Martin R. Maxey

Equation of motion for a small rigid sphere in a nonuniform flow

The forces on a small rigid sphere in a nonuniform flow are considered from first prinicples in order to resolve the errors in Tchen’s equation and the subsequent modified versions that have since appeared. Forces from the undisturbed flow and the disturbance flow created by the presence of the sphere are treated separately. Proper account is taken of the effect of spatial variations of the undisturbed flow on both forces. In particular the appropriate Faxen correction for unsteady Stokes flow is derived and included as part of the consistent approximation for the equation of motion.

Settling velocity and concentration distribution of heavy particles in homogeneous isotropic turbulence

The average settling velocity in homogeneous turbulence of a small rigid spherical particle, subject to a Stokes drag force, has been shown to differ from that in still fluid owing to a bias from the particle inertia (Maxey 1987). Previous numerical results for particles in a random flow field, where the flow dynamics were not considered, showed an increase in the average settling velocity. Direct numerical simulations of the motion of heavy particles in isotropic homogeneous turbulence have been performed where the flow dynamics are included. These show that a significant increase in the average settling velocity can occur for particles with inertial response time and still-fluid terminal velocity comparable to the Kolmogorov scales of the turbulence. This increase may be as much as 50% of the terminal velocity, which is much larger than was previously found. The concentration field of the heavy particles, obtained from direct numerical simulations, shows the importance of the inertial bias with particles tending to collect in elongated sheets on the peripheries of local vortical structures. This is coupled then to a preferential sweeping of the particles in downward moving fluid. Again the importance of Kolmogorov scaling to these processes is demonstrated. Finally, some consideration is given to larger particles that are subject to a nonlinear drag force where it is found that the nonlinearity reduces the net increase in settling velocity.

The gravitational settling of aerosol particles in homogeneous turbulence and random flow fields

The average settling velocity in homogeneous turbulence of a small rigid spherical particle, subject to a Stokes drag force, is shown to depend on the particle inertia and the free-fall terminal velocity in still fluid. With no inertia the particle settles on average at the same rate as in still fluid, assuming there is no mean flow. Particle inertia produces a bias in each trajectory towards regions of high strain rate or low vorticity, which affects the mean settling velocity. Results from a Gaussian random velocity field show that this produces an increased settling velocity.

Small-scale features of vorticity and passive scalar fields in homogeneous isotropic turbulence

Small‐scale structures of the vorticity and passive scalar fields have been examined by means of direct numerical simulations of homogeneous isotropic turbulence with 963 grid points and Rλ≊60. Both statistical and visual techniques have been used to examine the structure of certain quantities from the evolution equations for enstrophy and the scalar gradient. Tubelike regions of intense enstrophy contain large positive and sometimes large negative enstrophy production, and mostly moderate‐valued energy dissipation regions surround these tubes. The most intense regions of the scalar gradient are dissociated from the vortex tubes, and occur as large flat sheets. Within these sheets the scalar gradient production is large, the energy dissipation is small, and in their vicinity only moderate‐valued sheetlike enstrophy regions exist. The statistical techniques show that although activity in these intense regions is strong, on a volume normalized basis, by far the largest contributions to the terms in the evol...

The motion of small spherical particles in a cellular flow field

In an earlier paper, Maxey and Corrsin [J. Atmos. Sci. 43, 1112 (1986)] studied the motion of small aerosol particles settling under gravity through an infinite, periodic, cellular flow field subject to the effects of a Stokes drag force and inertia of the particles. Particle inertia was shown to have an important influence on the motion: No permanent suspension in the flow occurred, particles generally settled more rapidly than in still fluid, and the particle paths merged into isolated asymptotic trajectories. This study is continued for particles that are not necessarily much denser than the surrounding fluid but vary in density. Two basic responses are identified: an aerosol response for particles denser than the fluid, similar to that mentioned, and a bubble response for particles less dense. For both, particle accumulation is still a recurring feature. Results of numerical simulations are discussed, together with the stability of equilibrium points and the role of particle or fluid inertia.

Localized force representations for particles sedimenting in Stokes flow

Abstract A finite-valued force multipole expansion is developed to describe the dynamics of spherical particles sedimenting in Stokes flow. The lowest level of the method is a simple force coupling procedure to represent the dynamic coupling between the particle phase and the fluid phase. The length scale associated with the finite force envelope provides an additional parameter that when matched to the particle size gives consistent results for the particle velocity. Sedimentation velocities for particles in a cubic lattice are predicted with good accuracy up to volume fractions of 20% by this approximate method. The calculated kinetic energy dissipation by viscosity is correctly balanced with the release of potential energy. As a full expansion procedure it is possible to calculate exactly the flow surrounding the particles, using standard numerical procedures such as fast Fourier transforms in periodic domains. Examples for the settling of particle pairs and random suspension are also given.

Gravitational Settling of Aerosol Particles in Randomly Oriented Cellular Flow Fields

Abstract Statistics have been computed for the motion Of small particles settling under gravity in an ensemble of randomly oriented, periodic, cellular flow fields that are steady in time. The particles are small, spherical, and subject to a quasi-steady Stokes drag force from the flow. In the absence of particle inertia, a fraction of the particles may be suspended indefinitely, but inertia, however weak eventually causes all particles to settle out at a rate that over most parametric ranges is faster than in still fluid. More surprisingly, particles with small free fall velocity and weak inertia show a strong tendency to collect along isolated paths. Reducing inertia does not greatly alter this process, but only delays it.

Methods for evaluating fluid velocities in spectral simulations of turbulence

In studying particle motion with spectral similations of turbulence, it is necessary to evaluate the local fluid velocity at the instantaneous particle position. In general, this point does not coincide with a mesh point. A direct summation of the spectral Fourier series is the most accurate method but is very time consuming. Various approximate methods are tested and comparisons made of both their accuracy and the computational effort required.

Force-coupling method for particulate two-phase flow: stokes flow

In this paper we describe a force-coupling method for particle dynamics in fluid flows. The general principles of the model are described and it is tested on three different Stokes flow problems; a single isolated sphere, a pair of otherwise isolated spheres, and a single sphere in a channel. For sphere to sphere or sphere to wall distances larger than 1/4 of the sphere radius the force-coupling results compared well with analytical and accurate numerical values. For smaller distances the results agree qualitatively, but lubrication effects are not included and this leads to a quantitative discrepancy.

Numerical simulation of turbulent drag reduction using micro-bubbles

While turbulent drag reduction through the injection of micro-bubbles into a turbulent boundary layer is well established in experiments, there is a lack of corresponding supporting evidence from direct numerical simulations. Here we report on a series of numerical simulations of small bubbles seeded in a turbulent channel flow at average volume fractions of up to 8%. These results show that even for relatively large bubbles, an initial transient drag reduction can occur as bubbles disperse into the flow. Relatively small spherical bubbles will produce a sustained level of drag reduction over time.