multiphase flows in complex geometries: a uq perspective workshop/uqaw_2015_ta… · stochastic...

Multiphase flows in complex geometries:a UQ perspective

Matteo Icardi

Stochastic Numerics group & SRI-UQ center (KAUST)ICES (UT Austin)

SRI-UQ Annual WorkshopJanuary 6-9, KAUST

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Acknowledgements

KAUST

Ï R. Tempone, H. Höel, B. Saad, Q. Long

Ï SRI-UQ center andStochastic Numerics research group

POLITECNICO DI TORINO

Ï R. Sethi, T. Tosco (Groundwater Eng.)

Ï D. Marchisio,G. Boccardo (Chem. Eng.)

Ï C. Canuto, N. Quadrio (Math.)

Ï P. Asinari, D. Ceglia (Energy)

THE UNIVERSITY OF TEXAS AT AUSTIN

Ï S. Prudhomme (now at EP Montreal)

Ï M. Prodanovic (Petr. Eng.)

Ï I. Babuska (ICES)

A. Passalacqua, R. Fox (Iowa State Univ.), D. Tartakovsky (UCSD)2 / 34

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Motivation I: Data-driven UQ

Forward and inverse UQ techniques are mature enough to be applied in realcomplex engineering problems

Unfortunately in most applications, the main problem is defining/parametrizingthe input uncertainties (parametric uncertainties, error models)

MOMENTS

Sometimes parameters are dependent and only known through finite orderstatistics (mixed moments)

Part I: Quadrature-based moment methods

SAMPLES

Many uncertainties can hardly be parametrized (random geometry)

Part II: Multi-level Monte Carlo

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Motivation II: UQ as a physical closure problem

In multi-scale problems the concept of probability can be used for modelingpurposes. The forward UQ problem becomes a “closure” problem

REACTIONS: MICRO-MIXING PROBLEM

A+Bk−→ R SR =−kφAφB ⟨SR⟩ =−k⟨φAφB⟩ 6= −k⟨φA⟩⟨φB⟩

The chemical source term can be closed if we assume the existence of a jointprobability density function of the concentrations

⟨S(φ)⟩ =∫

S(ψ)n(ψ;x, t)dψ

Part II: Quadrature-based moment methods

MULTIPHASE FLOWS: MOMENTUM TRANSFER

How to model the drag force Fd acting on fluid-fluid or fluid-solid suspensionswhen shape, position, velocity and size of interphase vary in time and space ?Given a configuration space (probability space), statistics (mean, variance, ...) canbe computed

Part II: Multi-level Monte Carlo

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Multiscale models

Microscale modelFully resolved direct numerical simulation

Mesoscale modelPopulation balance equation

Boltzmann equationKinetic equation

(Lagrangian methods)

Macroscale modelMoment equations

Mixture modelTwo-fluid model

(Eulerian methods)

Density function closure

Moment transformMoment closure

Volume average

Reconstruction

5 / 34

PART I:Quadrature-based Moment Methods

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Method of Moments

Ï Important advantage is that the moments correspond to macroscopicquantities that have meaningful physical interpretations and are thereforedirectly measurable

Ï In many applications the NDF is not directly measured, but is inferred frommeasurements of integral quantities

Ï Two main issues arise when using MOM: number of moments to be trackedand closure problem

Ï The closure problem is the impossibility of writing a term as a function of afinite set of moments

Ï When the PDF equation contains transport in the phase space, we have acascade of equations (Mi depends on Mi+1)

Ï All non-polynomial integral source terms also need closure

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Closure problem

When applying the moment transform (i.e. integral)

THE CLOSURE PROBLEMS APPEAR IN THE FOLLOWING FORM:

I =∫Ωξ

g(ξ)n(ξ)dξ, (1)

where n(ξ) is the NDF to be approximated and g a generic function to be closed

PRESUMED-PDF APPROACHES

Ï A possible approach is to assume a functional form for the PDF/NDF

Ï This is reasonable when there is a fast relaxation towards a knownequilibrium

Ï How to find the best closure for a general case?

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Quadrature-based Moment Method

Ï The closure problem can be overcome by using the following quadratureformula: ∫

Ωξn(ξ)g(ξ)dξ≈

N∑α=1

wαg(ξα), (2)

where wα and ξα are the weights and the nodes/abscissas of the quadratureformula, and N is the number of nodes

GAUSSIAN QUADRATURE IS THE “BEST” CHOICE

The NDF is the weight function or measure; the moments

Mk = ⟨ξk⟩ =∫Ωξ

n(ξ)ξk dξ, k = 0,1,2, . . . (3)

are used to compute a Gaussian quadrature rule with a degree of accuracy of2N −1,.

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Quadrature-based moment methods

Ï For univariate problems things are simple: transport equations for the first2N moments are solved, the closure problem is overcome by using aquadrature approx.

Ï The quadrature is very accurate in approximating integrals of smoothfunctions in which the NDF appears (such as high-order moments and otherintegrals)

Ï The method implicitly assumes that the NDF is a summation of Dirac deltafunctions centered on the nodes and weighted by the weights of thequadrature approximation

Ï The quadrature approximation of order N is calculated from the first 2Nmoments with direct inversion algorithms: product-difference or Wheeleralgorithms

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Quadrature method of moments

Ï Let us consider the following example (pure growth process):

Dn

Dt= S =− ∂

∂ξ

(Gpn

), (4)

Ï Moment equations

DMk

Dt= k

∫ ∞

0⟨Gp|ξ⟩ξk−1n(ξ)dξ (5)

Ï The closure problem is overcome as follows:

DMk

Dt= k

N∑α=1

⟨Gp|ξα⟩(ξα)k−1wα (6)

Ï weights wα and nodes ξα are calculated from the PD or Wheeler algorithmfrom the moments

Ï This method is called Quadrature Method of Moments (the variables are themoments)

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Example: Boltzmann equation

Particle Velocity Distribution f (Up;x, t) =∫R

n(L,Up;x, t)dL

MODEL EQUATION

Ï scalar molecular speed ξ

Ï distribution of velocities f (ξ)

Ï no spatial derivatives

Ï only collisions Q(f , f )

Ï Hard-sphere model

Ï x = cos(ξ1∠n), y = cos(ξ2∠n)

Ï collision frequency |ξ2y−ξ1x|

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Homogeneous Isotropic Boltzmann equation

HIBE evolution equation:df

dt=Q(f , f )

Q = 2π2 a2∫ +∞

0ξ2

2

∫ +1

−1

∫ +1

−1

(f (ξ′1)f (ξ′2)− f (ξ1)f (ξ2)

) |ξ2y−ξ1x|dx dy dξ2

Let us introduce the even moments M2p =Φp (densityΦ0, energyΦ1, etc.). After

change of variable E = ξ2

2 and a few manipulations:

dΦp

dt= 4π

p2∫ ∞

0Q(f , f )Ep+1/2dE =

16π3p2∫ ∞

0

∫ ∞0

∫ +1

−1

∫ +1

−1|q|

[(C+

p )− (C−p )

]f (E)f (E∗)(EE∗)1/2dxdydE∗dE

q = yE1/2∗ −xE1/2, C+p =

[E(1−x2)+E∗y2

]p, C−

p = Ep

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Quadrature approximation for HIBE

COLLISIONAL INTEGRAL∫ ∞

0

∫ ∞

0

∫ +1

−1

∫ +1

−1|q|

[(C+

p )− (C−p )

]f (E)f (E∗)(EE∗)1/2dxdydE∗dE

M/2∑i=0

M/2∑j=0

∫ +1

−1

∫ +1

−1|q|

[(C+

p )− (C−p )

]wiwjdxdydE∗dE

Written in terms of pre-collisional quantities only → quadrature approximation

f (E) ≈M/2∑i=1

wi

4πp

2Eδ(E −Ei) ⇒ dΦp

dt≈ πp

2

M/2∑i=1

M/2∑j=1

wiwjΛij,p

at equilibrium Q = 0 → dΦp

dt = 0 BUT this is not guaranteed with the quadratureapproximation

Collisions drive the flows towards the wrong steady state

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Quadrature-Based Moment Method (QBMM)

dΦp

dt= πp

2

M/2∑i=1

M/2∑j=1

wiwjΛij,p = Sp

wi and Ei calculated fromΦp with inversion algorithms1

Ï QMOM fails to approach the equilibrium → source term must be corrected

Ï Static correction (QMOM+SC):

dΦp

dt= Sp −S

eqp

Ï Dynamic correction (QMOM+DC) weighted with a distance fromequilibrium:

dΦp

dt= Sp −

∣∣∣∣∣ Φp(t)−Φp(0)

Φeqp (t)−Φp(0)

∣∣∣∣∣h

Seqp , h = 1.5

1Product-Difference or Wheeler algorithms 14 / 34

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QMOM approximation vs DVM, Grad, oLBM

Let us consider a closed box of particles far from equilibrium (initial velocitydistribution NOT Gaussian):

QMOM approximation selecting M moments(M/2 nodes and weights)

COMPARISON WITH:Ï Discrete Velocities Method (DVM): reference results with 400 discrete

velocities - HomIsBoltz open-source Matlab code (Asinari, 2010)

Ï Grad expansion method (GM) of order M with generalized Laguerrepolynomials

Ï Quadrature approximation with M fixed Laguerre nodes → LatticeBoltzmann Method (LBM) with off-lattice prescribed velocities

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QMOM approximation vs DVM, Grad, oLBM – 1

M = 4Initial condition

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M = 4

Relaxation to equilibrium Rp = Φp−Φeqp

Φeqp

2nd energy moment 3rd energy moment

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M = 4Relative error onΦp


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HIBE steady state

Comparison of steady-state momentsM = 4 M = 6 M = 8

×108 Φ2 % Φ2 % Φ2 %Grad (exact) 1.0034 0.0 1.0034 0.0 1.0034 0.0

oLBM 1.0033 0.01 1.0034 0.003 1.0034 0.001QMOM 0.9704 3.29 0.9979 0.55 1.0022 0.12

QMOM+SC/DC 1.0034 0.0 1.0034 0.0 1.0034 0.0

×1011 Φ3 % Φ3 % Φ3 %Grad (exact) 1.4921 0.0 1.4921 0.0 1.4921 0.0

oLBM 1.4919 0.02 1.4921 0.004 1.4921 0.002QMOM 1.4707 1.44 1.4874 0.31 1.4910 0.08

QMOM+SC/DC 1.4921 0.0 1.4921 0.0 1.4921 0.0

×1014 Φ4 % Φ4 %Grad (exact) - - 2.8529 0.0 2.8529 0.0

oLBM - - 2.8527 0.005 2.8528 0.002QMOM - - 2.8302 0.80 2.8478 0.18

QMOM+SC/DC - - 2.8529 0.0 2.8529 0.0

×1017 Φ5 % Φ5 %Grad (exact) - - 6.6666 0.0 6.6666 0.0

oLBM - - 6.6662 0.006 6.6665 0.002QMOM - - 6.6244 6.34 6.6548 0.18

QMOM+SC/DC - - 6.6666 0.0 6.6666 0.0

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HIBE steady state

Quadrature error at the true equilibrium

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QBMM - multivariate case

The Generalized Population Balance Equation for M-dimensional phase space:

∂n

∂t+ ∂

∂x· (Upn

)= ∂

∂x·(D∂n

∂x

)− ∂

∂ξ· (Gn)+Q, (7)

After having defined a generic moments :

Mk ≡∫Ωξξ

k11 · · ·ξkM

M n(t,ξ)dξ (8)

equivalent notations Mk1,...,kM= Mk = M(k1, . . . ,kM ) = M(k), we solve the resulting

transport equations:

∂Mk∂t

+ ∂

∂x·(Uk

pMk

)− ∂

∂x·(Dk ∂Mk

∂x

)=

∫Ωξξk

[− ∂

∂ξ· (⟨Gp|ξ⟩n

)+Q

]dξ (9)

and the closure problem is overcome by using the quadrature approximation.

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Multivariate case

N(M +1) moments ( optimal moment set) are used to determine the quadratureapproximationBrute-force: direct solution of the following non-linear system (ill-conditioned)

Mki1,ki2,...,kiM= M(ki) =

N∑α=1

wα

M∏β=1

ξkiβ

β,α, 1 ≤ i ≤ N(M +1)

by employing the Newton-Raphson iterative schemeAlternatively the Tensor-Product QMOM can be used using marginals or theConditional QMOM introducing an order in the variables and using conditionalmoments

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QBMM - multivariate case

Bivariate Gaussian distr. with M = 9Direct (Newton) (diamond), Tensor product (circle) and Conditional (square)

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Conclusions

The idea of QBMM is similar to the “arbitrary Polynomial Chaos Expansion” (aPC,Oladyshkin and Nowak, 2012)

Ï Input random variables ξi known only by moments

Ï Moments induce orthogonal polynomials and quadrature rule

Ï Stochastic spectral and collocation approaches for computing the responsesurface

Ï Multidimensional correlated variables treated directly

Ï Forward propagation of uncertainty from moments of input to moments ofresponse, without passing through the PDF

Good alternative for correlated variables in moderate dimensions

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Future works

The main advantage in using only moments is the possibility of adaptivelyupdating the quadrature rule, when the underlying PDF changes in time/space

NON-LINEAR FOKKER-PLANCK

Ï Attar and Vedula, 2008; Otten and Vedula, 2011

Ï Drift-diffusion equations

xi = hi(x, t)+gi(x, t)Wi, Wij

Ï Applications to mixing in porous media

NON-LINEAR FILTERING AND BAYESIAN UPDATES

Ï Xu and Vedula, 2009, 2010

Ï Propagation step through Fokker-Planck

Ï Bayesian update based on quadrature or on EnKF

Ï Mixture of gaussians that preserves moments

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PART II:MLMC for dispersed multiphase flows

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Suspensions/dispersed flows

DNS of fluid-solid interface (particle-resolved simulations)are often unfeasible

WHY

Ï Limits in resolution

Ï Physical models too complex(e.g. particulate processes)

Ï We want a steady solution or2D/1D models

Ï Unknown particleshape/properties

EXAMPLES

Ï Fluidized beds, separators

Ï Environmental flows

Ï Pore-scale processes

Ï Long piping systems

We need effective closures for momentum transfer (e.g. drag forcecorrelations) that could lump together some of the complexities of thesystem.

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Drag correlations

Many different closures has been proposed:

Ï Isolated sphere: Stokes, Schiller-Neumann, Clift...

Ï High density/porous media: Ergun (Kozeny Carman), ...

Ï Periodic arrangements: Sangani and Acrivos, ...

Ï Granular flows: Gidaspow, Beetstra et al (2007), Tenneti et al (2011)

We generally seek a relation

Fd = Fd(Re,φ)

where Fd is the normalized equivalent mean drag force over a particleensemble and φ volume fraction.

Many systems (e.g., fluidized beds) sufferfrom high sensitivity to the drag law

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Stochastic interpretation

Is the problem deterministic or stochastic?

Ï Two-fluid (or mixture) models are derived with volume/ensemble averaging

Ï Deterministic averaging/homogenization limit exist for drag force in highdensity systems

∆p ∝ Fd ∝ µ

K⟨U⟩ (Darcy Eq with constant permeability)

Ï The ensemble/volume size however tends to infinity for low porosity

Ï Stochastic homogenization: the actual averaging ⟨·⟩ is not deterministic

1. What is the effect of particle random position and velocity on the drag?

2. What is the role/magnitude of drag force fluctuations over (and inside) theaveraging volume?

We seek Fd = Fd(Re,φ,θ) and F ′d = F ′

d(Re,φ,θ)

θ is a “granular temperature” and F ′d are the random drag fluctuations

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Statistical estimator

Even if we are interested only in the average drag on a fixed ensemble/volume size,the actual computations requires thousands 3D expensive CFD realizations

MLMC ALGORITHM

1. Each estimator, defined with prescribed tolerance and confidence, isinitialized with L levels

2. Each level is initialized with M` random realizations (samples)

3. For each sample, generate the geometry (packing algorithms), discretize(meshing), solve (finite volume) and extract the QoI (integrate) at level ` andlevel `−1

4. Re-assemble the multilevel estimator. If numerical or statistical error abovethreshold add respectively levels or samples and go back to 3

5. Estimate independently more times to double check accuracy criteria

ONGOING EXTENSIONS

Ï Combination of grid-based multilevel estimator with other discretizationparameters (domain size, tolerances, solvers) in the MIMC spirit

Ï High-order extrapolation of mean and variances using weighted sums ofmultiple levels

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Problem formulation

We developed fully automatic code to solve different stochastic PDE problems andcompute effective properties:

ADIMENSIONAL STEADY INCOMPRESSIBLE NAVIER-STOKES

u(x;ω) ·∇u(x;ω)+∇p(x;ω)− 1

Re∇· (∇u(x;ω)) = 0, x ∈ Γ(ω) ⊂ [0,1]d ,

∇·u = 0

no-slip: u = 0, ∇np = 0 on Ig

inlet: ∇nu = 0, p = pi(x) on Ii = x : x1 = 0

outlet: ∇nu = 0, p = po(x) = 0 on Io = x : x1 = 1

symmetry: ∇nut = 0, un = 0, ∇np = 0 on Is = x : x2,x3 = 0

The quantity of interest Q(u) is, in this case, the mean flux ⟨ux⟩ or, equivalently, the

drag force Fd = ∆p(1−φ)d2

18⟨ux⟩The algorithm is tested for creeping flow through random array of mono dispersedspheres. Extended to non-spherical, polydispersed, and moving2 particles.

2in local equilibrium 30 / 34

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Example: mean flux on mono dispersed spheres (φ= 5%)

Unstructured anisotropic mesh with non-uniform localized refinements

Computational savings: 158x faster than MC, (variance 132x)Mean: 0.113 Variance: 0.0136 CoV: 103%Error estimation: Numerical bias 0.4% (0.6%), Statistical 1.4% (3.1%)

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Example: mean flux on mono dispersed spheres (φ= 20%)

Unstructured anisotropic mesh with non-uniform localized refinements

Computational savings: 131x faster than MC, (variance 129x)Mean: 0.0124 Variance: 0.000165 CoV: 33%Error estimation: Numerical bias 0.5% (0.9%), Statistical 1.4% (3.0%)

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Why does the weak error stagnate?

This can be explained by the non-uniform refinement and the non-monotonenon-asymptotic behavior total discretization error

Deterministic convergence rate for simple regular packings

Convergence rate for pure diffusion (left) and Navier-Stokeswith porosity 0.47 (center) and 0.366 (right).

Voxelized mesh, Locally refined, Uniform

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Conclusions

ACHIEVEMENTS

Ï Developed automatic (and parallel) workflow for setup, meshing, simulationand post-processing for flow through random granular materials

Ï Assessed accuracy (convergence rates) and computed uncertainty(variations) for low-Re systems

Ï A-posteriori error control: both statistical and numerical

Ï Preliminary results for mono disperse spheres in Stokes regime

OPEN PROBLEMS AND ONGOING WORK

– High numerical accuracy (<1%) still unobtainable for large samples

Ï For new drag laws, huge amount of data for different Re and φ needed

Ï Influence and representativity of the packing algorithm

Ï Limit the effect of the boundary conditions (periodicity and lift estimation)

Ï How to use high-order moments computed by MLMC in Eulerian models?

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References (for Part I)

Ï D. Marchisio and R. Fox, “Computational Models for Polydisperse Particulateand Multiphase Systems” (Cambridge, 2012)

Ï Rodney Fox, “Computational Models for turbulent reacting flows”(Cambridge, 2003)

Ï H. Struchtrup, “Macroscopic Transport Equations for Rarefied Gas Flows”(Springer, 2005)

Ï Cercignani, “The Boltzmann equation and its applications” (Springer, 1988)

Ï W. Gautschi, “Orthoghonal Polynomials” (Oxford Sci., 2004)

Ï D. Ramkrishna, “Population Balances” (Academic Press, 2000)

Ï my PhD thesis (2012)

Ï O. Le Maitre and O. Knio, “Spectral Methods for Uncertainty Quantification”(Springer, 2010)

POSTERS: Flow, transport and diffusion in random geometries I and II

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BACKUP SLIDES

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Smoluchowski coagulation equation

Describe particle aggregation/coagulation

∂n(L, t)

∂t= 1

2

∫ L

0K (L− L, L)n(L− L, t)n(L, t)dL−

∫ ∞

0K (L, L)n(L, t)n(L, t)dL

Ï Not to be confused with the Smoluchowski equation (drift-diffusionequation)

Ï Integro-differential equation

Ï Number density n of particles with size L

Ï No spatial dependence

Ï RHS (Q) has a gain and a loss term written in terms of the aggregationfrequency K

Ï K is usually very complex and non-linear

Very interesting mathematical topic (stability, linearization, large-time asymptotics), see works of

Klemens Fellner (Graz)

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Population Balance Equation (PBE)

Extension of the Smoluchowski coagulation equation, very popular in ChemicalEngineering, Biology and Social Models

∂n

∂t+ ∂

∂x· (Un)+ ∂

∂ξ· (Gn) =Q

Ï n = n(ξ;x, t) number of particles per unit volume

Ï ξi generic internal variables (volume, size, area, composition, otherproperties)

Ï U given advection velocity field

Ï Q describe the generic particulate process with Birth (nucleation), Breakage(Fragmentation) and Aggregation (Coagulation)3

Q =Q0 +Q1(n)+Q2(n,n)

Ï G describe growth of particles (velocity in phase space ξ)

Ï Usually written for spatially inhomogeneous systems and coupled with CFDto simulate dispersed bubbles and particles

3with frequency kernels that can be seen as upscaled jump processes 36 / 34

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The physical basis of Boltzmann equation

Molecular encounters for hard-spheres potential

Ï d is the molecular diameter

Ï g = U−U∗ is the pre-collision velocitydifference

Ï g′ = U′−U′∗ is the post-collision velocitydifference

Number and momentum are conserved during a collision

m+m∗ = m′+m′∗; U+U∗ = U′+U′∗

as well as kinetic energy (in the case of elastic collisions)

U ·U+U∗ ·U∗ = U′ ·U′+U′∗ ·U′∗

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Boltzmann Equation (BE)

From the Liouville n-particles equation, assuming indistinguishable particle andthe "Stosszahlansatz" (molecular chaos4 hypothesis), the Boltzmann equation isobtained

∂n(U)

∂t+ ∂

∂x· (Un(U))+ ∂

∂U· (An(U)) =∫

R3

∫S+

[n(U′)n(U′∗)−n(U)n(U∗)

]β

(g,x

)dsdU∗

Ï A is the acceleration,

Ï S+ is the solid angle

Ï β(g,x

)is the collision kernel

Equations with similar structure: Vlasov (no collisions), Williams-Boltzmann sprayequation, ...

4particles are uncorrelated, two-particles PDF is product of one-particle PDF 38 / 34

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Boltzmann Equation and Equilibrium

Most of the studies and methods for BE are based on the concept of EquilibriumDistribution

Ï The equilibrium distribution feq is such that Q(feq, feq) = 0

Ï Also called "Maxwell-Boltzmann" or "Maxwellian" distribution, a Gaussian inthe standard case

Ï Fixing the first 2 moments (3 in case of NDF), only one Gaussian exist

Ï If one wants to study non-equilibrium phenomena, more moments must bestudied

Ï Grad’s moment method is based on small deviations from equilibrium

f ≈ feq(1+a1H1(v)+a2H2(v)+ . . . )

Ï Other methods (such as Lattice Boltzmann) rely on a linearized collision term(BGK, Bhatnagar-Gross-Krook approximation)

Q ≈ 1

τr(f − feq)

All these methods require the prior knowledge of the equilibrium distribution!

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The Method of Moments

Ï MOM originates in kinetic theory of gases5: n(t,x,U)

Ï The moment of order zero is related to the density:

ρ(t,x) ≡ mM0,0,0(t,x) = mÑ +∞

−∞n(t,x,U)dU

Ï The moment of order one is used to define the average velocity:

⟨U⟩ =(

M1,0,0

M0,0,0,

M0,1,0

M0,0,0,

M0,0,1

M0,0,0

)Ï By applying the MOM to the Boltzmann equation, an infinite system of

equations (cascade) appears

Ï Euler, Navier-Stokes and Burnett equations are obtained by with theChapman-Enskog method (asymptotic expansions at different orders)

5The same name however may refer to many different methods in different fields 40 / 34

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Flow regimes

The Boltzmann Equation is not only for rarefied gases!

Ï The flow regime of a granular gas depends on Knudsen number Kn = λL0

Ï The hydrodynamic description based on the Navier-Stokes-Fourier (NSF)equation is valid only for low Kn:

Ï Continuous regime (Kn < 0.01): NSF with no-slip BC.

Ï Slip regime (0.01 < Kn < 0.1): NSF and partial slip BC at walls.

Ï For Kn > 0.1: Full Boltzmann equation.

KnEuler N-S

Boltzmann equationCollisionless Boltzmann eq.

∞0 0.01 0.1 1 10010Adapted from: G. A. Bird (1994)

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The GPBE fluid-particle systems

Ï n(t,x,Up,ξp,Uf,ξf), represents the number of particles (per unit volume)with velocity equal to Up, internal coordinate ξp and see a continuous phasewith velocity Uf and internal coordinate ξf.

Ï The evolution of the NDF is dictated by the GPBE:

∂n

∂t+ ∂

∂x· (Upn

)+ ∂

∂Up·[(

Afp +Ap

)n]+ ∂

∂ξp· (Gpn

)+ ∂

∂Uf·[(

Apf +Af

)n]+ ∂

∂ξf· (Gfn

)=Q (10)

Ï phase-space velocity for particle velocity: Afp +Ap ( average particleacceleration → acceleration model (drag, lift, ...)

Ï phase-space velocity for particle internal coordinate: Gp (e.g. particle growthrate)

Ï Discontinuous jump term: Q (e.g. particle collision, aggregation, breakage,nucleation, etc.)

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History of quadrature-based moment methods

Ï McGraw (1997) introduced the Quadrature Method Of Moments (QMOM) inthe context of Population Balance Equation

Ï Marchisio and Fox (2005) developed the Direct Quadrature Method ofMoments (DQMOM) that became very popular in the Chemical Engineeringcommunity because very easy to implement in CFD codes

Ï Different extensions to multivariate cases have been proposed. The moregeneral one is the Conditional QMOM (CQMOM) (Yuan and Fox, 2011)

Ï Kernel density type reconstruction have been incorporated in QMOM withthe Extended QMOM (EQMOM) (Yuan, Laurent, Fox, 2012)

Ï Different applications have been studied: particulate processes, dispersedflows (gas-liquid, gas-solid, fluid-solid), turbulent micro-mixing, rarefiedgases, generic kinetic/Fokker-Planck equations, non-linear filtering, ...

Ï Recently proposed for Uncertainty Quantification (Passalacqua and Fox,2012; Attar and Vedula, 2012)

Ï A large number of variants have been proposed (FCMOM, TBMM,DuQMoGeM, SQMOM , ...)

All the closures based on Gaussian quadratures computed directly from momentstook the name of QUADRATURE-BASED MOMENT METHODS (QBMM)

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Quadrature-based moment methods

HOW DO WE COMPUTE THE GAUSSIAN QUADRATURE?

The N weights and N abscissas can be determined by solving the followingnon-linear system:

M0 =N∑α=1

wα,

...

M2N−1 =N∑α=1

wαξ2N−1α .

(11)

using the Newton-Raphson method, or any other non-linear equation solver (veryexpensive, ill-posed and very good initial guess needed)

Much more efficient are the product-difference or Wheeler algorithms

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Gaussian quadrature

A set of polynomials P0(ξ),P1(ξ), . . . ,Pα(ξ), . . . withPα(ξ) = kα,0xα+kα,1xα−1 +·· ·+kα,α, is said to be orthogonal in the integrationintervalΩξ, with respect to the weight function, if

∫Ωξ

n(ξ)Pα(ξ)Pβ(ξ)dξ

= 0 for α 6=β,

> 0 for α=β,(12)

and, of course, is said to be orthonormal if∫Ωξ

n(ξ)Pα(ξ)Pβ(ξ)dξ=

0 for α 6=β,

1 for α=β.(13)

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Gaussian quadrature

Any set of orthogonal polynomials Pα(ξ) has a recurrence formula relating anythree consecutive polynomials in the following sequence:

Pα+1(ξ) = (ξ−aα)Pα(ξ)−bαPα−1(ξ), α= 0,1,2, . . . (14)

with P−1(ξ) ≡ 0 and P0(ξ) ≡ 1 and where

aα =∫Ωξ

n(ξ)ξPα(ξ)Pα(ξ)dξ∫Ωξ

n(ξ)Pα(ξ)Pα(ξ)dξ, α= 0,1,2, . . . (15)

bα =∫Ωξ

n(ξ)Pα(ξ)Pα(ξ)dξ∫Ωξ

n(ξ)Pα−1(ξ)Pα−1(ξ)dξ, α= 1,2, . . . . (16)

One can calculate a0, then P1(ξ), then a1 and b1 and so on...

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Gaussian quadrature

Ï The coefficients aα and bα can be written in terms of the moments

Ï The coefficients necessary for the construction of a polynomial of order Ncan be calculated from the first 2N −1 moments of the NDF

Ï For example with M0, M1, M2 and M3, it is possible to calculate the followingcoefficients:

a0 = M1

M0,

a1 = M3M20 +M3

1 −2M2M1M0

M2M0 +M21 −2M2

1 M0,

b1 = M2M0 +M21 −2M2

1 M0

M20

,

(17)

which suffice for the calculation of the polynomial P2(ξ)

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Gaussian quadrature

GAUSSIAN QUADRATURE

The necessary and sufficient condition for the following formula:∫Ωξ

n(ξ)g(ξ)dξ=N∑α=1

g(ξα)wα+RN (g), (18)

to be a Gaussian quadrature approximation is that its nodes ξα coincide with theN roots of the polynomial PN (ξ) of order N orthogonal inΩξ with respect to theweight function n(ξ).

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Direct quadrature method of moments

Ï The fact that the closure problem is overcome with the quadratureapproximation: ∫

Ωξn(ξ)g(ξ)dξ≈

N∑α=1

wαg(ξα), (19)

Ï Is equivalent to the assumption that the NDF is as follows:

n(ξ) =N∑α=1

wαδ (ξ−ξα) , (20)

Ï Instead of tracking the evolution for the moments, the evolution of theweights and nodes in the quadrature approximation could be directlytracked: Direct quadrature method of moments

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Ï Assuming that the weights and nodes are differentiable in space/time thefollowing transport equation is obtained:

N∑α=1

δ(ξ−ξα)

(Dwα

Dt

)−

N∑α=1

δ′(ξ−ξα)

(wα

DξαDt

)= S(ξ), (21)

Ï If the weighted nodes (or weighted abscissas) ςα = wαξα are introduced:

N∑α=1

δ(ξ−ξα)

(Dwα

Dt

)−

N∑α=1

δ′(ξ−ξα)

(−ξα Dwα

Dt+ Dςα

Dt

)= S(ξ). (22)

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Ï We now define aα and bα to be the source terms:

Dwα

Dt= aα,

DςαDt

= bα. (23)

Ï Using these definitions Eq. (22) can be rewritten in a simpler form:

N∑α=1

[δ(ξ−ξα)+δ′(ξ−ξα)ξα

]aα−

N∑α=1

δ′(ξ−ξα)bα = S(ξ). (24)

Ï This equation can now be used to determine the unknown functions aα andbα by applying the moment transformation.

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Ï DQMOM can be applied for any independent set of moments (number ofmoments MUST be equal the number of unknown functions)

Ï Knowing that: ∫ +∞

−∞ξkδ(ξ−ξα)dξ= (ξα)k,∫ +∞

−∞ξkδ′(ξ−ξα)dξ=−k(ξα)k−1,

(25)

Ï The moment transform of Eq. (24) yields

(1−k)N∑α=1

ξkαaα+k

N∑α=1

ξk−1α bα = Sk, (26)

with k = k1,k2, . . . ,k2N .

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Ï The linear system in Eq. (26) can be written in matrix form:

Aα= d. (27)

where

α= [a1 · · · aN b1 · · · bN

]T =[

ab

], (28)

d = [Sk1

· · · Sk2N−1

]T, (29)

Ï The components of the matrix A are

aij =

(1−ki

)ξ

kij if 1 ≤ j ≤ N ,

kiξki−1j if N +1 ≤ j ≤ 2N .

(30)

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Ï If (as in QMOM) the first 2N integer moments are chosen (i.e., k = 0, . . . ,2N −1), thematrix of the linear system is

A =

1 · · · 1 0 · · · 00 · · · 0 1 · · · 1

−ξ21 · · · −ξ2

N 2ξ1 · · · 2ξN...

......

...2(1−N)ξ2N−1

1 · · · 2(1−N)ξ2N−1N (2N −1)ξ2N−2

1 · · · (2N −1)ξ2N−2N

.

(31)

Ï A does not depend on the weights wα and if the abscissas ξα are unique, then A will befull rank.

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Ï This method is called Direct quadrature method of moments and follows thisprocedure:

Ï The evolution equations for weights and nodes of the quadratureapproximation are solved:

Dwα

Dt= aα,

Dwαξα

Dt= bα. (32)

Ï The source terms are calculated by inverting the linear system and by usingthe following initial condition:

wα(0) = w0α, ςα(0) = w0

αξ0α for k = 1, . . . ,N . (33)

in turn calculated from the initial moments

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QMOM & DQMOM

Ï QMOM & DQMOM are very accurate in tracking the evolution of themoments of the NDF: 4-8 moments do the same job of many (e.g. 100) classesor sections (see for example the work of Marchisio et al., 2003 and Vanni,2000)

Ï The Wheeler algorithm is very robust (if the moments are realizable) and forparticular cases the Wheeler algorithm is successful when PD fails

Ï QMOM & DQMOM are identical for spatially homogeneous systems (if thenodes are distinct and if the problem is continuous in time)

Ï Important differences arise when treating spatially inhomogeneous systems(discussed next)

Ï In general increasing the number of nodes of the quadrature approximationand of moments to be tracked increases the accuracy

Ï Problems can appear when kernels and NDFs are discontinuous or whenthey are localized in the phase-space (e.g. fine dissolution)

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Product-difference algorithm

A smarter way is to employ the recursive relationship for the orthogonalpolynomials:

ξ

P0(ξ)P1(ξ)

...PN−2(ξ)PN−1(ξ)

=

a0 1b1 a1 1

. . .

. . . 1aN−1

P0(ξ)P1(ξ)

...PN−2(ξ)PN−1(ξ)

+

00...0

PN (ξ)

. (34)

The nodes of the quadrature approximation ξα, are the eigenvalues of thetridiagonal matrix appearing in the equation.This matrix is often re-written in terms of an equivalent tridiagonal symmetricmatrix!

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In fact the matrix can be made symmetric (preserving the eigenvalues) by a diagonalsimilarity transformation to give a Jacobi matrix:

J =

a0√

b1√b1 a1

√b2√

b2 a2√

b3√b3 a3

. . .

. . .. . .

. . .

. . . aN−2√

bN−1√bN−1 aN−1

(35)

This procedure transforms the ill-conditioned problem of finding the roots of a polynomialinto the well-conditioned problem of finding the eigenvalues and eigenvectors of atridiagonal symmetric matrix.The N weights can then be calculated as wα = M0ϕ

2α1 where ϕα1 is the first component of

the αth eigenvectorϕα of the Jacobi matrix.

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1. Construct the matrix P with components Pα,β:

Pα,β = P1,β−1Pα+1,β−2 −P1,β−2Pα+1,β−1

β ∈ 3, . . . ,2N +1 and α ∈ 1, . . . ,2N +2− j. (36)

2. where the first row of the matrix is:

Pα,1 = δα1 α ∈ 1, . . . ,2N +1, (37)

3. where δα1 is the Kronecker delta and where the components in the secondcolumn of P are

Pα,2 = (−1)α−1Mα−1 α ∈ 1, . . . ,2N . (38)

4. Calculate the coefficients of the continued fraction ζα:

ζα = P1,α+1

P1,αP1,α−1α ∈ 2, . . . ,2N . (39)

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1. The coefficients of the symmetric tridiagonal Jacobi matrix are then obtainedfrom sums and products of ζα:

aα = ζ2α+ζ2α−1 α ∈ 1, . . . ,N (40)

bα =−√ζ2α+1ζ2α α ∈ 1, . . . ,N −1. (41)

2. For example for N = 2 the P matrix is1 M0 M1 M0M2 − (M1)2 M0

(M3M1 − (M2)2)

0 −M1 −M2 − (M0M3 −M2M1)0 M2 M30 −M30

. (42)

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Extended QMOM

Ï only integral representation, no localized processes in phase space,smoothness of kernels β, G,...

Ï It fails in the computation of the entropy∫Ωξ

f log f dξ

Ï However smooth basis functions can be used instead of delta functions(Extended Quadrature Method of Moments, EQMOM)

Ï PDF reconstructed as a mixture of Gaussians (or other distributions) like theKernel Density Estimation

Ï The choice of basis functions can be done arbitrarily or with additionalunknown parameters (e.g., mean and variance of normal distribution)

Ï Additional parameters must be found by additional equations (i.e., moremoments must be solved)

- The inversion problems can become difficult to solve

An alternative is the reconstruction through Maximum Entropy Method or thereconstruction through the orthogonal polynomials

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Extended Quadrature-based moment methods

EQMOM for a dissolution problemGAUSSIAN DISTR. SOLUTION WITH N = 4 BETA DISTR.

−2 −1 0 1 2 3 40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

ξ

n(ξ)

0.1250.250.51

a

aYuan, C., Laurent, F., Fox, R.O. An extended quadrature method of moments for population balance equations (2012) Journal of Aerosol Science,51, pp. 1-23.

δσ = 1p2πσ

exp

(− (ξ−ξα)2

2σ2

);δσ = ξ

ξασ −1 (1−ξ)

(1−ξα)σ −1

B(ξα ,σ)

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Optimal moment set

OPTIMAL MOMENT SET

1. An optimal moment set consists of N(M +1) distinct moments.

2. An optimal moment set will result in a full-rank square matrix A for allpossible sets of N distinct, non-degenerate abscissas.

3. An optimal moment set includes all linearly independent moments of aparticular order γi before adding moments of higher order.

4. An optimal moment set must result in a perfectly symmetric treatment of theinternal coordinates.

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Optimal moment set

Table: Moments used to build a bivariate quadrature approximation (M = 2) for N = 2. In thiscase M0,3 is chosen as the third-order moment to saturate the degrees of freedom.

M(2,0)M(1,0)M(0,0) M(0,1) M(0,2) M(0,3)

Table: Moments used to build a bivariate quadrature approximation (M = 2) for N = 3. In thiscase M2,1, M1,2 and M0,3 are chosen among the third-order moments to saturate the degreesof freedom.

M(2,0) M(2,1)M(1,0) M(1,1) M(1,2)M(0,0) M(0,1) M(0,2) M(0,3)

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Optimal moment set

Table: Optimal moment set used to build a bivariate quadrature approximation (M = 2) forN = 4. Only when N1/M is an integer, there exists an optimal moment set (that fulfills thesymmetry requirement).

M(3,0) M(3,1)M(2,0) M(2,1)M(1,0) M(1,1) M(1,2) M(1,3)M(0,0) M(0,1) M(0,2) M(0,3)

Table: Optimal moment set used to build a bivariate quadrature approximation (M = 2) forN = 9.

M(5,0) M(5,1) M(5,2)M(4,0) M(4,1) M(4,2)M(3,0) M(3,1) M(3,2)M(2,0) M(2,1) M(2,2) M(2,3) M(2,4) M(2,5)M(1,0) M(1,1) M(1,2) M(1,3) M(1,4) M(1,5)M(0,0) M(0,1) M(0,2) M(0,3) M(0,4) M(0,5)

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Multivariate Conditional QMOM

Ï Methods based on conditional density functions:n(ξ1,ξ2) = n1(ξ1)f21(ξ2|ξ1) = n2(ξ2)f12(ξ1|ξ2)

Ï Univariate quadrature (N1) calculated from the first 2N1 −1:M0,0,...,0,0

...M2N1−1,0,...,0,0

PD/Wheeler−−−−−−−−−→

w1

...wN1

ξ1;1

...ξ1;N1

.

resulting for example in: n(ξ1,ξ2) =∑N1α1=1 wα1δ

(ξ1 −ξ1;α1

)f21(ξ2|ξ1)

Ï The generic moment becomes:

Mk1 ,k2=

Ïn(ξ1,ξ2)ξk1

1 ξk22 dξ1 dξ2

=N1∑α1=1

wα1ξk11;α1

∫f21(ξ2|ξ1;α)ξk2

2 dξ2

Ï Conditional moment:⟨ξ

k22

⟩α1

= ∫f12(ξ2|ξ1;α1 )ξk2

2 dξ2

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For each of these N1 nodes, 2N2 −1 conditional moments are calculated, andunivariate quadratures N2 are determined (in direction ξ2): Conditional QMOMor CQMOM

1;1ξξξξ1ξξξξ

2ξξξξ

1,1;2ξξξξ

(((( ))))1w

(((( ))))1;1w

2;1ξξξξ(((( ))))2w

11;1 −−−−Nξξξξ(((( ))))11 −−−−Nw

1;1 Nξξξξ(((( ))))

1Nw

2,1;2ξξξξ(((( ))))2;1w

1,1;2 2 −−−−Nξξξξ(((( ))))1;1 1 −−−−Nw

2,1;2 Nξξξξ(((( ))))1;1 Nw

1,2;2ξξξξ (((( ))))1;2w

2,2;2ξξξξ (((( ))))2;2w

(((( ))))1;2 2 −−−−Nw

2,2;2 Nξξξξ (((( ))))2;2 Nw

1,2;2 2 −−−−Nξξξξ

1,;2 1Nξξξξ (((( ))))1;1Nw

2,;2 1Nξξξξ (((( ))))2;1Nw

(((( ))))1; 21 −−−−NNw

21 ,;2 NNξξξξ (((( ))))21 ;NNw

1,;2 21 −−−−NNξξξξ

1,1;2 1 −−−−Nξξξξ(((( ))))1;11 −−−−Nw

2,1;2 1 −−−−Nξξξξ(((( ))))2;11 −−−−Nw

(((( ))))1;1 21 −−−−−−−− NNw

21 ,1;2 NN −−−−ξξξξ (((( ))))21 ;1 NNw −−−−

1,1;2 21 −−−−−−−− NNξξξξ

L

L

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Table: Moments used to build a bivariate quadrature approximation (M = 2) for N1 = N2 = 3using CQMOM with ξ2 conditioned on ξ1 (top) and ξ1 conditioned on ξ2 (bottom).

M(5,0)M(4,0)M(3,0)M(2,0) M(2,1) M(2,2) M(2,3) M(2,4) M(2,5)M(1,0) M(1,1) M(1,2) M(1,3) M(1,4) M(1,5)M(0,0) M(0,1) M(0,2) M(0,3) M(0,4) M(0,5)

M(5,0) M(5,1) M(5,2)M(4,0) M(4,1) M(4,2)M(3,0) M(3,1) M(3,2)M(2,0) M(2,1) M(2,2)M(1,0) M(1,1) M(1,2)M(0,0) M(0,1) M(0,2) M(0,3) M(0,4) M(0,5)

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Inhomogeneous Quadrature Method of Moments

Ï The spatially inhomogeneous GPBE operating on n(ξ) reads as follows:

∂n

∂t+ ∂

∂x· (⟨Up|ξ⟩n

)= ∂

∂x·(D(ξ)

∂n

∂x

)− ∂

∂ξ

(⟨Gp|ξ⟩n)+S (43)

Ï The application of the moment transform, Mk = ∫n(ξ)ξk dξ, generates several

closure problems:

∂Mk

∂t+ ∂

∂x·(Uk

pMk

)= ∂

∂x·(Dk ∂Mk

∂x

)+Sk (44)

where of course Ukp =

∫ ⟨Up|ξ⟩n(ξ)ξk dξMk

(similarly Dk) and

S k = ∫ (∂∂ξ

(⟨Gp|ξ⟩n)+S

)ξk dξ.

Ï The solution with QMOM is as usual ...

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Quadrature Method of Moments

Ï From M0,M1, . . . ,M2N−1 the Gaussian quadrature with N nodes isconstructed resulting in the following approximation

Mk(t,x) =∫Ωξ

n(t,x,ξ)ξk dξ≈N∑α=1

wα(t,x) (ξα(t,x))k (45)

Ï that can be used to overcome the different closure problems, for example:

Ukp(t,x) =

∫Ωξ

⟨Up|ξ⟩n(t,x,ξ)ξk dξ

Mk≈

∑Nα=1⟨Up|ξα⟩wαξ

kα

Mk(46)

Ï By using different velocities, Ukp(t,x), for the different moments, we can

describe mixing and segregation patterns

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Moreover if wα and ξα are continuous functions of space and time, DQMOM canalso be applied (slightly different now due to the diffusion term):

∂wα

∂t+ ∂

∂x· (⟨Up|ξα⟩wα

)= ∂

∂x·(D(ξα)

∂wα

∂x

)+aα (47)

∂wαξα

∂t+ ∂

∂x· (⟨Up|ξα⟩wαξα

)= ∂

∂x·(D(ξα)

∂wαξα

∂x

)+bα (48)

with α ∈ 1, . . . ,N and where the source terms are as usual determined by solvingthe following linear system:

(1−k)N∑α=1

ξkαaα+k

N∑α=1

ξk−1α bα = Sk +Ck, (49)

where

Ck = k(k−1)N∑α=1

ξk−2α Cα, Cα = wαD

(∂ξα

∂x· ∂ξα∂x

), (50)

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Spatial discretization and moment corruption

Ï A moment set is said to be valid or realizable, if the Hankel-Hadamard determinantsare all non-negative:

∆k,l =

∣∣∣∣∣∣∣∣∣∣

Mk Mk+1 . . . Mk+lMk+1 Mk+2 . . . Mk+l+1

......

......

Mk+l Mk+l+1 . . . Mk+l+l

∣∣∣∣∣∣∣∣∣∣≥ 0, k = 0,1, l ≥ 0 (51)

for k = 0,1 and l ≥ 0.

Ï For l = 1 we get:MkMk+2 −M2

k+1 ≥ 0 k = 0,1,2, . . . . (52)

which for k = 0 becomes the constraint of positive variance

Ï Equivalently this becomes:

ln(Mk)+ ln(Mk+2)

2≥ ln(Mk+1) k = 0,1,2, . . . ; (53)

or in other words convexity of the function ln(Mk) with respect to k

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Less stringent condition: convexity of function ln(Mk) withrespect to k

Ad-hoc correction algorithms are needed!

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Spatial discretization in QMOM

An alternative is to solve the transport equation for a generic moment Mk withdiscretization schemes that PRESERVE the moments ("Realizable schemes"):

∂Mk

∂t+

Up∂Mk

∂x= Sk −Up

∂Mk

∂x

After spatial discretization (finite-volume):

W P E

Ub

x

dMPk

dt= S

Pk − Up

∆x

(Me

k −Mwk

)

With first-order upwind Mek = MP

k and Mwk = MW

k

Spatial discretization schemes based on first-order upwind always result inVALID moments. Higher-order schemes (CDS, second-order upwind, QUICK,MUSCL) can result in INVALID moments.

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Spatial discretization in QMOM

Ï One solution would be to evaluate the moments at the faces Mek and Mw

kthrough the quadrature approximation

Ï We know the value of the moments at the center of the cells MWk , MP

k , MEk

W P E

Ub

x

Ï From these moments we can evaluate the corresponding weights wPα and

abscissas ξPα

Ï If weights at the center of the face are interpolated with pth-order spatialreconstruction and the abscissas are interpolated 1st-order spatial theresulting moments will be valid

Ï This allows to improve the numerical accuracy preserving the moments!

Ï Alternatively one can use semi-Lagrangian schemes (Attili and Bisetti, 2013)

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Spatial discretization in DQMOM

Ï In QMOM the governing equations are for the moments that are ‘conserved’variables

Ï In DQMOM the governing equations are for weights and weighted abscissasthat are ‘primitive’ variables

Ï When QMOM is used (if proper discretization schemes are used) only thetransported 2N moments are preserved and conserved (and their linearcombination)

Ï When DQMOM is used only weights and weighted abscissas are conservedand their linear combination: M0 and M1

Ï One disadvantage of DQMOM is that only two moments (of the 2N selected)are conserved and saved from numerical errors!

Ï Another way to look at the problem is to consider that the equations infinite-volume codes are solved with an error called numerical diffusionwhose coefficient is unknown!

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Ï The original DQMOM requires the solution of these equations:

∂wα

∂t+Up

∂wα

∂x= aα

∂wαξα

∂t+Up

∂wαξα

∂x= bα

m∂Mk

∂t+Up

∂Mk

∂x= Sk

Ï When the finite-volume discretization is applied:

dwPα

dt+ Up

∆x

(weα−Mw

k

) = aPα

d(wαξα)P

dt+ Up

∆x

((wαξα)e − (wαξα)w) = bP

α

mdMP

k

dt+ Up

∆x

(Me

k −Mwk

)= S

Pk

failing in conserving higher-order moments (k ≥ 2)

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Ï Solution: fully conservative version of DQMOM ( DQMOM-FC):

dMPk

dt= −Up

∆x

(Me

k −Mwk

)+S

Pk

mdwP

α

dt= −aP

c,α+aPα

d(wαξα)P

dt= −bP

c,α+bPα

now successfully conserving all the moments of the NDF

Ï Alternatively one can use semi-Lagrangian schemes (Attili and Bisetti, 2013)

Ï Summarizing if the equations for the moments have large physical diffusion termsDQMOM can be safely used (numerical diffusion will be smaller than real diffusion andthe solution will contain no discontinuities and no shocks)

Ï When only the equation for the moments do not contain any physical diffusion termthan DQMOM-FC should be used

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The convexity of the function ln(Mk) with respect to k can be easily verified bybuilding a difference table of ln(Mk).Example: VALID SET; moment of a Gaussian distribution (M0 = 1, M1 = 5, M2 = 26,M3 = 140, M4 = 778, M5 = 4450, M6 = 26140, M7 = 157400 )

k d0 = ln(Mk) d1 d2 d30 0 1.609 0.039 -0.00431 1.609 1.648 0.034 -0.00332 3.258 1.683 0.031 -0.00273 4.941 1.715 0.028 -0.00224 6.656 1.743 0.026 -0.00195 8.400 1.770 0.024 06 10.171 1.795 0 07 11.966 0 0 0

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The convexity of the function ln(Mk) with respect to k can be easily verified bybuilding a difference table of ln(Mk).Example: INVALID SET; moment of a Gaussian distribution (M0 = 1, M1 = 5,M2 = 25, M3 = 140, M4 = 778, M5 = 4450, M6 = 26140, M7 = 157400)

k d0 = ln(Mk) d1 d2 d30 0 1.609 0 0.1131 1.609 1.609 0.113 -0.1212 3.218 1.722 -0.007 0.0363 4.941 1.715 0.028 -0.0024 6.656 1.743 0.026 -0.0015 8.400 1.770 0.024 06 10.171 1.795 0 07 11.966 0 0 0

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Ï When using first-order upwind spatial discretization schemes and first-orderexplicit time discretization schemes the validity of the moment set should bepreserved

Ï In all the other cases it is very easy to CORRUPT the moment set and it isanyway safe to have algorithms that DETECT CORRUPTION AND CORRECTinvalid moment set

Ï If we transform the moment set so that d2 is positive, we are almost sure thatthe moment set is valid

Ï But how positive?

Ï The moments of a log-normal distribution have the smallest d3

n(ξ) = NT

σp

2πξexp

(−(

ln(ξ)−µ)2

2σ2

), (54)

Mk = NT exp

(kµ+ k2σ2

2

), (55)

Ï The log-normal distribution is the smoothest distribution!

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CORRECTION ALGORITHM BY MCGRAW

1. Build the difference table and check if d2 is negative

2. Identify the moment order k that causes the biggest change in d3

3. Change the moment (by multiplying it for a constant) in order to MINIMIZEd3

4. Go back to point 1

CORRECTION ALGORITHM BY WRIGHT

1. Build the difference table and check if d2 is negative

2. Replace the moments with those of a log-normal distribution with

µ= j

ij− i2ln

(Mi

M0

)+ i

ij− j2ln

( Mj

M0

)(56)

σ2 = 1

1− i/j

[2

j2ln

( Mj

M0

)− 2

ijln

(Mi

M0

)](57)

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QBMM approximation vs DVM, Grad, oLBM – 4

M = 4Normalized relaxation rate


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M = 4BGK-equivalent relaxation time νp(t) = dΦp

dt1

Φeqp −Φp


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Analytical equations for QBMM

dΦp

dt= πp

2

(M/2∑i=1

M/2∑j=1

wiwjΛ+ij,p −

M/2∑i=1

M/2∑j=1

wiwjΛ−ij,p

)

δij =√

Ej/Ei ; qij = q(x,y,Ei ,Ej); C+ij,p = C+

p (x,y,Ei ,Ej);

C−i,p = C−

p (Ei); Λ+ij,p = ∫ +1

−1∫ +1−1 |qij |C+

ij,p dx dy; Λ−ij,p = ∫ +1

−1∫ +1−1 |qij |C−

i,p dx dy

Λ+ij,p = 2E

pi

√E+

p∑α=0

(p

α

)p−α∑β=0

(p−αβ

)(−1)γij E

−β−αi Eα+ Eβ− 2

2β+1

1− r2α+2ij

2α+2+

r2α+2ij

2α+2β+3

+ 1

β+1

r2α+2ij

2α+1−

r2α+2ij

2α+2β+3

Λ−

ij,p = 2Epi

√E+

1+r2

ij

3

γij =

α ifEi ≥ Ej

β ifEi < EjE+ = max

(Ei ,Ej

); E− = min

(Ei ,Ej

); rij = min

(√EiEj

,

√EjEi

)

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Turbulent reactive flows

Ï Ns species denoted by capital letters A,B,C, . . . ,R,S

Ï Vector of concentrationsφ= (φ1, . . . ,φNs )

Ï Consider a simple reaction

A+Bk−→ R

Ï Transport equations for concentrations with source terms

SA =−kφAφB = SB

Ï In turbulent flows a RANS average (in time) or LES filter (in space) isperformed

⟨SA⟩ =−k⟨φAφB⟩ 6= −k⟨φA⟩⟨φB⟩

Ï In general the term ⟨S(φ)⟩ 6= S(⟨φ⟩)

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Ï The chemical source term can be closed if we assume the existence of a jointprobability density function of the concentrations

⟨S(φ)⟩ =∫

S(ψ)P(ψ;x, t)dψ

Ï Rigorously, when using LES we are dealing with a filtered-density function

P(ψ;x, t) =∫δ(ψ−φ(x, t))G(r−x)dr

where G is the LES filter

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Ï For the box filter this could be no more a PDF so we simply assume a PDF Pthat represents the spatial dishomogeneity in the cells such that∫

ψP(ψ;x, t)dψ= ⟨φ⟩

is the filtered scalar and∫(ψ2

α−⟨ψα⟩2)P(ψα;x, t)dψα = ⟨φ′2α ⟩

the scalar fluctuation

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PDF Transport Equation

Ï P can be solved in a advection-reaction-diffusion equation with Ns +4independent variables

Ï The turbulent diffusion term comes from the assumption that theconditional fluctuations ⟨U ′

i |ψ⟩P =−DT∂P∂xi

Ï Molecular mixing term needs to be modeled

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Mixing models

Ï The most simple and popular model for Eulerian simulations is the IEM

⟨D∇2φα|ψ⟩ =−Cφ

τ(ψ−⟨φ⟩)

Ï It is a linear relaxation of the scalars to the mean value with time scale τ and aparameter Cφ

Ï τ is chosen to be a turbulence time scale, 2kε for RANS and 2∆2

D+DTwhere ∆ is

the filter width

Ï Cφ is the scalar-to-mechanical time-scale ratio and it depends on the localSchmidt and Reynolds numbers (for gases at high Re Cφ ≈ 2)

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PDF discretization

Ï In the Quadrature Method of Moments (QMOM) the integrals areapproximated using a quadrature rule∫

g(ψ)P(ψ;x, t)dψ≈M∑

i=1g(φi)wi

whereφi are the abscissas and wi the weights of the quadrature rule

Ï For a given set of 2M moments, the M abscissas and weights can becalculated using inversion algorithms (Wheeler or Product-Difference)

Ï This means that we are approximating the exact PDF P with amulti-environment PDF f

fφ(ψ;x, t) =M∑

i=1wi(x, t)δ[ψ−φi(x, t)]

where δ is a multi-dimensional delta function

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DQMOM-IEM

Ï In the DQMOM transport equations for wi and wiψi are solved instead ofequations for µi

Ï M(1+N) equations with some constraints (e.g.∑M

i=1 wi = 1)

Ï The source term is such that the first M(1+N) moments are coherent withthe transported ones

Ï Let us consider a competitive reaction scheme simplified using a mixturefraction ξ and a reaction progress Y (linear combination of speciesconcentration). This results in N = M = 2 andφ= (ξ,Y )

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DQMOM-IEM

∂w1

∂t+U i

∂w1

∂xi− ∂

∂xi

(DT

∂w1

∂xi

)= 0,

w2 = 1−w1

∂w1ξ1

∂t+U i

∂w1ξ1

∂xi− ∂

∂xi

(DT

∂w1ξ1

∂xi

)

=Cφ

τw1w2[ξ2 −ξ1]+ DT

ξ1 −ξ2

(w1

∂ξ1

∂xi

∂ξ1

∂xi+w2

∂ξ2

∂xi

∂ξ2

∂xi

),

∂w2ξ2

∂t+U i

∂w2ξ2

∂xi− ∂

∂xi

(DT

∂w2ξ2

∂xi

)

=Cφ

τw1w2[ξ1 −ξ2]+ DT

ξ2 −ξ1

(w1

∂ξ1

∂xi

∂ξ1

∂xi+w2

∂ξ2

∂xi

∂ξ2

∂xi

),

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DQMOM-IEM

∂w1Y1

∂t+U i

∂w1Y1

∂xi− ∂

∂xi

(DT

∂w1Y1

∂xi

)

= w1S(ξ1 ,Y1)+Cφ

τw1w2[Y2 −Y1]

+ DT

Y1 −Y2

(w1

∂Y1

∂xi

∂Y1

∂xi+w2

∂Y2

∂xi

∂Y2

∂xi

),

∂w1Y2

∂t+U i

∂w1Y2

∂xi− ∂

∂xi

(DT

∂w1Y2

∂xi

)

= w2S(ξ2 ,Y2)+Cφ

τw1w2[Y1 −Y2]

+ DT

Y2 −Y1

(w1

∂Y1

∂xi

∂Y1

∂xi+w2

∂Y2

∂xi

∂Y2

∂xi

).

Only the progress variable Y has a source term S

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Poly-dispersed particles

Particle Size Distribution n(L;x, t) =∫R3

n(L,Up;x, t)dUp

St = τp

τf≈ ρpD2

18µτf

Particles of different size behave very differently with a non-linear relationbetween size and relative velocity

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Poly-dispersed particles

Quadrature-based Moment Method (QBMM)nodes and weights to approximate f (L)

2 moments → 1 node, 1 weightMean size

4 moments → 2 nodes, 2 weightsVolume fraction

6 moments → 3 nodes, 3 weights

8 moments → 4 nodes, 4 weights

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Real-space advection

Population balance equation:

∂t n+∂x⟨Up|L⟩n+: no growth term∂LnG =:

no aggregation,breakage,nucleation

Q(n,n)

Models for conditional velocity ⟨Up|L⟩ = Up(L)

PSEUDO-HOMOGENEOUS MODEL, St ¿ 1Particles flow with fluid velocity Up(L) = U

ALGEBRAIC MODELS (MIXTURE), St ≤ 1Up(L) is calculated with algebraic relations from the knowledge of U

DIFFERENTIAL MODELS (MULTI-FLUID), St > 1Up(L) is solved with a momentum balance equation

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Equilibrium Model (Ferry/Balachandar)

Ï similar to the classical Algebraic Slip Model

Ï expansion of Maxey-Riley equation for small particle response time τp

Up ≈ U+O(τp) ⇒ Up −U =−τp

(DU

Dt−g

)

Ï 1st order correction

Up −U =−τp

(I+τp∇UT

)−1(

DU

Dt−g

)Ï High Rep effects considered using a modified τ∗pÏ can be extended to take into account other forces and two-way coupling

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SGS model for particles

In Large Eddy Simulation (LES) we have only filtered velocity field U

APPROXIMATE DECONVOLUTION METHOD (ADM)To estimate the “real” unfiltered velocity we use an approximate inverse filter

U∗i =

Nc∑k=0

(I−G)k Ui

Ï G is a test filter

Ï Nc is the deconvolution order

Ï if Nc = 1 it is like an inverse dynamic model

Ï considered Nc = 1,5 with Gaussian weights

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PBE approximation

Ï Single momentum equation for the mixture

Ï Equilibrium model to calculate velocity Up(L)

Ï Population balance discretized with Quadrature-Based Moment Method(QBMM)

Ï Solved directly in terms of nodes Li and weights wi (DQMOM formulation)

SINGLE VELOCITY MODEL

All particles flow with an overall mean velocity Up(L) = Up(L)

MULTIPLE VELOCITIES MODEL

Each node (particle class) considered as a separate phase with its own relativevelocity

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Turbulent poly-dispersed channel flow

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multiphase flows in complex geometries: a uq perspective workshop/uqaw_2015_ta… · stochastic...

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