Formation and propagation of a shock wave due to
evaporation processes at imperfect interfaces
Yoshimoto Onishi, OOSHIDA Takeshi and Katsuhiko Tsubata
Department of Applied Mathematics and Physics, Tottori University, Tottori 680-8552, Japan
Abstract. Transient motions of a vapor due to evaporation processes from its plane condensed phase at
imperfect conditions have been considered numerically based not only on the Boltzmann equation of BGK
type but also on the formulation of fluid dynamic level, i.e., the Navier-Stokes equations subject to the
boundary conditions appropriate for evaporation and condensation derived earlier from the kinetic theory
analysis. The imperfect ness of the interface of the condensed phase has been taken into account in terms
of an adjustable parameter ac first introduced by Wort berg and his co-workes. The parameter, variously
called the condensation coefficient or condensation parameter or evaporation coefficient, may presumably
be associated with some kind of imperfectness of the interface but has nothing to do with the condensation
coefficient commonly defined at the level of kinetic theory. The results based on both of these systems
of equations agree quite well, describing even the process of establishment of the flow field as well as its
established state. Some of the results obtained are compared with the experimental results available for
Helium II by Furukawa and Murakami. The comparison between the present results and the experimental
ones shows that the incorporation of the adjustable parameter seems to work fairly well in some cases but
not in other cases. No decisive conclusion can yet be drawn. However, this parameter is simple and may
serve to incorporate to some extent ambiguous nature associated with the imperfectness of the interface.
INTRODUCTION
Flow problems involving phase-change processes are the ones to which the ordinary continuum-based fluid
dynamics is not directly applicable because of the existence of a nonequilibrium region called the Knudsen layer
in the vicinity of the condensed phase. The analysis for such problems, therefore, must necessarily be based
on kinetic equations because it is this nonequilibrium region that is responsible for the phase-change processes
to occur and its existence can never be neglected in any problems even in the continuum limit. However, if
this nonequilibrium region is small in its thickness, the flow field may well be described by the Navier-Stokes
equations. The problem arising then is that what kind of boundary conditions for fluid dynamic quantities are
to be specified at the condensed phase. The answer is to be given from the analysis of kinetic theory, and the
conditions for fluid dynamic quantities to satisfy at the condensed phase have already been given in general
terms (see e.g., [1]), although, of course, there are some restrictions on the values of the parameters involved
and hence on the applicability of the conditions. From a number of studies having so far been done in our
laboratory, these conditions are good enough when applied to the system of Navier-Stokes equations, giving
good agreement with the results based on the Boltzmann equation of BGK type [2]. Therefore, the system of
Navier-Stokes equations subject to these conditions at the condensed phases may be called the fluid dynamic
formulation [3], which is valid for problems with phase change processes involved. However, the experimental
verification of the numerical results on both of these systems of equations has not yet been done because of lack
in the experimental results appropriate for the comparison. A recent publication by Furukawa fe Murakami [4]
on experiments of evaporation processes of Helium II from its plane condensed phase has now made it possible
to make comparison between the two results, the experimental and the numerical.
Here we consider a simple flow field of a vapor due to evaporation and condensation processes from its
plane condensed phase based not only on the Boltzmann equation of BGK type [2] but also on the fluid
dynamic formulation just mentioned above. A new feature in the present study is that some imperfectness
of the interface of the condensed phase is taken into account in terms of a parameter ac variously called the
CP585, Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. A. Gallis
© 2001 American Institute of Physics 0-7354-0025-3/01/$18.00
591
condensation coefficient or condensation parameter or evaporation coefficient, which probably has first been
used by Wortberg and his co-workes [5] in their experimental and approximate analytical studies on this kind of
problems. So far the commonly used boundary condition at the interface for the velocity distribution function
of molecules leaving the condensed phase has been a Maxwellian distribution function corresponding to a
saturated state of the vapor at a temperature of its condensed phase, i.e., a distribution function characterized
by the temperature and velocity of the condensed phase, its number density being the saturated vapor number
density at that temperature. The introduction of ac, on the other hand, represents that a cetrain fraction 1 — ac
of molecules leaving the condensed phase may be subject to a velocity distribution of diffusive type holding
at a solid wall, namely, a Maxwellian distribution function characterized by the temperature and velocity of
the condensed phase but with the number density being determined by the no net mass flow condition across
the boundary surface. This parameter ac, which of course has nothing to do with the condensation coefficient
commonly defined at the level of kinetic theory, is assumed here to be associated with the imperfectness of the
interface of the condensed phase. Actually, it may happen that otc gives a certain measure of the imperfectness
of the interface or the imperfect accommodation of molecules at the interface. As to the boundary conditions
at the level of the fiuid dynamic formulation, on the other hand, these conditions have to be modified also
to incorporate ac, because they have been derived from an analysis of the Boltzmann equation of BGK type
subject to the commonly used equilibrium distribution function as the boundary condition at the interface (see
[1]). The method of inclusion of ac into the conditions has been proposed by Sone and Sugimoto [6] and we
have used the modified boundary conditions for the analysis to follow.
FORMULATION OF THE PROBLEM
Let a plane condensed phase be located at x = 0 and a half-space ( x > 0 ) be occupied by the stationary
vapor. Initially, the condensed phase and the vapor phase are in complete equilibrium at a temperature TO .
The pressure, density and number density of the vapor at this state are PQ, p$ and N$ , respectively. Suppose
that, at time t = 0, the temperature of the condensed phase is suddenly changed, i.e., TO —> TW - This leads to
the onset of phase change processes at the condensed phase, giving then rise to transient motions of the vapor
accompanied by shock waves and the contact regions (sometimes expansion waves involved). We investigate
the transient motions based not only on the Boltzmann equation of BGK type [2] but also on the Navier-Stokes
equations. The boundary conditions at the condensed phase for these two systems of equations will be specified
in accordance with the statement given in the Introduction.
Kinetic formulation
The Boltzmann equation of BGK type [2], which is adopted here as the governing system in kinetic theory
for the description of the motions of the vapor in the present one-dimensional problem, may be written as
= ^e-/),
(1)
N
1
£>X
(3)
(4)
where t is the time; x is the
the molecular velocity vector;
distribution characterized by
number density, the velocity,
coordinate measured from the original position of the membrane; (£x, £ y , £z ) is
/ is the molecular velocity distribution function, Fe being the local Maxwellian
the local fluid dynamic quantities; N, w, T, P and p are, respectively, the
the temperature, the pressure and the density of the gas; m is the molecular
592
mass; k is the Boltzmann constant and R = k/m the gas constant per unit mass of gas. vc is a constant
associated with the collision frequency ( Nvc is the local collision frequency) and, hence, can be calculated
either from the viscosity // or from the thermal conductivity A of the gas at a certain reference state, say, at
the initial equilibrium state, by the following relation
NM = ^ = \R^,
^0
*
(5)
^0
the suffix 0 being understood to indicate the quantities at the initial state. It may be noted here that, in the
BGK model equation, the relation A = (5/2)1? // holds and, hence, the Prandtl number is unity for this model
equation.
The initial condition for the distribution function / for the present problem is, at t = 0
^n
^
f x>0
for
6
-
The boundary condition for / at infinity (x —> oc) is exactly of the same form as the initial condition (6) for
all times and for £x < 0. The boundary condition at the condensed phase (x = 0), however, is
,
e
for
^ n at .
<•>«
w
*=
with
Nw = acNw + (l- ac)NB
(0 < ac < 1),
(8)
where NW is the number density for molecules emitted from the condensed phase. NW is the saturated vapor
number density at the temperature TW of the condensed phase to be determined by the Clapeyron-Clausius
relation as
w
>
=exp
""
p i
P$
{
rr
T\lw""
1 i
j
rr=
^*
9/ O x
The other quantity NB, on the other hand, is the number density for molecules reflected from the condensed
phase determined by the condition of no net mass flow across the phase boundary. Ng, therefore, is obtained
from the distribution function for incident molecules to the condensed phase (^ < 0) as
From this, it may be noted that NB is yet an unknown constant to be determined as part of the solution.
Fluid dynamic formulation
The governing system at the fluid dynamic level for the description of the motions of the gas in the present
problem is the set of the Navier-Stokes equations, which may be written as
8
d
= 0,
(11)
with
P = pRT,
(12)
T,
(13)
7-1
4
du
n
.£
T
1 = 1.
593
(14)
w
TABLE 1. Characteristic length and time scales, L and TO, corresponding to the
experimental conditions [4].
Case
A
B
C
T 0 (°K)
1.74
1.94
2.04
Po(Pa)
1315
2610
3512
c 0 (m/s)
75.91
79.40
81.01
L (/urn)
2.820293xlO~ 2
1.666244 x!0~2
1.331662 x!0~2
/uo(Pas)
4.46xlO~ 7
5.00xlO~ 7
5.27xlO~ 7
TO (/usec)
3.3916xlO~4
1.9157xlO~4
1.5006xlO~ 4
where e and h are the specific internal energy and enthalpy, respectively. cp is the specific heat at constant
pressure and 7 is the specific heat ratio (7 = 5/3 assumed). rxx is the viscous stress and q the heat flux.
ju and A are here functions of T, which is in consistence with the temperature-dependence of these transport
coefficients derived from the Boltzmann equation of BGK type.
The initial conditions for this set of equations are, at t = 0
u = 0,
P = PQ
T = T0,
for x > 0 .
(16)
As to the boundary conditions, the conditions at infinity is exactly the same as those in (16). The conditions
at the condensed phase (x = 0), on the other hand, may be specified as
P - Pw _
Pw
TW
u
4
1
2
'
(
}
-
where C| = —2.132039, d\ = —0.446749. These are the macroscopic conditions holding at the phase boundary
having been given earlier by the kinetic theory analysis [1]. For the corresponding expressions with the imperfectness of the condensed phase ac incorporated, however, the constant C| above should be replaced by
C% = C% — 2-y/TT (1 — a c )/a c , whereas d\ is left unchanged (see [6]). This modification in the constant C| will
be made where and when it becomes necessary.
RESULTS AND DISCUSSION
For the present problem, we introduce the length scale L and the time scale TO defined as
(2JRT0)1/2
P0
where CQ and /o are the sound speed and the molecular mean free path, respectively, at the initial state. They
are defined by CQ = (7 RT$)1^ and /o = (jUo/Po)(8^To/7r) 1 / 2 . It is mentioned that, since the length scale L
is of the order of the molecular mean free path JQ, the time scale TO here represents the mean collision time
of gas molecules at the initial state. In Table 1, the actual values of the length and time scales, L and TO,
corresponding to the experimental conditions done by Furukawa & Murakami [4] are listed just for reference.
With these length and time scales together with the fluid dynamic quantities at the initial state, both of these
systems of the governing equations, the Boltzmann equation of BGK type and the Navier-Stokes equations,
are appropriately nondimensionalized. The Prandtl number Pr defined by
AO
7 — 1 AO
(20)
which appear in the nondirnensional Navier-Stokes equations, has been set equal to unity here for simplicity
(cf. the Prandtl number is always unity for the Boltzmann equation of BGK type). Incidentally, it may be
noted that the Reynolds number Re defined by Re = p^c^L/jJL^ simply becomes Re = (27) 1 / 2 in this case.
594
1.01
1.000
100
X/L2QQ
0
100
X/L2QQ
100
X/L
1.000200
FIGURE 1. A transient flow of a vapor due to evaporation from its plane condensed phase with its different imperfectness, ac = 1.0 and ac = 0.72. TW/T0 = 1.00755, Pw/Po = 1.05083 and Pr = 1. Solid lines: Boltzmann. Dashed
lines: Navier-Stokes. • : experimental data measured ata;/L = 5 x ! 0 4 from the interface, the initial temperature and
pressure being TO = 1.74°K and PO = 1315Pa, respectively (Furukawa and Murakami [4]). The numbers in the graphs
indicate the values of time £/TO. The initial shock wave with its Mach number Ms = 1.00667 is from left to right. The
pressure and the flow velocity behind the shock wave are P/Po = 1.0168 and U/CQ =0.0100, respectively. In the pressure
distribution of the Boltzmann's solution, the Knudsen layer is clearly visible in the close vicinity of the interface.
Therefore, the parameters which characterize the present flow fields are, in addition to the Prandtl number
having already been set equal to unity here,
ac
Tw_
To'
Pw
(21)
where PW is the saturated vapor pressure at TW and is known for He II.
Numerical calculations have been carried out for various sets of the parameters in (21), based on both
governing systems of equations. First of all, we have confirmed that, for several sets of the parameters, the
595
results based on the Boltzmann equation agree quite well with those based on the fluid dynamic formulation or
simply the Navier-Stokes system. With this fact in mind, most of the calculations for various other sets of the
parameters are carried out based on the fluid dynamic formulation. Figure 1 shows one of the results for the
transient behavior of the relevant fluid dynamic quantities for Case A in Table 1 with different values of ac, i.e.,
ac = 1.0 and ac = 0.72. Good agreement can be seen between the results of the Boltzmann and those of the
fluid dynamic formulation. Incidentally, it is also recongnized that a wave or a region with steep gradients is
produced associated with the evaporation process occurring at the interface and the wave develops into a shock
wave and a contact region as time goes on. The time required for a shock wave to be fully established is about
t/TQ = 50, which corresponds approximately to an actual time t = O.OlT^usec for the He II experiment done by
Furukawa & Murakami [4]. Unfortunately, their experiments correspond only to the established states of the
unsteady flow fields and, therefore, no experimental data is available for the establishing process of the flow
fields, i.e., for the process of formation of waves at very early stages to the eventual shock waves and contact
regions. The contact region formed behind the shock wave develops gradually into a slightly wider region as
time goes on due to the diffusion effects involved within the region, for example, the region expands by about
po)^ ~ 0.14//m in both directions for a period of time £/TQ = 50 for the present experimental situation
(or in terms of the diffusion velocity, y/ PQ / (pat) ~ 0.109co, where CQ ~ 76m/s at TO = 1.74°K). Consequently,
the contact region with its left edge reached to the interface (because of the relatively large diffusion velocity
compared with the flow velocity) seems to be developing from the interface. Within this contact region, the
temperature and density have fairly large slopes as indicated in the figure. Furthermore, the velocity has a
hump (very small in this case), which was pointed out by Onishi et. al [7] as a new feature within the contact
region. At about this stage, the Knudsen layer has already established itself and its thickness is about four
times the length of the molecular mean free path at the initial state (see e.g., the pressure distribution). The
thickness of the established shock wave is about 50I/ = 1.41 jum and its Mach number is Ms = 1.00667, where
Ms = US/CQ, Us being the speed of the shock. As to the transient processes of the flow fields, there is almost
no qualitative difference between the results for ac = 0.72 and for ac = 1. The experimental data by Furukawa
& Murakami [4], which are reported to have been measured at a distance of x/L = 5 x 104 from the interface,
are also indicated by a symbol • in the figure. The agreement between the numerical and the experimental
results happens to be excellent in this case with ac = 0.72 but this is not always the case. Actually, there are
a number of cases in which the numerical and the experimental results are not in good agreement, whatever
values of ac may be. One of such cases is shown in Fig. 2, which corresponds to Case C of Table 1. The
experimental values are far larger than the numerical ones. No definite reason can be given at the moment for
this difference but one reason may be attributed to the errors involved in the experiments in this case and in
other cases which are not in good agreement with the numerical results. Of course, this may also be interpreted
to indicate that a single adjustable parameter such as ac introduced here is not sufficient to account for the
imperfectness or the ambiguous nature of the interface. However, a further study should be continued of the
roles and the effects of this parameter ac because the parameter is simple and, as indicated in Fig. 1, it might
serve in certain situations to account for the ambiguous and complex nature of the interface to some extent.
Finally, in order to see further the effects of ac on various fluid dynamic quantities at the calculation al level,
we have shown in Fig. 3 some of the numerical results for various ac, which have been calculated based on both
the kinetic and the fluid dynamic formulations. It is clearly seen that as the parameter ac becomes small, the
evaporation process weakens and hence a weaker shock wave is produced. This is clear from the nature and
the role of the parameter ac (see Eq. (8)). From the calculational results of this kind, it may become possible
to estimate, in turn, the imperfectness of the interface by comparison with the experiments.
The authors are grateful to Prof. Murakami and Dr. Furukawa at Tsukuba University for providing their
experimental data. This work was partially supported by the Grant-in-Aid (No. 10650175) from the Ministry
of Education in Japan.
REFERENCES
1. Onishi Y., and Sone Y., /. Phys. Soc. Japan 47, 1676-1685 (1979).
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3. Tanaka T., Onishi Y. and Miura H., Fluids Engineering Conference'99 held at Tot tori, Japan on October, 1999,
pp. 113-114 (in Japanese).
4. Furukawa T., and Murakami M., Rarefied Gas Dynamics, edited by R. Brun, R. Camp argue, R. Gatignol, J.C.
Lengrand, Cepadues-Editions, Toulouse, France, 1999, pp. 519-526.
596
1.02
1.01-
1.000
100
x/L
200
0
100
x/L
200
100
x/L
200
1.005
1.000FIGURE 2. A transient flow of a vapor due to evaporation from its plane condensed phase with its different imperfectness, ac = 1.0 and ac = 0.72. TW/T0 = 1.00658, Pw/Po = 1.03889 and Pr = 1. Solid lines: Boltzmann. Dashed
lines: Navier-Stokes. • : experimental data measured ata;/L = 5 x l 0 4 from the interface, the initial temperature and
pressure being TO = 2.04°K and PQ = 3512Pa, respectively (Furukawa and Murakami [4]). The numbers in the graphs
indicate the values of time £/TO. The initial shock wave with its Mach number Ms = 1.00699 is from left to right. The
pressure and the flow velocity behind the shock wave are P/Po = 1.0176 and U/CQ =0.0105, respectively. In the pressure
distribution of the Boltzmann's solution, the Knudsen layer is clearly visible in the close vicinity of the interface.
5. Mager R., Adomeit G., and Wortberg G., Rarefied Gas Dynamics: Physical Phenomena (Progress in Astronautics
and Aeronautics, Vol. 117), edited by E.P. Muntz, D.P. Weaver and D.H. Campbell, Washington B.C.: AIAA, 1989,
pp. 460-469.
6. Sone Y., and Sugimoto H., Adiabatic Waves in Liquid-Vapor Systems, edited by G.E.A. Meier and P.A. Thompson,
Berlin: Springer-Verlag, 1990, pp. 293-304.
7. Onishi Y., OOSHIDA T. and Umemura N., "On a small structure in velocity field within a contact region" to be
published in Phys. Fluids.
597
p_
Po
1.02
1.01
1.01
1.00
1.000
100
x/L
200
0
100
x/L
100
x/L
200
0
100
JC/L 200
200
1.000FIGURE 3. Effects of the parameter ac on a transient flow of a vapor at time I/TQ = 220 due to evaporation from
its plane condensed phase. TV/To = 1.00755, Pw/Po = 1.05083 and Pr = 1. Solid lines: Boltzmann. Dashed lines:
Navier- Stokes. • : experimental data measured at x/L = 5 x l 0 4 from the interface, the initial temperature and pressure
being TO = 1.74°K and PO = 1315Pa, respectively (Furukawa and Murakami [4]). In the pressure distribution of the
Boltzmann's solution, the Knudsen layer is clearly visible in the close vicinity of the interface.
598