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The purpose of this note is to present an enhancement to a Maxey–Riley theory proposed in recent years for the dynamics of inertial particles on the ocean surface [Beron-Vera et al., “Building a Maxey–Riley framework for surface ocean inertial particle dynamics,” Phys. Fluids 31, 096602 (2019)]. This updated model removes constraints on the reserve buoyancy, defined as the fraction of the particle volume above the ocean surface. The refinement results in an equation that correctly describes both the neutrally buoyant and fully buoyant particle scenarios.
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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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We investigate how turbulence in liquid affects the rising speed of gas bubbles within the inertial range. Experimentally, we employ stereoscopic tracking of bubbles rising through water turbulence created by the convergence of turbulent jets and characterized with particle image velocimetry performed throughout the measurement volume. We use the spatially varying, time-averaged mean water velocity field to consider the physically relevant bubble slip velocity relative to the mean flow. Over a range of bubble sizes within the inertial range, we find that the bubble mean rise velocity $$\left \langle v_z \right \rangle$$ decreases with the intensity of the turbulence as characterized by its root-mean-square fluctuation velocity, $u'$ . Non-dimensionalized by the quiescent rise velocity $$v_{q}$$ , the average rise speed follows $$\left \langle v_z \right \rangle /v_{q}\propto 1/{\textit {Fr}}$$ at high $${\textit {Fr}}$$ , where $${\textit {Fr}}=u'/\sqrt {dg}$$ is a Froude number comparing the intensity of the turbulence to the bubble buoyancy, with $$d$$ the bubble diameter and $$g$$ the acceleration due to gravity. We complement these results by performing numerical integration of the Maxey–Riley equation for a point bubble experiencing nonlinear drag in three-dimensional, homogeneous and isotropic turbulence. These simulations reproduce the slowdown observed experimentally, and show that the mean magnitude of the slip velocity is proportional to the large-scale fluctuations of the flow velocity. Combining the numerical estimate of the slip velocity magnitude with a simple theoretical model, we show that the scaling $$\left \langle v_z \right \rangle /v_{q}\propto 1/{\textit {Fr}}$$ originates from a combination of the nonlinear drag and the nearly isotropic behaviour of the slip velocity at large $${\textit {Fr}}$$ that drastically reduces the mean rise speed.more » « less
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A recent Maxey–Riley theory for Sargassum raft motion, which models a raft as a network of elastically interacting finite size, buoyant particles, predicts the carrying flow velocity to be given by the weighted sum of the water and air velocities (1−α)v+αw. The theory provides a closed formula for parameter α, referred to as windage, depending on the water-to-particle-density ratio or buoyancy (δ). From a series of laboratory experiments in an air–water stream flume facility under controlled conditions, we estimate α ranging from 0.02% to 0.96%. On average, our windage estimates can be up to nine times smaller than that considered in conventional Sargassum raft transport modeling, wherein it is customary to add a fraction of w to v chosen in an ad hoc piecemeal manner. Using the formula provided by the Maxey–Riley theory, we estimate δ ranging from 1.00 to 1.49. This is consistent with direct δ measurements, ranging from 0.9 to 1.25, which provide support for our α estimation.more » « less
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.48550/arXiv.2006.16577
