A Continuum Model for the Transition Regime:
Solutions of Instationary Problems
by the Moment Method
M. Torrilhon
ETH Zurich
Seminar for Applied Mathematics
CH-8092 Zurich, Switzerland
Abstract. The moment method was designed by Grad as continuum model that approximates Boltzmann’s
equation. Aim of this paper is to investigate empirically the approximation behavior, reliability and usability
of large systems of moment equations.
The paper relies on series expansion of the distribution function in the sense of Grad for the closure problem
of the transfer equations. Furthermore, it exploits the constitutive theory of extended thermodynamics to obtain a
structured way to derive large systems of moment equations. The problem of where to cut the infinite hierarchy
may be solved by assuming a reasonable, empirical convergence hypothesis, which proved to be practical in
many applications. Finally, the resulting systems are solved numerically for instationary test processes in one
dimension. Detailed solutions of instationary heat conduction at time scales of the mean free flight time will be
presented. The paper ends with a short discussion of further applications of the moment method.
MOMENT EQUATIONS
Boltzmann’s equation for rarefied gas flow is given by
∂f
∂f
+ ci
= S ( f, f )
∂t
∂ xi
(1)
where f (c, x, t) is the distribution function for the velocities of the particles. The moment method of kinetic gas theory
[1], [2] suggests to look at moments of the distribution function
Z
Fi1 ···in (x, t) = m ci1 · · · cin f (c, x, t) dc
(2)
rather than at the distribution function itself. The basic idea is that suffiently many moments describe the distribution
function in a certain point (x, t). Theorems from statistics support this idea. Once decided for particular moments,
Boltzmann’s equation (1) is substituted by transfer equations for these moments in order to describe the flow.
Some of the first moments are the density F = ρ, momentum density Fi = ρvi as well as energy density 21 Fii =
ρ ε + 12 v2 . Further moments include non-equilibrium quantities like heatflux and stress tensor. When working with
moments, it turns out to be practical to introduce the following compact multi-index notation: A moment is denoted
by
Z
FAs := Fk1 ···ks k1 ···ks i1 ···i A = m c2s c A f dc
(3)
if it has s traces and A free indices. Index A acts as multi-index in case of tensor contractions. In extension of notation
the index A is also used in (3) to indicate A velocity vectors ci with different indices. A certain set of variables is built
by choosing moments with different free indices and traces. For convenience, the index set
Iv := (s, A) : FAs is chosen as variable
(4)
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
FIGURE 1. Structure of the set of variables for the moment method if each moment is decomposed into its traces. A and s stand
for the number of free indices and the number of traces. The numbering of the boxes demonstrates how a regular system is built.
is introduced. Often several systems described by Iv(0) ⊂ Iv(1) ⊂ Iv(2) · · · are considered as successive approximations.
Unfortunately, there exists no ultimate theory of what moments shall be chosen. Some guidelines for the choice are
given by physicality as well as analytical and numerical comfort. In [3] and [4] some choices are discussed.
This paper will use so-called regular sets of variables as indicated in Fig. 1. In the figure each row describes the
decomposition of a moment with constant tensor rank into its traces. The number of free indices increases from right
to left. A regular system is built by successive rows and starting each row from the right with no gaps. Regular systems
have several advantages such as Galilei-invariance of the resulting field equations and simple form of the transfer
equations.
The system of transfer equations for moments as in (3) follows from integration of (1) and reads
s
∂t FAs + divFA+1
= PAs
with (s, A) ∈ Iv .
The right hand side is formed from production terms which are given by
Z
s
PA = m c2s c A S ( f, f ) dc.
(5)
(6)
If (5) is viewed in the sense of Fig. 1, the flux of one equation is always the variable of the box below. A closure
problem appears for the last variables, since then (s, A + 1) ∈
/ Iv is possible. Furthermore the right hand side must be
related to the variables.
The closure problem of transfer equations has been widely discussed. Several approaches may be found in [5],
[6]/[7], [8]/[9], and [10]. Much developement of the moment method has been made in the context of extended
thermodynamics [5]. Extended thermodynamics gives a general constitutive theory for any material and provides a
deep insight into both the physical and mathematical background of the moment method. See [5] also for further
applications of the moment method.
The original idea of Grad in [1], [2] is to assume a series expansion of the distribution function in the form
X
s 2s
f˜ (c, x, t) =
(7)
A c c A f M (c, ρ, v, T )
(s,A)∈Iv
with tensorial coefficients
s
A
depending on (x, t). Here, the Maxwell distribution
f M (c, ρ, v, T ) =
ρ/m
q
3
2π mk T
exp(−
(c − v)2
)
2 mk T
(8)
appears. As Grad states himself in [2], due to basic theorem of functional analysis it is most reasonable to assume a
series expansion to hold for any integrable distribution function f . Hence, a finite sum as in (7) should be an accurate
approximation. Nevertheless, mathematical statements concerning the convergence of the moment method towards
solutions of Boltzmann’s equation are missing. The coefficients sA are related to the moments by the equation
Z
s
r
FB A = m c2r c B f˜ sA dc.
(9)
For (r, B) ∈ Iv this yields equations to calculate the coefficients sA from the known variables FBs and thus f˜ in (7)
depends only on given moments. This solves the closure problem since any unknown moment may be calculated from
(9), once f˜ is given by known moments. To avoid cumbersome analytic calculations in the case of large systems
most of the constitutive theory may be algorithmized to suit computer algebra software [11]. At this stage extended
thermodynamics furnishes additional help to simplify the procedure, see [4]. Production terms are obtained by inserting
(7) into (6). Usually this leads to quadratic expressions in the variables [2] which are extremely complicated for large
moment systems. Instead, this paper uses productions terms calculated from the linearized collision operator for an
interaction potential of Maxwell type (see [4]).
Convergence hypothesis
Since the moment method may be viewed as series expansion, more variables will give more accurate results.
Lacking a rigorous theory of where to cut the infinte hierachy of transfer equations, the following empirical hypothesis
will be applied:
Hypothesis: Given a process, there exists a certain set of variable I v(?) , such that the results of systems with Iv ⊃ Iv(?)
(?)
do not differ significantly from the result of the system Iv . The solution is expected to be the solution of Boltzmann’s
equation.
Or, to put it in other words: Only if the enlargement of the set of variables does not change the result, the solution
may be trusted. Usually the process is characterized by a parameter, like Knudsen or Mach number, which gives
the degree of non-equilibrium. Higher non-equilibrium will require larger moment systems for a valid solution. The
hypothesis has been proven reliable in many applications, see e. g. [5], [12], [13], [14] and [15].
One-dimensional case
Most of the complexity of large systems of moments is introduced by the tensorial character of the moments. In one
space dimension the complications are reduced. Only the 1-components of the moments are considered as variables,
viz.
FAs := Fk1 ···ks k1 ···ks 1 · · · 1
(10)
| {z }
with A free indices equal to one. In order to write the resulting moment system in the canonical form
∂t u + ∂x f (u) = P
(11)
components of the variable vector u are defined by u i = u i(s,A) =
s traces and A free indices according to Fig. 1. Correspondingly, s (i ) gives the number of traces of variable i and A (i )
the number of free indices. The components of the flux function in (11) are given by f i = u i(s(i),A(i)+1) (i = 1, 2, ...N) ,
where N is the number of considered variables. For most i the flux consists of a certain variable u j 6=i from the box
below which is part of the set of variables. For the last fluxes with (s, A + 1) ∈
/ Iv the value i (s (i ) , A (i ) + 1) leads to
a number which is not in the set of variables. These u i with i > N are supplied by constitutive equations.
For dimensional reasons (see [4]) the constitutive relations may be put in the form
FAs . Here i (s, A) gives the number of the variable with
u i = ρ ϕ1(i) (v, θ) +
N
X
ϕ (i)
j (v, θ) u j
i>N
(12)
j =4
if density ρ, velocity vi = (v, 0, 0) and ”temperature” θ = mk T are written explictly. The coefficients ϕ (i)
j are polynomials defined by
m
bX
2c
(i)
(i)
ϕ j (v, θ) =
a j,k θ k v m−2k with m = A(i ) − A( j ) + 2(s(i ) − s( j)).
(13)
k=0
(i)
The coefficients a j,k are constants and follow from the constitutive theory by computer algebra software [11]. In
(12) non-equilibrium variables appear inside u 4 , u 5 , . . ., the first three equilibrium variables u 1 , u 2 , u 3 are substituted
by density ρ, velocity v and temperature θ. Furthermore, density and non-equilibrium variables enter (12) linearly,
velocity and temperature, however, occur in a non-linear way.
For the production terms in (11) there exists an expression analog to (12), see [4]. By using an appropriate
dimensionless representation all productions become proportional to the inverse of the Knudsen number which is
given by
r
8
λ0
4
with λ0 = 5ρ0 α
θ0
(14)
Kn =
x0
π
where λ0 is the mean free path of a reference state ρ0 and θ0 . The macroscopic length scale is given by x 0 . The constant
α follows from the Maxwell potential and is related to the viscosity µ of a Maxwell gas by µ = α1 θ. In the following
also the mean free flight time τ0 = λ0 ( π8 θ0 )−1/2 is used.
Numerical methods
The moment method yields systems of hyperbolic partial differential equations. For such systems highly developed
finite volume schemes are available in literature, see e. g. [16]. The results of this paper were obtained by the central
scheme proposed in [17] with ENO limiter and CFL = 0.485. Production terms are incorporated via a time step
splitting approach. Note, that the right hand side is given by algebraic expressions and, hence, may be considered
locally in a grid cell of the computational domain. Furthermore, only flux evaluations are needed inside the numerical
scheme which results in a quite robust implementation. This is a major advantage of the moment method over models
like Navier-Stokes-Fourier or Burnett which introduce crucial higher derivatives.
The test processes proposed in this paper use unbounded domains. The ghost cells required by the numerical method
have been filled with extrapolated values of the interior. Of course, boundary condition for higher moments form a
problem of the moment method. Most promising is a pure kinetic approach which deduces relations for the moments
from boundary conditions for the distribution function [18].
The results of the following sections show field quantities at certain times covering a certain range of the physical
domain. However, any numerical calculation has been performed in the interval x̂ ∈ [−1.5, 1.5] with 500 grid
√ points
and end time tˆend = 0.4. Dimensionless variables are given by x̂ = x/x 0 and tˆ = t v0 /x 0 , where v0 = θ0 is the
reference velocity. To construct solutions at different physical times, respectively in different physical domains, the
Knudsen number inside the productions in (11) has been chosen appropriately. An increasing Knudsen number assures
that the dimensionless time tˆend represents ever smaller physical times.
The duration of the calculations depends on the size of the system considered. On a Sun UltraSparc-II workstation
with 336 MHz processor the computations took approximately 6 min (500 time steps) for a system with 9 equations and
6 hours (1000 time steps) for 56 equations. No effort has been spent on parallelization or other forms of accelerations.
INSTATIONARY TEST PROCESSES
A special Cauchy-problem for the moment system (11) is constructed by supplying Riemann data in the form
equilibrium: T = T1 , p = p1 , v = 0
x <0
u (x, t = 0) =
equilibrium: T = T0 , p = p0 , v = 0
x >0
(15)
as initial conditions. By virtue of these conditions the gas is initially separated into two equilibrium domains which are
at rest but may have different temperatures and pressures. The values of T0/1 and p0/1 will be chosen in two different
variations. The aim is to calculate the time evolution of field quantities like density, temperature, velocity and pressure.
Most interesting in the solution is the short time behavior during the start-up phase of the process. In fact, for times in
the magnitude of several mean free flight times the flow is characterized by high Knudsen numbers.
As first process instationary heat conduction is considered. The initial conditions (15) are completed by T1 = 1.5T0
and p1 = p0 . Note, that the gas may flow and develop pressure pertubations in the solution, though initially the gas
is at rest and the pressure is homogeneous. For large times the solution will be governed by ordinary nonlinear heat
conduction. Nevertheless, for small times a complex interaction between diffusion and sound waves arises.
FIGURE 2. Approximation behavior of the moment method for an instationary process. The figure shows the density field in the
shocktube experiment for several systems of moment equations at two different times. Solutions that differ at small times start to
match at later times. Time and space are measured in mean free paths λ0 and mean free flight times τ 0 .
Another way to complete the Riemann data (15) is to choose p1 = 5 p0 and T1 = T0 as initial pressure and
temperature. This results in a typical shocktube experiment in which the gas is initially separated into a high and
low pressure domain. The wave pattern of the large time solution is well known and described, for instance, by Euler
equations. For small times the development of the shock and rarefaction wave will be observed.
The processes are calculated with several regular moment systems which have 9, 12, 16, 20, 25, 30, 36, 42, 49 and
56 moment equations. The systems have been formed by successively including a complete row of Fig. 1 in the set of
variables. Hence, in each system a tensorially complete moment is added. Due to space restrictions this paper shows
only the results for heat conduction. Readers interested in the shocktube experiment are referred to [14].
Free flight equation
For times below the mean free flight time the particles flow with almost no collisions. In this range of time a process
may be modeled by the free flight equation (1) with S ( f, f ) = 0, i. e. the collisionless Boltzmann equation. Formally
this equation describes the flow of a gas with Knudsen number K n → ∞. This equation is now used to describe the
very start-up phase of processes like in (15). The initial conditions are formulated by
(
x <0
f M (c, kp1 , 0, T1 )
m T1
(16)
f (c, x, t = 0) = f 0 (c, x) =
p0
f M (c, k , 0, T0 )
x >0
m T0
in terms of Maxwell distribution functions. The solution of the free flight equation in case of an one-dimensional
process is given by
(17)
f (c, x, t) = f 0 (c, x − c1 t) .
From this distribution function the short-time behavior of the moments may be obtained by integration.
Since the systems of moment equations are expected to approximate Boltzmann’s equation they should approximate
free flight solutions as well if K n → ∞, i. e. vanishing right hand sides in (11). Hence, results of the moment method
and results obtained via (17) should match for small times. Good agreement then gives additional confidence in the
approximation ability of the moment method.
FIGURE 3. Approximation residuum of several systems of moment equations measured by the ∞-norm of the difference of
solutions of successive systems. The graphs indicate convergence and justify the convergence hypothesis in an empirical way.
Moreover, averaging solutions of successive systems gives smaller residua.
Approximation behavior
According to the convergence hypothesis mentioned above the main objective in the use of moment equations is to
find systems whose solutions do not differ from those of larger systems. The different systems of moment equations
show a characteristic approximation behavior when calculating instationary processes. The degree of non-equilibrium
is given by the time at which the solution is considered, measured in mean free flight times. Thus, for smaller times
larger moment systems are required to obtain a valid solution. Furthermore, for the time evolution of the solution
of a particular moment system, there will be a certain time at which the solution becomes valid. Note, that this fact
also depends on the space scale chosen: Two solutions will show up more similiar at a coarse scale. Additionally the
solution might be valid in some domains but not in others. Fig. 2 tries to clarify this approximation behavior.
The figure shows the density field ρ of several moment systems for the shocktube experiment at two different times
t = 0.5τ0 and t = 2.5τ0 . The mean free flight time τ0 and mean free path λ0 are calculated with respect to the low
pressure domain. At t = 0.5τ0 the solutions of 25, 30, 36 and 42 equations show some agreement at least in the high
pressure region. The zoom, however, shows clear differences between the curves, hence, further enlargement of the
set of variables might be needed. At t = 2.5τ0 the agreement is much more advanced for the solutions of 36 and 42
moment equations, whereas the solutions of smaller systems do not match.
The non-valid solutions of moment equations exhibit several kinks, which are due to the hyperbolic waves of the left
hand side fluxes of the systems. During time evolution the production terms damp down these waves and the solutions
become smooth. In [14] and [19] the structure and relevance of the hyperbolic waves are discussed in more detail.
A close look at the lower graph in Fig. 2 shows that the two pairs of solutions of 25/36 and 30/42 equations have
very similar results. Several numerical evaluations like in [5] and [12] suggests that this is general property of moment
equations: Concerning the approximation behavior a (tensorially complete) system is always assigned to the next but
one (tensorially complete) system. This fact is also known from theory of Fourier series where approximations with
odd, resp. even, partial sums correspond. Guided by Fejèr-means of Fourier series this paper suggests to consider
averaged solutions of two successive moment systems in order to increase the approximation quality. Indeed, if the
difference of two successive solutions is taken as measure of the approximation residuum, Fig. 3 shows a decrease of
the residuum for averaged solutions. In the following averaged solutions will be denoted using (·, ·).
Results for heat conduction
Fig. 4 displays the time evolution of temperature, velocity, pressure and heat flux for (15) in the case of heat
conduction up to 25 mean free flight times. For the smallest time t = 1.0τ 0 the solution of the free flight equation is
shown. The solutions for larger times are obtained from systems of moment equations which are valid at the particular
time, i. e. an enlargement of the set of variables do not change the result. As may be read off the figure the averaged
solution of 12 and 16 moment equations are valid at 25τ0, whereas (49, 56) equations are needed for a time t = 2.5τ0 .
For this solution an increase of variables might be advisable since it is not completely smooth.
In the temperature field the start-up of the heat conduction is observed while the pressure field shows two pressure
waves travelling to the left and right. To some extent these sound disturbances are also present in the other fields. The
FIGURE 4. Time evolution of instationary heat conduction for the fields of temperature, velocity, pressure and heat flux. The
dashed curve shows the result of the free flight equation. The results of moment equations are chosen from valid systems.
heat flux drops from a maximal but finite value which is given by the free flight equation. The solution of the free flight
equation is self-similar, hence the graph is only scaled for smaller times and no further structure arises. Note that the
solution of the free flight equation is indeed matched by the results of the moment equations.
It turned out to be interesting to look at the temperature in the middle of the domain, i. e. at x = 0, as function of
time. The resulting curve is shown in Fig. 5. It is obtained by linear interpolation of results of valid moment systems at
certain times. The value at t = 0 is given by the solution of the free flight equation. The graph shows a non-monotone
behavior with a maximum at approximately 35 mean free flight times.
Moreover, the figure shows the corresponding curve calculated with the system of Navier-Stokes-Fourier with
viscosity and heat conduction of a Maxwell gas. That curve misses the non-monotonicity and reaches the curve of
the moment equations only for large times. This fact reflects the invalidity of the NSF system for time scales below
50-100 mean free flight times.
FIGURE 5. Time behavior of temperature at x = 0 calculated by the moment method and the system of NSF. The curves are
obtained from values of solutions in Fig. 4 by interpolation. The solution of NSF is valid only for sufficiently large times.
CONCLUSION
In this paper moment equations have been solved for instationary processes in a straight forward way. The solutions
show convergence in an empirical sense and, furthermore, they fit to solutions of the Boltzmann equation. The results
suggest that the moment method in principle supplies a reasonable and valid approximation to the Boltzmann equation
at any Knudsen number. Nevertheless, two drawbacks have to be mentioned: Firstly, large systems in multi-dimensions
will be very complicated to handle, hence the efficiency of many moment equations becomes doubtful. Secondly, the
equations of large systems seem to become numerically unstable and failures of the numerical methods have been
observed. It becomes clear that the moment method will never be able to completely substitute Boltzmann’s equation
in practise. In fact, hybrid approaches to rarefied flows like in [20] are most promising. The moment method could
act as a link between classical continuum models and direct solvers of the Boltzmann equation or DSMC methods.
Alltogether, it seems to be recommended to place emphasis on rather small systems, say up to 20-30 three dimensional
fields, which could be treated conveniently. As shown above, such a model would suffice to describe flows with higher
Knudsen numbers than classical models in a reasonable and reliable manner.
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