Numerical Simulations and Applications of Rarefied Gas Mixtures Flows

IntroductionParallel Implementation of DSBE

1D and 2D Multi-Component Gas Absorption Induced Flows

Numerical Simulations and Applications ofRarefied Gas Mixtures Flows

A. Frezzotti, G. P. Ghiroldi and L. Gibelli

Dipartimento di Matematica del Politecnico di MilanoPiazza Leonardo da Vinci, 32 - 20133 Milano, Italy

64th IUVSTA Workshop - May 16-19. 2011 - Leinsweiler, Germany

A. Frezzotti, G. P. Ghiroldi and L. Gibelli Numerical Simulations and Applications of Rarefied Gas Mixtures Flows



Outline

1 Introduction

2 Parallel Implementation of DSBE

3 1D and 2D Multi-Component Gas Absorption Induced Flows




Basic Equations

In many applications it is necessary to model flows of multi-componentrarefied gas flows.In this case flow properties are obtained by the solution of a system ofBoltzmann equations:

∂ fi∂ t

+ v◦ ∂ fi∂x

=N

∑i=1

Qij(fi, fj) (1)

In Eq. 1, fi(x,v|t) is the distribution function of the i− th component,Qij(fi, fj) is the ij collision integral whose form depends on molecularinteraction potential.For a hard sphere gas mixture, it takes the form:

Qij(fi, fj)=12

(di +dj

2

)2 ∫[f (x,v∗|t)f (x,w∗|t)− f (x,v|t)f (x,w|t)] |vr ◦ k̂|dwd2k̂

(2)




Kinetic Model for Mixtures

Following the arguments leading to the BGKW model, it is quite natural toreplace each collision term in the Boltzmann equations (1) with a relaxationterm:

∂ fi∂ t

+ v ·∇rfi =2

∑j=1

νij (Φij− fi) (3)

Φij(r,v|t) =nij(r|t)

[2πRiTij(r|t)]3/2 exp

{−[v−uij(r|t)]2

2RiTij(r|t)

}(4)

There is no unique way to define the quantities νij, nij, Tij and uij[Hamel (1965)], [McCormack (1973)].Eqs. (3) generally do not reduce to the one-component model when oneconsiders a fictitious mixture of mechanically identical components[Garzò,Santos and Brey (1989)], [Andries, Aoki and Perthame (2002)].




Numerical Methods

In view of the following discussions, it is convenient to divide numericalmethods used to solve Eq. (1) into three groups:(a) Particle methods(b) Semi-deterministic methods(c) Regular methods




Particle Methods

Methods in group (a) originate from the Direct Simulation Monte Carlo(DSMC) scheme proposed by G.A. Bird.The distribution function is represented by a number of mathematicalparticles which move in the computational domain and collideaccording to stochastic rules derived from the Boltzmann equation.Macroscopic flow properties are usually obtained by time averagingparticle properties. If the averaging time is long enough, then accurateflow simulations can be obtained by a relatively small number ofparticles.The method can be easily extended to deal with mixtures of chemicallyreacting polyatomic species.Application of particle schemes becomes problematic in the followingcases:

Flows resulting from small deviations from equilibriumUnsteady flowsMixtures containing very small amounts of one or more components




Regular & Semi-deterministic methods

Methods in groups (b) and (c) adopt similar strategies in discretizingthe distribution function on a regular grid in the phase space and inusing finite difference schemes to approximate the streaming term.However, they differ in the way the collision integral is evaluated.In semi-deterministic methods C (f , f ) is computed by Monte Carlo orquasi Monte Carlo quadrature methods.Standard integration schemes are used in deterministic methods whichare more often used in combination with kinetic model equations.Application of deterministic or semi-deterministic methods isparticularly useful in the following cases:

Flows resulting from small deviations from equilibriumUnsteady flowsMixtures containing very small amounts of one or more components

Since they produce accurate and noise-free solutions without thenecessity of time averaging




Deterministic & Semi-deterministic methods

Whatever method is chosen to compute the collision term, the adoptionof a grid in the phase space considerably limits the applicability ofmethods (b) and (c) to problems where particular symmetries reducethe number of spatial and velocity variables.As a matter of fact, a spatially three-dimensional problem wouldrequire a memory demanding six-dimensional phase space grid.When used to solve the full steady Boltzmann equation, computercodes based semi-deterministic methods do usually run more slowlythan DSMC codes.For the reasons listed above, the direct solution of the Boltzmannequation by semi-deterministic or deterministic methods has not beenconsidered a viable alternative to DSMC, not even for low speed and/orunsteady flows.However, as shown below the use of GPUs hardware might make suchmethod worth of consideration.




Essential Bibliography: Semi-deterministic methods

Aristov, V.V., & Tcheremissine, F.G. (1980).U.S.S.R. Comput. Math. Phys., 20, 208-225.

Frezzotti, A. (1991).Numerical study of the strong evaporation of a binary mixture.Fluid Dynamics Research, 8, 175-187.

Tcheremissine, F. (2005).Direct numerical solution of the Boltzmann Equation.RGD24 AIP Conference proceeding, 762, 677-685.

Baker, L.L., & Hadjiconstantinou, N.G. (2008).Variance-reduced Monte Carlo solutions of the Boltzmann equation forlow-speed gas flows: A discontinuous Galerkin formulation.Int. J. Numer. Meth. Fluids, 58, 381-402.




Mathematical & Numerical Formulation-1

Although not essential, in view of the applications to the study of low Machflows, it is convenient to rewrite Eq.(??) in terms of the deviational part ofthe distribution function, h(r,v|t), defined as

f (r,v|t) = Φ0(v) [1+ εh(r,v|t)] (5)

Φ0(v) is the Maxwellian at equilibrium with uniform and constant density n0and temperature T0, i.e.,

Φ0 =n0

(2πRT0)3/2 exp

(− v2

2RT0

)(6)

Such formulation allows reducing variance in the Monte Carlo evaluation ofthe collision integral and to capture arbitrarily small deviations fromequilibrium with a computational cost which is independent of themagnitude of the deviation.





Neglecting external forces and substituting Eq. (5) into Eq. (??), we obtain

∂h∂ t

+ v ·∇rh = Q(h,h) (7)

where the collision integral takes the form

Q(h,h) =σ2

2

∫Φ01[h∗+h∗1−h−h1 + ε (h∗h∗1−hh1)

]|k̂ · vr|dv1d2k̂ (8)

For Maxwell’s completely diffuse boundary condition, the distributionfunction of atoms emerging from walls is

Φ0 + ε Φ0 h = nw Φw (v−Vw) · n̂ > 0 (9)

In Eq. (9), n̂ is the inward normal and Φw is the normalized wall Maxwelliandistribution function

Φw(r,v) =1

(2πRTw)3/2 exp[− (v−Vw)

2

2RTw

](10)





The method of solution adopted to solve Eq. (7) is a semi-deterministicmethod in which finite difference discretization is used to evaluate thefree streaming term on the left hand side while the collision integral onthe right hand side is computed by a Monte Carlo technique.The three-dimensional physical space is divided into Nr = Nx×Ny×Nzparallelepipedal cells. Likewise, the three-dimensional velocity space isreplaced by a parallelepipedal box divided into Nv = Nvx ×Nvy ×Nvz

cells.h is represented by the arrayhi,j(t) = h(x(ix),y(iy),z(iz),vx(jx),vy(jy),vz(jz)|t)





The algorithm that advances hni,j = hi,j(tn) to hn+1

i,j = hi,j(tn +∆t) isconstructed by time-splitting the evolution operator into

a free streaming step, in which the right hand side of Eq. (7) isneglected

∂h∂ t

+ v ·∇rh = 0 (11)

a purely collisional step, in which spatial motion is frozen and only theeffect of the collision operator is taken into account by solving thehomogeneous relaxation equation

∂h∂ t

= Q(h,h) (12)





The free streaming step is discretized as

h̃n+1ix,iy;j = (1−Cux −Cuy)hn

ix,iy;j +Cux hnix−1,iy;bj +Cuy hn

ix,iy−1;j (13)

where Cux = vx(jx)∆t/∆x and Cuy = vy(jy)∆t/∆y are the Courantnumbers in the x and y directions, respectively.The collisional step is performed following the expression

dNi,j

dt≈

n20σ2π

Nc

Nc

∑l=1

[χj(v∗l )+χj(v∗1l)−χj(vl)−χj(v1l)

][h(vl)+h(v1l)+ ε h(vl)h(v1l)] |k̂ · vr| (14)

Nc is the number of samples in the sequence {(vl,v1l, k̂l), l = 1, . . . ,Nc},where vl and v1l are Gaussian velocity variates and the random vectors k̂l areuniformly distributed on the unit sphere




Correction Sub-step

Due to the discretization in the velocity space, momentum and energyare not exactly conserved.The numerical error is small but tends to accumulate during the timeevolution of the distribution function.The correction procedure proposed by Aristov and Tcheremissine(1980) has been adopted to overcome this difficulty.At each time step the full distribution function is corrected as follows:

Φ0,j(1+ ε hn+1

i,j)= Φ0,j

(1+ ε

˜̃hn+1i,j)[

1+A+B · v+Cv2] (15)

where the constants A,B and C are determined by mass conservation andmomentum and energy balance for individual species.




Alternative Correction scheme

An alternative correction scheme is presently under investigation. Thediscretized form of following operator is added to the collision term:

∂ fi∂ t

= afi +b◦ ∂ fi∂v

+ c∂

∂v[◦(v−u)fi] (16)

The coefficients a, b, and c are related to the mass, momentum andenergy error in the Monte Carlo evaluation of collision integrals andthey vanish when the velocity grid size goes to zero.The coefficients a, b, and c represent the intensity of a mass source, of avelocity independent force, and of a Gaussian thermostat.




Parallelization Strategy-1

Splitting the ime evolution of f into a free streaming and a collision stepallows breaking each one into a large number of independentcalculations which can be concurrently executed.Each atomic group with velocity vj is independently transported acrossthe spatial grid, during free streaming.The local structure of C (f , f ) allows the concurrent evaluation of thecollision integral at each spatial grid node.




Parallelization Strategy-2

The intrinsic parallelism of the algorithm can be exploited by thedevelopment of a program to be executed on a GPU consisting of a fewhundreds processors with a SIMD-like architecture.During each clock cycle, each core of the multiprocessor array executesthe same instruction but operates on different data.A computer program using GPU acceleration is organized into a serialprogram which runs on the host CPU and one or more kernels whichdefine the computation to be concurrently performed.Kernels are executed by threads, i.e identical tasks concurrentlyworking on each GPU core on different data.




Streaming kernel sketch

Thread Block 1

V1

V2Spatial GridSpatial Grid

Thread Block 2

h̃n+1ix,iy;j = (1−Cux −Cuy)hn

ix,iy;j +Cux hnix−1,iy;j +Cuy hn

ix,iy−1;j




Monte Carlo Collision Kernel Sketch

V

V

V1

2

1

2

V*

*

V

V

V1

2

1

2

V*

*

I

J

Thread JThread I

Spatial Grid

Velocity

Grid

dNi,j

dt≈

n20σ2π

Nc

Nc

∑l=1

[χj(v∗l )+χj(v∗1l)−χj(vl)−χj(v1l)

]×

[h(vl)+h(v1l)+ ε h(vl)h(v1l)] |k̂ · vr|




GPU-Parallel Performances

0 5000 10000 15000 20000N

r

0

100

200

300

400

500

S

Speed-Up Vs. Problem Dimension

(a)

4096 9216 16384 25600N

r

0

0,2

0,4

0,6

0,8

1

Ts,

Tc

Streaming and Collision Kernels Execution Times

202.6 s 397.7 s 648.3 s 1075.2 s

(b)

Figure: (a) Overall speed-up, S = TCPU/TGPU , versus the number of cells in thephysical space, Nr. δ = 1, Nv = 8000, Nc = 6144. (b) Relative time spent on thestreaming step (red bar) and on collision step (blue bar). The numbers above the barsrefer to the total execution time expressed in seconds. δ = 1, Nv = 8000, Nc = 6144.




GPU - Parallel Performances

0 5000 10000 15000 20000

Nr

0

50

100

150

200G

FL

OP

s

Streaming Kernel Perf.

Collision Kernel Perf.Overall Perf.

Measured GFlop Rates




Multi-Component Vapor Deposition

m1

p1

p1

p2

p2

m2

1−

1−

n n1 2

8u T 8

8 8

Τ w

vx∂ fi∂x

=2

∑j=1

Qij(fi, fj) i = 1,2

f∞i(v)=n∞i

(2πRiT∞)3/2exp(− (v−u∞x)2

2RiT∞

)

fi(0,v)=nwi

(2πRiTw)3/2exp(− v2

2RiTw

)vx > 0, nwi

√RiTw

2π=−(1−Pi)

∫vx<0

fi(0,v)dv

Problem: Determine the fluxes Ji = n∞i

√kBT∞

2miπφi(P1,P2,T∞/Tw,m1/m2,c∞), where

c∞ = n∞2n∞1+n∞2




Previous work/results

Applications: vapor deposition, getter pumps.The problem can be mapped on a vapor mixture condensation problemfor which particular cases have been considered in

S. Taguchi, K. Aoki and S. Takata(2003).Vapor flows condensing at incidence onto a plane condensed phase inthe presence of a noncondensable gas. I. Subsonic condensationPhysics of Fluids 15 689–705BGK-like model

S. Yasuda, S. Takata and K. AokiEvaporation and condensation of a binary mixture of vapors on a planecondensed phase: Numerical analysis of the linearized BoltzmannequationPhysics of Fluids 17 047105 (2005).




Analogy with Subsonic Condensation - 1

fw

ξ)(+

f 8( ξ)

n

T

u

88

8

n

T

w

w

fw( ξ)−

Hydrodyn.Region

L

Knudsen Layer





In the case of a single speciesWhen σe = 1 steady subsonic condensation (|M∞x| ≤ 1) is possible onlyif the upstream pressure ratio p∞/pw satisfies the relationship

p∞/pw = Fsub(M∞x,M∞t,T∞/Tw) (17)

For a monatomic gas, approximate values of the functionFsub(M∞x,M∞t,T∞/Tw) have been determined by numerical solution ofBGK model equation (Aoki et al. 1991) and DSMC simulations (Koganand Abramov, 1991).For a polyatomic gas, approximate values of the functionFsub(M∞x,M∞t,T∞/Tw) have been determined by numerical solution ofHolway model equation and DSMC simulations A Frezzotti, 2005





0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

M∞

1

2

3

4

5

6

7

8

9

10

11

12

p∞/p

w

Subsonic Condensation (DSMC)

Subsonic Condensation (BGKW)

Subsonic Condensation Polyatomic Gas (DSMC)

Gas Absorption (Pa=0.1-0.5)

Subsonic Condensation - Gas Absorption Analogy Single Monatomic Species - Hard Sphere Potential

In the case of vapor deposition an identical relationships holds if pw isreplaced with

pw← (1−Pa)√

2πRTwJ(i) J(i) =−∫

vnf (xw,v)dv (18)




Single Species Vapor Deposition

0 0,1 0,2 0,3 0,4 0,5

Pa

0

0,2

0,4

0,6

0,8

1

1,2

1,4

M∞

u∞

/(pa(RT

∞/2π)

1/2)

Single specie Absorption on a Planar SurfaceUpstream Mach Number Vs. Sticking Probability Hard Sphere Potential




Binary Mixture Pa1 = Pa2

0 0,05 0,1 0,15 0,2

Pa

0

0,02

0,04

0,06

0,08

U∞/(

R1T

∞)1

/2

x∞

=0.25

x∞

=0.5

x∞

=0.75

x∞

=1

x∞

=0

Mixture Mass Flux Vs Sticking Coefficientm

2/m=2, P

a1=P

a2




Binary Mixture Pa1 > Pa2

0 0,05 0,1 0,15 0,2 0,25

Pa1

0

0,01

0,02

0,03

0,04

0,05

0,06

U∞

/(R

1T

∞)1

/2

c∞

=0.25

c∞

=0.5

c∞

=0.75

m2/m

1=2.0 - P

a2=0.05

d1=d

2 - T

p/T

∞=1

0 0,05 0,1 0,15 0,2 0,25

Pa1

0

0,01

0,02

0,03

0,04

0,05

0,06

m2/m

1=1.0 - P

a2=0.05

d1=d

2 - T

p/T

∞=1




DSMC - DSBE Comparison

0 50 100 150 200

x/λ∞

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

DSMC n(x)/nref

DSMC c1(x)

DSMC c2(x)

DSBE n(x)/nref

DSBE c1(x)

DSBE c2(x)

Comparison of DSMC and DSBE Numerical Solutionsm

2/m

1=5; P

a1=P

a2=0.2; d

2/d

1=1; T

p/T

∞=1




DSMC - DSBE Comparison

0 50 100 150 200

x/λ∞

0,04

0,05

0,06

0,07

0,08

0,09

0,1

DSMC - Uhyd

(x)/(RT∞

)1/2

DSMC - U1(x)/(RT

∞)1/2

DSMC - U2(x)/(RT

∞)1/2

DSBE - Uhyd

(x)/(RT∞

)1/2

DSBE - U1(x)/(RT

∞)1/2

DSBE - U2(x)/(RT

∞)1/2

Comparison of DSMC and DSBE Numerical Solutionsm

2/m

1=5; P

a1=P

a2=0.2; d

2/d

1=1; T

p/T

∞=1




Flows with noncondensable components - I

0 100 200 300x/l

ref

1

1,2

1,4

1,6

1,8

2

t/tref

= 20

t/tref

= 220

t/tref

= 250

Time Evolution of the Unsteady Flow n

A/n

ref ( Γ = 0.83; P

A = 0.9 )

0 10 20 30 40 50x/l

ref

0

0,05

0,1

of a Binary Mixturen

B/n

ref ( Γ = 0,83; P

B= 0 )

mA

=mB=1; P

A=0.9, P

B=0




Flows with noncondensable components - II

0 0,2 0,4 0,6 0,8 1

PA

0

0,5

1

1,5

2J/

n∞(R

T∞)1

/2

Taguchi et al. (2003), BGK-like Kinetic Model

Direct Sol. BE - Γ = 0Direct Sol. BE - Γ = 0,83

Direct Sol. BE - Γ = 8.31

Comparison of Boltzmann Equation and Kinetic Model Prediction"Binary Mixture" ; m

A=m

B; d

A=d

B; P

A=0.9, P

B=0




Adsorption Induced Gas Flows in an infinite Plates Array

x

<−−Periodic b.c>−−Periodic b.c y

inflow

Hp

Ly

Lx

AbsorbingPlates




Mathematical and Numerical Tools

The following steady 2D Boltzmann equations for a single component and abinary mixture of hard sphere gases

vx∂ fi∂x

+ vy∂ fi∂y

=2

∑j=1

Qij(fi, fj) (19)

has been solved numerically by DSMC for the following parameters setting:Tp/T∞ = 1.0m2/m1 = 1,2,4 d1 = d2

λ∞/Hp = ∞,1.0,0.1; Lx/Hp = 5.0,1.0,0.5,0.2,0.1





0,008 0,04 0,2 1 5

Lx/H

p

0,001

0,01

0,1

U∞

/(R

1T

∞)1

/2

FMFK

N=1.0

KN

=0.1

Arraym

2/m

1=1 - P

1=P

2=0.1

0,008 0,04 0,2 1 5

Lx/H

p

0,001

0,01

0,1

FMFK

N=1.0

KN

=0.1

Arraym

2/m

1=1 - P

1=P

2=0.2





-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-300

-200

-100

0

100

200

3000.5

0.6

0.7

0.8

0.9

1





0 10 20 30 40 50 60 70 80 90 100y/λ

∞

-0,6

-0,4

-0,2

0

0,2 Hydrodynamic Velocity

Light Component

Light Component

X-Direction Averaged Y-Velocity Componentsm

2/m

1=4, P

a1=0.2, P

a2=0.05, c

∞=0.25, T

p/T

∞=1





0 10 20 30 40 50 60 70 80 90 100y/λ

∞

0

0,2

0,4

0,6

0,8

1

X- Averaged Overall Mixture density n(y)/n∞

Light Component Concentration

X-Direction Averaged Y-Velocity Componentsm

2/m

1=4, P

a1=0.2, P

a2=0.05, c

∞=0.25, T

p/T

∞=1




Conclusions

DSBE is a quite useful tool for the investigation of flows whereunfavorable signal to noise ratio makes the application of DSMCdifficult.For monatomic gas mixtures, the 1D and 2D CUDA version of theBoltzmann solver allows computing very low Mach number flows witha DSMC-like computational effort.Efforts in this direction should aim at reducing the memoryrequirements of such schemes.




Acknowledgments

The authors gratefully acknowledge:The support received from Fondazione Cariplo within the framework ofthe project Surface Interactions in micro/nano devices.The helpful discussions with Dr. Antonio Bonucci from SAES getters.

Thanks for your attention


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Numerical Simulations and Applications of Rarefied Gas Mixtures Flows

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