Abstract

Considered here is the motion of small particles beneath irrotational water waves. The added mass and inertial forces are shown to be an important role in the mean transport of particles. To leading order, particles are transported with a mean horizontal Stokes drift velocity and sediment with their terminal fall velocity. The combination of a settling velocity and a mean drift transports particles a finite distance forward from their point of release.

1. Introduction

An important environmental problem is how effective waves are in dispersing pollution such as oil or dredged material in coastal regions. The dispersive process depends critically on the type of flow generated by the waves, their amplitude, and the history of the flow. Linear water waves moving in the absence of wind stresses generate a mean streaming flow that may have an irrotational and viscous contribution (Longuet-Higgins 1953). For larger-amplitude waves, the boundary layer separates from the free surface, generating turbulence that diffuses beneath the waves (Hunt et al. 2001). As the wave amplitude increases further, the waves break, generating a large subsurface vortex that diffuses into the bulk of the fluid. The effect of wind stress is observed to also generate a growing boundary layer flow at the free surface, which ultimately breaks to generate a jetlike flow and a cellular flow pattern aligned with the wind stress (Melville et al. 1998). Other processes, such as Langmuir circulation cells (Leibovich 1983), are also important in redistributing heat and matter throughout the upper layer of the ocean (Thorpe 1984).

Stokes drift—a second-order drift velocity of fluid particles in the direction of wave propagation—is a major component of wave-induced transport. Stokes (1847) originally examined the transport of fluid particles by water waves propagating over an infinitely deep body of fluid, and he demonstrated that they execute circular orbits in addition to being transported by a constant drift velocity u2/c, where u is the average fluid speed beneath the waves moving with speed c. This second-order drift velocity also appears in many other areas of fluid mechanics, such as multiphase flows, and is intimately connected to Darwin’s drift (Eames and McIntyre 1999).

The main aim of this paper is to examine the transport of small particles and bubbles by progressive water waves. The mean transport of rigid particles beneath water waves has been studied by Grinshpun et al. (2000), who considered the action of a linear drag force on small particles close to neutral buoyancy. Dispersed material in the ocean (consisting of organic matter, larvae, bubbles, sediment) has a relative density comparable to or less than water. Typical values for the density of material that may be found in the ocean are 1076–1102 kg m−3 for larvae [which have diameters of 126–194 μm, estimated from the fall speed given by Krug and Zimmer (2004)), 873–973 kg m−3 for crude oil (depending on whether it originates from Texas or Mexico), ∼1500 kg m−3 for sediment, and zero for bubbles. The density of seawater is typically taken to be 1025 kg m−3. As such, other contributions to the force that governs particle dynamics, such as the added mass and inertial forces, are expected to be important. For neutrally buoyant particles, these additional contributions exactly cancel the contribution from viscous drag and are advected in the same manner as fluid particles. We develop a detailed analysis of how dense particles move on average and confirm the major aspects of our results numerically, before drawing our conclusions in section 5.

2. Mathematical model

We examine the motion of small, rigid, spherical particles of diameter d and density ρp, moving with velocity v in a flow u generated by waves. The force on particles moving in unsteady, inhomogeneous flows has been reviewed many times (see Hunt et al. 1995). The total force consists of a combination of drag and buoyancy, and because the density ratio ρp/ρ < O(1), the additional contributions from added mass and inertial forces are also important (see forces reviewed by Magnaudet and Eames 2000). When the Reynolds number, Rep = |vu|d/ν (where ν is the kinematic viscosity of water), based on the relative slip velocity of the particle to the fluid, is smaller than unity, the drag on the particles is viscously dominated and is proportional to the slip velocity. In combination, the particle dynamics are described, to leading order, by the combination of drag, buoyancy, added mass, and inertial forces, expressed as

 
formula

where Cm is the added mass coefficient and the value of Cm = 1/2 is prescribed for a spherical particle. The unity vector ŷ is directed vertically upward. Here, the particle response time is defined as

 
formula

to account for particle densities close to the density of water (ρ). The fall speed in a stagnant fluid is

 
formula

We define the Stokes number to be St = ωtp, based on the characteristic particle response time and the wave angular frequency. We apply a further restriction to the particle equation of motion, namely that the particles respond quickly to the changing flow; that is, the ratio of the particle response time to wave period T = 2π/ω is small, thus tp/T ≪ 1 or St ≪ 2π. This criterion ensures that history forces are negligible.

We consider how particles are transported by monochromatic waves of amplitude a, angular frequency ω, and wavenumber k, moving over an infinitely deep body of water. We restrict our attention to two-dimensional waves of steepness ak < 0.33 whose irrotational flow is described by

 
formula

where

 
formula

The wave height is ζ = a cos(kxωt) (see Lighthill 1978, p. 208). The speed of wave propagation is c = ω/k. For wavelengths greater than 2 cm, surface tension is not important and the dispersion relation is ω2 = gk. The analysis can be extended to include capillary forces by modifying the dispersion relation. The dimensionless ratio υ̃T = υT/c can be expressed in terms of St as υ̃T = (ρp/ρ − 1)St/(Cm + ρp/ρ), so that |υ̃T| < St.

Because we are studying the difference between the mean motion of small rigid particles and the mean motion of infinitesimal fluid particles, it is useful to recap Stokes drift velocity υSf (where the suffix f refers to fluid particles). The fluid particle Stokes drift velocity is

 
formula

The Stokes drift velocity can be calculated by following fluid particles (in the steady frame moving in the wave frame) using the method of Eames et al. (1994) or Longuet-Higgins (1986), but this requires Eulerian information about the periodic domain of the flow.

The particle response time increases with particle diameter. For the constraint St < 2π to apply, the particle diameter must be smaller than dm (i.e., d < dm), where the critical diameter is

 
formula

Figure 1a shows the critical diameter plotted for wavelengths less than 1 m and for density contrasts of ρp/ρ = 0, 1.5, and 2. For particles sedimenting beneath waves, the slip velocity is largely determined by the particle terminal velocity, and the particle Reynolds number is

 
formula

Figure 1b shows a contour plot of Rep for varying particle diameter and densities, with the limit Rep = 1 indicated. Also indicated in Fig. 1b are the approximate regions corresponding to crude oil, sediment, and larvae.

To study how particles move in an oscillatory flow field, we average the particle velocity over a wave period and define and as

 
formula

where v = (υx, υy) and X = (X, Y). The aim of the following calculations is to distinguish between the mean and oscillatory contributions to the mean particle motion.

3. Analytical calculation of particle dynamics

We employ a weakly inertial approximation St ≪ 1 to calculate how particles, released near the free surface, move on average. The weakly inertial approximation essentially converts Lagrangian information about v(t) into a field variable v(x, t) and provides a clear illustration of the differences between the particle and local fluid velocity. We have also explored an alternative approach using complex variables, and this yielded substantially the same results as the weakly inertial approximation.

a. Average particle velocity

For weakly inertial particles St ≪ 1 (e.g., Maxey 1987), the particle velocity can be calculated by a series expansion of (1). The particles’ sediment relative to the local fluid flow with an inertial correction, given by

 
formula

The acceleration of the fluid following the particles is

 
formula

Combining (10) and (11), we have

 
formula

The divergence of the particle velocity field is

 
formula

The divergence of the particle velocity field can be used as a diagnostic to determine whether inertial particle collisions are likely to occur (in the absence of Brownian motion) and has been used in the context of particles sedimenting in turbulence (Reeks 2005) and even in the production of warm rain in the atmosphere (Ghosh et al. 2005). From (13), the particle velocity field has a negative divergence for dense particles, indicating that the wave flow reduces the chance of inertial particle collision occurring for dense particles, but it increases the chance of collision for light particles.

From (12), weakly inertial neutrally buoyant particles (where ρp/ρ = 1 and υT = 0) move with the same velocity as fluid particles,

 
formula

In contrast, the neglect of added mass and inertial forces by Grinshpun et al. (2000) leads to

 
formula

Equation (15) is an incorrect result because it shows that neutrally buoyant particles are driven away from the free surface.

Substituting the flow field (4) into (12) gives the particle velocity

 
formula

where

 
formula

and the phase difference between the particle and fluid velocity is

 
formula

For St ≪ 1, dense particles move with a slightly faster orbital velocity than fluid particles moving at the same mean depth (𝒜/ak > 1), while light particles move slower than fluid particles (because 𝒜/ak < 1). This is because dense particles tend to execute a slightly larger circular trajectory than fluid particles (with the same mean depth), covering a farther distance in one wave period. On first inspection, we see that the pressure variation in the bulk of the fluid (p = −ρDu/Dt) is on average directed away from the free surface, which effectively increases the local buoyancy force on the dense particles increasing their fall speed. For small-amplitude waves, the vertical acceleration of the fluid is small, but near the crests of large-amplitude Stokes waves, it can be significantly larger than g (Longuet-Higgins 1986). The mean particle drift is calculated (to second order) using the first-order correction of the particle displacement to the particle velocity field. To first order, the particle position near the fixed point (X, Y) = (x0, y0) is

 
formula

To determine how the particles move on average, the particle velocity field is expanded about (x0, y0) to give

 
formula

(Batchelor 1967, p. 361) or

 
formula
 
formula

The particle velocity consists of orbital motion, with a mean horizontal drift velocity

 
formula

sedimentation with the terminal fall velocity, and a contribution due to the correlation between the sedimentation velocity and the orbital fluid flow. The mean horizontal drift velocity—which we refer to as the particle Stokes drift velocity—according to these estimates is of the same order as the fluid Stokes drift velocity υSf because 𝒜 ∼ ak.

On a longer time scale, we relax the constraint that x0, y0 are constant and include the horizontal drift and vertical settling (expressed as x0 = υSft, y0 = −υTt). Averaging (21) and (22), we obtain

 
formula
 
formula

For neutrally buoyant particles and fluid particles, the average velocity is

 
formula
 
formula

To leading order, neutrally buoyant particles and fluid particles are transported with the Stokes drift velocity [from (26)]. As a consequence of averaging the velocity over a wave period, there is a weak oscillatory component varying on the short time scale T and long time scale T/υSf because the particles do not return to their initial position. The long time scale arises from the time it takes for particles to be advected one wavelength. The weak oscillatory component decays much faster with distance y0 beneath the free surface (as e3ky0) than υSf (which decays as e2ky0). This is an important result for experimentalists—it means that the horizontal fluid Stokes drift velocity is not equal to the fluid particle velocity averaged over one wave period but rather requires Eulerian information about the wavelength.

For finite-sized particles, the average horizontal velocity is dominated by the particle Stokes drift velocity. The correlation between settling and orbital motion enters as an O(𝒜υT) contribution, while the averaging of the Stokes drift velocity introduces an O(𝒜3c) contribution, as described above, which dominates over the pressure contribution that scales as O(𝒜2υT). Thus, while the presence of a mean pressure gradient appears to be important in (16), the detailed analysis of the average particle velocity described above shows that it is not.

b. Mean particle transport

From the above analysis, the mean particle velocity is dominated by a horizontal particle Stokes drift velocity advecting the particle in the direction of the waves and the terminal fall velocity. The average particle position is expressed as

 
formula

where from (23), υSp = υSp(0)e2kY. Integrating (28), a particle released at the origin is transported to

 
formula

in time t. From (29), dense particles released at the free surface are permanently displaced a distance

 
formula

forward from their point of release. Light particles released far beneath the free surface are permanently transported a distance X (given above), before they rise to meet the free surface.

4. Numerical calculations

The equation of motion was integrated, and the positions (X, Y) at t = nT, denoted by Xn = X(nT), Yn = Y(nT), were calculated. The average particle velocity defined by (9) is calculated in terms of its Lagrangian displacements, through

 
formula

In the first instance, we consider tracer particles advected by the flow [Eqs. (13), (14)]. Figure 2a shows the trajectories of fluid particles being advected by the wavefield, with the typical looping trajectories beneath the wave (ak = 0.1). Figure 2b shows corresponding average velocity based on the fluid particle displacements. The integration was implemented using the Matlab routine ODE113 with an extremely high absolute tolerance of 10−15 to reduce accumulated numerical errors that would otherwise generate a spurious drift velocity. The horizontal and vertical fluid particle velocities were normalized by the fluid Stokes drift velocity at y = 0. The average drift velocity calculated from the particle displacement oscillates with a period T/(ak)2 ≈ 100T (for ak = 0.1), which corresponds to the time it takes for fluid particles to move one wavelength. This is captured by the analysis in section 3a. In Fig. 2a, the fluid particles are started at t = T/4 to ensure that Y = 0—this is only chosen here to demonstrate that Stokes drift velocity at y = 0 is recovered.

Figure 3 shows the trajectories of particles released beneath progressive waves at t = 0 for different particle densities (and therefore υT) for St = 0.02 and ak = 0.1 fixed. The orbital motion and settling of the particles are clearly observed, with the particles transported a finite distance forward. In Fig. 3a, dense particles are released at the origin—the densities of ρp/ρ ∼1.076 and 1.5 broadly correspond to the release of larvae and sediment. In Fig. 3b, light particles are released far beneath the waves (at a depth of 0.5λ) and correspond to crude oil and bubbles rising for ρp/ρ = 0.95 and 0, respectively. The numerical calculations were terminated in Fig. 3b when the particles first passed through the free surface. The relationship between particle diameter and wavelength (for a given value of ρp/ρ) is d = 18Stν/(Cm + ρp/ρ)(λ/2πg)1/2. Figure 4a shows the average horizontal particle velocity at t = nT, normalized by the fluid Stokes drift velocity at the surface, [υSf(0) = (ak)2c], and plotted as a function of the particle depth Yn. The horizontal particle drift velocity decays as the particle sediments, in the same manner as the Stokes drift velocity. The curves in Fig. 4a do not tend to unity (i) because of the presence of the third-order term O[(ak)3] (evaluated at t = nT) in (25) and (ii) because although the particles are released at the origin at t = 0, their average position over the first wave period is y = −λ(ak)/4π, sufficiently large so that the Stokes drift velocity is reduced. The difference between the mean vertical particle velocity and the fall velocity (nT) = υT is plotted in Fig. 4b. The mean velocity is dominated by the first three terms in (25), which arise from the averaging of the local flow velocity and the correlation between the fall velocity and the gradients of the fluid velocity. The predictions [Eqs. (24), (25)] quite accurately follow the numerical results of Fig. 4, confirming the detailed analysis.

The permanent-distance dense particles are transported forward from their point of release, are calculated numerically, and are compared to (30) in Fig. 5a. The accuracy of the analytical prediction of permanent-distance particles being transported increases as the sedimentation velocity decreases, as seen by the convergence of the numerical results to the analytical prediction (30), plotted as a dashed line. In Fig. 5b, the predicted permanent displacement (30) is plotted as a function of the particle fall velocity for wavelengths λ = 100, 200, and 300 cm and ak = 0.1. The permanent displacement increases significantly as the fall velocity decreases and with the square of the wave steepness (ak). Typical values of fall velocity, for larvae, are υT = 0.1–0.159 cm s−1 (Krug and Zimmer (2004)), which indicates that they are permanently transported a distance 100–600 cm and that this distance increases as the wave steepness increases.

5. Concluding remarks

We have studied analytically and numerically the motion of particles settling beneath progressive irrotational waves. We have shown that added mass and inertial forces must always be included in any description of the particle dynamics because the particle density is typically comparable to (or less than) water—their neglect leads to an analomous vertical drift.

A detailed analysis of the various contributions to the particle velocity averaged over a wave period shows that to leading order, particles are transported with a horizontal particle Stokes drift velocity and sediment with their terminal fall velocity. The particle Stokes drift velocity is slightly smaller than the fluid Stokes drift velocity for dense particles. Weakly inertial, neutrally buoyant particles move the same as fluid particles. Although the oscillatory motion beneath waves generates a near-surface vertical pressure gradient that slightly enhances sedimentation, this is masked (for waves with shallow slopes) by the larger contributions from the correlation between sedimentation and the orbital motion of the fluid and by averaging the orbital motion. Dense particles released near the surface are permanently transported a distance, which scales as O[(ak)2c/T].

The analysis presented is built on a simplified description of the force acting on the particles and the flow field. There are instances when additional forces such as a shear-induced lift may be important, for example, within the viscous boundary layers generated at the free surface. For large particles or long fast-moving waves, Rep and St are no longer small and corrections including a nonlinear drag and history force must be included. When the wave steepness increases, both the Stokes drift velocity and vertical pressure gradient increase significantly (Longuet-Higgins 1986) and are thus likewise expected to increase the mean drift motion of the particles.

Our main conclusion is that to leading order, the average motion of dense particles can be estimated from the particle Stokes drift velocity and their settling velocity, providing a practical method for estimating the influence of waves in transporting particles. Our future research will be to study these processes in the laboratory.

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Footnotes

Corresponding author address: Ian Eames, University College London, Torrington Place, London WC1E 7JE, United Kingdom