Study on Gas Kinetic Algorithm for Flows from Rarefied
Transition to Continuum
Zhihui Li and Hanxin Zhang
China Aerodynamics Research and Development Center
621000, Mianyang, Sichuan, China
Abstract. An unified numerical algorithm is presented for the flows from rarefied transition to continuum based on the nonlinear
Boltzmann model equation. Based on the discrete velocity ordinate method, the optimum Golden Section method is extended to
discretize velocity components, and then the simplified velocity distribution function equation adapted to various flow
regimes will be cast into hyperbolic conservation law form with nonlinear source term. The time-splitting method and the
NND finite difference method are employed to solve the equation. Four types of quadrature rules, such as the modified
Gauss-Hermite formula and the Golden Section number-theoretic integral method proposed by Hua L. K. and Wang Y.,
are developed to evaluate the macroscopic flow moments. To test the present method, the one-dimensional shock-tube
problem, the flow past two-dimensional circular cylinder and the flow over three-dimensional sphere with various
Knudsen numbers are simulated. The computational results are found in high resolution of the flow fields and good
qualitative agreement with the theoretical, DSMC, and experimental results.
INTRODUCTION
In order to study the aerodynamics from various flow regimes, the traditional way is to deal with them with
different methods, such as the Monte Carlo method for the rarefied flow and the N-S equation for the continuum
flow. Both the methods are totally different, and the computational results are difficult to be linked up smoothly with
altitude. In this study, it’s to be considered whether an unified numerical algorithm to predict the flows over the
complete spectrum of flow regimes can be developed.
The Boltzmann equation based on the kinetic theory of gases can describe the molecular transport phenomena
from various flow regimes and act as the main foundation for the study of gas dynamics[1]. However, the equation
describing flow characteristics in the view of micromechanics is a nonlinear integral-differential equation. It’s very
difficult to accurately solve the equation describing complex flows. On the other way round, based on the mass,
momentum and energy conservation laws, many scholars have put forward kinds of kinetic model equations
resembling to the various moments on the Boltzmann equation with applying the basic characteristics of molecular
movement and collision approximating equilibrium[2,3]. Based on the zeroth, first and second order approximations
of the Maxwellian distribution, the Euler, N-S and Burnett equations can be respectively deduced by the ChapmannEnskog procedure from the BGK kinetic model equation.
In the nineties, a new BGK type scheme applying the asymptotic expansion of the molecular velocity distribution
function to the Maxwellian distribution based on the flux conservation at the cell interface has been presented
according to the thoughts of the shock capturing difference method. The scheme has successfully simulated some
problems, such as one-dimensional shock wave structure and two-dimensional shock wave reflection[4] etc. In the
computation of the rarefied gas flows, the finite difference method solving two-dimensional BGK equation has been
set forth based on the applications of the reduced velocity distribution functions and the translation from BGKBoltzmann equation into partial differential equations[5,6].
In this study, the unified simplified velocity distribution function equation adapted to various flow regimes can
be presented based on the nonlinear Boltzmann equation. The discrete ordinate method[7] is applied to the
distribution equation in order to replace its continuous dependency on the velocity space, and then the equation will
be cast into hyperbolic conservation law form with nonlinear source term. The time-splitting method is used to split
up the relaxing procedure on the distribution function into the collision relaxation and the convection movement.
The non-oscillatory, containing no free parameters and dissipative (NND) finite difference method[8] is employed to
solve convection equations and the second order Runge-Kutta method is used to numerically simulate the colliding
relaxation equation. To improve the computational efficiency for various Mach number flows, four types of
quadrature rules, such as the modified Gauss-Hermite quadrature formula[7] and the Golden Section integral method
which is developed out of the original works of Hua L. K. and Wang Y.[9], are applied to the discrete velocity space
to evaluate the macroscopic flow parameters at each point in the physical space. To illustrate the feasibility of the
present numerical method, the one-dimensional Riemann shock-tube problem, the supersonic flow past circular
cylinder and the flow past three-dimensional sphere with various Knudsen numbers are simulated.
FOUNDATION OF THE SIMPLIFIED VELOCITY DISTRIBUTION EQUATION
Based on the Boltzmann equation[1] which describes the molecular transport phenomena from various flow
regimes in the view of micromechanics, the basic characteristics of molecular movement and collision
approximating equilibrium is used. The local equilibrium distribution function f N is taken as the expansion in
Hermite polynomials with the local Maxwellian distribution f M as the weighting function. The collision frequency
ν and the appropriate f N are related to the molecular transport, thermodynamic effect and molecular power law
from various flow regimes. The unified simplified velocity distribution function equation adapted to various flow
regimes can be presented with the non-dimensional form in the Cartesian coordinates[10].
H ∂f
∂f
+ V ⋅ H =ν ( f N − f )
∂r
∂t
H H
f N = f M ⋅ 1 + (1 − Pr )c ⋅ q 2c 2 T − 5 (5 PT 2 )
[
f M = n (πT )
32
ν = 8nT
1− χ
(5
[
exp − c 2
π Kn
)
(
T]
)
]
(1)
(2)
(3)
, K n = λ∞ L
(4)
H
H
Where f is the molecular velocity distribution function which depends on space r , molecular velocity V and time
H
t , Kn is the Knudsen number, Pr is the Prandtl number, c is the peculiar velocity of the molecule, χ is a constant
of the molecular power law. Each dimensionless quantity is referred to its equilibrium values at upstream ( n∞ , T∞ ).
The reference speed C∞ is 2 RT∞ and the reference time t∞ is L / c∞ , L is the reference length.
The macroscopic flow parameters, such as the number density, mean velocity, temperature, pressure, viscosity
stress and heat flux vector, can be determined by the moments of the distribution function on the velocity space[10].
H
H
n(r , t ) = ò f dV
H
H
nU i (r , t ) = ò Vi f dV
H
H
3
nT (r , t ) = ò c 2 f dV
2
P = nT
H
H
τ ij (r , t ) = 2 ò ci c j f dV − pδ ij
H
H
q i (r , t ) = ò c 2 ci f dV
(5)
(6)
(7)
(8)
(i、j = 1,2,3)
(9)
(10)
APPLICATI ON OF THE DISCRETE VELOCITY ORDINATE METHOD
Applying the discrete velocity ordinate method[7] to Eq.(1) for the ( Vx , V y , Vz ) velocity space, the unified
velocity distribution equation can be transformed into hyperbolic conservation equation with nonlinear source term.
∂Q ∂F x ∂F y ∂F z
+
+
+
=S
∂t
∂x
∂y
∂z
(
(11)
)
Where, Q = fσ ,δ ,θ , F x = Vxσ Q, F y = V yδ Q, F z = Vzθ Q, S = ν fσN,δ ,θ − fσ ,δ ,θ ; fσ ,δ ,θ and fσN,δ ,θ denote values of f and
f
N
at the discrete velocity points ( Vxσ , Vyδ , Vzθ ).
The selection of the discrete velocity points and the range of the discrete velocity space in the discrete ordinate
method are somewhat determined by the problem dependent. For various freestream Mach number flows, four types
of discrete velocity quadrature rules are used in the discrete ordinate method. One is the modified Gauss-Hermite
rule which is
ò
∞ −V 2
0 e
N
f (V )dV = å Wσ f (Vσ )
σ =1
(12)
Where, Vσ (σ = 1,L, N ) are the positive roots of Hermite polynomial of N th degree and Wσ are the
corresponding weights. The discrete velocity points and corresponding weights can be obtained by two ways. One
relies on the table of the Gauss-Hermite quadrature, the other is to solve the nonlinear Eqs. (12) and (13) in terms of
the decomposing principle.
ò
∞ −u 2 l
u du
0 e
1 æ l + 1ö
= Γç
÷
2 è 2 ø
(13)
The advantage of using Gauss-Hermite quadrature is its high accuracy, but for high freestream mach number
flows the number of discrete points needed to cover the appropriate discrete velocity range could become quite
large. The equally spaced three-point composite Newton-Cotes formulas and the Gauss-Legendre numerical
quadrarure rule whose integral nodes are the roots of the Legendre polynomial have been applied to this study. The
use of the above mentioned integration rules, although capable of treating high Mach number flows, is
computationally expensive.
The Monte-Carlo-Integration (MCI) technique has been exclusively used to evaluate multi-integrals[11]. It has
been noticed that the drawback to the MCI technique is the large consumption of CPU time, the MCI procedure
must be executed at each time step, and the selection of the discrete points is random in all directions. To simplify
the numerical integration over the discrete velocity space, in particular optimize the distribution of the discrete
velocity points, and improve the computational efficiency to multi-dimensional flows, the theory of the uniform
distribution and the algebraic number-theoretic method of the calculability and finality are tested to replace the
Monte Carlo method based on statistical experiment, and the Golden Section method is used to discretize the
velocity components, where the discrete velocity points in a direction are increasingly determined by fraction and
those in other direction are appropriately controlled by the principle of the Golden Section. Based on the intrinsic
relation between the asymptotic fraction (Fibonacci number) of the golden number (0.618) and the numerical
integral, the Golden Section number-theoretic integral formula[10] is developed to evaluate the macroscopic flow
moments on the discrete velocity space by applying the Hua-Wang’s method[9] that the multi-integral is
approximated by single summation. In particular, the double-integral can be evaluated by
1 1
ò 0 ò0
f ( x, y )dxdy ≈
1 Fm æç k ì Fm −1k ü ö÷
,í
å f
ý
Fm k =1 çè Fm î Fm þ ÷ø
(14)
Where, F0 = 0，F1 = 1 and Fm + 2 = Fm +1 + Fm is the Fibonacci number. The ternary-integral can be evaluated by
1 1 1
ò0 ò 0 ò 0
f ( x, y, z )dxdydz ≈
1 n æ k ì h1k ü ì h2 k ü ö
å f ç ,í
ý, í
ý÷
n k =1 çè n î n þ î n þ ÷ø
(15)
Where, (h1, h2 , n) can be taken as the extension of the Fibonacci number collection[9].
NUMERICAL ALGORITHM ON THE VELOCITY DISTRIBUTION EQUATION
To treat arbitrary geometry configuration, the body fitted coordinate is introduced, and the Eq.(11) in general
coordinates ( ξ ,η , ζ ) can be written as
∂U ∂F ∂G ∂H
+
+
+
=S
∂t
∂ξ ∂η ∂ζ
(16)
Where , U = JQ, F = U U , G = V U , H = W U , S = JS ; J = ∂ ( x, y, z ) / ∂ (ξ ,η ,ζ ) ; U = Vxσ ξ x + Vyδ ξ y + Vzθ ξ z ,
V = Vxσ η x + Vyδ η y + Vzθ η z , W = Vxσ ζ x + V yδ ζ y + Vzθ ζ z . The transformed coefficients A = ∂F ∂U , B = ∂G ∂U and
C = ∂H / ∂U of the Eqs. (16) have real eigenvalues a = U , b = V , c = W .
Based on the Taylor expanding with the second-order accuracy in time, the time-splitting method for the
unsteady equation has been developed by use of the second-order Runge-Kutta method and the NND-4(a)
scheme[8,12] which is two-stage scheme with second-order accuracy in time and space. For the first step, the Eqs.
∂U ∂t = S with non-linear source item can be solved by the Runge-Kutta method. For the second step, the
convection movement equation ∂U ∂t + c ∂U ∂ζ = 0 can be numerically solved by the NND-4(a) scheme. For the
third step, the Eqs. ∂U ∂t + b ∂U ∂η = 0 is also solved by the NND-4(a) scheme. For the fourth step, the Eqs.
∂U ∂t + a ∂U ∂ξ = 0 is equally solved by the NND-4(a) scheme.
Considering the simultaneously proceeding on the molecular movement and colliding relaxation in real gas, the
computing order of the previous and hind time steps is interchanged to couple the computation in the time-splitting
scheme. The finite difference second-order scheme has been constructed as
n +1
U ijk = LS (
Where,
∆t
∆t
∆t
∆t
∆t
∆t n
) Lζ ( ) Lη ( ) Lξ (∆t ) Lη ( ) Lζ ( ) LS ( )U ijk
2
2
2
2
2
2
(17)
U * = Ls (∆t )U n = U n + ∆t (1 − ν∆t / 2) S n
é
ù
c 2 ∆t 2
U ** = Lζ ( ∆t )U * = ê1 − c∆tδ ζ +
δ ζ 2 úU *
2
ë
û
2
2
é
ù
b ∆t
U *** = Lη ( ∆t )U ** = ê1 − b∆tδη +
δη 2 úU **
2
ë
û
2
2
é
ù
a ∆t
U n +1 = Lξ (∆t )U *** = ê1 − a∆tδ ξ +
δ ξ 2 úU ***
2
ë
û
Considering the basic feature of the molecular movement and collision approximating equilibrium, the time step
size ( ∆t ) in the computation should be less than the local mean collision time size ( ∆tc ).
∆t = CFL ⋅ min(∆tc , ∆t s ) ,
CFL = 0.95
(18)
Where, ∆ts = 1 max(ν / 2, U / ∆ξ , V / ∆η , W / ∆ζ ) ; ∆tc = 1 ν max .
In order to specify the interaction of the molecules with the solid surface, it is assumed that molecules which
strike the surface are subsequently emitted with a Maxwellian velocity distribution fully accommodating to the wall
temperature TW and velocity (UW ,VW ,WW ) . The density of molecules diffusing from the surface, nW , which is not
known previously, may be determined from the mass balance condition on the surface.
é (Vx − UW ) 2 + (V y − VW ) 2 + (Vz − WW ) 2 ù
nW
exp
ê−
ú
TW
(πTW )3 / 2
êë
ûú
H
H
H
1 π
nW = − ( )1 / 2 ò(VH ⋅nH <0 ) (V ⋅ n ) fdV
2 TW
fW =
H
(19)
(20)
Where n is the outward unit vector normal to the solid surface.
The distribution function for outgoing molecules through boundaries is determined by using the characteristicsbased boundary condition which is in accord with the upwind nature of the interior point scheme[13]. The distribution
function for incoming molecules through boundaries is assumed that there is no gradient along the outward direction.
The numerical algorithm for one- and two-dimensional gas flows can be found by the above mentioned
method[14].
NUMERICAL EXAMPLES AND RESULTS
To test the accuracy and efficiency of the present numerical method in solving the gas dynamical problems from
rarefied flow to continuum, the one-dimensional shock-tube problem, the flow past two-dimensional circular
cylinder and the flow past three-dimensional sphere with various Knudsen numbers are simulated.
Case I One-dimensional Riemann Shock-Tube Problem
Density
A diaphragm located at x=0.5 separates two regions, each in a constant equilibrium state at t=0. Here we consider
a case with initial states: ρ = 0.445, T = 13.21,U = 0.698 for 0.0 ≤ x ≤ 0.5 and ρ = 0.5, T = 1.9,U = 0.0 for 0.5 < x ≤ 1 . The
ratio of specific heats is 5/3 for a monatomic Maxwell gas. The computational results of non-dimensional density
profiles for Kn = 0.01,0.005,0.001 ,0.0001 at time t = 0.1314 are presented in Fig.1. To see the contribution of the
collision relaxation term in the
Boltzmann model equation to
the distribution function, we
1.2
test a numerical method which
is generated by neglecting the
nonlinear collision source term
rieman-exact--Density
and setting the distribution
cal. euler limit solution
1
function in convection terms
cal. Kn=0.0001
cal. Kn=0.001
equal to the equilibrium
cal. Kn=0.005
Maxwellian distribution. The
cal. Kn=0.01
solution obtained by such a
0.8
scheme is named as the
continuum Euler limit solution.
In Fig.1, the symbols ( )
0.6
denote the computed Kn = 0.01
results, the symbols (∇) denote
Kn = 0.005
the
computed
results, the symbols (<) denote
0.4
the computed Kn = 0.001 results,
the symbols (à) denote the
computed Kn = 0.0001 results,
0.2
the symbols (ο) denote the
0
0.25
0.5
0.75
1
computed Euler limit solution,
Position
the solid line denotes the
Riemann exact solution by
FIGURE 1. Density profiles of the shock-tube problem for various Kn numbers
using the Euler equations of gas
dynamics. The smaller is the
Knudsen number, the more crisp shock, rarefaction and contact profiles are given. The results appropriate to
Kn = 0.0001 are in good agreement with the Euler limit solution and the Riemann exact solution.
Case II Supersonic Flow past Circular Cylinder
The steady supersonic rarefied flows past a circular cylinder under different freestream Mach and Knudsen
numbers are computed. The computational results of the density contours are shown in Fig.2, respected to the states
of M ∞ = 1.8 , Pr = 1 , the ratio of the wall temperature to the total temperature TW / TO = 1 ,
Kn = 1,0.1,0.03,0.001( M ∞ = 4) . The obvious shock wave disturbing region doesn’t exist for the state Kn = 1 of fully
rarefied flow. It’s shown that the smaller is the Knudsen number, the more crisp is the front bow shock. For the near
continuum flow of Kn = 0.001 , the flow structures including the bow shock, stagnation region and near wake are
well captured. The computational results of the Mach number contours are shown in Fig.3 related to the state of
Kn = 0.0001 , M ∞ = 1.8 , Pr = 1 , TW / TO = 1 . Fig.4 shows an enlarged view of the streamline past the circular cylinder
for the foregoing case of Kn = 0.0001 , M ∞ = 1.8 , as is only the feature of the continuum flow. In Fig.5 and Fig.6, the
stagnation line profiles of gas density and flow velocity are shown together with the DSMC results[15] for two
FIGURE 2. Density contours of the supersonic flow past cylinder with various Kn numbers
Knudsen numbers ( Kn = 1,0.3) with the state of M ∞ = 1.8, TW TO = 1 , Pr = 1 , respectively. The solid line denotes
the computed ( Kn = 0.3 ) results, the symbols ( ) denote the DSMC ( Kn = 0.3 ) results, the dashed line denotes the
5
U/U00
R/R00
Kn=1
Density
1
4
0.8
3
0.6
2
Kn=0.1
1.2
Cal.
Kn=0.3
DSMC Kn=0.3
Cal.
Kn=1
DSMC Kn=1
Velocity
0.4
Cal.
Kn=0.3
DSMC Kn=0.3
0.2
Cal.
Kn=1
DSMC Kn=1
1
0
Kn=0.03
0
-12
-0.2
-10 -12
-8
-10
-6
-8
-4
-6
Kn=0.001,Mach=4
-2
-4
0
X/LMD00
-2
0
X/LMD00
FIGURE 5. Stagnation
FIGURE
line
6. density
Stagnation
profiles
line for
velocity
a cylinder
profiles for a cylinder
computed ( Kn = 1 ) results, the symbols ( ∆ ) denote the DSMC ( Kn = 1 ) results. In general good agreement between
the present computations and DSMC solutions can be observed.
Mach contours:20
Kn=0.0001,M00=1.8,Time=2.9,error=2.4e-3
Kn=0.0001,M00=1.8
Case III Supersonic Flow past Sphere with various
FIGURE 3. Mach number contours for Kn = 0.0001
FIGURE.4. Enlarged view of the streamlines past cylinder
(a) Pressure
(b) Mach number
Knudsen numbers
In Table 1, the comparisons between the calculated sphere drag coefficients and experimental data[16] are shown
for the cases of M ∞ = 2.0 , Pr = 0.72 , TW / TO = 0.7 , γ = 1.4 , Kn = 0.0126,0.02324,0.126,0.248,1.26 . The computed results
agree with the experimental data very well. The computational results of the density and Mach number contours in
the symmetry plane (xoz) are shown in Fig.7 for the forementioned case Kn = 0.0126 . It’s shown that the thickening
of the front bow shock and the dimmer recompression shock at the wake are the noticeable characteristics in the
rarefied transitional flow.
TABLE 1. Drag coefficients of sphere for the flow ( M ∞ = 2 ) with various Kn numbers
Knudsen ( Kn )
Cd (Cal.)
Cd (Exp.)
0.0126
1.7049
1.5997
0.02324
1.7467
1.6621
0.126
2.2472
2.1779
0.248
2.3771
2.3287
1.26
3.1902
3.1494
Error
6.58%
5.09%
3.18%
2.08%
1.30%
CONCLUDING REMARKS
The simplified molecular velocity distribution function equation describing microscopic transport phenomena is
transformed into the hyperbolic conservation equation with nonlinear source term by introducing the discrete
velocity ordinate method. The time-splitting method and the NND scheme are employed. Four types of quadrature
rules are developed to evaluate the macroscopic flow parameters. As a result, a simplified unified kinetic algorithm
from rarefied flow to continuum has been developed. The computations of one-, two-, and three-dimensional flows
from rarefied transition to continuum indicate that both high resolution of flow fields and good qualitative agreement
with theoretical, DSMC and experimental results can be obtained. The present method provides an economical and
efficient way that the gas dynamical problems from rarefied flow to continuum can be effectively simulated.
FIGURE 7. Pressure and Mach number contours in the symmetry plane (XOZ) past sphere for the state of Kn = 0.0126 , M ∞ = 2
(a) Pressure, (b) Mach number contours
ACKNOWLEDGMENTS
This work was supported by the National Nature Science Council of the People’s Republic of China under Grant.
No.19972008 and the national laboratory of CARDC on computation fluid dynamics in Beijing.
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