Simulation of Evaporation Flows from the Condensed
Phase with an Internal Structure
Based on the Fluid Dynamic Formulation
Y. Onishi , K. Yamada and S. Nakajima†
Department of Applied Mathematics and Physics, Tottori University, Tottori 680-8552, Japan
†
Institute of Space and Astronautical Science, Sagamihara, Kanagawa 229-8510, Japan
Abstract. Transient to virtually steady motions of a vapor due to evaporation and condensation processes from or onto the
plane condensed phase with a temperature field as its internal structure have been investigated based on the fluid dynamic
formulation, i.e., the formulation at the fluid dynamic level consisting of the Navier-Stokes equations and the macroscopic
conditions at the interface appropriate for evaporation and condensation problems derived earlier from the kinetic theory
analysis. In addition, the condition of the continuity of energy flow has to be imposed at the interface surface, which require
extra computational time because of the repetition of spacial integration of the governing equations. The behavior of the flow
field has been pursued for a long enough period of time in order to see how the flow field is established and approach its
final state. The results obtained here agree well with the corresponding ones from the Boltzmann equation of BGK type over
almost whole flow region execept, of course, a small region in the close vicinity of the condensed phase, i.e., the Knudsen
layer. From the present study in addition to the earlier works studied so far, the fluid dynamic fomulation gives satisfactory
results for motions of a gas phase associated with evaporation and condensation problems of practical interest, regardless of
whether the motions are steady or unsteady. Moreover, the treatment at the fluid dynamic level requires less knowledge on
kinetic theory and far less computational time compared to that at the kinetic level, which is its great advantage when dealing
with more practical problems of heat and mass transfer associated with phase changes.
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 close
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 the fluid dynamic quantities are to be specified at the interface of
the condensed phase. The answer is to be given at the level of kinetic theory, and the explicit conditions for the fluid
dynamic quantities to satisfy at the interface of the condensed phase have been given already in general terms from the
asymptotic analyses (see e.g., Sone & Onishi [1] for a linearized case and Onishi & Sone [2] for a weakly nonlinear
case) based on the Boltzmann equation of BGK type [3], although, of course, there are some restrictions on the range
of 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 as the boundary conditions, giving reasonably sound results for specific problems, transient or steady, in
the sense that they are in good agreement with those based on the Boltzmann equation of BGK type. Therefore, the
system of Navier-Stokes equations subject to these conditions at the interfaces of the condensed phase, which have
been called the fluid dynamic formulation [4], is surely valid for the description of various flow fields associated with
phase change processes, regardless of whether they are steady or unsteady, and can be used in place of the kinetic
equation. One may consult the studies, for example, on a half-space flow problem [5] and a cylindrical two-surface
problem [6], both of which are good examples where one can see the good agreement between the results based on
the Boltzmann equation of BGK type and the fluid dynamic formulation. Moreover, the results of the former problem
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
654
also agree fairly well with the corresponding experimental results done by Furukawa & Murakami [7].
The problems to which the fluid dynamic formulation has been applied so far are confined to those in which the
condensed phases have no internal structure because the corresponding studies at the level of kinetic theory were
not available for comparison. However, the analysis of a half-space problem in which the condensed phase has a
temperature field as its internal structure has recently been carried out by Onishi & Yamada [8] based on the Boltzmann
equation of BGK type and some new features associated with the effects of the internal structure or the finite thermal
conductivity of the condensed phase have been clarified. Here, instead of using the Boltzmann equation of BGK type,
we also treat almost the same problem based on the fluid dynamic formulation and confirm the validity and usefulness
of the formulation for two-phase systems of a vapor and its condensed phase with a temperature field as its internal
structure. The consideration of the internal structure of the condensed phase into the analyses of phase change problems
may cover wide range of subjects of practical importance in the field of heat and mass transfer. Moreover, the treatment
of the problems at the fluid dynamic level requires less knowledge on kinetic theory and far less computational time
compared to that at the kinetic level.
FORMULATION OF THE PROBLEM
Consider a two phase system of a vapor and its condensed phase which has a temperature field as its internal structure.
The condensed phase occupies a region D x 0 and the vapor the half-space x 0. Initially, the condensed phase
and the vapor phase are in complete equilibrium at a temperature T 0 . Let the pressure and density of the vapor at this
state be P0 and ρ0 , respectively. Suppose that at a certain time, say t 0, the temperature of the edge surface of the
condensed phase at x D is suddenly changed to Tc . Owing to the heat conduction within the condensed phase,
the temperature of the interface surface will soon be changed. This leads to the onset of phase change processes at the
interface of the condensed phase and a transient motion of the vapor occurs accompanied by a shock wave and the
contact region (sometimes expansion waves involved). For this analysis, the fluid dynamic formulation is applied to
the description of the motions of the gas phase, which may be written as
∂
∂t ρ
ρu
ρ e 12 u
2
∂ ∂x ρu
4 ∂u
ρu2 P µ
3 ∂x
1 2 4 ∂u ∂T
ρu h 2 u µ u λ
3
∂x
∂x
0
(1)
with
P ρ R T
h c pT
(2)
γ e γ γ 1 R T (3)
where t is the time; x is the coordinate with its origin at the interface surface of the condensed phase; ρ, u, T and P
are, respectively, the density, the velocity, the temperature and the pressure of the gas; e and h are the specific internal
energy and enthalpy, respectively. R is the gas constant per unit mass and c p is the specific heat at constant pressure,
γ being the specific heat ratio (γ 53 assumed). µ and λ are the viscosity and the thermal conductivity of the gas,
respectively, and they are here assumed to be the functions of T , i.e., µµ 0 T T0 and λλ0 T T0 , µ0 and λ0 being
the corresponding quantities at the initial state. The expressions for the constitutive relations are directly inserted into
the terms associated with the viscous stress and the heat flux in the above equations. For the temperature field within
the condensed phase, on the other hand, the equation of heat conduction with constant substance properties
∂T̃
∂t
ß ∂∂xT̃2 0
2
(4)
holds for D x 0 , where T̃ represents the temperature field within the condensed phase and
diffusivity, which is here assumed to be constant for simplicity.
The initial conditions for the above set of equations are, at t 0
u 0
T
T0
P P0
for x 0 ;
655
T̃
T0
for
ß
is its thermal
D x 0
(5)
As to the boundary conditions, the conditions at infinity ( x ∞ ) are exactly the same as those for x 0 in (5). The
condition for the temperature of the condensed phase at the edge surface is
T̃
Tc
x D
at
(6)
where Tc is the specified temperature of the edge surface of the condensed phase. The conditions at the interface
surface of the condensed phase (x 0), on the other hand, may be specified as
TW T̃
(7)
λc ∂∂xT̃ λ ∂T
∂x
∂u
1
4
hL u2 ρ u µ u 2
3
∂x
(8)
P PW
PW
C4 2R Tu 12
C4 2132039 (9)
T TW
TW
d4 2R Tu 12
d4 0446749 (10)
W
W
where λc is the thermal conductivity of the condensed phase (assumed constant). h L is the latent heat of vaporization
per unit mass. TW represents the temperature of the interface surface, which is yet unknown, and will eventually be
determined as part of the solution. PW is the saturated vapor pressure corresponding to the temperature T W of the
interface surface to be determined by the Clapeyron-Clausius relation as
PW
P0
exp Γ
T0
TW
1
Γ
hL
R T0
(11)
where Γ is a non-dimensional parameter associated with the latent heat of vaporization. The set of conditions (9)
and (10), which has been given earlier by Onishi & Sone [2] from a weakly nonlinear general analysis based on the
Boltzmann equation of BGK type [3], are crucial for problems associated with phase changes, giving the relations
between some of the fluid dynamic quantities holding at the phase boundary. Expression (8), which has been derived
from the consideration at the fluid dynamic level, represents the condition of the continuity of the energy flow across
the interface surface.
CHARACTERISTIC PARAMETERS
For the analysis of the present problem, let us introduce the length scale L and the time scale τ 0 taken as
L
π
2
l0 12
2
γ
c0
µ0
P0
τ0 L
2RT0 12
γ 12
2
L
c0
µP0 (12)
0
where c0 γRT0 12 is the sound speed of the gas at the initial state. l 0 is the mean free path of the gas molecules at the
initial state defined by l 0 µ0 P0 8RT0 π 12 . It may be mentioned that, since the length scale L is of the order of
the molecular mean free path l 0 , the time scale τ0 adopted here represents the mean collision time of gas molecules at
the uniform initial state. With the fluid dynamic quantities at the initial state together with these length and time scales,
the system of the governing equations and the initial and boundary conditions is appropriately nondimensionalized,
giving the following set of non-dimensional parameters characterizing the flow fields of the present problem:
λc
λ0
ß
ß0
Tc
T0
Γ
Pr c p
µ0
λ0
D
L
(13)
where ß0 is the thermal diffusivity of the gas at the initial state defined by ß 0 λ0 ρ0 c p . Pr is the Prandtl
number. Incidentally, we introduce here the Reynolds number Re and the Knudsen number Kn defined, respectively,
by Re ρ0 c0 Lµ0 and Kn l0 L. By noting that the relation Re 8γπ 12 Kn holds betwen the Reynolds number
and the Knuden number (based on the Boltzmann equation of BGK type [3]), it may be clear that the Reynolds number
Re simply becomes Re 2 γ 12 in this case.
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RESULTS
For a number of sets of the parameters in Eq. (13), the system of the governing equations at the fluid dynamic level has
been solved numerically, with a difference scheme applied, subject to the initial and boundary conditions specified. The
results obtained here agree not only qualitatively but also quantitatively with those by Onishi & Yamada [8] based on
the Boltzmann equation of BGK type except in the Knudsen layer where steep gradients of the fluid dynamic quantities
manifest themselves at the kinetic level. Moreover, the computational time required for the present calculations based
(a)
4000
T
T0
T
T0
1000
1.1
(b)
4000
1.1 1000
500
500
4000
1000
200
1.0
−2
4000
200
1.0
200
0
500
200
500
1000
2
x/L
4
−2
0
2
x/L
4
FIGURE 1. Transient temperature distributions in the close vicinity of the interface at x 0 (evaporation case). T T0 T̃ T0
for xL 0 and T T0 T T0 for xL 0. λc λ0 300, ß ß0 032, Tc T0 20, Pr 10 and DL 200. The numbers in
the figures indicate the time t τ0 . Case a : Γ 70. Case b : Γ 110. Dashed lines: the results by Onishi & Yamada [8] based
on the Boltzmann equation of BGK type.
TW −T
TW
PW −P
PW
0.3
0.06
0.2
Γ= 11
0.04
Γ= 7
Γ= 7
0.02
0.1
0.00
0
0.0
0
2000
t/ τ0
4000
2000
Γ= 11
t/τ0
4000
FIGURE 2. Time variations of the differences in temperature and pressure occurring at the interface at x 0 (evaporation case)
for two different values of Γ. λc λ0 300, ß ß0 032, Tc T0 20, Pr 10 and DL 200. T and P indicate the values
of the temperature and pressure of the vapor at x 0.
on the fluid dynamic formulation is far less than that for the Boltzmann equation of BGK type, although the comparison
may not be appropriate because the spacial mesh points and hence the time mesh of the latter case must be small enough
for the reasonable resolution of the Knudsen layer. The approximate computational time, for example, may be of the
order of a few hours of calculation on an ordinary Personal Computer for the former case whereas for the latter of the
657
order of a few hundred hours on a super computer such as Fujitsu-VPP800 system. Of course, the total computational
time for this kind of half-space problems depends on how long and how far we continue the calculation.
1.0
200
1.0
200
T
T0
200
500
200
T
T0
500
500
1000
1000
500
1000
4000
4000
4000
1000
0.9
−2
4000
(a)
0
2
(b)
0.9
x/L 4
−2
0
2
x/L 4
FIGURE 3. Transient temperature distributions in the close vicinity of the interface at x 0 (condensation case). T T0 T̃ T0
for xL 0 and T T0 T T0 for xL 0. λc λ0 300, ß ß0 032, Tc T0 05, Pr 10 and DL 200. The numbers in
the figures indicate the time t τ0 . Case a : Γ 70. Case b : Γ 110. Dashed lines: the results by Onishi & Yamada [8] based
on the Boltzmann equation of BGK type.
T−TW
TW
0.06
P−PW
PW
0.3
0.04
0.2
0.02
Γ= 7
Γ= 11
0.1
Γ= 11
0.0
0.00
0
Γ=7
2000
t/ τ0
4000
0
2000
t/τ0
4000
FIGURE 4. Time variations of the differences in temperature and pressure occurring at the interface at x 0 (condensation case)
for two different values of Γ. λc λ0 300, ß ß0 032, Tc T0 05, Pr 10 and DL 200. T and P indicate the values
of the temperature and pressure of the vapor at x 0.
Now, we show some of the results obtained here in the graphs where one can recognize the fairly good agreement
between the results of the present study and those based on the Boltzmann equation of BGK type. Figures 1 and 2 show
samples of the transient distributions of temperature and pressure in the immediate vicinity of the interface (Fig. 1)
and also the jumps in temperature and pressure at the interface (Fig. 2) for evaporation cases, whereas figures 3 and
4 show the corresponding ones for condensation cases. Of course, there can exist fairly large differences between the
two results owing to 1) the existence of the Knudsen layer near the interface at the kinetic level and 2) the restricted
validity of the conditions (9) and (10) for which P0 PW PW and T 0 TW TW should be satisfied. The
differences appearing in Figs. 1 and 3 are due mainly to the existence of the Knudsen layer (note that L is of the
658
order of the molecular mean free path), whereas those in Figs. 2 and 4 to the restricted validity of the boundary
conditions adpoted. The strengths of phase change processes at the interface for the present sets of the parameters
are slightly outside the range of validity of the conditions (9) and (10). Therefore, it may be expected that the degree
of the agreement between the two results should increase as the value of Γ increases or the ratio λ c λ0 decreases
because in each case the phase change processes become weaker (see Onishi & Yamada [8]). Finally, the comparison
of the transient distributions of the fluid dynamic quantities over the whole flow region is shown in Fig. 5 for an
evaporation case and in Fig. 6 for a condensation case. The enlarged portions of the distributions of the temperature
and pressure are also shown in Fig. 7 in order to see the fairly large variations in the close vicinity of the interface.
Good overall agreement between the two results, based on the fluid dynamic formulation and the Boltzmann equation
of BGK type, can be seen, which would indicate that the fluid dynamic formulation give the description of the flow
fields satisfactorily and can be used in place of the kinetic equation, especially for practical purposes.
ρ
ρ0
1.3
1.3
P
P0
1.2
1.2
4000
1.1
2000
1000
3000
4000
3000
1.1
2000
1000
1.0
1.0
0
2000
x/L
4000
1.1
0
2000
x/L
4000
u
c0
T
T0
4000
0.1
3000
2000
1000
1000
1.0
2000
3000
4000
0.0
0
2000
x/L
4000
0
2000
x/L
4000
FIGURE 5. Transient distributions of the fluid dynamic quantities (evaporation case). λc λ0 300, ß ß0 032, Tc T0 20,
Γ 110, Pr 10 and DL 200. The Mach number of the shock wave is about Ms 1080 (at about t τ0 4000). The
numbers in the figures indicate the time t τ0 . Dashed lines: the results by Onishi & Yamada [8] based on the Boltzmann equation
of BGK type. Note that ρρ0 x0 1265, PP0 x0 1256, T T0 x0 0993 and uc0 x0 0140 at t τ0 4000 in the
present analysis, wheras ρρ0 x0 1353, PP0 x0 1347, T T0 x0 0995 and uc0 x0 0131 in the work by Onishi &
Yamada [8].
659
1.0
P
P0
1.00
ρ
ρ0
4000
3000
4000
2000
3000
1000
2000
0.9
0.90
1000
0.82
0
2000
x/L
4000
0.8
0
2000
x/L
4000
0.0
1.00
T
T0
4000
u
c0
3000
4000
2000
3000
0.98
1000
2000
1000
0.96
−0.1
0.94
0
2000
x/L
4000
0
2000
x/L
4000
FIGURE 6. Transient distributions of the fluid dynamic quantities (condensation case). λc λ0 300, ß ß0 032, Tc T0 05,
Γ 110, Pr 10 and DL 200. The numbers in the figures indicate the time t τ0 . Dashed lines: the results by Onishi &
Yamada [8] based on the Boltzmann equation of BGK type. Note that ρρ0 x0 0837, PP0 x0 0847, T T0 x0 1011
and uc0 x0 0106 at t τ0 4000 in the present analysis, wheras ρρ0 x0 0822, PP0 x0 0818, T T0 x0 0995
and uc0 x0 0107 in the work by Onishi & Yamada [8]. Good agreement between the two results can be seen except in the
vicinity of the interface where the Knudsen layer exists at the kinetic level.
ACKNOWLEDGMENTS
This work was partially supported by the Grant-in-Aid for Scientific Research (No.14550153) from the Japan Society
for the Promotion of Science.
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1200
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T
T0
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P
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