A Moment Solution of Comprehensive Kinetic Model
Equation for Shock Wave Structure
K.Maeno*, R.Nagai*, H.Honma*, T.Arai** and A.Sakurai**
* Chiba University, ** Tokyo Denki University, Japan
Abstract. We consider the classic problem of a one-dimensional steady shock-wave solution of the Boltzmann kinetic
equation utilizing a new type of 13-moment approximation proposed by Oguchi (1998). The model, unlike previous ones,
expresses the collision term in an explicit function of the molecular velocity. We can thus obtain moment integrals
directly because of its explicit expression. The principal value is utilized for the moment integral to cope with the
singularity, and we can have five relations for five unknown functions to be determined as functions of the coordinate x.
These can be reduced to a first-order differential equation that can be solved to provide the familiar smooth monotonic
transition from the upstream supersonic state to the subsonic downstream state. Computed values of shock thickness for
various shock Mach numbers compare well with some existing results obtained by different models and methods to
certain Mach numbers beyond which no solution exists. Some problems concerning the existence of solution are
considered in Appendix.
INTRODUCTION
We consider the classic problem of a one-dimensional steady shock wave structure. Study of this problem has a
long history, beginning at almost the same time when research started on shock wave phenomenon. We can recall
such early works by Riemann [1] in Euler’s equation, Taylor [2] for weak shock solution of the Navier-Stokes
equation, and Becker [3] for an exact solution of the Navier-Stokes equation. We can recall also the first
measurement of shock profile by Talbot & Shermann [4] and many other early contributions; we will refer to some
of them below as they are needed. However, the current status now is far from satisfactory, and it is still necessary to
study its essential features. Besides its study based on continuum mechanics represented by the Navier-Stokes
equation, here we are interested in the approach by the Boltzmann kinetic model. Notice that the need to apply a
kinetic model to this problem was mentioned as early as Becker’s paper [3]. There are various different ways of
solving the Boltzmann kinetic equation. A recent standard one is Direct Simulation Monte Carlo (DSMC) (Bird [5]).
However, the classical way of seeking a distribution function close to a Maxwell distribution by perturbation
(Chapmann and Cowling [6]) is still useful for the problem we have now, to which we can apply a moment equation
derived from the basic Boltzmann equation. Here, we utilize a new type of 13-moment approximation (O-model)
proposed by Oguchi [7]. The model, unlike the previous ones originated by Grad [8], expresses the collision term in
an explicit function of the molecular velocity, where its dependency on spatial coordinates is given through
macroscopic quantities such as temperature, pressure, density, fluid velocity, stress tensor and thermal flux as
functions of the spatial coordinates to be determined. This fact enables us to examine directly the nature of the
singularity in the distribution function with respect to this particular problem caused by the vanishing molecular
velocity component (Beylich [9] ; Cercignani et al., [10]). First, we postulate a distribution function in which the
space derivative is bounded. It turns out that this leads to only a trivial result of a uniform flow without a shock layer.
Next, we allow the collision term to be finite at the crucial point. The O-model collision term can be utilized to
compute moments directly from integrals using its explicit expressions of the molecular velocity. The principal
value of the integral is introduced to cope with the singularity since it is only a simple pole. As a result, we can
derive five equations for five unknown functions of space coordinate x. These can be reduced to a first-order
differential equation that can be solved to provide the familiar smooth monotonic transition from its upstream
supersonic value to its subsonic downstream value. Computed values of shock-front thickness for various shock
Mach numbers compare well with results obtained by different models and methods to a certain Mach number
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
473
beyond which no solution exists. We consider the molecular velocity distribution function F(v, x, t ) of time t , the
molecular velocity v , and the spatial coordinates x whose components are given as v = (v x , v y , vz ) and x = ( x, y, z ) .
We use the Boltzmann kinetic equation for F in a new type of 13-moment approximation proposed by Oguchi [7],
which is given in the simplest case of the Maxwell molecular model as
Ê V 2 ˆ È 3 pij
qi Vi Ê 2 V 2 ˆ ˘
nn
∂F
∂F
(1)
V
V
exp
+ v◊
=+
- 1˜ ˙ ,
Í
i
j
Á
˜
∂t
∂x
pT ÁË 5 T
(pT )3/ 2
Ë T ¯ ÍÎ 4 pT
¯ ˙˚
with
1
2
V = v - u , V = V , n = n( x, t ) = Fdv , u = u( x,t ) =
Fvdv , T = T ( x, t ) =
FV 2 dv , p = p( x, t ) = nT ,
n
3n
pij
Ú
= p ( x, t ) = 2 FV V dv - d
Ú
ij
i j
ij p ,
Ú
qi = qi ( x, t ) =
Ú FV V dv ,
2
Ú
i
where i, j = 1, 2, 3 stand for x, y, z ; d ij is for the Kronecker d ; and the double index is for summation. All quantities
are in dimensionless form based on the representative collision frequency n 0 , the temperature T0 , the pressure p0 ,
the mean free path l0 and the density n0 as
(
)
n 0 t Æ t , x / l0 Æ x l0 = 2 RT0 / n 0 , F / {n0 (2 RT0 )3/ 2} Æ F , v / 2 RT0 Æ v , V / 2 RT0 Æ V ,
n / n0 Æ n , T / T0 Æ T , p / p0 Æ p , pij / p0 Æ pij , qi /
{
}
2 RT0 ◊ p0 Æ qi , n / n 0 Æ n .
SHOCK-WAVE SOLUTION
A one-dimensional steady-flow case of the shock structure problem reduces the above equation to dimensionless
equation
Ê V 2 ˆ È 3 pxx Ê 2 1 2 ˆ q x Vx Ê 2V 2 ˆ ˘
nn
∂F
(2)
exp
vx
=Á - T ˜ Í 4 pT Ë Vx - 2 V ¯ + pT Á 5T - 1˜ ˙ ∫ P ,
∂x
(pT )3/ 2
Ë
¯ ÍÎ
Ë
¯ ˙˚
2
where V = Vy 2 + Vz 2 and the relation pyy = pzz = -0.5 pxx from the symmetry are utilized, and
1
2
n = n( x ) = Fdv , u = u( x ) =
Fv x dv , T = T ( x ) =
FV 2 dv , p = nT ,
n
3n
Ú
Ú
Ú
2
pxx = pxx ( x ) = 2 FVx dv - p , q x = q x ( x ) = FV 2 Vx dv .
Ú
Ú
Here, we have the shock structure problem for which we consider a steady plane shock wave whose normal is in the
x-dimension, and the distribution function F becomes the Maxwell distribution function F± in the uniform subsonic
and supersonic regions as x Æ ±• or
Ê V2 ˆ
n±
F(v, x ) Æ F± ∫
exp
Á- T ˜ ,
(pT± )3/ 2
Ë ±¯
Vx = v x - u± , V 2 = (v x - u± ) + v y 2 + vz 2 as x Æ ±• ,
2
(3)
where n± , T± and u± are the values at x Æ ±• , They are related with each other through the Rankine-Hugoniot
condition as
Ï
Ô
n- u - = n+ u +
Ô
1
2 1
2
(4)
Ì n - u - + p- = n + u + + p + ,
2
2
Ô
Ôn- u- 3 + 5 p- u- = n+ u+ 3 + 5 p+ u+
2
2
Ó
where we assume the gas is ideal and consists of particles of a single kind so that the ratio of its specific heat is
g = 5 / 3 . We use the Mach number M of the flow field given as
474
M = M( x) =
6 u
and M- =
5 T
so that
6 u,
5 T-
(5)
u+ n- 3 + M- 2 p+ 5 M- 2 - 1
=
=
=
,
.
(6)
4
u - n+
p4 M- 2
We may use the quantities at x = -• as reference values to produce n- = p- = T- = 1, M- = 6 / 5u- .
We look for a solution F of eq.(2) subject to the boundary condition of eq.(3). eq.(2) is still of integro-differential
type as is the original Boltzmann kinetic equation, but its collision term is expressed in an explicit function of V with
its dependency on x being given through five moments, n , u , p (or T ), pxx and q x in eq.(2) as functions of x
only. Accordingly, we may derive relations between these moments by integrating eq.(2) over v with certain
weights, from which we may determine these. We can now obtain three relations by integrating eq.(2) over v , using
the weights 1, vx and v 2 and utilizing the conservation laws of mass, momentum and energy as
Ï
Ô
nu = C1
Ô
2 1
(7)
Ì nu + ( p + pxx ) = C2 ,
2
Ô
Ônu 3 + 5 pu + upxx + q x = C3
2
Ó
where
1
1
C1 = n- u- = n+ u+ , C2 = n- u- 2 + p- = n+ u+ 2 + p+ ,
2
2
5
5
3
3
C3 = n- u- + p- u- = n+ u+ + p+ u+ from eq.(4).
2
2
These relations as given in eq.(7) can conveniently be rewritten as
C
(8)
u= 1,
n
(9)
pxx = 2C2 - 2C1u - p ,
qx =
We also have
u2
3
(n - n- )(n+ - n) .
upxx - 4C1 Z , Z ∫ (u- - u)(u - u+ ) =
n- n+
2
(10)
(11)
p+ - p- = 2u- u+ (n+ - n- ) , pxx = q x = 0 at x = ±• .
We need two more equations to determine the five unknowns. For this, we have to see the nature of the singularity
of the solution F at v x = 0 in eq.(2), which is the kind appearing particularly in steady-flow problems of the
Boltzmann kinetic equation. Notice in this connection that the collision term in this model of eq.(2) is expressed
explicitly in variable V, so that we can directly examine the nature of the singularity through this explicit expression.
We allow P of eq.(2) to be finite at v x = 0 and ∂F / ∂x = P / v x to have a pole there. The principal value (PV)
is utilized for the moment integral to cope with this singularity.
Besides three moment equations for five quantities of ( n , u , p , pxx , q x ) given in eqs.(8), (9) and (10), we
consider two additional moment integrals given as
P
∂F
dv = ( PV )
dv ,
(12)
∂x
vx
Ú
Ú
V
∂F
Ú V ∂x dv = ( PV )Ú v
2
Using definitions for
2
x
Pdv .
n and p in eq.(2)
n=
We have from eqs.(12) and (13)
Ú Fdv , 2 p = Ú V
3
475
2
Fdv .
(13)
where
( 3)
Ï
È3
dn
nn
Ê I ( 2 ) - T I ( 0 ) ˆ + 2 q Ê I - 3 I (1) ˆ ˘
Ô
=p
Í
xx Ë
˜˙
¯ 5 x ÁË T
dx
2
2
p T (3/ 2 ) p ÍÎ 4
¯ ˙˚
ÔÔ
,
Ì
È3
T (2)
nn
2 Ê I (5) 1 (3) T (1) ˆ ˘
Ô d Ê3 ˆ
Ê
ˆ
(
4
)
2
(
0
)
Í
Ô dx Ë 2 p¯ = - p T (3/ 2 ) p Í 4 pxx Ë I + 2 I - T I ¯ + 5 q x ÁË T - 2 I - 2 I ˜¯ ˙
˙˚
ÔÓ
Î
I (n) = ( PV )
We have
I (n) = T n / 2
and
•
•
Ê V 2ˆ V n
expÁ - x ˜ x dv x .
-•
Ë T ¯ Vx + u
Ú
•
Ú-• exp(-z 2 )z n-1dz - uI (n-1) ÁËz =
Ê
(14)
Vx ˆ
˜
T¯
•
Ê V 2 ˆ dv x
sinh(2xh)
expÁ - x ˜
= 2 exp -x 2
exp -h 2
dh ∫ p I (x ) ,
h
Ë T ¯ Vx + u
-•
0
v
u
, h= x .
x=
T
T
Thus, eq.(14) is simplified to
Ï
dn
nn È 3
Ê 2 1ˆ I ¸ 2 Ï 2
2 Ê 2 3 ˆ I ¸˘
=(15)
Í upxx Ì-1 + Ë x - ¯ ˝ + q x Ìx - 1 - x Ë x - ¯ ˝˙ ∫ Apxx + Bq x ,
dx
pT Î 4
2 x˛ 5 Ó
2 x ˛˚
Ó
¸˘
dp 2 È 2
dn nn Ï 3
I 2
= Í u +T
+
(16)
Ì upxx + q x (xI - 1)˝˙ ∫ Cpxx + Dq x ,
dx 3 Î
dx
p Ó8
x 5
˛˚
from which we have the equation
dp Cpxx + Dq x
=
.
(17)
dn Apxx + Bq x
Notice the essential similarity of the relation between pxx and qx and dn/dx given in eq.(15) together with eqs.(8)-(10)
with the linear stress-strain relation of Newton’s law, which is one of the bases of the Navier-Stokes equation. Since
u , T , I , pxx and q x are all given by the previous three relations (8), (9) and (10) as functions of p and n , eq.(17)
is an ordinary differential equation for p(n) , n- £ n £ n+ , and we look for its solution p(n) subject to the boundary
condition:
p(n± ) = p± .
(18)
Since both pxx and q x should vanish at the two ends n = n± , which correspond to x = ±• as given in eq.(11),
eq.(17) becomes singular at n = n± as 0/0. This fact makes it difficult to start the solution from either n = n- or n+ .
We may start instead from a point n = n- , which is a little larger than n- with an adjustable p- value towards n+
to get p+ close enough to be p+ at a point n = n- ª n- .
The process is repeated for various M- values to a critical M- value of about 1.7, beyond which the process
breaks down. Results are shown in Fig. 1 as p-n curves for some M- values. It is remarkable that these curves are
almost straight lines. Inspired by this fact, we tried to construct an approximate solution of p(n) as
p = p = p- + 2u- u+ (n - n- ) .
(19)
which satisfies the boundary condition of eq.(18) obviously at n = n- and also at n = n+ , as seen in the RankineHugoniot condition given in eq.(11). We note here that this expression of eq.(19) corresponds to the Mott-Smith
solution (Mott-Smith [11] ; Sakurai [12]) in the sense of that it is an interpolating formula connecting boundary
conditions at two ends. We also have from eqs.(8), (9) and (10) that for p = p
2C
(20)
pxx = 1 Z , q x = - C1 Z ,
u
and
Ê dp Cp + Dq x ˆ
2C - uD
,
S ∫ Á - xx
= 2 u- u+ ˜
2 A - uB
Ë dn Apxx + Bq x ¯ p = p
I ( 0 ) = ( PV )
Ú
( )Ú
(
)
476
( )
FIGURE 1. Numerical solution of p(n) for M- = 1.05, 1.2 and 1.5.
We find that S is small in the entire range of n- £ n £ n+ and accordingly demonstrates the validity of the
approximations of p = p as given in eq.(19). Notice also that the factor Z is canceled out there, as are the
singularities at n = n± in eq.(17). In fact, for the limiting weak-shock case of M- Æ 1 , we have
2C - uD ˆ
5
S Æ S- = (2u- u+ ) M- =1 - Ê
ª - 1.76 ª -0.1 ,
Ë 2 A - uB ¯ M =1 3
and the starting value of dp/dn at n = n- for small M- - 1 is found to be 2 n- u- 2 , which coincides exactly with the
value of 2u- u+ in the limit of M- Æ 1 to 2u- 2 = 2( 5 / 6 )2 = 5 / 3 .
Using the p(n) function given above in eq.(19), the function n(x), -• < x < • can be derived from eq.(15)
combined with boundary conditions of n( ±•) = n± as given in eq.(3). eq.(15) is now reduced by eq.(20) to
dn
nn 1 Ï11
u
2 Ê 4 9 2 15 ˆ I (x ) ¸
= Apxx + Bq x =
,
Ì + 2x - Ë 2x + x - ¯
˝C1 Z , x =
dx
pT 5 Ó 2
2
4
x ˛
T
which can be rewritten as
dn
= E(n - n- )(n+ - n) ,
dx
C 3k 1 Ï11
2 Ê 4 9 2 15 ˆ I (x ) ¸
E∫ 1
Ì + 2x - Ë 2x + x - ¯
˝,
x ˛
5n- n+ pT Ó 2
2
4
where we have used eq.(10) and the Maxwell molecular model for n to yield
32
n = kn , k =
.
15 p
(
Then the mean free path of hard-sphere molecule l0 = 16 m 0 / 5n0 m 2pRT0
(21)
(22)
)
of upstream flow is selected as
standard of length. Since T=p/n, u = C1 / n , p = p- + 2u- u+ (n - n- ) from eqs.(2), (8) and (19), and x=u/ T , E can
be regarded as a given function of n. We can thus express eq.(21) in an integrated form as
n
dn
1
x=
, n0 = n(0) = (n- + n+ ) ,
(23)
n0 E ( n - n- )( n+ - n)
2
where we have used boundary conditions n( ±• )= n± .
As can be observed in eq.(23), as long as E is bounded positive, this provides the density function n(x),
-• < x < • in the familiar shock layer profile of a monotonic transition of n from n- at x= -• to n+ at x= +• . In
Ú
477
fact, we can see this feature more explicitly in the limiting case of a weak shock wave for which we have
approximately E = E- for small M- - 1 and
n
dn
1
1
n - nx=
log
=
(24)
E- n+ n0 (n - n- )(n+ - n) E- (n+ - n- )
n+ - n
with
¸
k 5 Ï 43 25 6
E- = ( E ) x =-• =
I ( 5 / 6 )˝
Ì 6 6 Ó 6 18 5
˛
from
n- = p- = T- = 1, u- = 5 / 6
of eq.(6), (23) or
Ú
[
]-1.
n = n- + (n+ - n- ) 1 + exp{- E- (n+ - n- ) x}
Shock thickness t given by the maximum slope of dn/dx defined as
1
Ï dn ¸
-1
= max Ì ˝(n+ - n- )
dx
t
Ó ˛
(25)
(26)
for M- Æ 1 , from eq.(21)
E
1
-1
= E- max{(n - n- )(n+ - n)}(n+ - n- ) = - (n+ - n- ) Æ 0.377( M- - 1)
t
4
(I( 5 / 6 )=1.0819).
(27)
In general, we can find a profile by integrating eq.(23) over n as long as E is bounded positive. This is violated at
about M- = 1.82 , to which E in eq.(21) becomes zero, and the solution does not exist beyond this M- values of
1.82. This value can be compared with the corresponding Grad’s [8] value of M- =1.65, beyond which his 13moment solution does not exist, and for which the theoretical limit of M- =1.85 for the convergence of the Hermite
polynomial expansion solution was given by Holway [13].
FIGURE 2. Numerical solution of the reduced distribution function G ( M- = 1.15).
The solution F considered above can have a singularity of a pole at vx = 0. A purely numerical solution is given
by a step-by-step integration of an unsteady flow equation, for which the term ∂F / ∂t is retained and starts from a
discontinuous step function type condition. An example of the solution reaching a steady state after many steps is
shown in Fig. 2, where we can indeed notice a pole-like singular nature at vx=0. The details of this numerical study
will be published soon in another paper. We added some standard theoretical data such as by Taylor (1/t =
0.311( M- -1)) and the Navier-Stokes solution by Becker (1/t = 0.37( M- -1)). These can be compared directly with
478
our result given in eq.(27) as 1/t = 0.377( M- -1). These comparisons show the consistency of our approach with the
conventional approaches.
CONCLUSION
We attempted to practically apply the new 13-moment approximation model to the shock structure problem and
obtained some satisfactory results. In particular, the obtained shock profile has familiar features, and its thickness is
reasonably consistent with other data acquired by various methods. The explicit expression of the collision term of
this model based on the variable of molecular velocity is found to be useful, especially in studying the nature of a
singularity caused by vanishing molecular velocity in the smoothness question concerning the molecular velocity
distribution function.
ACKNOWLEDGMENTS
The authors are indebted to Prof. Oguchi (Prof. Emeritus of the University of Tokyo) for his useful advice in
theoretical model equation and to the Japanese Ministry of Education for the support of a Grant-in-aid (No.
09305015).
APPENDIX. REMARK ON THE EXISTENCE PROBLEM
We have eqs.(17), (18) for p(n) as
dp Cpxx + Dq x
=
.
(A1)
dn Apxx + Bq x
As noticed above, eq.(A1) becomes singular as 0/0 at two ends of n = n± , since both pxx and qx should vanish there,
which correspond to x = ±• . We examine the nature of the solution there. First, we utilize eq.(17) combined with the
Rankine-Hugoniot relation of eq.(7) to have
pxx = 2C2 - 2C1u - p = -( p - p+ ) - 2C1 (u - u+ ) = -( p - p- ) - 2C1 (u - u- ),
3
q x = upxx - 4C1 Z , Z ∫ (u- - u)(u - u+ ) ,
2
Ê C + 3 uDˆ p - 4C DZ
1
xx
dp
u 2 dp Ë
2 ¯
==
,
3
dn
C1 dn Ê
A + uBˆ pxx - 4C1 BZ
Ë
2 ¯
and transform the above equation (A1) into the following two different forms for p(u) with the independent variable
u, u+ £ u £ u- as
Ê C + 3 uDˆ p - p + 2C u - u + 4C DZ
(
±)
1(
±)
1
d ( p - p± )
C1 Ë
2 ¯
=- 2
, p(u± ) = p± .
(A2)
d (u - u ± )
u Ê A + 3 uBˆ ( p - p ) + 2C (u - u ) + 4C DZ
±
1
±
1
Ë
2 ¯
Each of these equations in (A2) represents a singular initial value problem of the type
dy cx + dy
, ( x, y) = (0, 0) , (say),
=
dx ax + by
After some algebra following the standard procedure of dealing with this type of singular initial value problem of a
differential equation [14], we have found that the nature of the solution near u = u+ is of node type while it is of
saddle point type near u = u-. Accordingly (i) we have two regular solution curves starting from (p-, u-) of which one
stretches towards to an area near (p+, u+) while (ii) any solution curve near u = u+ approaches to the point (p+, u+).
{
}
{
}
479
(iii) Further, solution of (A2) passing a point ( p˜ , u˜ ) of u+ < ũ < u-, p < p̃ < p+ is regular and unique in u+ < u < u- as
far as the magnitude of the parameter M- is close to 1 or M--1 is small enough, since from eq.(6)
5
3 M- 2 - 1 p+
u+
= 1 + ( M- 2 - 1) ,
= 1,
4
p4 M- 2
ufrom which we have (a, b, c, d) above are almost constant to the present case of eq.(A2).
Combine all three facts of (i), (ii), (iii) above, we can see that there exist a unique smooth solution of eq.(A1)
starting from (p-, n-) and ending at (p+, n+).
Once these five moments u, p, n, pxx , q x are given as functions of x , higher moments can be obtained from these
five moments above since we have simply from eq.(2),
∂F
w H (i ) v x
dV = w H (i ) PdV = 0 for i > i0 ( = 5, say) ,
∂x
where H (i ) i = 0,1,.... are the Hermite Polynominals H (i ) (V) [8] of with the weight function w . So that we have
formally
Ú
Ú
F=
•
Âa
(i )
H (i ) ,
(A3)
i=0
which satisfies the boundary condition of eq.(3).
Alternatively, the solution F can be obtained simply by integrating both sides of eq.(2) over x from x = -• to
x to yield
v x ( F - F- ) =
It is not certain here that we have
x
Ú-• Pdx .
•
Ú-• Pdx = vx ( F+ - F- )
to satisfy the boundary condition of F at x = • . However, at least
requirement for pxx and q x Æ 0 as x Æ ±• .
Ú
•
-•
Pdx can be finite because of the
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