Discrete unified gas kinetic scheme for all Knudsen number flows. III. Binary gas mixtures of Maxwell molecules.

TLDR: The proposed DUGKS is an effective and reliable method for binary gas mixtures in all flow regimes based on the Andries-Aoki-Perthame kinetic model and is compared with those from other reliable numerical methods.

Abstract: Recently a discrete unified gas kinetic scheme (DUGKS) in a finite-volume formulation based on the Boltzmann model equation has been developed for gas flows in all flow regimes. The original DUGKS is designed for flows of single-species gases. In this work, we extend the DUGKS to flows of binary gas mixtures of Maxwell molecules based on the Andries-Aoki-Perthame kinetic model [P. Andries et al., J. Stat. Phys. 106, 993 (2002)JSTPBS0022-471510.1023/A:1014033703134. A particular feature of the method is that the flux at each cell interface is evaluated based on the characteristic solution of the kinetic equation itself; thus the numerical dissipation is low in comparison with that using direct reconstruction. Furthermore, the implicit treatment of the collision term enables the time step to be free from the restriction of the relaxation time. Unlike the DUGKS for single-species flows, a nonlinear system must be solved to determine the interaction parameters appearing in the equilibrium distribution function, which can be obtained analytically for Maxwell molecules. Several tests are performed to validate the scheme, including the shock structure problem under different Mach numbers and molar concentrations, the channel flow driven by a small gradient of pressure, temperature, or concentration, the plane Couette flow, and the shear driven cavity flow under different mass ratios and molar concentrations. The results are compared with those from other reliable numerical methods. The results show that the proposed scheme is an effective and reliable method for binary gas mixtures in all flow regimes.

Figures

FIG. 8. Schematic of the plane Couette flow.

FIG. 12. Velocity profiles across the cavity center at Re = 400.

FIG. 4. Schematic of the channel flow.

FIG. 3. Structure of a Ma1 = 3.0 shock wave in a binary gas mixture with mA/mB = 0.5 and aBB/aAA = 1: (a) χ A 1 = 0.1; (b) χ A 1 = 0.9.

FIG. 9. Velocity profiles in the Couette flow for the Ne-Ar mixture with C0 = 0.5 at (a) δ = 0.1, (b) δ = 1 and (c) δ = 10.

TABLE III. Wall time (in seconds) and iteration steps of reaching steady states. 24 cores are used in the DUGKS and UGKS, and 48 cores are used in DSMC.

Full text

Discrete unified gas kinetic scheme for all Knudsen flows. III.
Binary mixtures of Maxwell molecules
Yue Zhang,
1
Lianhua Zhu,
1
Ruijie Wang,
2
and Zhaoli Guo
1, ∗
1
State Key Laboratory of Coal Combustion,
School of Energy and Power Engineering,
Huazhong University of Science and Technology, Wuhan 430074, China
2
School of Power and Energy, Northwestern Polytechnical University, Xi’an 710072, China
(Dated: November 22, 2019)
Abstract
Recently a discrete unified gas kinetic scheme (DUGKS) in finite-volume formulation based on
the Boltzmann model equation is developed for gas flows in all flow regimes. The original DUGKS
is designed for flows of single-species gases. In this work, we extend the DUGKS to flows of
binary gas mixtures of Maxwell molecules based on the AAP kinetic model [P. Andries et al., J.
Stat. Phys. 106, 993 (2002)]. A particular feature of the method is that the flux at each cell
interface is evaluated based on the characteristic solution of the kinetic equation itself, thus the
numerical dissipation is low in comparison with that using direct reconstruction. Furthermore, the
implicit treatment of the collision term enable the time step to be free from the restriction of the
relaxation time. Unlike the DUGKS for single-species flows, a nonlinear system must be solved to
determine the interaction parameters appearing in the equilibrium distribution function, which can
be obtained analytically for Maxwell molecules. Several tests are performed to validate the scheme,
including the shock structure problem under different Mach numbers and molar concentrations, the
channel flow driven by small gradient of pressure, temperature or concentration, the plane Couette
flow, and the shear driven cavity flow under different mass ratios and molar concentrations. The
results are compared with those from other reliable numerical methods. The results show that the
proposed scheme is an effective and reliable method for binary gas mixtures in all flow regimes.
Keywords: binary gas mixtures, Maxwell molecules, Boltzmann model equation, kinetic scheme
∗
Corresponding author: zlguo@hust.edu.cn
1

I. INTRODUCTION
Rarefied gas mixtures flows exist widely in nature and practical applications, such as
chemical reactions, evaporation-condensation and the Micro-Electro-Mechanical-System
(MEMS). The rarefaction degree of gas flows is normally characterized by the Knudsen
number (Kn), which is defined as the ratio of the mean free path of gas molecules to the
characteristic length of the system. The conventional fluid dynamics models, such as the Eu-
ler equations and the Navier-Stokes equations, are valid for continuum flows (Kn < 0.001),
but for flows with relative large Kn, non-equilibrium effects will appear and continuum
models become invalid [1].
Alternatively, the Boltzmann equation can be used to describe the gas mixtures flows in
all regimes. But it is difficult to obtain the accurate solutions of the Boltzmann equation
directly due to the complicated collision term. Conventionally, the Direct Simulation Monte
Carlo (DSMC) method was employed to investigate non-equilibrium behaviors of the rarefied
gas mixtures in many studies, e.g., [2–5], which is a prevailing numerical technique for
simulating moderate and highly rarefied gas flows. However, the streaming and collision
processes of the DSMC are decoupled, such that the time step and mesh size are limited by
the molecular collision time and the mean free path, respectively [6]. This limitation leads
to expensive computational costs for continuum and near-continuum flows. It is noted that
some efforts have been made to reduce these difficulties [7, 8]. Besides the DSMC method,
some deterministic numerical methods for the Boltzmann equation, such as [9–12], have been
applied to gas mixtures flows with simple geometries. These deterministic methods can offer
accurate solutions of the full Boltzmann equation, but are usually rather complicated and
computationally expensive.
Some efforts have been devoted to simplify the full Botlzmann equation for gas mixtures
by replacing the full collision operator with certain simplified models. Compared with the
single-species kinetic model equations, the non-unitary mass ratio between different molec-
ular species increases the difficulty. One of such models is the McCormack model [13] which
linearizes the nonlinear collision term under the assumption that the systems only slight
deviate from equilibrium; it is noted that extension to nonlinear problems has also been
made recently [14, 15]. Another simplified model is the so called AAP model [16] in which
the collision term is modeled by a single Bhatnagar-Gross-Krook (BGK) [17] operator con-
2

sidering both self-collision and cross-collision effects. Owing to its simple formulation, the
AAP model has been applied to a number of rarefied mixture flows [18–21].
Based on the kinetic models, some numerical schemes have been developed, such as the
lattice Boltzmann method (LBM) [22–24] and the discrete velocity methods (DVM) [25–27].
Particularly, a unified gas kinetic scheme (UGKS) for binary gas mixtures of hard sphere
molecules and Maxwell molecules has been constructed [28, 29] for all flow regimes based on
the AAP model. The original UGKS is designed for single-species gas flows covering differ-
ent flow regimes [30, 31], which is a finite-volume scheme for the discrete velocity Boltzmann
model equation. A distinctive feature of the UGKS is that the reconstruction of the nu-
merical flux is based on the local analytical characteristic solution of the kinetic equation
rather than interpolation, such that the numerical dissipation is small. Furthermore, the
semi-implicit discretization of the collision term in UGKS enables it to be unified stable,
i.e., the time step is not limited by the mean-collision time. The UGKS also has the nice
asymptotic preserving (AP) property [32], i.e., it solves the Navier-Stokes equations in the
continuum flow regime.
Recently, another unified kinetic method, i.e., the discrete unified gas kinetic scheme
(DUGKS) [33, 34] was developed for single-species gas flows covering different flow regimes.
The DUGKS shares all the advantages of the UGKS, but some apparent differences also
exist between the two schemes. Firstly, they achieve the characteristic solution by different
approach: the UGKS uses an analytical temporal-spatial integral solution of the governing
equation, while the DUGKS uses a discrete characteristic solution which is much simpler
than the analytical integral one. Secondly, in the UGKS, flow variables are required to
be updated first to evaluate explicitly the implicit treatment of the collision term, while
the DUGKS removes the implicitness by introducing a new distribution function that is
tracked in implementation. The above differences make the DUGKS more efficient than the
UGKS [28]. Compared with the LBM, the computational cost of the DUGKS is somewhat
expensive with the same uniform mesh, but is less expensive if a nonuniform mesh is em-
ployed [35]. The DUGKS has already been applied successfully to flows of single-species
gases from continuum to rarefied regimes [36–39]. Recently, some extensions of DUGKS
to complex flows have also been made. For example, a DUGKS for two-phase flows was
proposed based on the phase-field theory [40]. Another possible extension is for flows in
porous media, by making use of unstructured meshes like the finite-volume LBM [41, 42].
3

The aim of this work is to extend the DUGKS to binary gas mixtures of Maxwell molecules
based on the AAP model. The remaining part of this paper is organized as follows. Section 2
will introduce the AAP model for binary gas mixtures. In Sec. 3, the DUGKS will be
constructed based on the AAP model, and in Sec. 4 several numerical tests are performed.
Finally, a brief summary is given in Sec. 5.
II. THE AAP MODEL FOR GAS MIXTURES
The Boltzmann equation for a binary gas mixture of species A and B can be written
as [43],
∂f
α
∂t
+ ξ · ∇f
α
= Q
α
(f, f), (1)
with
Q
α
(f, f) =
X
α=A,B
Q
αβ
(f
α
, f
β
), Q
αβ
(f
α
, f
β
) =
Z
R
3
Z
B
+
(f
0
α
f
0
β
∗
−f
α
f
β
∗
)B
αβ
(N·V , |V |)dξ
∗
dN,
(2)
where the Greek letters α and β will be used symbolically to represent the gas species, i.e.,
{α, β} = {A, B}; f
α
≡ f
α
(x, ξ, t) represents the distribution function of species α with
particle velocity ξ at position x and time t in 3-dimensional physical space; Q
α
(f, f) is the
Boltzmann collision operator for species α, B
αβ
(N · V , |V |) is the collision kernel which
is decided by the intermolecular force between species α and β, ξ and ξ
∗
are pre-collision
velocities, N is a unit vector and B
+
is the semi-sphere defined by N · V = 0, where V is
the relative velocity
V = ξ − ξ
∗
. (3)
From conservation laws of momentum and energy:





m
α
ξ + m
β
ξ
∗
= m
α
ξ
0
+ m
β
ξ
0
∗
,
m
α
|ξ|
2
+ m
β
|ξ
∗
|
2
= m
α
|ξ
0
|
2
+ m
|
ξ
0
∗
|
2
,
(4)
the post-collision velocities ξ
0
and ξ
0
∗
can be written as





ξ
0
= ξ −
2m
αβ
m
α
N[(ξ − ξ
∗
) · N ],
ξ
0
∗
= ξ
∗
+
2m
αβ
m
β
N[(ξ − ξ
∗
) · N ],
(5)
4

with the reduced mass being
m
αβ
=
m
α
m
β
(m
α
+ m
β
)
, (6)
in which m
α
and m
β
are the molecular masses of species α and β, respectively. Without
loss of generality, we assume m
A
< m
B
.
Furthermore, the macroscopic quantities of species α, such as the molecular number
density n
α
, mass density ρ
α
, flow velocity u
α
, the total energy E
α
, and the internal energy

α
are calculated as the moments of distribution function f
α
:
ρ
α
=
Z
f
α
dξ, n
α
= ρ
α
/m
α
, (7a)
ρ
α
u
α
=
Z
ξf
α
dξ, (7b)
ρ
α
E
α
=
1
2
Z
ξ
2
f
α
dξ =
1
2
ρ
α
u
2
α
+ 
α
, (7c)

α
=
1
2
Z
|c
α
|
2
f
α
dξ, (7d)
where c
α
= ξ −u
α
. The mass density ρ
m
, number density n
m
, flow velocity u
m
, energy E
m
and internal energy 
m
of the mixture can then be obtained as
ρ
m
=
X
α=A,B
ρ
α
, n
m
=
X
α=A,B
n
α
, (8a)
ρ
m
u
m
=
X
α=A,B
ρ
α
u
α
, (8b)
ρ
m
E
m
=
X
α=A,B
ρ
α
E
α
=
1
2
ρ
m
|u
m
|
2
+ 
m
. (8c)
The AAP model is a relaxation approximation of the full Boltzmann equation in Eq. (1)
∂f
α
∂t
+ ξ · ∇f
α
= Ω
α
(f, f) =
f
∗
α
− f
α
τ
α
, (9)
where α = A or B with
f
∗
α
= ρ
α

m
α
2πk
B
T
∗
α

3
2
exp

−
m
α
2k
B
T
∗
α
(ξ − u
∗
α
)
2

, (10)
in which k
B
is the Boltzmann constant. The parameters u
∗
α
and T
∗
α
are introduced to
recover the correct inter-species transfer of momentum and energy due to the collisions
5

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Discrete unified gas kinetic scheme for all knudsen flows. iii. binary mixtures of maxwell molecules" ?

In this work, the authors extend the DUGKS to flows of binary gas mixtures of Maxwell molecules based on the AAP kinetic model [ P. Andries et al., J. Stat. Phys. 106, 993 ( 2002 ) ]. Furthermore, the implicit treatment of the collision term enable the time step to be free from the restriction of the relaxation time. 

Q2. What have the authors stated for future works in "Discrete unified gas kinetic scheme for all knudsen flows. iii. binary mixtures of maxwell molecules" ?

Further development of the method based on more accurate kinetic models such as the ellipsoidal models [ 59, 60 ] or the McCormack model will be studied in the future work. 

Q3. What is the AP property of the DUGKS?

Due to the coupling of particle transport and collision in the reconstruction of the interface distribution function, the DUGKS has the asymptotic preserving (AP) property [30, 32]. 

Q4. How many velocities are distributed in the velocity space?

The velocity space is discretized by Newton-Cotes quadrature with 101 ve-locity points distributed uniformly in [−8 √ 2kBT1/m, 8 √ 2kBT1/m]. 

Q5. How many cores are used in the DSMC solver?

Both the DUGKS and UGKS are run with 24 cores using OpenMP programming, while the DSMC solver is run with 48 cores using MPI programming. 

Q6. How is the velocity space for each species discretized?

The velocity space for each species is discretized using the Newton-Cotes rule, with 101×101 velocity points distributed uniformly in [−4 √ 2RαT0, 4 √ 2RαT0] × [−4 √ 2RαT0, 4 √ 2RαT0]. 

Q7. How much does the rarefaction parameter affect the shear stress?

The rarefaction parameter δ is related to the Reynolds number asδ = RekBT0 mAUwv0 , (58)whereRe = ρ0UwHµA , ρ0 = n0mA. (59)In the simulations, the authors take Uw = 0.1v0. 

Q8. What is the distribution function at the interface?

Once the distribution function φ̄α at the interface is known, the original distribution function φα can be obtained according to Eq. (34), i.e.,φα(xb, ξ, tn+1/2) = 2τα2τα + s φ̄α(xb, ξ, tn + s) +s2τα + s φ∗α(xb, ξ, tn + s). 

Q9. How is the computational efficiency of the DUGKS and UGKS evaluated?

In order to evaluate the computational efficiency, the authors also measure the computing time of the DUGKS, UGKS and the DSMC method using the lid-driven cavity flow case with of the Ne-Ar mixture. 

Q10. What is the translational kinetic temperature for the Ne-Ar mixture?

The translational kinetic temperature [6] is considered in this case, which is defined by3 2 kBTtr = 1 2 ∑ α (nα/n)mαc ′2 α , (60)where c′2α can be expressed asc′2α = u 2 α + 3RαTα − u2m. (61)For the Ne-Ar mixture with a small mass ratio, the DUGKS results agree excellently with the DSMC solutions in all cases. 

Q11. what is the boltzmann equation for a binary gas mixture?

The Boltzmann equation for a binary gas mixture of species A and B can be writtenas [43], ∂fα ∂t + ξ ·∇fα = Qα(f, f), (1) with Qα(f, f) = ∑ α=A,B Qαβ(fα, fβ), Qαβ(fα, fβ) = ∫ R3 ∫ B+ (f ′αf ′ β∗−fαfβ∗) 

Q12. What is the difference between the two models?

For δ = 10, the velocity difference of Ne between the DUGKS and the McCormack model is less than 2.3% and that of Ar is less than 1% as C0 varies from 0.1 to 0.9. 

Q13. What is the difference between the two UGKS?

It can be seen that the DUGKS results agree well with the benchmark data, suggesting that the AAP model satisfies the indifferentiability principle,which requires that the total distribution function f = ∑α=A,B fα satisfies the single-speciesBGK equation when the two species are the same. 

Q14. What is the difference between the two DUGKS?

As for the cavity flow, the proposed DUGKS results agree well with the DSMC solutions in all flow regimes when the mass ratio is small, but clear deviations appear in the near-continuum regime with large mass ratio, which can be attributed to the relaxation approximation of the collision operator. 

Q15. What is the effect of the rarefaction parameter on the shear stress?

The influence of the rarefaction parameter δ on the shear stress is demonstrated in Fig. 11 with δ ranging from 0.01 to 80 and C0 = 0.5 for the Ne-Ar and He-Xe mixtures.