Transient to steady evaporation and condensation
of a vapor between the cylindrical phases
- Navier-Stokes and Boltzmann solutions Yoshimoto Onishi, OOSHIDA Takeshi and Tomohiro Tanaka
Department of Applied Mathematics and Physics, Tottori University, Tottori 680-8552, Japan
Abstract. Motions of a vapor, transient to steady, due to evaporation and condensation processes occurring
at the cylindrical condensed phases have been considered numerically based not only on the Boltzmann
equation of BGK type but also on the formulation for phase-change problems at ordinary fluid dynamic level,
in other words, the Navier-Stokes equations with the boundary conditions at the interface for evaporation
and condensation derived earlier from a general asymptotic analysis of the weakly nonlinear version of the
Boltzmann equation of BGK type. The results based on both of these systems of governing equations agree
quite well, describing even the early processes of establishment of the flow fields as well as their transient
and final states. The production of the waves, shock waves and contact regions, their propagation and
interaction with the condensed phases are clearly seen on both systems of equations. From this fact, the
usefulness of the fluid dynamic formulation can be recognized and it can be used for analyses of problems
associated with phase changes in place of the Boltzmann equation of BGK type, although of course there are
some restrictions on its applicability because of the parameters involved having restricted ranges of values.
INTRODUCTION
Problems of evaporation and condensation and the associated transient and steady flows have a wide application in engineering, especially in mass, heat and energy transfer systems in aero-space engineering which
require detailed and accurate information of the flow fields. So far, various flow problems due to evaporation and condensation have been studied intensively. Among others, a flow problem between the cylindrical
condensed phases is one of the intriguing problems in the sense that it involves an expanded or a contracted
region through which the flow occurs and also it can easily be tested experimentally. We, therefore, consider
motions of a vapor between the coaxial cylindrical condensed phases on which the phase change processes are
occurring. The problems of this kind are the problems to which the ordinary continuum-based fluid dynamics
is not directly applicable because of a nonequilibrium region called the Knudsen layer always existent even in
the continuum limit in the close vicinity of the condensed phase. It is this layer that is responsible for the
phase-change processes to occur at the interfaces. Naturally, the analyses for such problems must necessarily
be based on kinetic equations. In the continuum limit, however, the Knudsen layer is small in its thickness,
and, therefore, the flow field may well be described by the Navier-Stokes equations if appropriate boundary
conditions are applied to the interface. These conditions, which are to be given from the kinetic theory analysis, are now available. Actually we adopted as the appropriate boundary conditions for the Navier-Stokes
equations the macroscopic conditions which have been derived in general terms by Onishi and Sone [1] from
the weakly nonlinear analysis of the Boltzmann equation of BGK type [2]. Of course, some restrictions on the
applicability of these macroscopic conditions are unavoidable because of the restricted ranges of values of the
parameters involved in the weakly nonlinear analysis. Here this system of the Navier-Stokes equations with the
boundary conditions given from the kinetic theory may be termed the fluid dynamic formulation. A number of
studies having so far been done in our laboratory show that the fluid dynamic formulation gives good results
for transient and steady flow fields which are in almost complete agreement with those based on the Boltzmann
equation of BGK type [2], not only qualitatively but also quantitatively. Our purpose is to show, in addition to
the qualitative and quantitative nature of the present cylindrical problem, that the fluid dynamic formulation
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
575
works well in this problem and it can be used in place of the Boltzmann equation. In fact, we carried out the
numerical analysis of the present problem based on the two governing systems, the kinetic formulation and the
fluid dynamic formulation, and compared the results between the two systems. Furthermore, the comparison
has also been made with the analytical solutions, linearized (Onishi [3]) and weakly nonlinear (Onishi [4])
solutions, based on the kinetic equation for binary gas mixtures proposed by Hamel [5] by taking in those
analyses a limit of noncondensable gas being zero.
FORMULATION OF THE PROBLEM
Consider a vapor between the coaxial cylindrical condensed phases. Let the radii of the inner and outer
cylindrical phases be RI and R%, respectively. Initially, the vapor and its condensed phases are in complete
equilibrium at a temperature TO . The pressure, density and number density of the vapor at this state are
PO, po and NQ , respectively. Suppose that, at time t = 0, the temperatures of the cylindrical condensed
phases are suddenly changed, i.e., from TO to TI and T2, respectively. This leads to the onset of phase change
processes at the condensed phases, 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 fluid dynamic formulation or simply
the Navier-Stokes equations. The condition for the distribution function at the condensed phase is of diffusive
type commonly used, namely, molecules leaving the condensed phase are subject to a Maxwellian distribution
corresponding to the saturated state at the temperature of the condensed phase.
Kinetic formulation
The Boltzmann equation of BGK type [2] in terms of the cylindrical coordinates (r, 0, z) may be written
down for the present problem with axial symmetry as
777
(3)
J — 00 J — 7T J 0
(4)
where t is the time; r is the radial coordinate; / is the distribution function; £ is the magnitude of the vector £j_
(€ — l£_i_l)j which is the component of the molecular velocity vector £ perpendicular to the z axis; a is an angle
between the radial vector r and the component molecular velocity vector £j_; £ has its components (£ r , £$, £ z )
in the directions of the coordinate axes with the relations £r = £ cos a, £# = £ sin a (0 < £ < oc, —TT < a < TT).
Nj w, p, T and P are, respectively, the number density, the mean flow velocity in the r direction, the density,
the temperature and the pressure. Fe is the local Maxwellian distribution characterized by the local fluid
dynamic quantities. R = k/m is the gas constant per unit mass, k being the Boltzmann constant and m the
molecular mass. z/c is a constant (Nvc : local collision frequency) related to the transport coefficients of the
gas, for example,
NM=*
= 2IA
^0
AO
(5)
fj,Q and AQ being the viscosity and thermal conductivity of the gas, respectively, at the initial equilibrium state.
The initial condition for the distribution function / is everwhere (Ri < r < J?2)
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The boundary conditions for the distribution function /, which are of diffusive type for evaporation and
condensation, may be specified as follows:
at the inner surface ( r = RI ) and
at the outer surface (r = ^ 2 ) ? where NI and A^ are the number densities for molecules leaving the inner
and outer condensed phases, respectively, and they are here given the values of the saturated vapor number
densities corresponding to the temperatures of the condensed phases. Actually, NI and N% are calculated from
the Clapeyron-Clausius relation as
Pi (i = I , 2) being the saturated vapor pressure corresponding to the temperature Ti of the condensed phase.
hi, is the latent heat of vaporization per unit mass.
After the distribution function / has been determined, we may be able to calculate the normal stress crrr
and the energy flux E as
CO
/
-|
/»OQ
/
£3 cos2 a f d£ da
(10)
« 2 coBa(^ + ^)
(11)
-OoJ — 7T«/0
/»OO
=~™
^
/»7T
/
/»7T
/
/»OO
/
J-OoJ-TTJO
The heat flux Q, if needed, can be found from
(12)
For the actual numerical calculations, the following new functions g , h , G and H
have been introduced here. In addition, it has been taken into account that the distribution function / is an
even function of a and, hence, g and h are also the even functions of a.
Fluid dynamic formulation
The governing system at the fluid dynamic formulation is the set of the Navier-Stokes equations and, for the
present problem with axial symmetry, it may be written in terms of the cylindrical coordinates as
di
pu
1 9
rpu
+ -—
2
.r{pu(h + |w ) - rrru + q}.
with
577
1
Tee-P=Q,
0
J
(14)
(15)
7-1
. 9 T
«=-**'
A T
X-0=T0>
where e and h are the specific internal energy and enthalpy, respectively, 7 being the specific heat ratio
(7 = 5/3 here). rrr and TQQ are the viscous stresses and q is the heat flux. // and A are the local viscosity
and thermal conductivity, respectively, and are here assumed to be the functions of T, the temperaturedependence of which is consistent with that of the transport coefficients derived from the Boltzmann equation
ofBGK type.
The initial conditions for the present problem are, at t = 0
« = 0,
P = P0,
T = TQ
(R1<r<R2).
(18)
The boundary conditions at the fluid dynamic formulation are the macroscopic conditions having been derived
earlier from the weakly nonlinear analysis in kinetic theory [1] (see also Sone & Onishi [6] for the linearized
analysis) and, for the present problem, they are given as follows: at the cylindrical condensed phases
(Q* = -2-132039),
(19)
K = -0-446749),
(20)
where the upper sign with i = 1 applies to the inner surface at r = RI and the lower with i = 2 to the outer
surface at r = R2. Pi is the saturated vapor pressure at T,- of the condensed phase as before.
Characteristic parameters
We now introduce various characteristic quantities here. Let RI be the characteristic length and -\/2RTo be
the characteristic velocity. The characteristic time TO then become TO = (7/2) 1 / 2 (^i/co), CQ being the sound
speed at the initial state defined by CQ = (7!? To) 1 / 2 , which is also taken as another characteristic velocity
whenever convenient. With these characteristic scales together with the fluid dynamic quantities at the initial
state, the governing two systems, kinetic and fluid dynamic formulations, are appropriately nondimensionalized.
We then have the following nondimensional parameters
_ p0cQRl
,,
jJiQ
7
'
~
(~
(7
fl/jp
\
'
—n
Ij AO
R2
TI_
TI
fi» '
HI
T
IQ '
T
IQ '
hL
OT1 '
H!Q
^
'
which characterize the present flow fields, where Re and Pr are the Reynolds number and the Prandtl number,
respectively. The Reynolds number Re here is related to the Knudsen number Kn as
/o
I -
l
°
--
-
- --
'
(22)
/o being the mean free path at the initial state. Once these parameters in Eq. (21) are specified, the fluid
dynamic quantities are determined as the unique functions of x/R± and t/TQ. It may be noted that the
specification of F means the specification of the saturated vapor pressures PI/PQ and PZ/PQ corresponding
to the given temperature ratios Ti/To and T2/To, respectively. Incidentally, a brief mention is made of the
parameters which characterize the steady flows for the present problem. At the steady state, the history of
the initial condition is completely disappears and, therefore, the parameters characterizing the present steady
flows become
J
.
P, = fr ,
HI
,
12
,
f = r.J-2 (23,
where the tilde parameters are newly defined based on the condition associated with the outer condensed phase.
578
1.5
r/Rj
2.0
1.0
O.OOh
1.0
FIGURE 1. Transient distributions of the fluid dynamic quantities due to an evaporation process at the inner cylindrical condensed phase for R2/Ri = 2.0, Ti/To = 1.005, T2/T0 = 1.0, T = 11.0 (Pi/P0 = 1.056, P2/Po = 1.0), Kn = 0.005
(Re = 412.026) and Pr = 1. The numbers in the graphs indicate the time £/TQ. Solid lines: Boltzmann. Dashed
lines: Navier-Stokes. Dot-dashed lines: weakly nonlinear analytical solution (in the limit of zero noncondensable gas)
at steady state by Onishi [4] (see the lines at t/ro = 135.0, which may not be clearly visible). The figures on the right
represent further time developments of the quantities of the corresponding figures on the left.
579
p_
Po
1.02
T/R!
2.0
__
1.00
2.0
1.0
1.5
110.0
70.0
^'2.0
SOU) 30.0
20.0
10.0
1.00
1.5
r/Rj
2.0
1.0
1.5
2.0
0.00
1.0
FIGURE 2. Transient distributions of the fluid dynamic quantities due to an evaporation process at the outer cylindrical condensed phase for R2/Ri = 2.0, 7\/T0 = 1.0, T2/T0 = 1.005, T = 11.0 (Pi/P0 = 1.0, P2/Po = 1.056), Kn = 0.005
(Re = 412.026) and Pr = 1. The numbers in the graphs indicate the time £/TQ. Solid lines: Boltzmann. Dashed
lines: Navier-Stokes. Dot-dashed lines: weakly nonlinear analytical solution (in the limit of zero noncondensable gas)
at steady state by Onishi [4] (see the lines at t/ro = 110.0, which may not be clearly visible). The figures on the right
represent further time developments of the quantities of the corresponding figures on the left.
580
RESULTS AND DISCUSSION
The calculations have been carried out with a difference scheme applied to the governing systems for a
number of sets of the parameters listed in Eq. (21). Here the Prandtl number Pr appearing in the NavierStokes system has been set equal to unity in order to be consistent with the kinetic formulation because the
Prandtl number is always unity in the Boltzmann equation of BGK type. The distributions of various fluid
dynamic quantities including the mass and energy flows for the transient and final steady states have been
obtained based on the two governing systems, the Boltzmann and the Navier-Stokes equations. Two typical
samples of a number of the results obtained are shown in Figs. 1 and 2, which show the transition processes of
the flow fields between the cylindrical condensed phases to their final states caused by the evaporation at the
inner and at the outer condensed phases, respectively. Good agreement can be seen between the results of the
two governing systems in both cases. Also in the figures, some dot-dashed lines are plotted, which represent a
weakly nonlinear analytical solution at steady state for a vapor-gas mixture by Onishi [4] having taken a limit
of noncondensable gas being zero.
Figure 1 is one of the cases of outward flows due to a relatively small increase of the temperature of the inner
condensed phase. Owing to the onset of evaporation at the inner condensed phase, the region of high pressure
(or compression) is formed almost all of a sudden in the vicinity of the inner condensed phase and is propagated
outward as a shock wave to the wider undisturbed region. This shock wave, weakening as it propagates into
the diverging region, reaches the outer condensed phase at about t/r® ~ 0.9 and increases the pressure of the
vapor there, causing then the condensation process at the outer condensed phase. Owing to the slight excess
of the mass flow behind the shock wave over the mass flow due to the condensation at the outer condensed
phase, a weak compression region is produced in its vicinity and starts propagating as a very weak reflected
shock wave, too weak to be clearly visible in the figures, toward the inner condensed phase in order to adjust
the mass flows at both the condensed phases, evaporation at the inner and condensation at the outer phases.
The reflected shock wave, interacting with the inner condensed phase, produces another shock wave possibly
with a contact region accompanied, and the secondary produced wave and region propagate toward the outer
condensed phase. Interaction processes of this kind go on within the flow region: weak shock waves produced
propagate back and forth in the flow region, interacting with the condensed phases and contact regions, and
eventually fade away. During the period, the initial and later produced contact regions, moving with the local
flow velocity, approach the outer condensed phase and then the flow field assumes almost steady state. The
formation of the steady profiles of the temperature and density is considerably slower than that of the pressure
and velocity which are not affected by the existence of the contact regions.
Figure 2, on the other hand, is one of the cases of inward flows due to an evaporation process at the outer
condensed phase. It can be seen that a shock wave, which is initially produced and is sent out from the
outer condensed phase, is propagating toward the inner condensed phase, strengthening as it goes through
the converging region of the flow field. On reaching the inner condensed phase, the shock wave causes a
condensation process there. This condensation process is not strong enough to accommodate the mass flow
behind the shock wave and, hence, a compression region is formed, producing a reflected shock wave (weak in
this case) toward the outer condensed phase. The weak shock waves and the contact regions produced having
the interaction between themselves and with the boundaries eventually bring the flow field to its final steady
state, the behavior of which is qualitatively the same as that depicted in Fig. 1. Turning our attention to
the accuracy of the numerical results, the solution to the kinetic formulation in this case becomes bad and is
getting worse as the time development of calculation proceeds and this unfavorable behavior manifests itself
fairly strongly in the density and temperature, but not so appreciably in the pressure, velocity and total mass
flow. The reason for this may be attributted to the fact that the flow strengthens as it moves toward the inner
condensed phase and the numerical errors are likely to be accumulated during the course of calculation. We
should have taken smaller meshes in the relevant variables (compared with the outward flow case), especially
in a and £, but doing this requires a huge amount of calculational time, which we could not afford. Therefore,
we omitted putting the kinetic results in the temperature and density distributions in order to avoid unsightly
and mixed up graphs. However, the analytical kinetic results at steady state [4] are plotted, which will suffice.
Lastly, we show the transient and steady behavior of the total mass flow in Fig. 3, where the corresponding
analytical solution at steady state by Onishi [4] is also plotted by dot-dashed lines. These dot-dashed lines,
which are straight lines of course, are not clearly distinguishable from the other two lines because the agreement
of the results based on the two governing systems and of the analysis is very good, although the line for the
inward flow is slightly deviated from the numerical results of the both governing systems. However, the
results based on the two governing systems may be considered to be very good not only qualitatively but also
581
o.oor/Rt
0.03
\rPu\
2.0
1.0
r/Rt
0.03
2.0
1.6
\rPu\
0.02
0.01
0.00
1.0
1.0
0.00
1.0
110.0
1.5
2.0
FIGURE 3. Transient total mass flows due to evaporation and condensation between the cylindrical condensed phases
for R2/Ri = 2.0, Kn = 0.005 (Re = 412.026) and Pr = 1. Two figures above (outward flow case): Ti/To = 1.005,
T2/T0 = 1.0, T = 11.0 (Pi/Po = 1.056, P2/Po = 1.0). Two figures below (inward flow case): Ti/To = 1.0, T2/T0 = 1.005,
F = 11.0 (Pi/Po = 1.0, P2/Po = 1.056). The numbers in the graphs indicate the time t/ro. Solid lines: Boltzmann.
Dashed lines: Navier-Stokes. Dot-dashed lines: weakly nonlinear analytical solution (in the limit of zero noncondensable
gas) at steady state by Onishi [4] (see the lines at t/ro = 135.0 above and £/TO = 110.0 below, respectively). The figures
on the right represent further time developments of the quantities of the corresponding figures on the left.
quantitatively. Incidentally, it may be noted that the transition of the mass flow to its steady state is quite
fast; at about t/T$ ~ 40.0, the mass flow virtually assumes its steady state.
Finally, a brief mention is made about strong evaporation and condensation cases. A numerical analysis of
the present problem with strong evaporation and condensation has been done by Onishi et al. [7] based on the
Boltzmann equation of BGK type, one of the features of which is the formation of standing shock wave between
the cylindrical phases. However, we could not deal with strong cases at the fluid dynamic level because the
fluid dynamic formulation for strong evaporation and condensation cases is not yet available.
This work was partially supported by the Grant-in-Aid (No. 10650175) from the Ministry of Education in
Japan.
REFERENCES
1.
2.
3.
4.
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Onishi Y., Rarefied Gas Dynamics: Theory and Simulations (Progress in Astronautics and Aeronautics, Vol. 159),
edited by B.D. Shizgal and D.P. Weaver, Washington D.C.: AIAA, 1994, pp. 537-547.
5. Hamel B.B., Phys. Fluids 8, 418-425 (1965).
6. Sone Y. and Onishi Y., J. Phys. Soc. Japan 44, 1981-1994 (1978).
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Japan on March, 1995 (in Japanese).
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