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Andrew Bragg

Associate Professor of Civil and Environmental Engineering
Civil and Environmental Engineering
2459 CIEMAS, Durham, NC 27708
Office hours By appointment.  

Selected Publications


Gas Transfer Across Air-Water Interfaces in Inland Waters: From Micro-Eddies to Super-Statistics

Journal Article Water Resources Research · November 1, 2024 In inland water covering lakes, reservoirs, and ponds, the gas exchange of slightly soluble gases such as carbon dioxide, dimethyl sulfide, methane, or oxygen across a clean and nearly flat air-water interface is routinely described using a water-side mean ... Full text Cite

Understanding the effect of Prandtl number on momentum and scalar mixing rates in neutral and stably stratified flows using gradient field dynamics

Journal Article Journal of Fluid Mechanics · August 27, 2024 Recently, direct numerical simulations (DNS) of stably stratified turbulence have shown that as the Prandtl number is increased from 1 to 7, the mean turbulent potential energy dissipation rate (TPE-DR) drops dramatically, while the mean turbulent kinetic ... Full text Cite

Heterogeneous Land-Surface Effects on TKE and Cloud Formation: Statistical Insights From LES Cases

Journal Article Journal of Geophysical Research: Atmospheres · June 28, 2024 This manuscript investigates the impact of land-surface heterogeneity on atmospheric processes by comparing 92 large-eddy simulation cases over the Southern Great Plains, leveraging high-resolution spatially heterogeneous and homogeneous land-surface field ... Full text Cite

Investigating the parametric dependence of the impact of two-way coupling on inertial particle settling in turbulence

Journal Article Journal of Fluid Mechanics · May 16, 2024 Tom et al. (J. Fluid Mech., vol. 947, 2022, p. A7) investigated the impact of two-way coupling (2WC) on particle settling velocities in turbulence. For the limited parameter choices explored, it was found that (i) 2WC substantially enhances particle settli ... Full text Cite

A Two-Column Model Parameterization for Subgrid Surface Heterogeneity Driven Circulations

Journal Article Journal of Advances in Modeling Earth Systems · May 1, 2024 Earth system models currently struggle to account for the complex effects that land surface heterogeneity can have on land-atmosphere interactions. There have been attempts to include the impact of this heterogeneity on the atmosphere, but they ignore the ... Full text Cite

Estimation of Particle Dispersion Characteristics in a Fluidized Bed with the Binary Mixture of Geldart A and B Types Using the Optical Fiber Probe Method

Journal Article Industrial and Engineering Chemistry Research · May 1, 2024 This study proposes a novel method for experimentally assessing particle dispersion characteristics within a binary-mixture fluidized bed. Verification experiments were conducted, demonstrating satisfactory agreement between the measured results obtained f ... Full text Cite

A Reinterpretation of Phenomenological Modeling Approaches for Lagrangian Particles Settling in a Turbulent Boundary Layer

Journal Article Boundary-Layer Meteorology · April 1, 2024 It has long been known that under the right circumstances, inertial particles (such as sand, dust, pollen, or water droplets) settling through the atmospheric boundary layer can experience a net enhancement in their average settling velocity due to their i ... Full text Cite

Detailed characterization of extreme clustering at near-contact scales in isotropic turbulence

Journal Article Journal of Fluid Mechanics · March 7, 2024 Recent measurements of inertial particles in isotropic turbulence (Hammond & Meng, J. Fluid Mech., vol. 921, 2021, A16) revealed surprising extreme clustering of particles at near-contact separations (r), whereby the radial distribution function, g(r), gro ... Full text Cite

Reduced Sediment Settling in Turbulent Flows Due To Basset History and Virtual Mass Effects

Journal Article Geophysical Research Letters · November 28, 2023 The behavior of suspended particles in turbulent flows is a recalcitrant problem spanning wide-ranging fields including geomorphology, hydrology, and dispersion of particulate matter in the atmosphere. One key mechanism underlying particle suspension is th ... Full text Cite

Fate of bubble clusters rising in a quiescent liquid

Journal Article Journal of Fluid Mechanics · October 16, 2023 We use experiments to study the evolution of bubble clusters in a swarm of freely rising, deformable bubbles. A new machine learning-Aided algorithm allows us to identify and track bubbles in clusters and measure the cluster lifetimes. The results indicate ... Full text Cite

A Note on Modeling the Effects of Surfactants on Bubble-Induced Turbulence

Journal Article Chemical Engineering and Technology · September 1, 2023 Recent experimental results reveal that with rising concentration of surfactants, the generated bubble-induced turbulence (BIT) increases as well, although the bubbles rise slower. Herein, it is shown that this phenomenon has not be considered so far in th ... Full text Cite

Effects of surfactants on bubble-induced turbulence

Journal Article Journal of Fluid Mechanics · August 30, 2023 We use experiments to explore the effect of surfactants on bubble-induced turbulence (BIT) at different scales, considering how the bubbles affect the flow kinetic energy, anisotropy and extreme events. To this end, high-resolution particle shadow velocime ... Full text Cite

Logarithmic scaling of higher-order temperature moments in the atmospheric surface layer

Journal Article International Journal of Heat and Fluid Flow · August 1, 2023 A generalized logarithmic law for high-order moments of homogeneous passive scalars is proposed for turbulent boundary layers. This law is analogous to the generalized log law that has been proposed for high-order moments of the turbulent longitudinal velo ... Full text Cite

Lagrangian model for passive scalar gradients in turbulence

Journal Article Journal of Fluid Mechanics · June 5, 2023 The equation for the fluid velocity gradient along a Lagrangian trajectory immediately follows from the Navier-Stokes equation. However, such an equation involves two terms that cannot be determined from the velocity gradient along the chosen Lagrangian pa ... Full text Cite

Stochastic model for the residence time of solid particles in turbulent Rayleigh-Bénard flow

Journal Article Physical Review Fluids · February 1, 2023 The Pi Chamber generates moist turbulent Rayleigh-Bénard flow in order to replicate steady-state cloud conditions. We take inspiration from this setup and consider a particle-laden, convectively driven turbulent flow using direct numerical simulation. The ... Full text Cite

Asymptotic closure model for inertial particle transport in turbulent boundary layers

Journal Article Physical Review Fluids · January 1, 2023 Transport equations for heavy inertial particles in turbulent boundary layers may be derived from an underlying phase-space probability density function (PDF) equation. These equations, however, are unclosed, and the standard closure approach is to use a q ... Full text Cite

The effect of tilt on turbulent thermal convection for a heated soap bubble

Journal Article Physics of Fluids · October 1, 2022 We use direct numerical simulation (DNS) to explore the effect of tilt on two-dimensional turbulent thermal convection on a half-soap bubble that is heated at its equator. In the DNS, the bubble is tilted by an angle δ ∈ [0 °, 90 °], the Rayleigh number is ... Full text Cite

How does two-way coupling modify particle settling and the role of multiscale preferential sweeping?

Journal Article Journal of Fluid Mechanics · September 25, 2022 For one-way coupled (1WC) flows, Tom & Bragg (J. Fluid Mech., vol. 871, 2019, pp. 244-270) advanced the analysis of Maxey (J. Fluid Mech., vol. 174, 1987, pp. 441-465), which applied to weakly inertial particles, to particles of arbitrary inertia, and the ... Full text Cite

Analysis of scale-dependent kinetic and potential energy in sheared, stably stratified turbulence

Journal Article Journal of Fluid Mechanics · September 10, 2022 Budgets of turbulent kinetic energy (TKE) and turbulent potential energy (TPE) at different scales in sheared, stably stratified turbulence are analysed using a filtering approach. Competing effects in the flow are considered, along with the physical mecha ... Full text Cite

Examining Parameterizations of Potential Temperature Variance Across Varied Landscapes for Use in Earth System Models

Journal Article Journal of Geophysical Research: Atmospheres · April 27, 2022 Earth system models (ESMs) and mesoscale models have come to employ increasingly complex parameterization schemes for the atmospheric boundary layer, requiring surface boundary conditions for numerous higher order turbulence statistics. Of particular inter ... Full text Cite

An experimental study on the multiscale properties of turbulence in bubble-laden flows

Journal Article Journal of Fluid Mechanics · April 10, 2022 The properties of bubble-laden turbulent flows at different scales are investigated experimentally, focusing on the flow kinetic energy, energy transfer and extreme events. The experiments employed particle shadow velocimetry measurements to measure the fl ... Full text Cite

A Co-Spectral Budget Model Links Turbulent Eddies to Suspended Sediment Concentration in Channel Flows

Journal Article Water Resources Research · March 1, 2022 The vertical distribution of suspended sediment concentration remains a subject of active research given its relevance to a plethora of problems in hydraulics, hydrology, ecology, and water quality control. Many of the classical theories developed over the ... Full text Cite

Hydrodynamic interactions and extreme particle clustering in turbulence

Journal Article Journal of Fluid Mechanics · February 25, 2022 Expanding recent observations by Hammond & Meng (J. Fluid Mech., vol. 921, 2021, A16), we present a range of detailed experimental data of the radial distribution function (r.d.f.) of inertial particles in isotropic turbulence for different Stokes number, ... Full text Cite

Dimension reduced turbulent flow data from deep vector quantisers

Journal Article Journal of Turbulence · January 1, 2022 Analysing large-scale data from simulations of turbulent flows is memory intensive, requiring significant resources. This major challenge highlights the need for data compression techniques. In this study, we apply a physics-informed Deep Learning techniqu ... Full text Cite

A Physics-Informed Vector Quantized Autoencoder for Data Compression of Turbulent Flow

Conference Data Compression Conference Proceedings · January 1, 2022 Analyzing large-scale data from simulations of turbulent flows is memory intensive, requiring significant resources. This major challenge highlights the need for data compression techniques. In this study, we apply a physics-informed Deep Learning techniqu ... Full text Cite

Settling strongly modifies particle concentrations in wall-bounded turbulent flows even when the settling parameter is asymptotically small

Journal Article Physical Review Fluids · December 1, 2021 We explore the role of gravitational settling on inertial particle concentrations in a wall-bounded turbulent flow. While it may be thought that settling can be ignored when the settling parameter Svvs/uτ is small (vs is Stokes settling velocity, uτ is flu ... Full text Cite

Scale-dependent anisotropy, energy transfer and intermittency in bubble-laden turbulent flows

Journal Article Journal of Fluid Mechanics · November 25, 2021 Data from direct numerical simulations of disperse bubbly flows in a vertical channel are used to study the effect of the bubbles on the carrier-phase turbulence. We developed a new method, based on an extension of the barycentric map approach, that allows ... Full text Cite

Assessment of the validity of a log-law for wall-bounded turbulent bubbly flows

Journal Article International Journal of Heat and Fluid Flow · October 1, 2021 There has been considerable discussion in recent years concerning whether a log-law exists for wall-bounded, turbulent bubbly flows. Previous studies have argued for the existence of such a log-law, with a modified von Kármán constant, and this is used in ... Full text Cite

Semi-Coupling of a Field-Scale Resolving Land-Surface Model and WRF-LES to Investigate the Influence of Land-Surface Heterogeneity on Cloud Development

Journal Article Journal of Advances in Modeling Earth Systems · October 1, 2021 Contemporary Earth system models mostly ignore the sub-grid scale (SGS) heterogeneous coupling between the land surface and atmosphere. To aid in the development of coupled land and atmosphere SGS parameterizations for global models, we present a study of ... Full text Cite

Mechanisms governing the settling velocities and spatial distributions of inertial particles in wall-bounded turbulence

Journal Article Physical Review Fluids · June 1, 2021 We use theory and direct numerical simulations (DNSs) coupled with point particles to explore the average vertical velocities and spatial distributions of inertial particles settling in a wall-bounded turbulent flow. The theory is based on the exact phase- ... Full text Cite

Exploring the turbulent velocity gradients at different scales from the perspective of the strain-rate eigenframe

Journal Article Journal of Fluid Mechanics · January 1, 2021 Expressing the evolution equations for the filtered velocity gradient tensor (FVGT) in the strain-rate eigenframe provides an insightful way to disentangle and understand various processes such as strain self-amplification, vortex stretching and tilting, a ... Full text Cite

Turbulence and heat transfer on a rotating, heated half soap bubble

Journal Article Journal of Fluid Mechanics · January 1, 2021 We use direct numerical simulations to study the two-dimensional flow of a rotating, half soap bubble that is heated at its equator. The heating produces buoyancy and rotation generates Coriolis forces in the fluid. However, owing to the curved surface of ... Full text Cite

The Conservative Pressure Hessian and the Free Fluid Particle Model

Conference Springer Proceedings in Physics · January 1, 2021 In this work we aim to understand the interplay between deformation and energy of a fluid particle in a turbulent flow, focusing on the role of the pressure Hessian. A new decomposition of the Hessian is proposed, defining a conservative Hessian which cont ... Full text Cite

Fluctuation theorem and extended thermodynamics of turbulence.

Journal Article Proceedings. Mathematical, physical, and engineering sciences · November 2020 Turbulent flows are out-of-equilibrium because the energy supply at large scales and its dissipation by viscosity at small scales create a net transfer of energy among all scales. This energy cascade is modelled by approximating the spectral energy balance ... Full text Cite

Scalewise Return to Isotropy in Stratified Boundary Layer Flows

Journal Article Journal of Geophysical Research: Atmospheres · August 27, 2020 Anisotropic turbulence is ubiquitous in atmospheric and oceanic boundary layers due to differences in energy injection mechanisms. Unlike mechanical production that injects energy in the streamwise velocity component, buoyancy affects only the vertical vel ... Full text Cite

Explicit algebraic relation for calculating Reynolds normal stresses in flows dominated by bubble-induced turbulence

Journal Article Physical Review Fluids · August 1, 2020 Two new algebraic turbulence models for flows dominated by bubble-induced turbulence (BIT) are presented. They combine different elements of existing models that are considered superior to their alternatives. Both models focus on the core region of a chann ... Full text Cite

Local analysis of the clustering, velocities, and accelerations of particles settling in turbulence

Journal Article Physical Review Fluids · March 1, 2020 Using three-dimensional Voronoi analysis, we explore the local dynamics of small, settling, inertial particles in isotropic turbulence using direct numerical simulations. We independently vary the Taylor Reynolds number Reλ[90,398], Froude number Fraη/g[0. ... Full text Cite

Is vortex stretching the main cause of the turbulent energy cascade?

Journal Article Journal of Fluid Mechanics · January 1, 2020 In three-dimensional turbulence there is on average a cascade of kinetic energy from the largest to the smallest scales of the flow. While the dominant idea is that the cascade occurs through the process of vortex stretching, evidence for this is debated. ... Full text Cite

Symmetry transformation and dimensionality reduction of the anisotropic pressure Hessian

Journal Article Journal of Fluid Mechanics · January 1, 2020 The dynamics of the velocity gradient tensor in turbulence is governed in part by the anisotropic pressure Hessian, which is a non-local functional of the velocity gradient field. This anisotropic pressure Hessian plays a key dynamical role, for example in ... Full text Cite

The influence of reynolds and froude number on the local distribution of settling, inertial particles in turbulence

Conference AIAA Scitech 2020 Forum · January 1, 2020 Particle-laden turbulent flows appear in a diverse range of engineering systems and natural phenomena and have been widely investigated in many theoretical, numerical and experimental studies. In this study, we employ Direct Numerical Simulations (DNS) to ... Full text Cite

Multiscale fluid-particle thermal interaction in isotropic turbulence

Journal Article Journal of Fluid Mechanics · December 25, 2019 We use direct numerical simulations to investigate the interaction between the temperature field of a fluid and the temperature of small particles suspended in the flow, employing both one- and two-way thermal coupling, in a statistically stationary, isotr ... Full text Cite

Corrigendum: Multiscale preferential sweeping of particles settling in turbulence (Journal of Fluid Mechanics (2019) 871 (244-270) DOI: 10.1017/jfm.2019.337)

Journal Article Journal of Fluid Mechanics · August 10, 2019 In Tom & Bragg (2019) the ordinate axis label of figure 5 was incorrect. The correct version of this figure is shown below. (Formula Present). ... Full text Cite

Multiscale preferential sweeping of particles settling in turbulence

Journal Article Journal of Fluid Mechanics · July 25, 2019 In a seminal article, Maxey (J. Fluid Mech., vol. 174, 1987, pp. 441-465) presented a theoretical analysis showing that enhanced particle settling speeds in turbulence occur through the preferential sweeping mechanism, which depends on the preferential sam ... Full text Cite

Influence of Reynolds number on the motion of settling, bidisperse inertial particles in turbulence

Journal Article Physical Review Fluids · May 1, 2019 Using direct numerical simulations, we examine the effects of the Taylor Reynolds number, Rλu′λ/ν (where u′, λ, and ν denote the fluid root-mean-squared velocity, the Taylor microscale and the fluid kinematic viscosity, respectively), and Froude number, Fr ... Full text Cite

Enhanced and suppressed multiscale dispersion of bidisperse inertial particles due to gravity

Journal Article Physical Review Fluids · March 1, 2019 Using direct numerical simulation (DNS), we investigate how gravity modifies the multiscale dispersion of bidisperse inertial particles in isotropic turbulence. The DNS has a Taylor Reynolds number Rλ=398, and we simulate Stokes numbers (based on the Kolmo ... Full text Cite

Modulation of fluid temperature fluctuations by inertial particles in turbulence

Conference Springer Proceedings in Physics · January 1, 2019 We investigate the effect of the thermal two-way coupling between the fluid temperature field and point-like particles on the temperature statistics in homogeneous and isotropic steady turbulence by means of direct numerical simulations (DNS). Results show ... Full text Cite

Extremes, intermittency, and time directionality of atmospheric turbulence at the crossover from production to inertial scales

Journal Article Physical Review Fluids · September 1, 2018 The effects of mechanical generation of turbulent kinetic energy and buoyancy forces on the statistics of air temperature and velocity increments are experimentally investigated at the crossover from production to inertial range scales. The ratio of an app ... Full text Cite

Small-scale dynamics of settling, bidisperse particles in turbulence

Journal Article Journal of Fluid Mechanics · March 25, 2018 Mixing and collisions of inertial particles at the small scales of turbulence can be investigated by considering how pairs of particles move relative to each other. In real problems the two particles will have different sizes, i.e. they are bidisperse, and ... Full text Cite

Effects of Reynolds number and Stokes number on particle-pair relative velocity in isotropic turbulence: A systematic experimental study

Journal Article Journal of Fluid Mechanics · March 25, 2018 The effects of Reynolds number and Stokes number on particle-pair relative velocity (RV) are investigated systematically using a recently developed planar four-frame particle tracking technique in a novel homogeneous and isotropic turbulence chamber. We co ... Full text Cite

On the theory of drainage area for regular and non-regular points.

Journal Article Proceedings. Mathematical, physical, and engineering sciences · March 2018 The drainage area is an important, non-local property of a landscape, which controls surface and subsurface hydrological fluxes. Its role in numerous ecohydrological and geomorphological applications has given rise to several numerical methods for its comp ... Full text Cite

Fluid particles only separate exponentially in the dissipation range of turbulence after extremely long times

Journal Article Physical Review Fluids · March 1, 2018 In this paper, we consider how the statistical moments of the separation between two fluid particles grow with time when their separation lies in the dissipation range of turbulence. In this range, the fluid velocity field varies smoothly and the relative ... Full text Cite

Particle-pair relative velocity measurement in high-Reynolds-number homogeneous and isotropic turbulence using 4-frame particle tracking velocimetry

Journal Article Experiments in Fluids · February 1, 2018 The radial relative velocity (RV) between particles suspended in turbulent flow plays a critical role in droplet collision and growth. We present a simple and accurate approach to RV measurement in isotropic turbulence—planar 4-frame particle tracking velo ... Full text Cite

Is the kinetic equation for turbulent gas-particle flows ill posed?

Journal Article Physical review. E · February 2018 This paper is about the kinetic equation for gas-particle flows, in particular its well-posedness and realizability and its relationship to the generalized Langevin model (GLM) probability density function (PDF) equation. Previous analyses, e.g. [J.-P. Min ... Full text Cite

Irreversibility inversions in two-dimensional turbulence

Journal Article Physical Review Fluids · February 1, 2018 In this paper, we consider a recent theoretical prediction [Bragg et al., Phys. Fluids 28, 013305 (2016)PHFLE61070-663110.1063/1.4939694] that for inertial particles in two-dimensional (2D) turbulence, the nature of the irreversibility of the particle-pair ... Full text Cite

Analysis of the forward and backward in time pair-separation probability density functions for inertial particles in isotropic turbulence

Journal Article Journal of Fluid Mechanics · November 10, 2017 In this paper we investigate, using theory and direct numerical simulations (DNS), the forward in time (FIT) and backward in time (BIT) probability density functions (PDFs) of the separation of inertial particle pairs in isotropic turbulence. In agreement ... Full text Cite

Developments and difficulties in predicting the relative velocities of inertial particles at the small-scales of turbulence

Journal Article Physics of Fluids · April 1, 2017 In this paper, we consider the development of theoretical models to predict the relative velocities of inertial particles in isotropic turbulence. In particular, we use our recently developed theory for the backward-in-time (BIT) relative dispersion of ine ... Full text Cite

Model of non-stationary, inhomogeneous turbulence

Journal Article Theoretical and Computational Fluid Dynamics · February 1, 2017 We compare results from a spectral model for non-stationary, inhomogeneous turbulence (Besnard et al. in Theor Comp Fluid Dyn 8:1–35, 1996) with direct numerical simulation (DNS) data of a shear-free mixing layer (SFML) (Tordella et al. in Phys Rev E 77:01 ... Full text Cite

The effect of Reynolds number on inertial particle dynamics in isotropic turbulence. Part 2. Simulations with gravitational effects

Journal Article Journal of Fluid Mechanics · June 10, 2016 In Part 1 of this study (Ireland et al., J. Fluid Mech., vol. 796, 2016, pp. 617-658), we analysed the motion of inertial particles in isotropic turbulence in the absence of gravity using direct numerical simulation (DNS). Here, in Part 2, we introduce gra ... Full text Cite

Forward and backward in time dispersion of fluid and inertial particles in isotropic turbulence

Journal Article Physics of Fluids · January 1, 2016 In this paper, we investigate both theoretically and numerically the Forward-In-Time (FIT) and Backward-In-Time (BIT) dispersion of fluid and inertial particle-pairs in isotropic turbulence. Fluid particles are known to separate faster BIT than FIT in thre ... Full text Cite

The effect of Reynolds number on inertial particle dynamics in isotropic turbulence. Part 1. Simulations without gravitational effects

Journal Article Journal of Fluid Mechanics · January 1, 2016 In this study, we analyse the statistics of both individual inertial particles and inertial particle pairs in direct numerical simulations of homogeneous isotropic turbulence in the absence of gravity. The effect of the Taylor microscale Reynolds number, R ... Full text Cite

On the relationship between the non-local clustering mechanism and preferential concentration

Journal Article Journal of Fluid Mechanics · September 3, 2015 'Preferential concentration' (Squires & Eaton, Phys. Fluids, vol. A3, 1991, pp. 1169-1178) refers to the clustering of inertial particles in the high strain, low-rotation regions of turbulence. The 'centrifuge mechanism' of Maxey (J. Fluid Mech., vol. 174, ... Full text Cite

Mechanisms for the clustering of inertial particles in the inertial range of isotropic turbulence

Journal Article Physical Review E - Statistical, Nonlinear, and Soft Matter Physics · August 27, 2015 In this paper, we consider the physical mechanism for the clustering of inertial particles in the inertial range of isotropic turbulence. We analyze the exact, but unclosed, equation governing the radial distribution function (RDF) and compare the mechanis ... Full text Cite

New insights from comparing statistical theories for inertial particles in turbulence: II. Relative velocities

Journal Article New Journal of Physics · January 1, 2014 Part I of this two-part series compared two theories for the radial distribution function (RDF), a statistical measure of the clustering of inertial particles in isotropic turbulence. In Part II, we will contrast three theoretical models for the relative v ... Full text Cite

New insights from comparing statistical theories for inertial particles in turbulence: I. Spatial distribution of particles

Journal Article New Journal of Physics · January 1, 2014 In this paper, we contrast two theoretical models for the spatial clustering of inertial particles in isotropic turbulence, one by Chun et al (2005 J. Fluid Mech. 536 219) and the other by Zaichik et al (2007 Phys. Fluids 19 113308). Although their predict ... Full text Cite

Drift-free kinetic equations for turbulent dispersion.

Journal Article Physical review. E, Statistical, nonlinear, and soft matter physics · November 2012 The dispersion of passive scalars and inertial particles in a turbulent flow can be described in terms of probability density functions (PDFs) defining the statistical distribution of relevant scalar or particle variables. The construction of transport equ ... Full text Cite

Particle transport in a turbulent boundary layer: Non-local closures for particle dispersion tensors accounting for particle-wall interactions

Journal Article Physics of Fluids · October 3, 2012 Continuum equations derived from a probability density function kinetic equation contain dispersion tensors that describe the interaction between inertial particles and the underlying turbulent flow in which they are transported. These tensors require clos ... Full text Cite