|arXiv (Cornell University)|Oct 8, 2012

Particle Dynamics in Fluid Flows10 references3 citations

TL;DR

This paper analytically and numerically investigates Stokes drift for inertial particles in deep-water surface gravity waves, showing that particle inertia induces a second-order correction to the horizontal Stokes drift and a first-order vertical drift unrelated to gravity. The vertical drift, which enhances sedimentation even in the absence of gravity, is a novel result with implications for particle transport and sedimentation in coastal and oceanic environments.

ABSTRACT

We study the effect of surface gravity waves on the motion of inertial particles in an incompressible fluid. Using the multiple-scale technique, we perform an analytical calculation which allows us to predict the dynamics of such particles; results are shown for both the infinite- and finite-depth regimes. Numerical simulations based on the velocity field resulting from the second-order Stokes theory for the surface elevation have been performed, and an excellent agreement with the analytical predictions is observed. Such an agreement seems to hold even beyond the formal applicability of the theory. We find that the presence of inertia leads to a non-negligible correction to the well-known horizontal Stokes drift; moreover, we find that the vertical velocity is also affected by a drift. The latter result may have some relevant consequences on the rate of sedimentation of particles of finite size. We underline that such a drift would also be observed in the (hypothetical) absence of the gravitational force.

Motivation & Objective

• To understand how particle inertia modifies Stokes drift in deep-water surface gravity waves.
• To investigate the effect of inertia on both horizontal and vertical particle transport in wave fields.
• To determine whether a vertical drift arises even in the absence of gravitational forces.
• To validate analytical predictions with kinematic numerical simulations beyond the perturbative regime.

Proposed method

• Perturbation expansion in the Stokes number (St) to derive corrections to the Stokes drift velocity for inertial particles.
• Use of the linearized, irrotational, incompressible velocity field of a Stokes wave in deep water (u ∝ e^{kz} cos(kx - ωt)).
• Derivation of Lagrangian trajectories for particles with finite density (characterized by β = ρ_p/ρ_f) and Stokes number St.
• Computation of mean drift velocities by averaging particle displacement over one Lagrangian period.
• Kinematic numerical simulations of particle trajectories under wave forcing to test analytical predictions.
• Comparison of analytical results with simulations across varying wave steepness (ε) and particle inertia (St).

Experimental results

Research questions

• RQ1 How does particle inertia modify the classical horizontal Stokes drift velocity in deep-water waves?
• RQ2 Does a vertical drift emerge for inertial particles in wave fields, even when gravity is absent?
• RQ3 What is the dependence of the drift velocity on the particle Stokes number and density contrast?
• RQ4 How accurate are perturbative analytical predictions when extended beyond the weak-inertia regime?
• RQ5 Can kinematic simulations reproduce the analytical drift behavior for moderate to strong wave steepness?

Key findings

• A second-order correction to the horizontal Stokes drift velocity arises due to particle inertia, with a sign depending on the particle's density relative to water.
• A first-order vertical drift velocity emerges in the absence of gravity, acting in the same direction as sedimentation and proportional to the Stokes number St.
• The vertical drift modifies the effective sedimentation velocity, potentially accelerating particle settling in natural systems.
• Numerical simulations confirm analytical predictions with excellent agreement, even for St = O(1) and wave steepness ε = 0.33, beyond the perturbative limit.
• The ratio of vertical to horizontal drift velocity in the absence of gravity follows a non-trivial analytical form involving St and β, with good agreement in simulations.
• The total horizontal displacement of heavy particles diverges in the absence of gravity due to the persistent vertical drift, indicating unbounded net transport.