Deterministic Hybrid Computation of Rarefied Gas
Flows
Tomoki Ohsawa and Taku Ohwada
Department of Aeronautics and Astronautics, Graduate School of Engineering, Kyoto University
Abstract. A deterministic hybrid method is developed on the basis of the BGK equation. The conventional
finite difference scheme for the BGK equation is combined with the finite volume method for the NS equation
derived from the BGK equation by the Chapman-Enskog expansion. The numerical test is carried out in
the shock tube problem and the leading edge problem. The two solutions are connected smoothly without
any special treatment.
1
INTRODUCTION
The numerical analysis of rarefied gas flows for small Knudsen numbers requires huge amount of computation. The deviation of distribution function of the gas molecules from the local Maxwellian is of the order
of the Knudsen number and it exerts an appreciable influence on the variation of the distribution function
through the molecular collision (the collision term is multiplied by the inverse of the Knudsen number).
Thus, the numerical error should be much smaller than the magnitude of the deviation and the computation should be performed with high accuracy. The hybrid particle-continuum approach, which employs
the DSMC in the nonequilibrium region and CFD in the nearly equilibrium region, has been attracting a
lot of interests as an efficient method for a rarefied gas flow with nearly continuum regimes. On the other
hand, the deterministic computation of the Boltzmann equation or its model equations such as the BGK
equation is a counterpart to the DSMC computation in the numerical study of rarefied gas flows and the
reliable computation in 2D and 3D cases is extremely expensive even when the model equation is employed.
Thus the curtailment of the computational cost is one of the challenging subjects in this approach. The
main difficulty of the particle-continuum approach lies in the interface between the kinetic solution and the
fluid-dynamic solution; the fluid-dynamic solution is polluted by the statistical noise of the DSMC solution.
The deterministic hybrid approach, which is free from the statistical noise, seems to be quite promising.
In the present paper, a deterministic hybrid method is developed on the basis of the BGK equation. The
conventional finite difference scheme for the BGK equation is employed as the kinetic solver and the finite
volume method for the NS equation derived from the BGK equation by the Chapman-Enskog expansion
[1, 2] is employed as the CFD solver. The numerical test is carried out in the shock tube problem and the
leading edge problem.
2
NOTATION AND BASIC EQUATION
The main notation is summarized. The ρ0 and T0 are, respectively, the reference density and temperature
of the system under consideration; Xi = Lxi are the Cartesian coordinates, where L is the characteristic
length of the system; l0 is the mean
√ free path of gas
√ molecules in the equilibrium state at rest with density
ρ0 and temperature T0 ; k = ( π/2)(l0 /L) = ( π/2)Kn, where Kn is the Knudsen number of the system; L(2RT0 )−1/2 t is the time, where R is the specific gas constant; (2RT0 )1/2 ζi is the molecular velocity;
ρ0 (2RT0 )−3/2 f (xi , ζi , t) is the distribution function of the gas molecules; ρ0 ρ, (2RT0 )1/2 ui , and T0 T are the
density, flow velocity, and temperature of the gas, respectively.
We develop the deterministic hybrid method on the basis of the BGK equation
∂f
∂f
ρ
+ ζi
= (f0 − f ),
∂t
∂xi
k
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
931
(1)
ρ
(ζi − ui )2
exp
−
,
T
(πT )3/2




1
ρ
 f dζ.
 ρui  = 
ζi
2
3ρT /2
(ζi − ui )
f0 =
3
(2)
(3)
DETERMINISTIC HYBRID APPROACH
We describe the construction of the hybrid method in the case where the physical quantities are independent of x2 and x3 and u2 = u3 = 0.
3.1
BGK solver
In the nonequilibrium region, we employ the standard finite difference scheme for the BGK equation. In
the spatially one-dimensional case with u2 = u3 = 0, the BGK equation is reduced to
∂Φ
∂Φ
ρ
+ ζ1
= (Φ̃ − Φ),
∂t
∂x1
k
∞ ∞
χ
1
Φ=
=
f dζ2 dζ3 ,
φ
ζ2 2 + ζ3 2
−∞ −∞
(ζ1 − u1 )2
ρ
χ0
1
Φ̃ =
exp −
=
,
φ0
T
T
(πT )1/2
∞
∞
1 ∞
2
χ dζ1 , u1 =
ζ1 χ dζ1 , T =
[(ζ1 − u1 )2 χ + φ] dζ1 .
ρ=
ρ −∞
3ρ −∞
−∞
(4)
(5)
(6)
(7)
Let the mesh point for x1 and that for ζ1 be denoted by x(i) (x(a) < x(b) for a < b) and ζ (j) , respectively,
and let t(n) = n∆t be the time level. We express the physical quantities for (x1 , ζ1 , t) = (x(i) , ζ (j) , t(n) ) using
the subscripts i and j and superscript n, i.e., ρni = ρ(x(i) , t(n) ), Φnij = Φ(x(i) , ζ (j) , n(n) ), and so on. In the
present study, we employ the following standard implicit finite difference formula:
− Φnij
Φn+1
ρni
n
ij
+ ζ (j) ∇Φn+1
(Φ̃ij − Φn+1
=
ij
ij ),
∆t
k
(8)
where ∇Φn+1
is the standard finite difference expression for ∂Φ(x(i) , ζ (j) , t(n+1) )/∂x1 ; the backward differij
ence formula (i, i − 1, and i − 2) is employed for ζ (j) > 0 and forward one (i, i + 1, and i + 2) is done for
ζ (j) < 0. The values of the macroscopic variables at the mesh point are obtained by the numerical integration
with respect to ζ1 . The Φn+1
for ζ (j) > 0 is solved in order of increasing x(i) and the order is reversed for
ij
ζ (j) < 0.
3.2
NS solver
In the nearly equilibrium region, we employ the finite volume method for the Navier-Stokes equation
derived from the BGK equation by the Chapman-Enskog expansion.[1, 2] In this method the time evolution
of the macroscopic variables h = t(ρ, ρu1 , 3ρT /2 + ρu1 2 ) at x1 = x(i) are computed by
hn+1
= hni −
i
1
(F i+1/2 − F i−1/2 ).
− x(i)
x(i+1)
(9)
The F i+1/2 is the numerical flux defined by
−
F i+1/2 = F +
i+1/2 + F i+1/2 ,
932
(10)
F±
i+1/2 =
t(n+1)
t(n)
ζ1 ≷0
ζ1 ψ f (x(i+1/2) ∓ 0, ζi , t) dζ dt,
(11)
where x(i+1/2) = (x(i) + x(i+1) )/2, ψ = t(1, ζ1 , ζj 2 ). The f (x(i+1/2) ∓ 0, ζi , t) is an approximate solution of
the following Cauchy problem:
∂f
∂f
+ ζ1
= J 0,
∂t
∂x1
C 2 ∂u1
5 ∂T
0
−1/2
2
2
+T
C1 C −
J = 2 C1 −
f0 ,
3
∂x1
2 ∂x1
f (x1 , ζi , t(n) ) = f0 + kf1 ,
(12)
(13)
(14)
where Ci = (ζi − ui )/T 1/2 (u2 = u3 = 0), C = (Cj 2 )1/2 , and f1 is the Chapman-Enskog Navier-Stokes
distribution function and is given by f1 = −J 0 /ρ for the BGK equation. The explicit form of f (x(i+1/2) ∓
0, ζi , t) is given by
∂f0
(i+1/2)
(i+1/2)
(n)
(n)
0
(x(i+1/2) ± 0, ζi , t(n) ). (15)
± 0, ζi , t) = (f0 + kf1 )(x
± 0, ζi , t ) + (t − t ) J − ζ1
f (x
∂x1
At the beginning of each time step, each conservative variable is approximated by a piecewise linear distribution that takes the exact values at x1 = x(i) and allows the discontinuity at x1 = x(i+1/2) . The integration
in Eq. (11) is carried out in advance and the numerical flux is computed only from the macroscopic data in
the actual computation.
3.3
Domain decomposition
The computational domain is decomposed into the continuum (nearly equilibrium) region and the kinetic
(nonequilibrium) region at the beginning of each time step. The point x(i) is judged to belong to the
continuum region if the breakdown parameter measured by a prescribed method is less than a threshold;
it is judged to belong to the kinetic region otherwise. The breakdown parameter may oscillate around the
threshold due to the numerical error and some small regions that contain few mesh points may appear. In
order to avoid the isolation of small regions, the continuum region with one or two mesh points is re-judged
to be the kinetic region and the kinetic region with one or two mesh points is done to be the continuum
region.
In the present study we employ two breakdown parameters. The first breakdown parameter is the local
Knudsen number based on the density gradient Knρ = (l0 /Lρ)(∂ρ/∂x1 ) [breakdown parameter-1], which
is widely used in particle-continuum approach. The second breakdown parameter
is the deviation of the
distribution function from the local Maxwellian [breakdown parameter-2], i.e., |χ − χ0 | dζ1 , which is a
more faithful measure but its computation is more expensive. This parameter is employed for the purpose
of the comparison.
If x(i) belongs to the continuum region at t = t(n) and is judged to belong to the kinetic region at
are created according to f = f0 + kf1 . If x(i) belongs to the kinetic
t = t(n+1) , the kinetic data Φn+1
ij
region at t = n∆t and is judged to belong to the continuum region at t = t(n+1) , the kinetic data Φn+1
are
ij
n+1
discarded and only the macroscopic data hi
are stored.
3.4
Interface between two solutions
We describe the interface between the fluid-dynamic solution and the kinetic solution in the case where
x(i) for i < ic belong to the continuum region and those for ic ≤ i belong to the kinetic region. Since the
numerical flux of the present NS solver splits into two parts, two solutions can be connected at x1 = x(ic −1/2)
in principle; the numerical flux F −
ic −1/2 can be computed from the solution of the BGK equation; the
distribution function for ζ1 > 0 at x1 = x(ic −1/2) is recovered from the NS solution as f = f0 + kf1 and
is employed as the boundary condition for the BGK equation. On the other hand, the matching of two
solutions should be done in the region where both of the solutions are valid. If this principle holds around
933
x1 = x(ic −1/2) , we have much freedom in the design of the interface. In the present study, we employ the
following two methods for the interface;
Method-A : The numerical flux F +
ic −1/2 is computed from the macroscopic data in the continuum region.
−
The numerical flux F ic −1/2 is computed from f (x(ic −1/2) , ζi , t(n) ), which is obtained as the average
of f (x(ic −1) , ζi , t(n) ) and f (x(ic ) , ζi , t(n) ). The former distribution is recovered from the macroscopic
data in the continuum region as f = f0 + kf1 and the latter is computed by the BGK solver. The
macroscopic data at t = t(n+1) in the continuum region are computed by Eq. (9) and the kinetic data
n+1
(j)
> 0 are recovered from the macroscopic data hn+1
as f = f0 + kf1 . These
Φn+1
i
ic −2,j and Φic −1,j for ζ
n+1
newly obtained kinetic data are employed by the BGK solver (8) to compute Φn+1
ic ,j , Φic +1,j , · · · .
Method-B : The macroscopic data hnic and hnic +1 are computed from the data Φnic ,j and Φnic +1,j . The
numerical flux F ic −1/2 is computed from these macroscopic data in the same way as in the case of
other cell boundaries in the continuum region. The rest of the computation is the same as in the case
of Method-A.
4
4.1
NUMERICAL TESTS
Shock tube problem
As the first test case we consider the one-dimensional unsteady flow in a shock tube. Initially, the gas is
at rest and is in a constant state given by ρ0 and T0 for X1 > 0 and in a constant state given by ρ1 and T1
for X1 < 0. The mean
√ free path in the equilibrium state at rest with ρ0 and T0 is taken as the characteristic
length L, i.e., k = π/2. The computational condition is as follows. The domain for x1 is limited to (−D, D)
and D is large enough so that no disturbance arrives at the computational boundaries x1 = ±D during the
computation. The width of the x1 mesh is uniform and is 0.1. The domain for ζ1 is limited to −10 ≤ ζ1 ≤ 10
and is divided
√ into 200 nonuniform intervals. The computation was carried out for ∆t = 0.01 (the unit of
time is l0 / 2RT0) and four methods for the interface, (Method-A, breakdown parameter-1), (Method-B,
breakdown parameter-1), (Method-A, breakdown parameter-2), (Method-B, breakdown parameter-2), were
tested. The initial kinetic region was −2 ≤ x1 ≤ 2 and the threshold of the breakdown parameter-1 and
that of the breakdown parameter-2 were both 0.003.
Figures 1 and 2 show the distributions of density ρ, flow velocity u1 and temperature T at t = 50 for
(ρ1 /ρ0 , T1 /T0 ) = (8, 10/8) [Sod’s test case] and (ρ1 /ρ0 , T1 /T0 ) = (3, 10/3). The hybrid computation for each
method of interface was carried out successfully and the differences are almost invisible in these figures.
The result of the hybrid computation is compared with the BGK solution and the excellent agreement is
confirmed. In these figures, the kinetic regions at t = 50 are shown as the rectangles (Method-B, breakdown
parameter-1). In order to check the history of hybrid computation, we define the characteristic function
C(x1 , t(n) ) in such a way that C(x1 , t(n) ) = 1 in the cell (x(i−1/2) , x(i+1/2) ) if x(i) belongs to the kinetic region
at t = t(n) and C(x1 , t(n) ) = 0 otherwise. Since the transitions of C(x1 , t(n) ) for these methods of interface
are similar, only that for (Method-B, breakdown parameter-1) is shown in Fig. 3 [(ρ1 /ρ0 , T1 /T0 ) = (8, 10/8)].
The distribution function in the form of f = f0 + kf1 is employed in the interface of two solutions.
This distribution function contains the derivatives of the macroscopic variables. If the gap between two
solutions is not far smaller than the cell size, the inclusion of the derivative leads to instability and spurious
oscillations; neither instability nor spurious oscillation occurs when the local Maxwellian f0 is employed
instead of f0 + kf1 . The threshold of breakdown parameter should be chosen so that it is much smaller than
the cell size. The efficiency of the hybrid computation is confirmed from the comparison of the total CPU
time. Owing to the hybrid computation (breakdown parameter-1), the cost is cut down to about 30% of
that of full BGK computation.
934
9.0
3.5
Hybrid
BGK
Kinetic Region
6.0
2.5
5.0
4.0
2.0
3.0
1.5
2.0
1.0
1.0
0.0
−100
−50
0.5
0
x1
50
100
0.8
0.7
0.6
−100
−50
0
x1
50
100
0
x1
50
100
1.0
Hybrid
BGK
Kinetic Region
Hybrid
BGK
Kinetic Region
0.8
0.5
0.6
u1
u1
Hybrid
BGK
Kinetic Region
3.0
ρ
ρ
8.0
7.0
0.4
0.4
0.3
0.2
0.2
0.1
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1.0
0.9
0.8
0.7
−100
−50
0.0
0
x1
50
100
−100
−50
3.5
Hybrid
BGK
Kinetic Region
Hybrid
BGK
Kinetic Region
3.0
2.5
T
T
0.0
−100
2.0
1.5
1.0
−50
0.5
0
x1
50
100
Fig. 1: Comparison of the result of hybrid computation
with the BGK solution for (ρ1 /ρ0 , T1 /T0 ) = (8, 10/8).
Solid line indicates the BGK solution, open circles indicate that of hybrid computation plotted with every 10
points out of the actual computational grids, and rectangles drawn with dashed line indicate the kinetic regions.
−100
−50
0
x1
50
100
Fig. 2: Comparison of the result of hybrid computation
with the BGK solution for (ρ1 /ρ0 , T1 /T0 ) = (3, 10/3).
Solid line indicates the BGK solution, open circles indicate that of hybrid computation plotted with every 10
points out of the actual computational grids, and rectangles drawn with dashed line indicate the kinetic regions.
0
10
C(x1 , t)
1
20
30
0
−100
−50
t
40
0
x1
50
50
100
Fig. 3: Transition of the domain decomposition. The kinetic region splits into three main parts due to the appearance
of uniform equilibrium regions between the expansion wave and the contact discontinuity and between the contact
discontinuity and the shock.
935
4.2
Leading edge problem
x2
x2
x2
As a 2D test case, we consider a steady rarefied gas flow past a flat plate at zero angle of attack. This
problem is analyzed numerically on the basis of the BGK equation and the diffuse reflection boundary
condition by taking account of the propagation of the discontinuity of the distribution function into the
gas from the leading and trailing edges [3] and the accurate solution, which deserves to be the standard
solution, is available. We analyze the problem for
3
Hybrid
the same equation and boundary condition using the
BGK
2.5
hybrid method. The length of the plate is taken as
1.05
1.2
the reference length L and the plate is located at
2
the position (−L/2 ≤ X1 ≤ L/2, X2 = 0). The
ρ = 1.01
1.01
density and temperature at upstream condition are
1.5
taken as the reference values ρ0 and T0 , respectively.
1
1.4
We investigate the case where the upstream Mach
0.9
number is 1.5, the Knudsen number based on the
0.5
length of the plate is 0.05, and the temperature of
0
the plate is equal to T0 . The computational domain
-2
-1
0
1
2
3
x1
for space is the rectangle (−5.199 ≤ x1 ≤ 5.092,
3
0 ≤ x2 ≤ 12). Our main interest in the present nuHybrid
1.1
merical test is the confirmation of the validity of the
BGK
2.5
1.05
simple interface, 2D version of Method-A in Sec. 3.4,
and the adaptive mesh refinement, adaptive domain
2
decomposition, and special treatment of the disconT = 1.01
1.5
tinuity of the distribution function around the leading and trailing edges are not employed. A nonuni1.2
1
1.01
form grid system in x1 x2 plane is employed; the x1
mesh is precise around x1 = ±1/2, and so is the
0.5
x2 mesh near x2 = 0; there are 273 × 161 grids.
0
This grid system is coarser than that employed in
-2
-1
0
1 1.1
2
3
x1
Ref. [3]. The kinetic region is fixed to the rectan3
gle (−0.903 ≤ x1 ≤ 1.527, 0 ≤ x2 ≤ 0.5). The
Hybrid
BGK
computational domain for ζ1 and ζ2 is limited to
2.5
1.4
(−5 ≤ ζ1 ≤ 6.988, −5 ≤ ζ2 ≤ 5) and a nonuniform
2
grid system is employed; there are 115 × 101 grids
1.3
in ζ1 ζ2 plane.
Mc = 1.49
1.5
The isograms of the density ρ, temperature T ,
1.49
and
Mach number Mc defined by Mc =
1
the local
1.0
6/5(u1 2 +u2 2 )1/2 T −1/2 are shown in Fig. 4, where
0.5
the BGK solution of Ref. [3] is shown for comparison. Figure 5 and 6 show the distributions of
0
-2
-1
0
1
2
3
ρ, u1 , u2 , and T along the lines x1 = const and
x1
x2 = const. In these figures, the BGK solution [3]
and the NS solution under the slip boundary condi- Fig. 4: Isograms of the density, temperature, and local
tion on the plate are shown for comparison. While Mach number. ρ = 0.75 + 0.05m (m = 0, . . . , 4), 1.01,
the fair agreement between the hybrid solution and and 1.05 + 0.05m (m = 0, . . . , 17); T = 1.01 and 1.05 +
the full BGK solution is confirmed at downstream, 0.05m (m = 0, . . . , 9); Mc = 1.49 and 1.4 − 0.1m (m =
0, . . . , 12). Solid lines indicate the BGK solution, dashed
the discrepancy is appreciable especially around the
lines indicate the Hybrid solution, and dash-dot lines indileading edge and in the oblique shock-layer-like re- cate the interface.
gion arising from the leading edge. Nevertheless the
present hybrid method yields a better result than the NS solution under the slip boundary condition. Although the accuracy of the present hybrid computation is not very satisfactory, the 2D version of the simple
interface works quite well; the BGK solution and the NS solution are smoothly connected.
Incidentally, the slip boundary condition in the present case (BGK, Tw = T0 ; Tw : temperature of the
936
plate) is
u1 = −
k0 k ∂u1
,
ρ ∂x2
u2 = 0,
T =1+
d1 k ∂T
,
ρ ∂x2
(16)
where k0 and d1 are slip (jump) coefficients and are given by
k0 = −1.01619,
d1 = 1.30272,
(17)
for the BGK equation.[4] The quantities of jump (slip) are proportional to the local mean free path for ρ
and T at the boundary; they are multiplied by (Tw /T0 )1/2 in the case of Tw /T0 = const = 1.
1.6
1.35
Kinetic
1.5
Hybrid
BGK
NS
Continuum
Kinetic
x1 = −0.244
1.25
x1 = −0.244
1.4
Hybrid
BGK
NS
Continuum
1.30
T
ρ
1.20
1.3
1.15
x1 = 0.244
1.2
x1 = 0.244
1.10
1.1
1.05
1.0
1.00
0
0.5
1
1.5
x2
2
2.5
3
0
1.4
0.20
1.2
0.5
1
Kinetic
1.5
x2
2
2.5
Continuum
Hybrid
BGK
NS
x1 = 0.244
0.15
x1 = −0.244
1.0
3
x1 = −0.244
u2
u1
0.8
0.10
0.6
0.4
Kinetic
Continuum
x1 = 0.244
0.05
Hybrid
BGK
NS
0.2
0.0
0.00
0
0.5
1
1.5
x2
2
2.5
3
0
0.5
1
1.5
x2
2
2.5
3
Fig. 5: Comparison of the hybrid solution, BGK solution, and Navier-Stokes solution under the slip boundary condition along the lines x1 = −0.244 and x1 = 0.244. Solid lines indicate the BGK solution, dashed lines indicate
the Hybrid solution, and dash-dot lines indicate the NS solution under the slip boundary condition. The vertical
dash-dot line indicates the interface.
5
CONCLUDING REMARKS
We have carried out the deterministic hybrid computations in the shock tube problem and the leading edge
problem. The BGK solution and the NS solution were connected smoothly without any special treatment.
The deterministic hybrid method yielded excellent results in the shock tube problem and a fair success was
achieved in the leading edge problem. Thus, the deterministic hybrid approach is worth being studied further
towards an efficient and reliable numerical method for complex flows with nonequilibrium regions and nearly
equilibrium regions.
937
2.2
1.6
2.0
1.8
Hybrid
BGK
NS
x2 = 0.004
Continuum
Kinetic
1.5
1.4
Continuum
Hybrid
BGK
NS
x2 = 0.004
Continuum
Kinetic
Continuum
1.3
x2 = 1.009
1.4
x2 = 1.009
T
ρ
1.6
1.2
1.2
1.1
1.0
1.0
0.8
0.6
−2
−1
0
1
x1
2
0.9
−2
3
1.4
0.20
1.2
0.15
−1
0
x1
1
2
Hybrid
BGK
NS
x2 = 1.009
x2 = 1.009
1.0
Continuum
0.10
Kinetic
Continuum
0.05
Continuum
Kinetic
Continuum
u2
u1
0.8
3
0.6
0.00
0.4
−0.05
0.2
0.0
−2
Hybrid
BGK
NS
x2 = 0.004
−1
0
x1
1
2
x2 = 0.004
−0.10
3
−0.15
−2
−1
0
x1
1
2
3
Fig. 6: Comparison of the hybrid solution, BGK solution, and Navier-Stokes solution under the slip boundary condition along the lines x2 = 0.004 and x2 = 1.009. Solid lines indicate the BGK solution, dashed lines indicate the
Hybrid solution, and dash-dot lines indicate the NS solution under the slip boundary condition. The vertical dash-dot
lines indicate the interface.
6
ACKNOWLEDGMENTS
The present study is supported by Grant in Aid for Scientific Research No.14550150 from the Japan
Society for the Promotion of Science. Professor Kazuo Aoki of Kyoto University is acknowledged for providing
the detailed data of BGK solution in the leading edge problem.
REFERENCES
1. Ohwada, T., “Boltzmann schemes for the compressible Navier-Stokes equations,” in Rarefied Gas Dynamics: AIP
Conference Proceedings No. 585, edited by Bartel, T.J. and Gallis, M.A., (AIP, 2001), pp.321-328.
2. Ohwada, T., “On the construction of kinetic schemes,” J. Compt. Phys. 177, 156-175 (2002).
3. Aoki, K., Kanba, K., and Takata, S., “Numerical analysis of a supersonic rarefied gas flow past a flat plate,” Phys.
Fluids 9, 1144-1161 (1997).
4. Sone, Y. and Onishi, Y., “Kinetic theory of evaporation and condensation—Hydrodynamic equation and slip
boundary condition,” J. Phys. Soc. Jpn. 44, 1981-1994 (1978).
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