Tutorial: Multi-Species Lattice Boltzmann Models and Applicationsstatistics.roma2.infn.it/~dsfd2010/TUTORIALS/Asinari.pdf · 2010. 7. 13. · Boltzmann equation by using the properties

Tutorial: Multi-Species Lattice Boltzmann Modelsand Applications

Pietro Asinari, PhD

Dipartimento di Energetica, Politecnico di Torino, Torino, Italy,e-mail: [email protected],

home page: http://staff.polito.it/pietro.asinari

19th Discrete Simulation of Fluid DynamicsCNR and University of Rome, "Tor Vergata", Rome, Italy, on July 5-9, 2010

Pietro Asinari, PhD (Politecnico di Torino) Multi-Species LB Models Rome, Italy, on July 5-9, 2010 1 / 51

[email protected]://staff.polito.it/pietro.asinari

Abstract of this tutorial

The aim of this tutorial is to discuss a numerical scheme based on theLattice Boltzmann Method (LBM) for gas mixture modeling, which fullyrecovers Maxwell-Stefan diffusion model in the continuum limit, withoutthe restriction of the mixture-averaged approximation [1]. Themixture-averaged approximation is used to express the molarconcentration gradient by means of a Fickian expression and a properdiffusion coefficient, both designed in such a way to be consistent withthe Maxwell-Stefan formulation only in some limiting cases.Unfortunately (a) this approximation yields to Fickian diffusioncoefficients depending on both the fluid transport properties and thelocal flow and (b) it is not valid in general. The key idea for avoidingthese problems is to start from a Bhatnagar-Gross-Krook-type kineticmodel for gas mixtures, recently proposed by Andries, Aoki andPerthame [2] (the so-called AAP model).


Outline of this tutorial

1 Kinetic theory of rarefied gas mixturesFull Boltzmann equationsExchange relation for momentum

2 Lattice Boltzmann solvers and applicationsSingle-relaxation-time Lattice Boltzmann schemeAndries-Aoki-Perthame (AAP) modelLBM formulation by variable transformation

3 Validation and simple applications


Kinetic theory of rarefied gas mixtures

Outline Compass





Kinetic theory of rarefied gas mixtures Full Boltzmann equations

Full Boltzmann equations for gas mixtures

The simultaneous Boltzmann equations for a mixture withoutexternal force can be written as [3, 4, 5, 6]:

∂tfσ + ξ ·∇fσ + aσ ·∇ξfσ = Qσ =∑ς

Qσς , (1)

where Qσς = Qςσ, ς 6= σ, is the cross collision term for twodifferent species σ (with mass mσ) and ς (with mass mς ).Obviously, for an N -component system, there will be N suchequations. In general, the collision term is

Qσς=̇

∫R3ξ

∫g·n


Collision kernel and post–collision velocities

The weight Kσς(g,n) = Kςσ(g,n) is a volumetric particle flux orcollision kernel. In the following, we will discuss only the case ofMaxwell molecules [7], namely Kσς(|g · n|/|g|), where the collisionkernel is independent of the relative velocity.In the previous equation, the post–collision test and field particlevelocities ξ′ and ξ′∗ are given by

ξ′ = ξ +2µσςmσ

(g · n) n, (3)

ξ′∗ = ξ∗ −2µσςmς

(g · n) n, (4)

which means that there are many possible outcomes (ξ′, ξ′∗) froma given pair of incoming (test and field) particle velocities (ξ, ξ∗),depending on the impact direction n.The reduced mass µσς is defined as

µσς = mσmς/(mσ +mς). (5)



Indifferentiability Principle and H–theorem

Indifferentiability Principle: when all the masses mσ andcross-sections Kσς are identical, the total distribution f =

∑σ fσ

obeys the single species Boltzmann equation.H–theorem: in the case of a mixture, the entropy inequality(H–theorem) is ∑

σ

∫R3ξQσ log(fσ)dξ ≤ 0. (6)

These properties of the full Boltzmann equations for mixtures willguide the design of BGK–type simplified models and consequentlyof the Lattice Boltzmann solvers.


Kinetic theory of rarefied gas mixtures Exchange relation for momentum

Exchange relation for momentum (1 of 3)

The momentum equation is

∂t(ρσuσ) +∇ ·Πσ = ρσaσ +∫R3ξmσξQσdξ. (7)

Taking into account the symmetries of the elementary collisionyields (see for example Eq. (48) in the Ref. [8])∫

R3ξmσξQσdξ =

∑ς

〈∫g·n



Let us express the surface element dn in Eq. (10) by using g aspolar axis, φ as polar angle and θ as azimuthal angle.Hence Kσς(|g · n|/|g|) = Kσς(cosφ) = Kσς(φ) and consequently∫

R3ξmσξQσdξ =

∑ς

2µσς

〈∫g·n



Consequently

1

2

∫ 2π0

∫ π0Kσς(φ) |g| cosφ

00

cosφ

sinφdφdθ = χσς g. (12)where

χσς =1

2

∫ 4π0

Kσς(φ) (cosφ)2 dn. (13)

Coming back to Eq. (10) yields∫R3ξmσξQσdξ =

∑ς

2µσςχσς 〈g〉σς = p∑ς

Bσςyσyς(uς−uσ), (14)

where

Bσς =2µσςχσς n

2

p= Bςσ, (15)

where n is the total particle number density, p is the pressure andyσ is the molar concentration, as it will be discussed next.


Lattice Boltzmann solvers and applications

Outline Compass





Lattice Boltzmann solvers and applications Andries-Aoki-Perthame (AAP) model

Simplified kinetic models

The full Boltzmann equations for gas mixtures are much moreformidable to analyze than the equation for a single-speciessystem [3, 4, 5, 6].A popular approach is to derive simplified model Boltzmannequations which are more manageable to solve. Numerous modelequations are influenced by Maxwell’s approach to solve theBoltzmann equation by using the properties of the Maxwellmolecule and the linearized Boltzmann equation [12].The simplest model equations for a binary mixture is that by Grossand Krook [13, 12], which is an extension of thesingle-relaxation-time model for a pure system — the celebratedBhatnagar-Gross-Krook (BGK) model.Many other models exist: Sirovich [14, 15], Hamel [16, 17], ...



Basic consistency constraints

1 The Indifferentiability Principle, which prescribes that, if aBGK-like equation for each species is assumed, this set ofequations should reduce to a single BGK-like equation, whenmechanically identical components are considered, i.e. the singlefluid description should be recovered [18].

2 The relaxation equations for momentum and temperature shouldbe as close as possible to those derived by means of the fullBoltzmann equations [19].

3 All the species should tend to a target equilibrium distributionwhich is a Maxwellian, centered on a proper macroscopic velocity,common to all the species.

4 The non-negativity of the distribution functions for all the speciesshould be satisfied.

5 A generalized H theorem for mixtures should hold.



Simplified AAP model

Let us consider a simplified version of the AAP model, proposedby Andries, Aoki, and Perthame [2], which is based on only oneglobal (i.e., taking into account all the species ς) operator for eachspecies σ, namely

∂t̂fσ + ξ ·∇̂fσ = λσ[fσ(∗) − fσ

], (16)

where

fσ(∗) =ρσ

(2πϕσ/3)exp

[−3 (ξ − u

∗σ)

2

2ϕσ

], (17)

and

u∗σ = uσ +∑ς

m2

mσmς

BσςBmm

xς(uς − uσ). (18)



Properties of simplified AAP model

The target velocity can be easily recasted as

u∗σ = u+∑ς

(m2

mσmς

BσςBmm

− 1)xς(uς − uσ). (19)

If mσ = m for any σ, then (Property 1)

u∗σ = u+∑ς

(m2

mm

BmmBmm

− 1)xς(uς − uσ) = u. (20)

Clearly (Property 2)

∑σ

xσu∗σ = u+

∑σ

∑ς

(m2

mσmς

BσςBmm

− 1)xσxς(uς − uσ) = u.

(21)



Diffusive scaling

In the following asymptotic analysis [20], we introduce thedimensionless variables, defined by

xi = (lc/L) x̂i, t = (UTc/L) t̂. (22)

Defining the small parameter � as � = lc/L, which corresponds tothe Knudsen number, we have xi = � x̂i.Furthermore, assuming U/c = �, which is the key of derivation ofthe incompressible limit [20], we have t = �2 t̂. Then, AAP model isrewritten as

�2∂fσ∂t

+ � ξi∂fσ∂xi

= λσ[fσ(∗) − fσ

]. (23)

In this new scaling, we can assume ∂αfσ = ∂fσ/∂α = O(fσ) and∂αM = ∂M/∂α = O(M), where α = t, xi and M = ρσ, qσi whereqσi = ρσuσi.



Regular Knudsen expansion

Clearly the solution of the BGK equation depends on �. Thesolution for small � is investigated in the form of the asymptoticregular expansion

fσ = f(0)σ + �f

(1)σ + �

2f (2)σ + · · · . (24)

ρ and qσi are also expanded:

ρσ = ρ(0)σ + �ρ

(1)σ + �

2ρ(2)σ + · · · , (25)

qσi = �q(1)σi + �

2q(2)σi + · · · , (26)

since the Mach number is O(�), the perturbations of qσi starts fromthe order of �. Consequently

fσ(∗) = f(0)σ(∗) + �f

(1)σ(∗) + �

2f(2)σ(∗) + · · · , (27)

Regular expansion means ∂αf(k)σ = O(1) and ∂αM (k) = O(1).



Asymptotic analysis of AAP model

Collecting the terms of the same order yields

f (k)σ = f(k)σ(∗) − g

(k)σ , (28)

g(0)σ = 0, (29)

g(1)σ = τσ∂Sf(0)σ(∗), (30)

g(2)σ = τσ[∂tf(0)σ(∗) + ∂Sf

(1)σ(∗) − τσ∂

2Sf

(0)σ(∗)], (31)

· · · ,

where ∂S = ξi∂/∂xi and τσ = 1/λσ.The previous coefficients of the regular expansion allows one toderive the macroscopic equations recovered by the AAP model.



Tuning the single species relaxation frequency

Taking the first order moments of g(1)σ yields

λσρ(0)σ [u

∗(1)σ − u(1)σ ] = ∇p(0)σ , (32)

where p(k)σ = ϕσρ(k)σ /3.

If λσ is selected as λσ = pBmm/ρ, then the previous expressionbecomes

1/p(0)∇p(0)σ =∑ς

Bσς yσyς [u(1)ς − u(1)σ ], (33)

which clearly proves that the leading terms of the macroscopicequations recovered by means of the AAP model are consistentwith Maxwell-Stefan model.


Lattice Boltzmann solvers and applications LBM formulation by variable transformation

D2Q9 lattice

Let us define the AAP model for a set of discrete velocities,

�2∂fσ∂t

+ �Vi∂fσ∂xi

= λσ[fσ(∗) − fσ

], (34)

where Vi is a list of i-th components of the velocities in theconsidered lattice and f = fσ(∗), fσ is a list of discrete distributionfunctions (change in the notation !!) corresponding to thevelocities in the considered lattice.Let us consider the two dimensional 9 velocity model, which iscalled D2Q9, namely

V1 =[

0 1 0 −1 0 1 −1 −1 1]T, (35)

V2 =[

0 0 1 0 −1 1 1 −1 −1]T. (36)



Rule of computation for the list

The components of the molecular velocity V1 and V2 are the listswith 9 elements. Before proceeding to the definition of the localequilibrium function fσ(∗), we define the rule of computation for thelist.Let h and g be the lists defined by h = [h0, h1, h2, · · · , h8]T andg = [g0, g1, g2, · · · , g8]T . Then, hg is the list defined by[h0g0, h1g1, h2g2, · · · , h8g8]T . The sum of all the elements of the listh is denoted by 〈h〉, i.e. 〈h〉 =

∑8i=0 hi.

Then, the (dimensionless) density ρσ and momentum qσi = ρσuσiare defined by

ρσ = 〈fσ〉, qσi = 〈Vifσ〉. (37)



Continuous equilibrium moments

Let us introduce the following function

fe(ρ, ϕ, u1, u2) =ρ

(2πϕ/3)exp

[−3 (ξ − u)

2

2ϕ

]. (38)

Let us define 〈〈·〉〉 =∫ +∞−∞ · dξ1dξ2 and the generic equilibrium

moment mpq = 〈〈fe ξp1ξq2〉〉.

All the equilibrium moments appearing in the Euler system ofequations are the following: m00, m10, m01, m20, m02, m11. Due tothe lattice deficiencies, only the equilibrium moments m21, m12must be considered in addition, for recovering the desiredNavier-Stokes system of equations. The moment m22 is arbitrarilyselected in order to complete the set of discrete moments.



Simplified continuous equilibrium moments

Collecting the previous results yields

m̄c(ρ, ϕ, u1, u2) = ρ [1, u1, u2,

u21 + ϕ/3, u22 + ϕ/3, u1u2,

u1 u22 + u1ϕ/3, u

21 u2 + u2ϕ/3,

ϕ (u21u22 + u

21ϕ/3 + u

22ϕ/3 + ϕ/9)]

T .

The previous analytical results involve high order terms (like u1 u22)which are not strictly required, in order to recover the macroscopicequations we are interested in.

mc(ρ, ϕ, u1, u2) = ρ [1, u1, u2,

u21 + ϕ/3, u22 + ϕ/3, u1u2,

u1/3, u2/3,

(u21 + u22)/3 + ϕ/9]

T



Design of discrete local equilibrium

On the selected lattice, the discrete integrals mσ(∗), correspondingto the previous continuous ones, can be computed by means ofsimple linear combinations of the discrete equilibrium distributionfunction fσ(∗) (still unknown), namely mσ(∗) = Mfσ(∗) where M isa matrix defined as

M = [1, V1, V2, V21 , V

22 , V1V2, V1V

22 , V

21 V2, V

21 V

22 ]T . (39)

We design the discrete local equilibrium such asmσ(∗) = mc(ρσ, ϕσ, u

∗σ1, u

∗σ2), or equivalently

fσ(∗) = M−1mc(ρσ, ϕσ, u

∗σ1, u

∗σ2). In particular the latter provides

the operative formula for defining the local equilibrium andconsequently the scheme.



Discrete operative formula

Eq. (34) is formulated for discrete velocities, but it is stillcontinuous in both space and time.Since the streaming velocities are constant, the Method ofCharacteristics is the most convenient way to discretize space andtime and to recover the simplest formulation of the LBM scheme.Applying the second-order Crank–Nicolson yields

f+σ = fσ + (1− θ)λσ[fσ(∗) − fσ

]+ θ λ+σ

[f+σ(∗) − f

+σ

], (40)

where θ = 1/2.The previous formula would force one to consider quitecomplicated integration procedures [21]. A simple variabletransformation has been already proposed in order to simplify thistask [22].



Variable transformation

(Step 1) Let us apply the transformation fσ → gσ defined by

gσ = fσ − θ λσ[fσ(∗) − fσ

]. (41)

(Step 2) Let us compute the collision and streaming step leadingto gσ → g+σ by means of the modified updating equation

g+σ = gσ + λ′σ

[fσ(∗) − gσ

], (42)

where λ′σ = λσ/(1 + θλσ).(Step 3) Finally let us come back to the original discretedistribution function g+σ → f+σ by means of

f+σ =g+σ + θ λ

+σ f

+σ(∗)

1 + θ λ+σ. (43)



Problem for mixtures

In case of mixtures, the problem arises from the (Step 3), whichrequires both λ+σ and f

+σ(∗), depending on the updated

hydrodynamic moments at the new time step.Since the single component density is conserved, Eq. (41) yields

ρ+σ = 〈g+σ 〉, (44)

consequently it is possible to compute p+σ , ρ+, p+ and λ+σ .However this is not the case for the single component momentum,because this is not a conserved quantity and hence the first ordermoments for g+σ and f+σ differ [23], namely

〈Vi g+σ 〉 = ρ+σ u+σi − θ λ+σ ρ

+σ (u

∗+σi − u

+σi) =

= ρ+σ u+σi − θ p

+∑ς

Bσς y+σ y

+ς (u

+ςi − u

+σi). (45)



Solution: solving locally a linear system of equations

In the general case, Eq. (45) can be recasted as

〈Vi g+σ 〉 = q+σi − θ λ+σ

∑ς

χσς (x+σ q

+ςi − x

+ς q

+σi), (46)

where q+σi = ρ+σ u

+σi and

χσς =m2

mσmς

BσςBmm

. (47)

Finally, grouping together common terms yields

〈Vi g+σ 〉 =

[1 + θ λ+σ

∑ς

(χσς x+ς )

]q+σi − θ λ

+σ x

+σ

∑ς

(χσς q+ςi). (48)

Clearly the previous expression defines a linear system ofalgebraic equations for the unknowns q+σi.


Validation and simple applications

Outline Compass






Single-relaxation-time MixLBM code



Basic algorithm

The proposed numerical code is formulated not in the standardway (and it is quite inefficient from the computational point ofview).Even though it is not an efficient implementation, the proposedformulation is much more similar to any other explicit finitedifference (FD) scheme.This offers some advantages:

1 it makes easier to implement hybrid schemes, i.e. to mix up kineticand conventional schemes on the same discretization;

2 it makes easier to compare the LBM scheme with other FDschemes, mainly in terms of updating rule;

3 it makes easier to implement simple boundary conditions, based onthe concept of local equilibrium.

Anyway the basic sequence of collision and streaming step ispreserved.



Schematic view



Main loop

fd(1:nx,1:ny,0:8,1:species)

do t = 1,nt,1do i = 1,nx,1

do j = 1,ny,1call UpdateLatticeData(...,f(:,:));do s = 1,species,1

fd_new(i,j,:,s) = f(:,s);call HydrodynamicMoments(...);

enddoenddo

enddofd(:,:,:,:) = fd_new(:,:,:,:);

enddo



UpdateLatticeData loop

do k=0,8,1iI = i + Incr(k,1); jI = j + Incr(k,2);do s=1,species,1

! BCs rs,uxs,uys in I(i+,j+)enddodo s=1,species,1call EquilibriumDistribution(...,feq(:))lambda(s)=...;do ik=0,8,1fc(ik)=f(ik,s)+lambda(s)*(feq(ik)-f(ik,s));enddof_new(BB(k),s) = fc(BB(k));

enddoenddo



UpdateLatticeData loop with variable transformation

do k=0,8,1do s=1,species,1call EquilibriumDistribution(...,feq(:))lambda(s)=...;TRANSFORMATION f(:,s) -> g(:,s)do ik=0,8,1gc(ik)=g(ik,s)+lambda’(s)*(feq(ik)-g(ik,s));enddog_new(BB(k),s) = gc(BB(k));

enddoenddoBACK-TRANSFORMATION g_new(:,s) -> f_new(:,s)COMPUTE CONSERVED MOMENTSSOLVE LINEAR SYSTEM FOR NON-CONSERVED MOMENTS



Ternary mixture

In case of ternary mixture Eq. (14) reduces to

n∇y1 = B12y1k2 +B13y1k3 − (B12y2 +B13y3)k1, (49)

n∇y2 = B21y2k1 +B23y2k3 − (B21y1 +B23y3)k2, (50)

n∇y3 = B31y3k1 +B32y3k2 − (B31y1 +B32y2)k3. (51)

The molecular weights are mσ = [1, 2, 3], the internal energiesare [eσ = 1/3, 1/6, 1/9] and consequently ϕσ = [1, 1/2, 1/3].The theoretical Fick diffusion coefficient is Dσ = α/mσ, whereα ∈ [0.002, 0.8] and the theoretical Maxwell–Stefan diffusionresistance is given by [28]

Bσς = β

(1

mσ+

1

mς

)−1/2, β ∈ [5, 166]. (52)



Solvent test case

A component of a mixture is called solvent if its concentration ispredominant in comparison with the other components of themixture [10].Let us suppose that, in our ternary mixture, the component 3 is asolvent. In particular, the initial conditions for the solvent test caseare given by

p1(0, x) = ∆p

[1 + tanh

(x− L/2δx

)]+ ps, (53)

p2(0, x) = ∆p

[1− tanh

(x− L/2δx

)]+ ps, (54)

p3(0, x) = 1− 2 (∆p+ ps), (55)

where clearly p(0, x) =∑

σ pσ = 1 and ∆p = ps = 0.01.



Solvent test case: simplified transport coefficients

Hence y3 ∼= 1 and consequently y1 ∼= 0 and y2 ∼= 0. Under theseassumptions, Eqs. (49, 50) reduce to

∇y1 = −B13y1(u1 − v) = B13y1(v − u1), (56)

∇y2 = −B23y2(u2 − v) = B23y2(v − u2), (57)

Consequently the measured diffusion resistances are given by

B∗13 =1

D∗1=

∂y1/∂x

y1(v − u1), (58)

B∗23 =1

D∗2=

∂y2/∂x

y2(v − u2), (59)

where, in this test, the Maxwell–Stefan model reduces to the Fickmodel.



Solvent test case: Fick model



Solvent test case: Maxwell–Stefan model



Dilute test case

A component of a mixture is said dilute if its concentration isnegligible in comparison with the other components of the mixture[10].Let us suppose that, in our ternary mixture, the component 1 isdilute. In particular, the initial conditions for the dilute test case aregiven by

p1(0, x) = ∆p

[1 + tanh

(x− L/2δx

)]+ ps, (60)

p2(0, x) = ∆p

[1− tanh

(x− L/2δx

)]+ ps +

+(1− r) (1− 2 ∆p), (61)p3(0, x) = r (1− 2 ∆p)− 2 ps, (62)

where p(0, x) =∑

σ pσ = 1, ∆p = ps = 0.01, r = 1/2.



Dilute test case: Maxwell–Stefan model



Non-Fickian test case: Stefan tube

It is essentially a vertical tube, open at one end, where the carrierflow licks orthogonally the tube opening [10]. In the bottom of thetube is a pool of quiescent liquid. The vapor that evaporates fromthis pool diffuses to the top.

p1(0, x) = p1(0, 0)1

2

[1− tanh

(x− L/2δx

)]+ ps, (63)

p2(0, x) = p2(0, 0)1

2

[1− tanh

(x− L/2δx

)]+ ps, (64)

p3(0, x) = [1− p3(0, 0)]1

2

[1 + tanh

(x− L/2δx

)]+ p3(0, 0),

(65)

where the constant ps = 10−4 has been introduced for avoiding todivide per zero.



Stefan tube


Conclusions

In the present tutorial, a LBM scheme for gas mixture modeling,which fully recovers Maxwell–Stefan diffusion model in thecontinuum limit, without the restriction of the macroscopicmixture-averaged approximation, was discussed.As a theoretical basis for the development of the LBM scheme, arecently proposed BGK-type kinetic model for gas mixtures [2]was considered. This essentially ties the LBM development to therecent progresses of the BGK-type kinetic models and opens newperspectives.In the reported numerical tests, the proposed scheme producesgood results on a wide range of relaxation frequencies.


References I

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References II

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References III

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[13] E. P. Gross and M. Krook.Models for collision processes in gases: Small-amplitude oscillations of chargedtwo-component systems.Phys. Rev., 102:593–604, 1956.

[14] L. Sirovich.Kinetic modelling of gas mixtures.Phys. Fluids, 5:908–918, 1962.

[15] L. Sirovich.Mixtures of Maxwell molecules.Phys. Fluids, 9:2323–2326, 1966.

[16] B. B. Hamel.Kinetic models for binary mixtures.Phys. Fluids, 8:418–425, 1965.


References IV

[17] B. B. Hamel.Two-fluid hydrodynamic equations for a neutral, disparate-mass, binary mixtures.Phys. Fluids, 9:12–22, 1966.

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References V

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[23] S. Arcidiacono, I. V. Karlin, J. Mantzaras, and C. E. Frouzakis.Lattice boltzmann model for the simulation of the multicomponent mixtures.Phys. Rev. E, 76:046703–1—046703–11, 2007.

[24] P. Asinari.Multiple-relaxation-time lattice boltzmann scheme for homogeneous mixture flows withexternal force.Phys. Rev. E, 77:056706, 2008.

[25] P. Asinari.Lattice Boltzmann scheme for mixture modeling: Analysis of the continuum diffusionregimes recovering Maxwell-Stefan model and incompressible Navier-Stokes equations.Phys. Rev. E, 80:056701, 2009.

[26] H. Grad.Theory of rarefied gases.In F. Devienne, editor, Rarefied Gas Dynamics, pages 100–138. Pergamon, London, 1960.


References VI

[27] P. Asinari and T. Ohwada.Connection between kinetic methods for fluid-dynamic equations and macroscopicfinite-difference schemes.Comput. Math. Appl., 58:841, 2009.

[28] W.E. Stewart R.B. Bird and E.N. Lightfoot.Transport Phenomena.John Wiley & Sons, New York, 1960.


Kinetic theory of rarefied gas mixturesFull Boltzmann equationsExchange relation for momentum

Lattice Boltzmann solvers and applicationsSingle-relaxation-time Lattice Boltzmann schemeAndries-Aoki-Perthame (AAP) modelLBM formulation by variable transformation

Validation and simple applicationsAppendix

Documents

Tutorial: Multi-Species Lattice Boltzmann Models and Applicationsstatistics.roma2.infn.it/~dsfd2010/TUTORIALS/Asinari.pdf · 2010. 7. 13. · Boltzmann equation by using the properties