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
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
Introduction Target Applications Application: polydisperse multiphase flows Bubble columns Jet break up Brown-out Spray flames Volcanos Power stations R. O. Fox (ISU & ECP) Quadrature-Based Moment Methods WMMKTII 2014 3 / 45
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
Introduction Kinetic-based models Overview of kinetic modeling approach Microscale Model Direct numerical simulation Kinetic theory + density function closures Mesoscale Model Volume or ensemble averages Kinetic equation + closures for “fluctuations” Euler-Lagrange models Moments of density + moment closures Macroscale Model Hydrodynamic description Euler-Euler models Mesoscale model incorporates more microscale physics in closures! R. O. Fox (ISU & ECP) Quadrature-Based Moment Methods WMMKTII 2014 5 / 45
Introduction Kinetic-based models Types of mesoscale transport (kinetic) equations Population balance equation (PBE): n ( t , x , ξ ) � � ∂ n ∂ t + ∂ ∂ x i [ u i ( t , x , ξ ) n ] + ∂ ∂ξ j [ G j ( t , x , ξ ) n ] = ∂ D ( t , x , ξ ) ∂ n + S ∂ x i ∂ x i with known velocity u , acceleration G , diffusivity D and source S Kinetic equation (KE): n ( t , x , v ) ∂ n ∂ t + ∂ ∂ x i ( v i n ) + ∂ ∂ v i [ A i ( 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 + ∂ ∂ x i ( v i n ) + ∂ ∂ v i [ A i ( t , x , v , ξ ) n ] + ∂ ∂ξ j [ G j ( 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
Introduction Kinetic-based models Moment transport equations PBE: M k = � ξ k n d ξ ∂ M k + ∂ �� � ξ k − 1 Gn d ξ + ∂ �� ξ k D ∂ n � � � ξ k un d ξ ∂ x d ξ ξ k S d ξ = k + ∂ t ∂ x ∂ x KE: M k = � v k n d v ∂ M k + ∂ M k + 1 � � v k − 1 An d v + v k C d v = k ∂ t ∂ x GPBE: M kl = � v k ξ l n d v d ξ ∂ M kl + ∂ M k + 1 l � � � v k − 1 ξ l An d v d ξ + l v k ξ l − 1 Gn d v d ξ + v k ξ l C d v d ξ = k ∂ t ∂ x Terms in red will usually require mathematical closure R. O. Fox (ISU & ECP) Quadrature-Based Moment Methods WMMKTII 2014 7 / 45
Introduction Kinetic-based models Closure with moment methods Density 6-D solver Kinetic Equation n(t,x,v) Reconstructed density n * (t,x,v) Integrate over Integrate over phase space phase space Closure using quadrature Reconstruct using quadrature Moments Moment Equations 3-D solver M(t,x) Close moment equations by reconstructing density function R. O. Fox (ISU & ECP) Quadrature-Based Moment Methods WMMKTII 2014 8 / 45
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
Quadrature-Based Moment Methods Gauss quadrature in 1-D (real line) Gauss quadrature in 1-D (real line) The formula N � � g ( v ) n ( v ) d v = n α g ( v α ) + R N ( g ) α = 1 is a Gauss quadrature iff the N nodes v α are roots of an N th -order orthogonal polynomial P N ( v ) ( ⊥ with respect to n ( v ) ) Recursion formula for P N ( v ) : α = 0 , 1 , 2 , . . . P α + 1 ( v ) = ( v − a α ) P α ( v ) − b α P α − 1 ( v ) , Inversion algorithm (QMOM) for moments M k = v k n ( v ) d v : � { M 0 , M 1 , . . . , M 2 N − 1 } hard = ⇒ { a 0 , a 1 , . . . , a N − 1 } , { b 1 , b 2 , . . . , b N − 1 } easy ⇒ { n 1 , n 2 , . . . , n N } , { v 1 , v 2 , . . . , v N } = R. O. Fox (ISU & ECP) Quadrature-Based Moment Methods WMMKTII 2014 10 / 45
Quadrature-Based Moment Methods Szegö quadrature on unit circle Szegö quadrature on unit circle If n ( φ ) is periodic on the unit circle: � π N � g ( e i φ ) n ( φ ) d φ = n α g ( e i φ α ) + R N ( g ) − π α = 1 is a Szegö quadrature iff the N nodes z α = e i φ α are zeros of an N th -order para-orthogonal polynomials B N ( z ) Trigonometric moments: � π � π 1 1 2 ( z n + z − n ) n ( φ ) d φ, � sin ( n φ ) � = 2 ( z n − z − n ) n ( φ ) d φ � cos ( n φ ) � = − π − π are natural choice for reconstruction Except for special case [ n ( φ ) symmetric wrt 0], no fast inversion algorithm is available to find n α and φ α R. O. Fox (ISU & ECP) Quadrature-Based Moment Methods WMMKTII 2014 11 / 45
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: N d M � � S ( v ) n ( v ) d v ≈ d t = n α S ( v α ) α = 1 where M = { M 0 , M 1 , . . . , M 2 N − 1 } and S is “source term” Exact if S is polynomial of order ≤ 2 N − 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 n ∗ ( v ) = � n α δ ( v − v α ) α = 1 ⇒ realizable if n α ≥ 0 for all α = R. O. Fox (ISU & ECP) Quadrature-Based Moment Methods WMMKTII 2014 12 / 45
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 = { M ijk : i , j , k ∈ 0 , 1 , . . . } , find n α and v α such that N � 1 v j � 1 α v j v i 2 v k 3 n ( v ) d v = n α v i 2 α v k M ijk = 3 α α = 1 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
Quadrature-Based Moment Methods Quadrature in multiple dimensions Brute-force QMOM (2-D phase space) Given 3 n 2 bivariate optimal moments ( n = 2): M 00 M 01 M 02 M 03 M 10 M 11 M 12 M 13 M 20 M 21 M 30 M 31 Solve 12 moment equations: 4 � | n α | u i α v j α = M ij α = 1 to find { n 1 , . . . , n 4 ; u 1 , . . . , u 4 ; v 1 , . . . , v 4 } 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
Quadrature-Based Moment Methods Quadrature in multiple dimensions Conditional QMOM (2-D phase space) Conditional density function and conditional moments (2-D) � � V k | U = u � = v k f ( v | u ) d v n ( u , v ) = f ( v | u ) n ( u ) ⇒ = 1-D adaptive quadrature for U direction ( n = 2) � U k � = M k 0 , k ∈ { 0 , 1 , 2 , 3 } = ⇒ find weights ρ i , abscissas u i Solve linear systems for conditional moments � V k | u i � : � � V k � � ρ 1 � � � V k | u 1 � � � � M 0 k � ρ 2 for k ∈ { 1 , 2 , 3 } = = � V k | u 2 � � UV k � ρ 1 u 1 ρ 2 u 2 M 1 k In principle, CQMOM controls 10 of 12 optimal moments: M 00 M 01 M 02 M 03 M 10 M 11 M 12 M 13 M 20 M 30 R. O. Fox (ISU & ECP) Quadrature-Based Moment Methods WMMKTII 2014 15 / 45
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