2020 1.1 Microscopic Versus Macroscopic Descriptions

Moment Closures & Kinetic Equations 1. Introduction C. P. T. Groth c©2020

1. Introduction

Coverage of this section:

I Microscopic Versus Macroscopic Descriptions

I Moment Closure Methods

I Exemplar Kinetic Theories

I Brief History of Moment Closure Methods

I Notation

I Some Suggested References

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1.1 Microscopic Versus Macroscopic Descriptions

Example: system of gaseous molecules in a room

I Microscopic description: The microscopic description in thiscase is given by a full prescription of all molecules, includingtheir number and instantaneous positions, velocities, andinternal energies. This would of course require great deal ofinformation as the number of molecules can be very large.Under STP conditions, the number density of air is

n ≈ 2.5× 1025molecules/m3 .

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I Macroscopic description: The macroscopic description hereignores the molecular nature of the gas and makes acontinuum assumption (a mathematical idealization formodelling the collective response, or state, of discrete systems)in which a relatively small number of intensive and extensivemacroscopic quantities can be used to describe the system.

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I Macroscopic description: Under conditions ofthermodynamic equilibrium, this macroscopic descriptionreduces to just two intensive macroscopic quantities: e.g., thegas pressure, p, and temperature, T . For practical engineeringapplications, such a description is obviously far moreaccessible and workable.

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⇐⇒


I Kinetic theory: Provides a mathematical framework forrelating transport phenomena in the microscopic andmacroscopic descriptions.

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1.1.1 Kinetic Descriptions of Transport Phenomena

⇐⇒

I Kinetic-based models and theories provide microscopic descriptions ofcomplex transport phenomena

I Can be useful in the following three roles:I provide a means for evaluating transport coefficients of conventional

macroscopic continuum-based mathematical descriptions;I provide a means for determining the various way and manner in

which conventional macroscopic continuum-based mathematicaldescriptions become invalid or fail; and

I provide a means for constructing improved mathematicaldescriptions of complex transport phenomena beyond theconventional macroscopic continuum-based mathematicaldescriptions.

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1.1.2 Statistical Nature of Theories

⇐⇒

I Kinetic-based models and theories adopt a statistical approachfor describing the system of interest

I For example in the case of the gas-filled room, rather thantracking the individual instantaneous motion of all moleculesin the room, the many microscopic states of the system arerepresented in terms of a probability density function (PDF)with a number of continuous random variables (e.g.,translational velocity, ~v , of molecules) for which the sum ofthe probabilities of all allowed states is unity.

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1.1.3 Statistical Nature of Theories

⇐⇒

I Transport equations for the PDF are generally of anintegro-differential nature and involve high dimensionality;nevertheless, the statistical approach is considerably lesscostly than directly tracking particles!

I Macroscopic quantities associated with the microscopicdescription can be found using the PDF.

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1.2 Moment Closure Methods

1.2.1 Approximate Solution Method

I Moment closure methods essentially provide a means ofconstructing approximate solutions to the governing kineticequation

I Generally involve approximating the PDF by some assumedform involving a number of free parameters, the latter whichcan be related to selected macroscopic quantities or momentsassociated with the PDF solution

I Rather than solving the kinetic equation directly, solutions areinstead sought to the transport equations for the moments

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1.2 Moment Closure Methods

1.2.2 Complexity Reduction via Dimensionality Reduction

I The approximate solutions offered by moment closuremethods also provide a means of complexity reduction for theproblem of interest

I The problem of solving the kinetic equation is transformed toone of solving the moment equations with control ofcomplexity provide by the assumed form for the PDF and themoments of interest

I The complexity reduction is achieved via dimensionalityreduction (i.e., by reducing the number of independentvariables associated with the problem)

I Dimensionality reduction is commonly used in machinelearning and information theory; here, it is part of procedurefor constructing the approximate solutions

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1.3 Exemplar Kinetic Theories

I Boltzmann Equation — kinetic equation describing thermalnon-equilibrium transport of gas and plasma species

I Williams-Boltzmann Equation — kinetic equationdescribing transport of disperse liquid sprays

I Radiative Transfer Equation (RTE) — kinetic equationdescribing transport of radiative energy through participatingmedia

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1.3.1 Boltzmann Equation – Non-Equilibrium Gases

Transonic Airflow Past a NACA0012 Micro-Airfoil: Ma = 0.8, T = 257K,

ρ = 1.161× 10−4Kg/m3, chord = 0.04m, Kn = 0.017.

I Non-equilibrium gaseous flows (non-equilibrium of translational,rotational, and vibrational energies)

I Thermal non-equilibrium effects can significantly influence momentumand heat transfer

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1.3.2 Williams-Boltzmann Equation – Disperse Sprays

I Disperse multi-phase flows and sprays subject to droplet breakup andatomization, evaporation and growth, collisions and interactions, as wellas aerodynamic forces

I Treatment of polydisperse and polykinetic behaviour as well as particletrajectory crossings (PTCs) required for accurate representation ofdisperse spray transport

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1.3.3 Radiative Transfer Equation (RTE)

I Radiative transfer equation (RTE) describes radiative heat transfer viathe propagation of light at various frequency and accounts for theemission, absorption, and scattering produced by the backgroundparticipating media

I Angular distribution of the radiative energy flux can range from isotropicdistributions to distributions associated with collimated beams and thecrossing of the latter

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1.4 Brief History of Moment Closure Methods

History of kinetic theory and moment closure methods dates backmore than 150 years:I 1858 – Rudolf Julius Emanuel Clausius – introduced the

concept of the mean free pathI 1859 – James Clerk Maxwell – introduced the concept of the

velocity distribution and recognized equipartition principle ofmean molecular energy – combined with the mean free path,derived formulae for the transport coefficients of the gas(viscosity, thermal conductivity and diffusion coefficient)

I 1872 – Ludwig Boltzmann – derived the Boltzmann equationsand H theorem

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Maxwell’s Desk, Cavendish Laboratory, Cambridge University, 2013

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I 1887 – Eugen von Lommel – radiative transfer equation (RTE)

I 1916-17 – Sydney Chapman – Chapman-Enskog method –perturbative expansion technique providing expressions fortransport coefficients

I 1917 – David Enskog – Chapman-Enskog method –perturbative expansion technique providing expressions fortransport coefficients

I 1917 – James Jean – spherical harmonics approximation forradiation transport

I 1935 – David Burnett – second-order Chapman-Enskogsolutions

I 1949 – Harold Grad – Grad moment closure method

I 1957 – Edwin Jaynes – showed relationship of maximumentropy interpretation of thermodynamics to more generalBayesian/information theory

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I 1958 – Forman Williams – Williams-Boltzmann equation forpolydisperse sprays

I 1987 – Wolfgang Dreyer – entropy maximization and momentclosures

I 1993 – Ingo Muller and Tomasso Ruggeri – extendedthermodynamics and maximum entropy moment closures

I 1996 – David Levermore – mathematical theory and hierarchyof maximum entropy closures

I 1997 – Robert McGraw – quadrature method of moments(QMOM) for aerosol dynamics

I 1998 – Michael Junk – identified singular behaviour ofmaximum entropy closures

I 1999 – Bruno Dubroca and Jean-Luc Feugeas – maximumentropy closures for radiative transfer

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I 2002 – Michael Frenklach – method of moments withinterpolative closure (MOMIC) for aerosol dynamics

I 2003 – Daniele Marchisio and Rodney Fox – direct quadraturemethod of moments (DQMOM)

I 2004 – Henning Struchtrup – order of magnitude analysis

I 2004 – Manuel Torrilhon and Henning Struchtrup –regularized Grad moment closures

I 2010 – Christophe Chalons, Rodney Fox, and Marc Massot –extended quadrature method of moments (EQMOM)

I 2011 – Cansheng Yuan and Rodney Fox – conditionalquadrature method of moments (CQMOM)

I 2012 – Cansheng Yuan, Frederique Laurent, and Rodney Fox– extended quadrature method of moments (EQMOM)

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I 2013 – McDonald and Groth – interpolative-based maximumentropy closures (1D)

I 2013 – McDonald and Torrilhon – interpolative-basedmaximum entropy closures (1D & 3D)

I 2017 – Pichard et al. – interpolative-based maximum entropyclosure for radiative transfer

I 2019 – Forgues et al. – Gaussian closure for poly-disperse,poly-kinetic multi-phase flows

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1.5 Notation

Vector and tensor notation is used extensively throughout thecourse and the notation adopted here is therefore briefly reviewed.

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1.5 NotationExpression Vector Notation Tensor Notation

scalars π, c π, c(zeroth-order tensor)

operations

(+, −, ×, /) e.g., π − c ,π

cπ − c ,

π

c

vectors ~a, ~x ai , xi(3D space) (first-order tensor,

it is taken that i ∈ {1, 2, 3})

addition ~b = ~a + ~x bi = ai + xi = aj + xk

vector productsinner product ~a · ~x =

∑i aixi = c aixi = c

(scalar result)∑

i aixi = a1x1 + a2x2 + a3x3 (Einstein notation: sum implied)

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1.5 Notation

1.5.1 Einstein Summation ConventionEinstein summation convention: repetition of an index in any termdenotes a summation of the term with respect to that index overthe full range of the index (i.e., 1, 2, 3).Thus, for the inner product

aixi =3∑

i=1

aixi = a1x1 + a2x2 + a3x3

the sum is implied and need not be explicitly expressed. Note thatusing matrix-vector mathematical notation, the inner product oftwo 3× 1 column vectors, a and x, can be experssed as

aTx = [a1 a2 a3]

x1

x2

x3

= a1x1 + a2x2 + a3x3

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1.5 Notation

Expression Vector Notation Tensor Notation

cross product ~a× ~x = ~r =

∣∣∣∣∣ ~i ~j ~ka1 a2 a3

x1 x2 x3

∣∣∣∣∣ εijkajxk = ri

(vector result) ~r =(a2x3 − a3x2)~i

−(a1x3 − a3x1)~j

+(a1x2 − a2x1)~k

εijk = permutation tensor

(sum over j & k implied)

outer product ~a~x = ~a⊗~x =

~~J aαxβ = Jαβ(dyadic result, (second-order tensor,

vector of vectors) 9 elements,

6 elements for symmetric tensor)

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1.5 Notation

1.5.2 Dyadic Quantity: A Vector of Vectors

In vector notation, a dyadic quantity,~~d is essentially a ‘vector of

vectors’ as defined by the outer product:

~~d = ~u~v

It is equivalent to the second-order tensor, dij ,

dij = uiuj

using tensor notation. In this case using matrix-vector notation,the outer product of two 3× 1 column vectors, u and v, can beexperssed as

uvT =

u1

u2

u3

[v1 v2 v3] =

u1v1 u1v2 u1v3

u2v1 u2v2 u2v3

u3v1 u3v2 u3v3

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1.5 NotationExpression Vector Notation Tensor Notation

dyads~~d = ~u~v dij = uiuj

dyad-vector products~~A · ~x = ~b Aαβxβ = bα

(vector result) equivalent to Ax = b

high-order tensors~~~Q Qijk

(third-order tensor,

27 elements, 10 symmetric)

~~~~R Rijkl

(fourth-order tensor,

81 elements, 15 symmetric)

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1.5 Notation


contracted quantities ~h hi = qijj(contacted 3rd-order tensor,

vector)

~~P Pij = Rijkk

(contacted 4th-order tensor,

second-order tensor, dyad)

p p = Riikk

(double contacted tensor,

scalar quantity)

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1.5 Notation

1.5.3 Permutation Tensor

The permuation tensor, εijk , is a third-order tensor that isintroduced for defining cross products with the following propertiesfor its elements:

ε123 = ε231 = ε312 = 1 , even permutations

ε213 = ε321 = ε132 = −1 , odd permutations

ε111 = ε222 = ε333 = 0 , repeated indices

ε112 = ε113 = ε221 = ε223 = ε331 = ε322 = 0 , repeated indices

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1.5 Notation

1.5.4 Kronecker Delta Tensor

The Kronecker delta tensor, δij , is a second-order tensor that isdefined as follows:

δij =

{1 , for i = j0 , for i 6= j

The Kronecker delta tensor is equivalent ot the identity dyad,~~I

and the 3× 3 indentity matrix, I, in matrix-vector mathematicalnotation given by

I =

1 0 00 1 00 0 1

Note also that

δii = trace(I) = 3

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1.5 Notation

1.5.5 ε− δ Indentity

The following identity relates the permutation and Kronecker deltatensors:

εijkεist = δjsδkt − δjtδks

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1.5 Notation


differential operators

gradient ~V = ~∇φ Vi =∂φ

∂xi

divergence c = ~∇ · ~a c =∂ai∂xi

~u · ~∇φ ui∂φ

∂xi

curl ~g = ~∇× ~a gi = εijk∂ak∂xj

vector derivative~~P = ~∇~B Pij =

∂Bj

∂xi

Laplacian c = ∇2φ = ~∇ · ~∇φ c =∂2φ

∂xi∂xi

~a = ∇2 ~A = ~∇ · ~∇~A ai =∂2Ai

∂xj∂xj

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1.6 Some Suggested References

I An Introduction to Thermodynamics, the Kinetic Theory ofGases, and Statistical Mechanics, by F. W. Sears,Addison-Wesley, 1950.

I Molecular Flow of Gases, by G. N. Paterson, John Wiley andSons, 1956.

I The Mathematical Theory of Non-Uniform Gases, by S.Chapman and T. G. Cowling, Cambridge University Press,1960.

I An Introduction to the Kinetic Theory of Gases, by J. H.Jeans, Cambridge University Press, 1962.

I Introduction to Physical Gas Dynamics, by W. G. Vincentiand C. H. Kruger, John Wiley and Sons, 1965.

I Flow Equations for Composite Gases, by J. M. Burgers,Academic Press, 1969.

I Rarefied Gas Dynamics, by M. N. Kogan, Plenum Press, 1969.

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I An Introduction to the Theory of the Boltzmann Equation, byS. Harris, Holt, Rinehart, and Winston, 1971.

I Fundamentals of Maxwell’s Kinetic Theory of a SimpleMonatomic Gas, Treated as a Branch of Rational Mechanics,by C. Truesdell and R. G. Muncaster, Academic Press, 1980.

I Molecular Nature of Aerodynamics, by G. N. Patterson,UTIAS, 1981.

I Entropy Optimization Principles with Applications, by J. N.Kapur and H. K. Kesavan, Academics Press, Boston, 1992.

I Extended Thermodynamics, by I. Muller and T. Ruggeri,Springer-Verlag, 1993.

I The Mathematical Theory of Dilute Gases, by C. Cercignani,R. Illner, M. Pulvirenti, Springer-Verlag, 1994.

I Gaskinetic Theory, by T. I. Gombosi, Cambridge UniversityPress, 1994.

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I Molecular Gas Dynamics and the Direct Simulation of GasFlows, by G. A. Bird, Oxford Science Publications, 1995.

I Rational Extended Thermodynamics, by I. Muller and T.Ruggeri, Springer-Verlag, 1998.

I Ionospheres: Physics, Plasma Physics, and Chemistry, by R.W. Schunk and A. F. Nagy, Cambridge University Press, 2000.

I Radiative Heat Transfer, by M. F. Modest, Academic Press,New York, 2003.

I Macroscopic Transport Equations for Rarefied Gas Flows, byH. Struchtrup, Springer-Verlag, Berlin, 2005.

I Computational Models for Polydisperse Particulate andMultiphase Systems, by D. L. Marchisio and R. O. Fox,Cambridge University Press, Cambridge, 2013.

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2020 1.1 Microscopic Versus Macroscopic Descriptions

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