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Abstract Because of their buoyancy, rigidity, and finite size, inertial particles do not obey the same dynamics as fluid parcels. The motion of small spherical particles in a fluid flow is described by the Maxey–Riley equations and depends nonlinearly on the velocity of the fluid in which the particles are immersed. Fluid velocities in the ocean often have a strong small-scale turbulent component which is difficult to observe or model, presenting a challenge to predicting the evolution of distributions of inertial particles in the ocean. To overcome this challenge, we assume that the turbulent velocity imposes a random force on particles and consider a stochastic analog of the Maxey–Riley equations. By performing a perturbation analysis of the stochastic Maxey–Riley equations, we obtain a simple and accurate partial differential equation for the spatial distribution of particles. The equation is of the advection–diffusion type and handles the uncertainty introduced by unresolved turbulent flow features. In several numerical test cases, distributions of particles obtained by solving the newly derived equation compare favorably with distributions obtained from Monte Carlo simulations of individual particle trajectories and with theoretical predictions. The advection–diffusion form of our newly derived equation is amenable to inclusion within many existing ocean circulation models. Significance StatementWe introduce a new model for describing spatial distributions of small rigid objects, such as plastic debris, in the ocean. The model takes into account the effects of finite particle size and particle buoyancy, which cause particle trajectories to differ from fluid parcel trajectories. Our model also represents small-scale turbulence stochastically.
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On the Maxey-Riley equation of motion and its extension to high Reynolds numbers
The inertial response of a particle to turbulent flows is a problem of relevance to a wide range of environmental and engineering problems. The equation most often used to describe the force balance is the Maxey-Riley equation, which includes in addition to buoyancy and steady drag forces, an unsteady Basset drag force related to past particle acceleration. Here we provide a historical review of how the Maxey-Riley equation was developed and how it is only suited for studies where the Reynolds number is less than unity. Revisiting the innovative mathematical methods employed by Basset (1888), we introduce an alternative formulation for the unsteady drag for application to a broader range of particle motions. While the Basset unsteady drag is negligible at higher Reynolds numbers, the revised unsteady drag is not.
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- Award ID(s):
- 1841870
- PAR ID:
- 10323192
- Date Published:
- Journal Name:
- ArXivorg
- ISSN:
- 2331-8422
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.48550/arXiv.2006.16577
