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A rarefied gas flow with dsmc
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rarefied gas ,dsmc,A rarefied gas flow induced by a temperature field Numerical analysis of the flow between two coaxial elliptic cylinders with different uniform temperatures
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Pergamon
Computers Math. Applic.
Vol. 35, No. 1/2, pp. 15-28, 1998
Copyright©1998 Elsevier Science Ltd
Printed in Great Britain. All rights reserved
0898-1221/98 $19.00 % 0.00
Plh S0898-1221(97)00255-1
A Rarefied Gas Flow Induced by a
Temperature Field: Numerical Analysis of the
Flow between Two Coaxial Elliptic Cylinders
with Different Uniform Temperatures
K.
AOKI,
Y. SONE AND Y. WANIGUCHI
Department of Aeronautics and Astronautics, Graduate School of Engineering
Kyoto University, Kyoto 606-01, Japan
Abstract--A steady flow of a rarefied gas induced by a temperature field is investigated, on the
basis of kinetic theory, for the case where the temperature of each boundary is uniform (i.e., where the
flow caused by the nonuniformity of the boundary temperature, such as the thermal transpiration
flow, wnishes). More specifically, a rarefied gas confined in the gap between two coaxial elliptic
cylinders at rest with different uniform temperatures is considered, and the steady gas flow induced
in the gap is analyzed numerically by the direct simulation Monte Carlo method for a wide range
of the Knudsen number. The flow patterns, together with the density and temperature fields, are
obtained, and the features of the flow are clarified.
Keywords--Rarefied gas flows, Kinetic theory of gases, Boltzmann equation, Direct simulation
Monte Carlo method, Thermal creep flow.
1. INTRODUCTION
In a rarefied gas, in contrast to the Navier-Stokes gas, a steady flow can be induced by a steady
temperature field even when there is no external force. For small Knudsen numbers, the features
of such flows have been clarified on the basis of the asymptotic theory [1-5], a general theory
describing the steady behavior of the gas at small Knudsen numbers, derived systematically from
the Boltzmann equation. According to the results, in addition to the well-known thermal creep
flow [6-8] induced along a boundary with a nonuniform temperature, the thermal stress slip flow
(TSS flow for short) [2,9,10] and the nonlinear thermal stress flow (NTS flow) [4,5,11] are caused
in the gas. The TSS flow is induced over a boundary along which the temperature gradient of
the gas normal to the boundary is not uniform, and the NTS flow occurs in the gas where the
distance between isothermal surfaces varies along them. Thus, both flows can be induced in
the case where the thermal creep flow vanishes, i.e., where the temperature of each boundary is
uniform. The NTS flow, which is of the first order of the Knudsen number, is negligible when the
temperature variation in the system is small. On the other hand, the TSS flow occurs even when
the temperature variation is small, though it is of the second order of the Knudsen number. Thus,
the TSS flow is important in a small system as in micr0machines, since the large temperature
difference in a small distance hardly occurs.
The above flows for small Knudsen numbers are classified by the local temperature field. In
contrast, for nonsmall or intermediate Knudsen numbers, the features of the induced flow are not
characterized by the local temperature but are directly affected by the overall properties, i.e.,
the configuration of the system. In this situation, therefore, systems with various configurations
should be investigated in order to clarify the features of the flow. However, it should be noted
Typeset by ~4A4S-TEX
15
16 K. AOKI
et al.
that, if the temperature of a boundary is not uniform, a flow usually occurs along it. The thermal
transpiration flow [12] and the thermophoresis of an aerosol particle [13] are typical examples of
the flow. This flow makes it difficult to observe other types of flow caused by the direct effect
of the configuration of the system. Therefore, for the purpose of clarifying the properties of the
latter types of flow, we should investigate systems where the temperature of each boundary is
uniform.
In the present study, therefore, we consider such a system, i.e., a system where a uniformly
cooled (or heated) body is placed in a rarefied gas confined in a closed vessel with a uniform
temperature. More specifically, we consider a rarefied gas confined in the gap between two
coaxial elliptic cylinders with different uniform temperatures and investigate the steady gas flow
induced in the gap on the basis of kinetic theory. The analysis is carried out numerically by the
direct simulation Monte Carlo (DSMC) method [14,15], and the behavior of the gas is clarified
for a wide range of the Knudsen number.
Now we should mention our previous works closely related to the present problem. In [16], a
similar problem (a gas between noncoaxial circular cylinders) was investigated, in the case where
the temperature difference between the body and the vessel is small (linearized problem), by an
accurate finite-difference analysis of the linearized Boltzmann-Krook-Welander (BKW) equation,
and the behavior of the flow was clarified for the whole range of the Knudsen number. The
case with a large temperature difference (nonlinear problem) was studied in [17], where a body
with sharp edges (flat plate) was considered with special interest in the effect of the edges on
the flow. The behavior of the gas was analyzed for a wide range of the Knudsen number by the
DSMC method, and it was shown that a fairly intense flow is induced by the effect of the sharp
edges. The relation between these previous results and the flow in the present problem (nonlinear
problem with a smooth boundary) will also be discussed.
2. PROBLEM
Let us consider a rarefied gas in the two-dimensional domain between two coaxial elliptic
cylinders at rest. Let the surface of the outer cylinder (pipe) be given by
(X1/al) 2 + Xg = L 2,
(al > 0, L > 0), and that of the inner cylinder by
(X1/ao) 2 + (X~/bo) 2 = L ~, (0 < ao < al, 0 <
b0 < 1), where (X1, X2, )(3) is the rectangular space coordinate system with the Xa axis along
the common axis of the cylinders. Further, we assume that the inner cylinder is kept at a uniform
temperature To and the outer cylinder is kept at another uniform temperature T1. (See Figure 1.)
We investigate the steady behavior of the gas for a wide range of the Knudsen number on the
basis of kinetic theory under the following assumptions.
(i) The gas molecules are hard spheres of a uniform size and undergo complete elastic collisions
among themselves.
(ii) The gas molecules make diffuse reflection on the surfaces of the inner and outer cylinders.
We now summarize the additional main notations used in this paper: ~ = (~1, ~2, ~3) is the
molecular velocity,
f(X1, X2, ~)
is the velocity distribution function of the gas molecules, p is
the density of the gas, v = (vl, v2, v3) is its flow velocity (vs = 0), T is its temperature, P0 is the
average density of the gas over the domain, m is the mass of a gas molecule, dm is its diameter,
lo = (vf21rd2po/m)-1
is the mean free path of the gas molecules in the equilibrium state at rest
with density P0 and temperature
To, Kn = lolL
is the Knudsen number, and R is the gas constant
per unit mass.
3. BASIC EQUATION AND BOUNDARY CONDITION
The present problem, which is time-independent and spatially two-dimensional, is symmetric
with respect to the X1 and )(2 axes. Therefore, we can analyze the problem only in the first
quadrant by imposing the specular reflection condition on the symmetry axes in the gas.
Numerical Analysis of the Flow
X2L~///////Z//~7//// ....
0 ao L al L
Xl
17
with
Pw = - "~w
.n<O ~- nf d~l d~2 d~3, (6)
To, (on the inner cylinder),
Tw = T1, (on the outer cylinder),
where n is the unit normal vector, pointing into the gas, to the boundary.
The symmetry condition on the X1 and X2 axes in the domain of the gas
(aoL < X1 "< alL,
X2 = 0 and X1 = O, boL < X2 < L)
is expressed as
f(X1, X2, ~)
"~
f(X1, X2, ~ -
2(~. n)n), (~. n > 0), (7)
where n = (0, 1, 0) on the X1 axis and n = (1, 0, 0) on the X2 axis.
The macroscopic variables are expressed by the moments of f, e.g.,
p = / f d~l d~2 d~3,
v = - ~f d~l d~2 d~3, (8)
P
T = (3-~p) f (t~- v) " ('- v)f d~l d'zd~3,
where the range of integration is the whole space of ~.
Figure 1. Rarefied gas between two elliptic cylinders (the first quadrant).
The Boltzmann equation in the present situation is written as follows [5,18]:
Of Of
~ l -ff--~l + ~ 2 -~2 = J ( f , f ) ,
(1)
g(f , f) = lm
f (f' f"
- f f.)Sd~(ct)
d~l. d~2. d~3., (2)
with
S = I(~. - ~)'a[d2m
2 ' (3)
f = f(Xl, X2, ~), f. = f(Zl, Z2, ~.),
f' = f(X1, X2,
~'), f.~ =
f(Xl,
X2,
~'.),
(4)
¢'
= + - ala, = - -
where t~ is a unit vector, dfl(a) is the solid angle element around c~, ~, = (~1., ~a., ~a.) is the
variable of integration corresponding to ~, and the integration is carried out over the whole space
of a and that of ~..
The boundary condition on the inner and outer cylinders is given as
~:2 ~_C2.~_c2X
Pw ~1 ~2 ~3
f = (2~rRTw)3/2
exp 2~: ]' (~.n > 0), (5)
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