Modeling Approaches and Computational Methods for Particle-laden Turbulent Flows

Front Cover
Modeling Approaches and Computational Methods for Particle-laden Turbulent Flows
Copyright
Contents
Contributors
About the editors
Preface
References
Acknowledgment
1 Introduction
1.1 Physical description
1.2 Scope
1.3 Deterministic descriptions
1.3.1 Complete deterministic description
1.3.2 Filtered fluid description: Euler–Lagrange approach
1.3.3 Filtered fluid and particle description: Euler–Euler approach
1.4 Statistical descriptions
1.4.1 Field-based description
1.4.1.1 Second-moment and probability density function closures
1.4.2 Point-based description
1.4.2.1 Collision-resolved statistical description
1.4.2.2 Collision-modeled statistical description
Boltzmann–Enskog equation for particle-laden flow
1.5 Important non-dimensional quantities
1.5.1 Particle volume fraction
1.5.2 Mass loading
1.5.3 Particle Reynolds number
1.5.4 Particle Stokes number
1.5.5 Knudsen number and Mach number
1.6 Multiscale nature of turbulent particle-laden flows
1.6.1 Microscale physics
1.6.2 Mesoscale physics
1.6.3 Macroscale physics
1.7 Outline of the book
1.7.1 Phenomenology and flow physics
1.7.2 Model-free particle-resolved simulations at the microscale
1.7.3 Euler–Lagrange models and simulations
1.7.4 Euler–Euler models and simulations
Nomenclature
References
2 Particle dispersion and preferential concentration in particle-laden turbulence
2.1 Introduction
2.2 Particle dispersion
2.2.1 Fluid elements
2.2.2 Inertial particles
2.2.3 Dispersion in inhomogeneous shear flows
2.3 Preferential concentration of particles by turbulence
2.3.1 Experimental and numerical observations
2.3.2 Physical mechanisms of preferential concentration
2.3.2.1 Centrifuge mechanism
2.3.2.2 Non-local clustering mechanism
2.3.2.3 Sweep-stick mechanism
2.3.3 Quantifying preferential concentration
2.3.3.1 Radial distribution function
2.3.3.2 Box counting methods
2.3.3.3 Voronoi tessellation
2.3.4 Practical significance
2.3.4.1 Gravitational settling
2.3.4.2 Two-way and four-way coupling
2.3.4.3 Convective heat and mass transfer
2.3.4.4 Radiation heat transfer
2.3.4.5 Electric fields
2.4 Turbophoresis
2.4.1 Definition and physical mechanism
2.4.2 Modeling implications
References
3 Physics of two-way coupling in particle-laden homogeneous isotropic turbulence
3.1 Introduction
3.1.1 Classification map for turbulent flows laden with dispersed particles
3.1.2 Resolution challenges for DNS of turbulent flows laden with particles whose dp < η
3.2 Particle-laden flows with dp < η
3.2.1 First experiments demonstrating the two-way coupling effects in turbulent flows (dp < η)
3.2.2 Physics of two-way coupling in homogeneous isotropic turbulence
3.2.2.1 Particles in a vortex
3.2.2.2 Particles in decaying isotropic turbulence
Decay rate of TKE in the physical space
Dissipation rate of TKE in the physical space
Decay rate of TKE in the spectral space
3.2.3 Why is DNS of forced particle-laden isotropic turbulence not appropriate for two-way or four-way coupling?
3.3 Particle-laden flows with dp > η
3.3.1 Physics of two-way coupling in particle-laden homogeneous isotropic turbulence (dp > η)
3.3.1.1 Is the Stokes number an appropriate indicator for turbulence modulation for particles with dp>η?
3.3.1.2 Physical space
Turbulence kinetic energy
Dissipation rate of TKE
Two-way coupling rate of change of TKE
Effects of particle rotation
3.3.1.3 Spectral space
3.A Governing equations
3.A.1 Particles with dp<η
3.A.2 Particles with dp>η
3.B Equations of conservation of linear and angular momenta for a solid particle moving in an incompressible fluid
References
4 Coagulation in turbulent particle-laden flows
4.1 Introduction
4.2 Geometric collision kernel
4.2.1 Dynamic and kinematic descriptions of the geometric collision kernel
4.2.2 Examples of geometric collision kernels
4.2.3 Turbulent collision of inertial particles – theoretical considerations
4.2.4 Turbulent collision of inertial particles – numerical simulations
4.2.5 Turbulent collision of inertial particles – parameterizations
4.3 Collision efficiency
4.4 Modeling the evolution of particle size distribution
4.5 A specific application: turbulent collision coalescence of cloud droplets and its impact on warm rain precipitation
4.6 Summary and outlook
Acknowledgments
References
5 Efficient methods for particle-resolved direct numerical simulation
5.1 Introduction
5.2 The immersed boundary method in Navier–Stokes-based solvers
5.2.1 Direct forcing immersed boundary method
5.2.2 How and where to distribute Lagrangian marker points?
5.2.3 What level of accuracy can be achieved with the immersed boundary method?
5.2.4 Further refinements
5.2.5 Describing the motion of rigid particles
5.3 Distributed Lagrange multiplier methods
5.4 Boltzmann equation-based mesoscopic methods
5.4.1 From the Boltzmann equation to Navier–Stokes–Fourier
5.4.2 The on-grid lattice-Boltzmann method
5.4.3 The generalized off-grid DUGKS method
5.4.4 Treating the moving solid–fluid interfaces
5.4.5 Remarks
5.5 Reference datasets
5.5.1 Single particle settling in unbounded ambient fluid
5.6 Comparing PR-DNS methods: a difficult exercise
5.7 Conclusion and outlook
Acknowledgment
References
6 Results from particle-resolved simulations
6.1 Introduction
6.2 PR-DNS of dense fluidized systems for drag force parameterizations based on dynamic simulations
6.3 PR-DNS of unbounded flows in the dilute regime
6.3.1 Settling in initially ambient fluid
6.3.2 Finite-size particles in turbulent background flow
6.4 PR-DNS of wall-bounded shear flows
6.4.1 Vertical plane channel flow
6.4.2 Horizontal plane channel flow
6.4.2.1 Focusing of heavy particles in low-speed streaks
6.4.2.2 Sediment transport
6.4.2.3 Sediment pattern formation
6.5 Conclusions and outlook
References
7 Modeling of short-range interactions between both spherical and non-spherical rigid particles
7.1 Introduction
7.2 Motion of a non-spherical rigid body
7.3 Geometric description of a non-spherical rigid body and the problem of collision detection of non-spherical rigid bodies
7.4 Non-collisional short-range hydrodynamic interactions: lubrication in dilute regime
7.5 Methods for Lagrangian tracking of non-spherical rigid bodies with collisions
7.5.1 Governing equations
7.5.2 Hard-sphere event-driven model for binary collisions
7.5.3 Time-driven rigid body dynamics and non-smooth contact dynamics for multibody collisions with persistent contacts
7.5.4 Soft-sphere time-driven molecular dynamics for multibody collisions with persistent contacts
7.5.4.1 Contact force models
In translation
In rotation
7.5.4.2 Time integration schemes in DEM
Integration of the translational motion
Integration of the angular motion
7.6 Efficient and parallel implementation of granular dynamics solvers and their parallel coupling to the fluid solver
7.7 Test cases
7.7.1 A single sphere bouncing on a horizontal plane
7.7.2 A single cylinder bouncing on a horizontal plane
7.7.3 Bedload transport of sediment in laminar open-channel flow
7.8 Outlook
References
8 Improved force models for Euler–Lagrange computations
8.1 Introduction
8.2 Undisturbed quantities
8.3 Stochastic effects in Euler–Lagrange simulation for unresolved fields
8.4 Fluid equations for dilute flows modeled with the Euler–Lagrange method
8.5 Particle equation of motion
8.6 Eulerian–Lagrangian data transfer
8.7 Correction schemes for the undisturbed quantities
8.7.1 The distinction between interpolated and undisturbed fluid velocity in numerical simulation
8.7.2 Unification involving discrete Green's functions
8.7.3 Gaussian-based corrections
8.7.4 Non-Gaussian-based corrections
8.7.5 Concluding remarks on undisturbed fluid velocity models
8.8 Summary and future directions
8.9 Discussion questions
Acknowledgments
References
9 Deterministic extended point-particle models
9.1 Motivation to go beyond the point-particle model
9.2 Neighbor influence
9.3 Undisturbed flow prediction
9.3.1 Superposable wake
9.4 Deterministic particle force prediction using the PIEP model
9.5 Beyond pairwise approximation using machine learning
9.6 Concept and statement of the force coupling method
9.6.1 FCM equations
9.7 FCM results for individual particles
9.7.1 Stokes flow
9.7.2 Unsteady Stokes flow
9.7.3 Finite Reynolds number
9.8 Examples of FCM applications
9.9 Comments
References
10 Stochastic models
10.1 Motivation for stochastic models
10.1.1 Scope
10.2 Dispersion of inertial particles from a point source
10.2.1 Uniform flow
10.2.2 Effect of gravitational settling
10.2.3 Homogeneous shear flow
10.2.3.1 Extension to general flows at finite slip Reynolds number
10.2.4 Sources of randomness
10.2.4.1 Fluid-phase turbulence
10.2.4.2 Flow perturbation by neighboring particles
10.2.5 Presumed PDF methods
10.2.6 Stochastic models of dispersion
10.3 Lagrangian particle description
10.3.1 PDF transport equation for physical particles
10.3.1.1 Particle position PDF
Mean squared displacement and the variance of particle position
10.3.1.2 Particle velocity PDF
10.3.2 Deterministic Lagrangian particle models
10.3.3 Stochastic Lagrangian particle models
10.3.4 Stochastic differential equations
10.3.5 Model PDF evolution equation
10.3.6 A rational approach to building stochastic models
10.4 Challenges in modeling turbulent particle-laden flow
10.5 Models for inertial particles in turbulence
10.5.1 Position Langevin
10.5.2 Velocity Langevin
10.5.2.1 TKE model: Peirano et al. [73]
10.5.2.2 TKE model: Pai and Subramaniam [71]
10.5.2.3 PTKE model: Tenneti et al. [105]
10.5.3 Acceleration Langevin
10.5.3.1 PTKE model: Lattanzi et al. [51]
10.6 Numerical considerations
10.6.1 Interpolation of Eulerian fields to Lagrangian particles
10.6.2 Numerical integration of SDEs
10.6.2.1 Random number generation
10.6.3 Estimation of Eulerian source terms from Lagrangian particle data
10.7 Summary and extensions
10.A Details of numerical integration of SDEs
10.B Fast and slow variables
References
11 Volume-filtered Euler–Lagrange method for strongly coupled fluid–particle flows
11.1 Strongly coupled fluid–particle flows
11.2 Microscale description
11.3 Volume-filtering
11.3.1 Notation and definitions
11.3.2 Some useful identifies
11.3.3 Continuity
11.3.4 Momentum
11.3.5 General scalar
11.3.6 Particle treatment
11.4 Closure modeling
11.5 Numerical implementation
11.5.1 Time integration of the fluid phase
11.5.2 Lagrangian treatment of the particle phase
11.5.3 Two-way coupling
11.5.4 Granular temperature evaluation
11.6 Application to the study of strongly coupled particle-laden flows
11.6.1 Homogeneous cluster-induced turbulence
11.6.2 Particle-laden channel flow
11.7 Extensions
11.7.1 Spray atomization
11.7.2 Compressible flows
11.8 Concluding remarks
References
12 Quadrature-based moment methods for particle-laden flows
12.1 Introduction
12.2 The kinetic equation and its generalization
12.2.1 Generalized kinetic equation
12.2.2 The collision integral and linearized collision operators
12.2.3 Dimensionless form of the kinetic equation and Knudsen effects
12.2.4 Velocity distribution in gas–particle flows with inertial particles
12.3 Generalities on moment methods
12.3.1 Moment definition and evolution equations
12.3.2 Moment realizability and challenges of the solution of moment conservation equations
12.4 Quadrature-based moment closures
12.4.1 Quadrature approximation of the NDF
12.4.2 Conditional quadrature method of moments for joint size–velocity NDF
12.4.3 Higher-order velocity closures
12.4.4 Closure of the advection flux and its numerical implementation
12.5 Anisotropic Gaussian closure for monodisperse flows
12.5.1 Formulation of anisotropic Gaussian closure
12.5.2 Closure of the spatial fluxes
12.5.3 Solution algorithm
12.6 Anisotropic Gaussian closure for polydisperse flows
12.6.1 Size–velocity kinetic equation
12.6.2 Joint size–velocity moments and their balance equations
12.6.3 Application of Gaussian quadrature and AG closure
12.7 Closure
References
13 Eulerian–Eulerian modeling approach for turbulent particle-laden flows
13.1 Introduction
13.2 Derivation of the Eulerian–Eulerian model for fluid–solid flows
13.2.1 Averaging over an indicator function
13.2.2 Averaging over particles
13.3 Probability density function
13.3.1 Deriving the governing equations using PDFs
13.3.2 The flow of particle properties
13.3.3 Influence of collisions
13.3.4 Expressing f(2) in f
13.3.5 Solving the probability density function in the non-ideal state: the Enskog approach
13.3.5.1 The second approximation to f for a slightly inelastic granular material
13.4 Closure relations
13.4.1 Turbulence modeling and modulation
13.4.2 Momentum exchange
13.4.3 Interfacial work
13.4.4 Constitutive relations
13.4.4.1 Solid phase
13.4.4.2 Fluid phase
Eddy viscosity
13.4.5 k-ϵ model for fluid–solid flow
13.4.6 Solid-phase fluctuating kinetic energy
13.5 Outlook and conclusions
References
14 Multiscale modeling of gas-fluidized beds
14.1 Introduction
14.1.1 Fluidized beds
14.1.2 Fluidization regimes
14.1.3 Underlying challenges
14.1.3.1 Interphase drag force
14.1.3.2 Interparticle forces
14.1.3.3 Large separation of scales
14.2 Multiscale modeling
14.2.1 Particle-resolved direct numerical simulation
14.2.1.1 Homogeneous drag correlations
14.2.1.2 Heterogeneous drag models
14.2.1.3 Non-spherical drag correlations
14.2.1.4 Gas–liquid–solid interactions
14.2.2 Euler–Lagrange models
14.2.2.1 Assessment of drag closures
14.2.2.2 Fluidization of cohesive particles
14.2.2.3 Fluidization of non-spherical particles
14.2.3 Euler–Euler models
14.2.3.1 Kinetic theory-based TFM
14.2.3.2 Continuum modeling of hydrodynamics in gas-fluidized beds
14.3 Outlook
Acknowledgment
References
15 Future directions
15.1 Future directions
15.2 Mapping the high-dimensional parameter space
15.2.1 Role of machine learning
15.3 Discovery and quantification of flow physics
15.4 Theoretical challenges
15.5 Modeling needs
15.5.1 Higher-order closure models from microscale PR–DNS
15.5.2 From scale-specific to scale-aware models
15.5.3 Particle structure
15.5.4 Relaxing the assumption of scale separation
15.6 Need for collaborative efforts
References
Index
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