Quadrature-Based Moment Methods for Kinetic Models Rodney O. Fox - - PowerPoint PPT Presentation

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Quadrature-Based Moment Methods for Kinetic Models Rodney O. Fox - - PowerPoint PPT Presentation

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Quadrature-Based Moment Methods for Kinetic Models Rodney O. Fox Department of Chemical and Biological Engineering Iowa State University, Ames, Iowa, USA & cole Centrale Paris, France Workshop on Moment Methods in Kinetic Theory II

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Quadrature-Based Moment Methods for Kinetic Models

Rodney O. Fox

Department of Chemical and Biological Engineering Iowa State University, Ames, Iowa, USA & École Centrale Paris, France

Workshop on Moment Methods in Kinetic Theory II Fields Institute, University of Toronto October 14-17, 2014

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 1 / 45

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Introduction Target Applications

Application: polydisperse multiphase flow

continuous phase disperse phase size distribution finite particle inertia collisions variable mass loading multiphase turbulence

Bidisperse gas-particle flow (DNS of S. Subramaniam)

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 2 / 45

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Introduction Target Applications

Application: polydisperse multiphase flows

Bubble columns Power stations Brown-out Volcanos Jet break up Spray flames

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 3 / 45

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Introduction Target Applications

Modeling challenges

Strong coupling between continuous and disperse phases Wide range of particle volume fractions (even in same flow!) Inertial particles with wide range of Stokes numbers Collision-dominated to collision-less regimes in same flow Granular temperature can be very small and very large in same flow Particle polydispersity (e.g. size, density, shape) is always present

Need a modeling framework that can handle all aspects!

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 4 / 45

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Introduction Kinetic-based models

Overview of kinetic modeling approach

Microscale Model

Direct numerical simulation

Macroscale Model

Hydrodynamic description Euler-Euler models

Mesoscale Model

Kinetic equation Euler-Lagrange models Volume or ensemble averages + closures for “fluctuations” Kinetic theory + density function closures Moments of density + moment closures

Mesoscale model incorporates more microscale physics in closures!

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 5 / 45

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Introduction Kinetic-based models

Types of mesoscale transport (kinetic) equations

Population balance equation (PBE): n(t, x, ξ)

∂n ∂t + ∂ ∂xi [ui(t, x, ξ)n] + ∂ ∂ξj [Gj(t, x, ξ)n] = ∂ ∂xi

  • D(t, x, ξ) ∂n

∂xi

  • + S

with known velocity u, acceleration G, diffusivity D and source S Kinetic equation (KE): n(t, x, v)

∂n ∂t + ∂ ∂xi (vin) + ∂ ∂vi [Ai(t, x, v)n] = C

with known acceleration A and collision operator C Generalized population balance equation (GPBE): n(t, x, v, ξ)

∂n ∂t + ∂ ∂xi (vin) + ∂ ∂vi [Ai(t, x, v, ξ)n] + ∂ ∂ξj [Gj(t, x, v, ξ)n] = C

with known accelerations A, G and collision/aggregation operator C

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 6 / 45

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Introduction Kinetic-based models

Moment transport equations

PBE: Mk =

  • ξkn dξ

∂Mk ∂t + ∂ ∂x

  • ξkun dξ
  • = k
  • ξk−1Gn dξ + ∂

∂x

  • ξkD∂n

∂x dξ

  • +
  • ξkS dξ

KE: Mk =

  • vkn dv

∂Mk ∂t + ∂Mk+1 ∂x = k

  • vk−1An dv +
  • vkC dv

GPBE: Mkl =

  • vkξln dvdξ

∂Mkl ∂t + ∂Mk+1l ∂x = k

  • vk−1ξlAn dvdξ + l
  • vkξl−1Gn dvdξ +
  • vkξlC dvdξ

Terms in red will usually require mathematical closure

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 7 / 45

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Introduction Kinetic-based models

Closure with moment methods

Kinetic Equation Moment Equations Moments M(t,x)

Integrate over phase space Closure using quadrature Reconstruct using quadrature

Close moment equations by reconstructing density function

Density n(t,x,v) Reconstructed density n*(t,x,v)

Integrate over phase space 6-D solver 3-D solver

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 8 / 45

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Quadrature-Based Moment Methods Basic idea

Quadrature-based moment methods (QBMM)

Basic idea: Given a set of transported moments, reconstruct the number density function (NDF)

Things to consider: Which moments should we choose? What method should we use for reconstruction? How can we extend method to multivariate phase space? How should we design the numerical solver for the moments?

We must be able to demonstrate a priori that numerical algorithm is robust and accurate!

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 9 / 45

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Quadrature-Based Moment Methods Gauss quadrature in 1-D (real line)

Gauss quadrature in 1-D (real line)

The formula

  • g(v)n(v) dv =

N

  • α=1

nαg(vα) + RN(g) is a Gauss quadrature iff the N nodes vα are roots of an Nth-order

  • rthogonal polynomial PN(v) (⊥ with respect to n(v))

Recursion formula for PN(v): Pα+1(v) = (v − aα)Pα(v) − bαPα−1(v), α = 0, 1, 2, . . . Inversion algorithm (QMOM) for moments Mk =

  • vkn(v)dv:

{M0, M1, . . . , M2N−1} hard = ⇒ {a0, a1, . . . , aN−1}, {b1, b2, . . . , bN−1}

easy

= ⇒ {n1, n2, . . . , nN}, {v1, v2, . . . , vN}

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 10 / 45

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Quadrature-Based Moment Methods Szegö quadrature on unit circle

Szegö quadrature on unit circle

If n(φ) is periodic on the unit circle:

π

−π

g(eiφ)n(φ) dφ =

N

  • α=1

nαg(eiφα) + RN(g)

is a Szegö quadrature iff the N nodes zα = eiφα are zeros of an Nth-order para-orthogonal polynomials BN(z) Trigonometric moments:

cos(nφ) = π

−π

1 2(zn + z−n)n(φ) dφ, sin(nφ) = π

−π

1 2(zn − z−n)n(φ) dφ

are natural choice for reconstruction Except for special case [n(φ) symmetric wrt 0], no fast inversion algorithm is available to find nα and φα

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 11 / 45

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Quadrature-Based Moment Methods Quadrature method of moments

1-D quadrature method of moments (QMOM)

Use Gaussian quadrature to approximate unclosed terms in moment equations:

dM dt =

  • S(v)n(v)dv ≈

N

  • α=1

nαS(vα)

where M = {M0, M1, . . . , M2N−1} and S is “source term” Exact if S is polynomial of order ≤ 2N − 1 Provides good approximation for most other cases with small N ≈ 4 Complications arise in particular cases (e.g. spatial fluxes) In all cases, moments M must remain realizable for moment inversion N.B. equivalent to reconstructed N-point distribution function:

n∗(v) =

N

  • α=1

nαδ(v − vα)

= ⇒ realizable if nα ≥ 0 for all α

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 12 / 45

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Quadrature-Based Moment Methods Quadrature in multiple dimensions

Quadrature in multiple dimensions

No method equivalent to Gaussian quadrature for multiple dimensions! Given a realizable moment set M = {Mijk : i, j, k ∈ 0, 1, . . . }, find nα and vα such that Mijk =

  • vi

1vj 2vk 3n(v)dv = N

  • α=1

nαvi

1αvj 2αvk 3α

What moment set to use? If M corresponds to an N-point distribution, then method should be exact Avoid brute-force nonlinear iterative solver (poor convergence, ill-conditioned, too slow, . . . ) Algorithm must be realizable (i.e. non-negative weights, . . . ) Strategy: choose an optimal moment set to avoid ill-conditioned systems

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 13 / 45

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Quadrature-Based Moment Methods Quadrature in multiple dimensions

Brute-force QMOM (2-D phase space)

Given 3n2 bivariate optimal moments (n = 2):

M00 M01 M02 M03 M10 M11 M12 M13 M20 M21 M30 M31

Solve 12 moment equations:

4

  • α=1

|nα|ui

αvj α = Mij

to find {n1, . . . , n4; u1, . . . , u4; v1, . . . , v4}

Problem: iterative solver converges slowly (or not at all) Problem: system is singular for (nearly) degenerate cases

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 14 / 45

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Quadrature-Based Moment Methods Quadrature in multiple dimensions

Conditional QMOM (2-D phase space)

Conditional density function and conditional moments (2-D)

n(u, v) = f(v|u)n(u) = ⇒ Vk|U = u =

  • vkf(v|u) dv

1-D adaptive quadrature for U direction (n = 2)

Uk = Mk0, k ∈ {0, 1, 2, 3} = ⇒ find weights ρi, abscissas ui

Solve linear systems for conditional moments Vk|ui:

ρ1 ρ2 ρ1u1 ρ2u2 Vk|u1 Vk|u2

  • =

Vk UVk

  • =

M0k M1k

  • for k ∈ {1, 2, 3}

In principle, CQMOM controls 10 of 12 optimal moments:

M00 M01 M02 M03 M10 M11 M12 M13 M20 M30

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 15 / 45

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Quadrature-Based Moment Methods Quadrature in multiple dimensions

Conditional QMOM (cont.)

1-D adaptive quadrature in V direction for each i:

Vk|ui, k ∈ {0, 1, 2, 3} = ⇒ find weights ρij, abscissas vij

Adaptive quadrature sets some ρij = 0 if subset of conditional moments are not realizable Reconstructed density:

n∗(u, v) =

i

  • j ρiρijδ(u − ui)δ(v − vij)

Conditioning on V = vi uses 10 of 12 optimal moments: M00 M01 M02 M03 M10 M11 M20 M21 M30 M31 Union of two sets = ⇒ optimal moment set Extension to higher-dimensional phase space is straightforward

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 16 / 45

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Quadrature-Based Moment Methods Quadrature in multiple dimensions

Optimal moment set

Moments needed for all CQMOM permutations = ⇒ Optimal moment set N = 4 nodes in 2-D M00 M10 M20 M30 M01 M11 M21 M31 M02 M12 M03 M13 12 moments N = 9 nodes in 2-D M00 M10 M20 M30 M40 M50 M01 M11 M21 M31 M41 M51 M02 M12 M22 M32 M42 M52 M03 M13 M23 M04 M14 M24 M05 M15 M25 27 moments

Only optimal moment set is transported

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 17 / 45

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Quadrature-Based Moment Methods Quadrature in multiple dimensions

Examples of 2-D quadrature

(a) ρ = 0 and N = 4 (b) ρ = 0 and N = 9 (c) ρ = 0.5 and N = 4 (d) ρ = 0.5 and N = 9

QBMM approximations for bivariate Gaussian with ρ = 0 (top) and ρ = 0.5 (bottom) for N = 4 (left) and N = 9 (right). Brute-force QMOM (green diamond) Tensor-product QMOM (blue circle) CQMOM (red square)

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 18 / 45

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Quadrature-Based Moment Methods CQMOM on unit circle

CQMOM on unit circle

For n(φ) on the unit circle, define x = cos φ and y = sin φ Conditional pdf is known exactly n(x, y) = n(x)f(y|x)

f(y|x) = w1(x)δ

  • y −
  • 1 − x2
  • + w2(x)δ
  • y +
  • 1 − x2
  • with one unknown w1(x) (w2 = 1 − w1)

Apply QMOM with 2N moments cosn φ to find nα and xα = cos φα CQMOM requires conditional moment y|xα to find w1(xα) Apply CQMOM with N moments sin φ cosn φ to find w1α = w1(xα) Gauss/Swegö quadrature for symmetric ndf, otherwise fast, realizable reconstruction

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 19 / 45

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Quadrature-Based Moment Methods CQMOM on unit circle

Example of CQMOM on unit circle

8-pt Gauss quadrature 16-pt CQMOM quadrature Exactly reproduces trigonometric moments up to N = 8

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 20 / 45

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Quadrature-Based Moment Methods Extended quadrature

Extended quadrature method of moments (EQMOM)

Can we improve reconstructed distribution using kernel density functions?

n(v) =

N

  • i=1

niδσ(v, vi)

with N weights ni ≥ 0, N abscissas vi but only one spread parameter σ ≥ 0 Gaussian (−∞ < v < +∞):

δσ(v, vi) ≡ 1 √ 2πσ2 exp

  • −(v − vi)2

2σ2

  • Beta (0 < v < 1): with λi = vi/σ and µi = (1 − vi)/σ

δσ(v, vi) ≡ vλi−1(1 − v)µi−1 B(λi, µi)

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 21 / 45

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Quadrature-Based Moment Methods Extended quadrature

EQMOM algorithm

2N + 1 moments of n(v) (denote by mk) for beta-EQMOM:

m0 = m∗ m1 = m∗

1

m2 = 1 1 + σ (σm∗

1 + m∗ 2 )

m3 = 1 (1 + 2σ)(1 + σ)

  • 2σ2m∗

1 + 3σm∗ 2 + m∗ 3

  • m4 =

1 (1 + 3σ)(1 + 2σ)(1 + σ)

  • 6σ3m∗

1 + 11σ2m∗ 2 + 6σm∗ 3 + m∗ 4

  • ≡ m†

2N(σ)

m = A(σ)m∗ m∗ = A(σ)−1m with m∗

k ≡ N i=1 nivk i (i.e. QMOM moments)

Given mk for k = 0, . . . , 2N

1

Guess σ

2

Solve for m∗ for k = 0, . . . , 2N − 1

3

Solve for ni and vi using 1-D quadrature with m∗

k for k = 0, . . . , 2N − 1

4

Compute m∗

2N and resulting estimate m† 2N

5

Iterate on σ until m2N = m†

2N

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 22 / 45

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Quadrature-Based Moment Methods Extended quadrature

Example: beta-EQMOM with N = 2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 4 6 8 10 12 14 16

ξ n(ξ)

1 0.1 0.01 0.001

n1 = n2 = 1/2, ξ1 = 1/3, ξ2 = 2/3 for different values of σ

First 2N moments always exact with max σ : m2N ≥ m†

2N(σ)

Converges to exact NDF as N → ∞ (Gavriliadis and Athanassoulis 2002)

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 23 / 45

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Quadrature-Based Moment Methods Extended quadrature

Closure with EQMOM

Unclosed integrals (given ni, vi and σ):

  • g(v)n(v)dv =

N

  • i=1

ni

  • g(v)δσ(v, vi)dv

Use Gaussian quadrature with known weights wij and abscissas vij:

  • g(v)δσ(v, vi)dv =

Mi

  • j=1

wijg(vij) where Mi can be chosen arbitrarily large to control error

Dual-quadrature representation of EQMOM:

n(v) =

N

  • i=1

Mi

  • j=1

niwijδ (v − vij) (Mi = 1 when σ = 0) = ⇒ exact for polynomials of order ≤ 2N

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 24 / 45

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Quadrature-Based Moment Methods EQMOM on unit circle

EQMOM on unit circle

For n(φ) on the unit circle, define x = cos φ and y = sin φ , and

n(x, y) =

N

  • α=1

nαδσφ(x, xα)fα(y|x)

where the kernel density δσφ(x, xα) is periodic wrt φ Conditional pdf is known exactly, but with constant weights:

fα(y|x) = w1αδ

  • y −
  • 1 − x2
  • + w2αδ
  • y +
  • 1 − x2
  • with w2α = 1 − w1α

Apply EQMOM with 2N + 1 moments cosn φ to find nα, xα and σφ Apply CQMOM with N moments sin φ cosn φ to find w1α

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 25 / 45

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Quadrature-Based Moment Methods EQMOM in multiple dimensions

Multivariate EQMOM

Example: 2-D case ⇒ Extended CQMOM

n(u, v) = n(u)f(v|u) =

N

  • α=1

nαδσu(u, uα)  

Nα

  • β=1

nαβδσv,α(v, vαβ)  

with N abscissas uα, N = N

α=1 Nα weights wαβ = nαnαβ ≥ 0 and N

abscissas vαβ, but only one parameter σu ≥ 0 and N parameters σv,α Define moments:

Mij =

  • uivjn(u, v) du dv =

N

  • α=1

Nα

  • β=1

wαβm(α)

1,i m(αβ) 2,j

where m(α)

1,i ≡

  • uiδσu(u, uα) du

m(αβ)

2,j

≡

  • vjδσv,α(v, vαβ) dv

are known functions of the EQMOM parameters

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 26 / 45

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Quadrature-Based Moment Methods EQMOM in multiple dimensions

Algorithm for 2-D ECQMOM

1-D EQMOM for moments in u:

Mi0 =

N

  • α=1

nαm(α)

1,i

for i = 0, . . . , 2N = ⇒ nα, uα and σu

Use CQMOM to find conditional moments Vjα ≡ Nα

β=1 nαβm(αβ) 2,j

from the bivariate moments (i.e. solve linear system):

N

  • α=1

nαm(α)

1,i Vjα = Mij

for i = 0, . . . , N − 1

For each α, apply 1-D EQMOM to conditional moments:

{1, Vα, . . . , V2Nαα} = ⇒ nαβ, vαβ and σv,α

Uses the extended optimal moment set

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 27 / 45

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Quadrature-Based Moment Methods EQMOM in multiple dimensions

Extended optimal moment set

All CQMOM permutations = ⇒ Extended optimal moment set N = 4 nodes in 2-D M00 M10 M20 M30 M40 M01 M11 M21 M31 M41 M02 M12 M03 M13 M04 M14 16 moments N = 9 nodes in 2-D M00 M10 M20 M30 M40 M50 M60 M01 M11 M21 M31 M41 M51 M61 M02 M12 M22 M32 M42 M52 M62 M03 M13 M23 M04 M14 M24 M05 M15 M25 M06 M16 M26 33 moments

Only extended optimal moment set is transported

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 28 / 45

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Quadrature-Based Moment Methods ECQMOM on unit sphere

ECQMOM on unit sphere

For n(φ, θ) on the unit sphere, ECQMOM reconstruction is

n(φ, θ) =

N

  • α=1

nαδσθ(θ, θα)  

Nα

  • β=1

nαβδσφ,α(φ, φαβ)  

with periodic kernel density functions for θ ∈ [0, π] and φ ∈ [−π, π] Define z = cos θ and conditional pdf f(φ|z) Apply EQMOM for 2N + 1 moments zn to find nα, θα and σθ Use CQMOM to find trigonometric moments involving φ conditioned on zα = cos θα For each α, apply EQMOM on unit circle to conditional moments to find nαβ, φαβ, σφα and w1αβ

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 29 / 45

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Quadrature-Based Moment Methods Summary of QBMM

Summary of QBMM

Extended optimal moment set = ⇒ reconstruct NDF with fast, robust algorithm NDF must be realizable and moment-inversion algorithm must be robust

CQMOM is always realizable by construction EQMOM gives a smooth NDF with low computational cost

Current “best” moment-inversion algorithms:

1-D phase space = ⇒ EQMOM Multivariate phase space = ⇒ multivariate ECQMOM

Dual-quadrature representation used to close source terms Given smooth NDF, high-order kinetic-based transport solvers can be derived to ensure realizability

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 30 / 45

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Kinetic-Based Finite-Volume Methods

Kinetic-based finite-volume methods (KBFVM)

Given a set of extended optimal moments, solve

∂Mkl ∂t + ∂Mk+1l ∂x = k

  • vk−1ξlAn dvdξ + l
  • vkξl−1Gn dvdξ +
  • vkξlC dvdξ

where RHS is closed using QBMM: ∂Mkl ∂t + ∂Mk+1l ∂x =

N

  • α=1

nα

  • kvk−1

α

ξl

αAα + lvk αξl−1 α Gα + vk αξl αCα

  • Things to consider:

How do we discretize the spatial fluxes? How do we update the moments in time? How can we ensure that the moments are always realizable?

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 31 / 45

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Kinetic-Based Finite-Volume Methods Kinetic-based spatial fluxes

Kinetic-based spatial fluxes

Spatial fluxes can use kinetic formulation: e.g. ∂tM00 + ∂xM10 = 0 M10 = Q−

10 + Q+ 10

=

−∞

u

  • n∗(u, v)dv
  • du +

∞ u

  • n∗(u, v)dv
  • du

Using reconstructed n∗, downwind and upwind flux components are Q−

10 = N

  • α=1

nαuαI(−∞,0) (uα) Q+

10 = N

  • α=1

nαuαI(0,∞) (uα) where IS(x) is the indicator function for the interval S

Kinetic-based fluxes are always hyperbolic

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 32 / 45

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SLIDE 33

Kinetic-Based Finite-Volume Methods Kinetic-based spatial fluxes

Finite-volume method: definitions

1-D advection problem: ∂M ∂t + ∂F(M) ∂x = 0 where M =

  • K(v)n(v)dv and F(M) =
  • vK(v)n(v)dv

Finite-volume representation of moment vector: Mn

i ≡ 1

∆x xi+1

xi

M(tn, x)dx Finite-volume formula: Mn+1

i

= Mn

i − λ

  • G
  • Mn

i+ 1

2,l, Mn

i+ 1

2,r

  • − G
  • Mn

i− 1

2,l, Mn

i− 1

2,r

  • where G (Ml, Mr) =
  • v+K(v)nl(v)dv +
  • v−K(v)nr(v)dv
  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 33 / 45

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SLIDE 34

Kinetic-Based Finite-Volume Methods Kinetic-based spatial fluxes

Realizability and spatial fluxes

Flux functions: given Mn

i define G (Ml, Mr) to achieve high-order

spatial accuracy but keep Mn+1

i

realizable! Discrete distribution function: Define Mn+1

i

≡

  • K(v)hi(v)dv

and finite-volume formula can be written as hi(v) = λ|v−|nn

i+ 1

2,r + λv+nn

i− 1

2,l + nn

i − λ|v−|nn i− 1

2 ,r − λv+nn

i+ 1

2,l

(black part ≥ 0, red part can be negative) Sufficient condition for realizable moments: hi(v) ≥ 0 for all v and i

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 34 / 45

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SLIDE 35

Kinetic-Based Finite-Volume Methods Quasi-high-order realizable schemes

Realizable, high-order, spatial fluxes

First order: nn

i− 1

2,r = nn

i+ 1

2,l = nn

i so that

h = λ|v−|nn

i+1 + λv+nn i−1 + (1 − λ|v−| − λv+) nn i

⇒

1 |v−|+v+ ≥ λ

Moments are realizable, but scheme is diffusive ... Quasi-higher order: Let nn

i = α ρn α,iδ(v − vn α,i)

and define

nn

i− 1

2 ,r =

α ρn α,i− 1

2 ,rδ(v − vn

α,i)

nn

i+ 1

2 ,l =

α ρn α,i+ 1

2 ,lδ(v − vn

α,i)

so that

h = λ|v−|nn

i+ 1

2 ,r + λv+nn

i− 1

2 ,l +

α

  • ρn

α,i − λ|v−|ρn α,i− 1

2 ,r − λv+ρn

α,i+ 1

2 ,l

  • δ(v − vn

α,i)

= ⇒ minα  

ρn

α,i

|v−

α,i|ρn α,i− 1 2 ,r+v+ α,iρn α,i+ 1 2 ,l

  ≥ λ

Use high-order, finite-volume schemes only for the weights

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 35 / 45

slide-36
SLIDE 36

Kinetic-Based Finite-Volume Methods Realizable time-stepping schemes

Realizable time-stepping schemes

First-order explicit:

Mn+1

i

= Mn

i − λ

  • G
  • Mn

i+ 1

2 ,l, Mn

i+ 1

2 ,r

  • − G
  • Mn

i− 1

2 ,l, Mn

i− 1

2 ,r

  • is realizable

Second-order Runga-Kutta (RK2) is not realizable RK2SSP:

M∗

i = Mn i − λ

  • G
  • Mn

i+ 1

2 ,l, Mn

i+ 1

2 ,r

  • − G
  • Mn

i− 1

2 ,l, Mn

i− 1

2 ,r

  • M∗∗

i

= M∗

i − λ

  • G
  • M∗

i+ 1

2 ,l, M∗

i+ 1

2 ,r

  • − G
  • M∗

i− 1

2 ,l, M∗

i− 1

2 ,r

  • Mn+1

i

= 1 2 (Mn

i + M∗∗ i )

is realizable

Achieve second order in space and time on unstructured grids

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 36 / 45

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SLIDE 37

Kinetic-Based Finite-Volume Methods Example

Bubbly flow

Loading movie.. .

Quasi-second-order realizable finite-volume scheme on unstructured mesh

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 37 / 45

slide-38
SLIDE 38

Kinetic-Based Finite-Volume Methods Summary of KBFVM

Summary of KBFVM

When solving moment transport equations, we must guarantee realizability First-order FV methods are realizable, but too diffusive Standard high-order FV methods lead to unrealizable moments Kinetic-based flux functions can be designed to be realizable Use dual-quadrature representation with high-order spatial reconstruction High-order time-stepping schemes are also possible KBFVM provide robust treatment of shocks/discontinuous solutions

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 38 / 45

slide-39
SLIDE 39

Conclusions Final remarks

Final remarks

Mesoscopic models have direct link with underlying physics and result in a kinetic equation QBMM solves kinetic equation by reconstructing distribution function from moments Reconstruction requires realizable moments Numerical schemes must ensure that moments are always realizable QBMM on unit sphere can be used for radiation transport

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 39 / 45

slide-40
SLIDE 40

Conclusions Collaborators and funding

Principal collaborators and funding

École Centrale Paris: C. Chalons,

  • F. Laurent, M. Massot, A. Vié

Iowa State University: V. Vikas,

  • Z. J. Wang, C. Yuan

Oakridge National Lab: C. Hauck US Department of Energy (Grant DE-FC26-07NT43098) US National Science Foundation (Grant CFF-0830214) Marie-Curie Senior Fellowship

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 40 / 45

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SLIDE 41

Conclusions The end

Thanks for your attention! Questions?

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 41 / 45

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SLIDE 42

Related References

Related references I

CHALONS, C., FOX, R. O. & MASSOT, M. 2010 A multi-Gaussian quadrature method of moments for gas-particle flows in a LES

  • framework. Proceedings of the Summer Program 2010. Center for Turbulence Research, Stanford, pgs. 347–358.

CHALONS, C., FOX, R. O., LAURENT, F., MASSOT, M. & VIÉ, M. 2011 A multi-Gaussian quadrature method of moments for simulating high-Stokes-number turbulent two-phase flows. Annual Research Briefs 2011. Center for Turbulence Research, Stanford,

  • pgs. 309–320.

CHENG, J. C. & FOX, R. O. 2010 Kinetic modeling of nanoprecipitation using CFD coupled with a population balance. Industrial & Engineering Chemistry Research 49, 10651–10662. CHENG, J. C., VIGIL, R. D. & FOX, R. O. 2010 A competitive aggregation model for Flash NanoPrecipitation. Journal of Colloid and Interface Science 351, 330–342.

DE CHAISEMARTIN, S., FRÉRET, L., KAH, D., LAURENT, F., FOX, R. O., REVEILLON, J. & MASSOT, M. 2009 Eulerian models

for turbulent spray combustion with polydispersity and droplet crossing. Comptes Rendus Mecanique 337, 438–448. DESJARDINS, O., FOX, R. O. & VILLEDIEU, P. 2006 A quadrature-based moment closure for the Williams spray equation. Proceedings of the Summer Program 2006. Center for Turbulence Research, Stanford, pgs. 223–234. DESJARDINS, O., FOX, R. O. & VILLEDIEU, P. 2008 A quadrature-based moment method for dilute fluid-particle flows. Journal of Computational Physics 227, 2514–2539. FOX, R. O. 2008 A quadrature-based third-order moment method for dilute gas-particle flows. Journal of Computational Physics 227, 6313–6350. FOX, R. O. 2009 Higher-order quadrature-based moment methods for kinetic equations. Journal of Computational Physics 228, 7771–7791.

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 42 / 45

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SLIDE 43

Related References

Related references II

FOX, R. O. 2009 Optimal moment sets for the multivariate direct quadrature method of moments. Industrial & Engineering Chemistry Research 48, 9686–9696. FOX, R. O. 2012 Large-eddy-simulation tools for multiphase flows. Annual Review of Fluid Mechanics 44, 47–76. FOX, R. O., LAURENT, F. & MASSOT, M. 2008 Numerical simulation of spray coalescence in an Eulerian framework: direct quadrature method of moments and multi-fluid method. Journal of Computational Physics 227, 3058–3088. FOX, R. O. & VEDULA, P. 2010 Quadrature-based moment model for moderately dense polydisperse gas-particle flows. Industrial & Engineering Chemistry Research 49, 5174–5187. ICARDI, M., ASINARI, P., MARCHISIO, D. L., IZQUIERDO, S. & FOX, R. O. 2012 Quadrature-based moment closures for non-equilibrium flows: hard-sphere collisions and approach to equilibrium. Journal of Computational Physics, (in press). KAH, D., LAURENT, F., FRÉRET, L., DE CHAISEMARTIN, S., FOX, R. O., REVEILLON, J. & MASSOT, M. 2010 Eulerian quadrature-based moment models for dilute polydisperse evaporating sprays. Flow, Turbulence, and Combustion 85, 649–676. MARCHISIO, D. L. & FOX, R. O. 2005 Solution of population balance equations using the direct quadrature method of moments. Journal of Aerosol Science 36, 43–73. MEHTA, M., RAMAN, V. & FOX, R. O. 2012 On the role of gas-phase chemistry in the production of titania nanoparticles in turbulent flames. Chemical Engineering Science, (submitted). MEHTA, M., SUNG, Y., RAMAN, V. & FOX, R. O. 2010 Multiscale modeling of TiO2 nanoparticle production in flame reactors: Effect of chemical mechanism. Industrial & Engineering Chemistry Research 49, 10663–10673. PASSALACQUA, A. & FOX, R. O. 2011 Advanced continuum modeling of gas-particle flows beyond the hydrodynamic limit. Applied Mathematical Modelling 35, 1616–1627.

  • R. O. Fox (ISU & ECP)

Quadrature-Based Moment Methods WMMKTII 2014 43 / 45

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SLIDE 44

Related References

Related references III

PASSALACQUA, A. & FOX, R. O. 2011 An iterative solution procedure for multi-fluid gas-particle flow models on unstructured grids. Powder Technology 213, 174–187. PASSALACQUA, A. & FOX, R. O. 2012 Simulation of mono- and bidisperse gas-particle flow in a riser with a third-order quadrature-based moment method. Industrial & Engineering Chemistry Research 52, 187-198. PASSALACQUA, A., FOX, R. O., GARG, R. & SUBRAMANIAM, S. 2010 A fully coupled quadrature-based moment method for dilute to moderately dilute fluid-particle flows. Chemical Engineering Science 65, 2267–2283. PASSALACQUA, A., GALVIN, J. E., VEDULA, P., HRENYA, C. M. & FOX, R. O. 2011 A quadrature-based kinetic model for dilute non-isothermal granular flows. Communications in Computational Physics 10, 216–252. SUNG, Y., RAMAN, V. & FOX, R. O. 2011 Large-eddy simulation based multiscale modeling of TiO2 nanoparticle synthesis in turbulent flame reactors using detailed nucleation chemistry. Chemical Engineering Science 66, 4370–4381. VIKAS, V., WANG, Z. J., PASSALACQUA, A. & FOX, R. O. 2011 Realizable high-order finite-volume schemes for quadrature-based moment methods. Journal of Computational Physics 230, 5328–5352. VIKAS, V., HAUCK, C. D., WANG, Z J. & FOX, R. O. 2013 Radiation transport modeling using extended quadrature methods of

  • moments. Journal of Computational Physics 246, 221–241.

VIKAS, V., WANG, Z. J. & FOX, R. O. 2013 Realizable high-order finite-volume schemes for quadrature-based moment methods applied to diffusion population balance equations. Journal of Computational Physics 249, 162–179. VIKAS, V., YUAN, C. WANG, Z. J. & FOX, R. O. 2011 Modeling of bubble-column flows with quadrature-based moment methods. Chemical Engineering Science 66, 3058–3070.

  • R. O. Fox (ISU & ECP)

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SLIDE 45

Related References

Related references IV

YUAN, C. & FOX, R. O. 2011 Conditional quadrature method of moments for kinetic equations. Journal of Computational Physics 230, 8216–8246. YUAN, C., LAURENT, F. & FOX, R. O. 2012 An extended quadrature method of moments for population balance equations. Journal

  • f Aerosol Science 51, 1–23.
  • R. O. Fox (ISU & ECP)

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