7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Vapor Flows with Phase Changes Induced by Sound Waves
M. Inaba*, T. Yanot, M. Watanabe* and S. Fujikawa*
Division of Mechanical and Space Engineering, Hokkaido University, Sapporo 0608628, Japan
Department of Mechanical Engineering, Osaka University, Suita 5650871, Japan
inaba@mechme.eng.hokudai.ac.jp
Keywords: evaporation coefficient, GaussianBGK Boltzmann equation, asymptotic theory
Abstract
Vapor flows in a onedimensional space between an oscillating plate and a vaporliquid interface is theoretically
analyzed by applying the asymptotic theory of GaussianBGK Boltzmann equation for Kn < M < 1, where Kn
is the Knudsen number and M is the typical Mach number. The result shows that the gas region consists of the
following three regions, i.e., the Knudsen layer, the thermal boundary layer and the isentropic sound region. For the
case where the evaporation coefficient a is of the order of unity and the frequency of plate is high, no shock waves
is formed in the system even at a resonance condition, in contrast to the case of resonant gas oscillation in a closed tube
Introduction
When a net condensation or evaporation occurs at a
vaporliquid interface, the vapor adjacent to the interface
is not in a local equilibrium state because the molecules
impinging onto and those leaving from the interface are
governed by different laws. The vapor near the inter
face should be treated by the Boltzmann equation with
the kinetic boundary condition at the interface (Cercig
nani 2000; Sone 2002, 2006). However, since we don't
know the real molecular scattering law at the interface,
the available kinetic boundary condition remains a math
ematical model, and hence we cannot predict the va
por flow caused by the phase change in real situations.
In a recent molecular dynamics study (Ishiyama et al.
2004), the kinetic boundary condition is obtained in a
form of Maxwellian multiplied by a factor including
the socalled evaporation coefficient in the case that the
net mass flux across the interface is sufficiently small,
i.e., nonequilibrium effects are sufficiently weak. In the
present study, we theoretically analyze a vapor flow with
phase changes induced by sound waves, which satis
fies the restriction of weak nonequilibrium state because
the variations of pressure, temperature and velocity in
sound waves are usually sufficiently small, and thereby
we can adopt the Maxwellian with a factor including
the evaporation coefficient as the kinetic boundary con
dition. Using the asymptotic analysis for small Knud
sen numbers, we obtain the detailed structure of flow
field and the solution of sound field with phase change
a (cos nt* 1)
Oscillating Condensed
Plate Phase
Polyatomic Vapor
2a
Sound Wave
Figure 1: Schematic of model.
in a form containing the evaporation coefficient as a pa
rameter. This will be utilized to determine the values
of evaporation coefficient through the measurement of
sound field (Nakamura et al. 2010) in our future work.
Problem
Figure 1 shows the schematic of a sound wave in this
study. We consider a onedimensional space between an
oscillating plate (sound source) and a planer surface of
a thin liquid layer. The space is filled with a polyatomic
vapor which is initially in an equilibrium state at rest
with the density po and the temperature To. The origin
of the spatial coordinate X1 is set at the initial location
of the sound source surface. At a time t* = 0, the sound
source begins a harmonic oscillation with amplitude a
and angular frequency 2. The variation of macroscopic
quantities (pressure, velocity, temperature and so on) in
duced by the sound source propagates as a sound wave
in the vapor. The sound wave is partially reflected by the
liquid surface, and a quasisteady gas oscillation may be
formed after a sufficiently large time has passed.
Especially, we focus on the nonlinear resonant gas os
cillation with phase changes excited in the space. The
behavior of resonant gas oscillation of plane waves in a
closed tube bounded by a oscillating plate and a rigid
wall was theoretically studied by, e.g., Chester (1964);
Keller (1976); Goldshtein et al. (1996). According to
these analysis, a shock wave is formed in the system
under a large Reynolds number and its nondimensional
wave amplitude at a quasisteady state is of the order
of square root of Mach number of sound source. In this
study, to describe such a resonant gas oscillation, we set
the nondimensional parameters as satisfying the follow
ing relation:
M2 Mp Kn= 2 < 1, (1)
2
where M is the typical Mach number, Mp is the Mach
number of sound source defined by aQl/, /RTo, Kn is
the Knudsen number defined by oQI/2RTo, R is the
gas constant, 7 is the specific heat ratio and {o is the
mean free path of gas molecules at the initial tempera
ture. Under the above c' li i Reynolds number Re finds
a large value ( O(e 1) )with the aid of von Karman re
lation,
M oc Re Kn.
In a vapor flow which has a large Reynolds number,
there appears a thin layer, called the thermal boundary
layer, near the boundary. In this layer the variation of the
state is strongly anisotropic, i.e., the length scale of the
variation in the direction normal to the boundary shrinks
by the factor of inverse of square root of Knudsen num
ber. There also appears a nonequilibrium region, called
the Knudsen layer, adjacent to the boundary. In the fol
lowing section, we therefore carry out the asymptotic
analysis for small Knudsen numbers taking into account
the thermal boundary layer and the Knudsen layer. The
details of solution of resonant sound with phase changes
are discussed.
Governing equations and boundary conditions
We assume the polyatomic vapor is governed by the
polyatomic version of the GaussianBGK Boltzmann
equation (Andries et al. 2000) shown below. The sev
eral model Boltzmann equations can describe the be
havior of polyatomic molecules and some can adjust the
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Prandtl number and bulk viscosity (Morse 1964; Hol
way 1966) to their known experimental values. How
ever, the Boltzmann H theorem has been proven only for
the GaussianBGK Boltzmann equation. Its nondimen
sional form suitable for the analysis of onedimensional
problem can be written as
Dy D0 1 1+P ](3)
at 9i e62 (1 + T)P
D y D 1 1+P [ (
at+ 1 [ 7 + Tr6e + 1Tre]
at 19X 1 2 (1 + 7)1
E(1+ 4') e
73/2 V/det)S,(
x exp [((i Ui),,s.
u)] (5)
6 n V170f(1 + )1, (6)
2 2 2R r (6)
where xi /2RTI / is the distance from the origin,
t 1 is the time, (i2RTo is the ith component of
molecular velocity (i 1, 2, 3), p is the viscosity, po is
the viscosity at the initial temperature, B is the constant,
F is the Gamma function, det is the determinant and
1
E exp(q). (7)
4 and y are the nondimensional form of distribution
function f, which are given by
0 />00
f dl I/ 12 f dl
4' 1, y 1, (8)
/ fodl 12/ fodl
where I is the internal energy parameter, n is the inter
nal degrees of freedom and fo is the Maxwellian at the
initial equilibrium state.
The nondimensional forms of macroscopic variables
are given by
u /JE d
S(1 + w)t, C _) E dC
(2+)i,= ( )d 2,
(1 + W)Tint (7P 0) E dC
(1 W )u,
(11)
(12)
(1+ 2 (/C(l EdC 2(1 + w .
Qi Ci PEd( + ; I E C dC
3 + n
Pui (1 + W)UiUJ,
5+n
2
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
3 n1
T Ttr + Tint,
3 + n 3 + n
P = W + T + WT,
(15) The widely used boundary condition at the vapor
liquid interface for (1 < ue is given by (Cercignani et
(16) at 1985)
Tl = OT + (1 )Tint, (17)
Sj = [1 (1 0)v(1 + w)1 + (1 0)(1 v)Tt,
+0T] 8ij + (1 o)veij, (18)
where po(l + w) is the density, ua2RT is the ve
locity, To(1 + Tt,) and To(1 + Tit) are the tempera
tures associated with translational and internal motions
of molecules, respectively, po [6ij + (1 + '1 .] is the
stress tensor, po is the initial pressure, po2RToQi is
the heat flux, To(1 + T) is the temperature, po(1 + P)
is the pressure, T(1l + T,el) is the relaxation tempera
ture, po(l + w)Sij is the corrected stress tensor, 86 is
Kronecker's delta and dC dC1 d(2 d(s. 0 and v are
chosen so that the Prandtl number Pr and the bulk vis
cosity Pb given by
S 1 Pb 2n 1 v+v
Pr = (19)
1 v+u po 3+n 30
may agree with their known experimental values for gas.
The ratio of the bulk viscosity and viscosity is assumed
to be 0.73, which is a value for N2 at 293 K (Prangsma
et al. 1973). The specific heat ratio 7 is given by
n+5
7 + (20)
n + 3
The boundary condition at the sound source for (1 >
u,s is defined as follows:
P = .ws, Y = Yws atxi = Mp(cost 1). (21)
Ows and Yws are assumed to be a diffuse reflection con
dition given by
1 ,rws
E(1 + Ow) /(1 + a ,7
73/2 (1 + T )3/
71 + oT
1 + wa, 1 .
1 w 2 C
21 + r ,s
2exp( 1+%sj
(22)
2 exp [ ( (.)2
e 1 + w
(23)
(C1 uws) (1+
awsi a Uws 6ili
where u,s = /2 Mp sin t is the nondimensional
oscillating speed of sound source, To(1 + Tw,) is the
temperature of sound source surface which is assumed
to be constant temperature, i.e., Tw, 0.
0 = Owe, Y = Lwe at xi = L
we and Ywe contain the evaporation coefficient a, and
condensation coefficient ac as follows:
1 + 0c + owa e + (1 + a) J e
E(1 + ) _3/2(1 + T W)3/2
x exp (j1 , .) (27)
1 1 + ]w I
1+ ae ac + aewwf + (1 ac)awE
E(1 + Y) i73/2(1 + T7,)1/2
x exp (j1 +T ,) (28)
L 1 + Te\
1+ 2wL 2
((1 ue) (1 + p)EdC,
(29)
where po(1 + cc ) is the saturated vapor density at the
temperature of liquid surface To(1 + Twe), uae is the
moving velocity of liquid surface. The saturated vapor
pressure po (1 + P) is given by the linearized Clausius
Clapeyron equation, P, = HLTw, where HLRTo is
the latent heat. We assume ac = ae (say a), which
is the case in equilibrium states (Ishiyama et al. 2004;
Kobayashi et al. 2008).
The liquid is governed by the incompressible Navier
Stokes set of equations:
due d P
=9 ,
thz Otz
dTe d2T
0,
Ot Oz2 '
where '/Keo/I z is the spatial coordinate normal to
the vaporliquid interface pointed to the liquid, KLO is
the heat diffusivity of the liquid for the initial tempera
ture, ceoue is the velocity of liquid, po +, .... ,, Pe is the
pressure, To(1 + Te) is the temperature, pee is the ini
tial density of liquid, ceo is the initial speed of sound.
The boundary conditions on the rigid wall (z = D
D* V /Ko) are given by
u,(D,t)=0, T~(D,t)=0. (31)
From Eqs. (30) and (31), we can easily find ue 0.
6)E d, With the aid of continuity conditions of fluxes though
the vaporliquid interface, the boundary conditions at the
(24) vaporliquid interface (z = 0) may be given as follows:
Mass: Uw, = pu,
Momentum: Pf = ^ (11,
zL
Energy: A
ib
L* '
V2RTo'
HL 1
Q1, (34)
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
equations up to the second order:
where
Po
= ,
Peo
V2RTo
5 + n eo r K/
2/ Ao V2 Keo
HRTo is the enthalpy of vapor, A0 and Aeo are the ther
mal conductivities of vapor and liquid for the initial tem
perature and K0 is the thermal diffusivity of vapor for
the initial temperature.
Asymptotic theory for small Knudsen numbers
We shall carry out a asymptotic analysis of the
boundaryvalue problem Eqs. (3)(6), (21) and (26) for
small Knudsen numbers, following Sone (2006).
First, putting aside the boundary condition (21) and
(26), we seek a moderately varying solution Os and ys
to Eqs. (3) and (4) satisfying a0 s/0xi O(qs) and
a 's/ 0i = O(Ys) in the form of a power series of e:
's = Osie+ s2 +... (37)
YS = 1si1 + s .2 + (38)
where these series start from the first order in c since
qs and 's are assumed to be O(c). Note that the co
efficients of expansion Os, and Y's, are quantities of
the order of unity. Corresponding to the expansions,
the macroscopic variables ws, Uis, Ts, ... are also ex
panded in e:
hs hsic +hs2 +... (39)
where h represents w, ui, T, etc.
Substituting the expansions (37) and (38) into the
GaussianBGK Boltzmann equation (3) and (4), and ar
ranging the same order quantities in c, then we get a
sequence of integral equation for Osm and 'sm. In
the leading order of approximation, we can obtain local
Maxwellians as solutions of linear homogeneous inte
gral equations for Osi and 'sl:
si = wsi + 2Ci(lis + I
Ysi = Wsi + 2C1a1s1 +
3
) Ts1,
2/
I s .
In the subsequent orders, from the solvability conditions
for linear inhomogeneous integral equations, we obtain
the following series of the Euler set of partial differential
+ u 0,
9uisi 1 apgi 0
at 2 ax1
3 + n Psi 5 + n Ouisl
2 5t 2 3 1
OS9g2 +>ULS2 9iwsluls1
at 0271 0ss 1
9t 9xi 9xI
OUis2 1 9Ps2 9csluisi
at 2 9xi a t
3 + n 9Ps2 5 + n 9UlS2
2 9t 2 Oxi1
aau si 5+ n Psiulst
at 2 ax1
9x1
We shall remark that if we set n 0 then Eqs. (44) and
(47) become the same equations as those derived from
the BKW equation, and viscous and thermal conductiv
ity effects appear in O(c3).
Next, we consider the correction of the distribution
functions in the thermal boundary layer 9T and YT:
T(y,(,t) = 9 s, yTT(Y, i,t) = 0 s, (48)
lim T= lim OT = 0, (49)
yoo yoo
where y is the new coordinate variable which is a
stretched coordinate normal to the boundary for describ
ing macroscopic variables in the thermal boundary layer
(xi cy near the sound source or 3x1 L cy near the
interface). The corrections OT and OT are also expanded
in power series of e:
T = T16 + 9T262 +
OT = T16 + 'T262 +
According to these expansions, the macroscopic vari
(40) ables in the thermal boundary layer are also expanded
in e:
(41)
hT = hTi + hT2e2 + (52)
The resulting fluiddynamictype equations in the ther
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
mal boundary layer are
where
dUlT1
dy
DU1T2
0y
aU1T1
at
0 = C4 + 2 1 
0, PT 0,
Dy
8wyTI
dWT1
at
1 OPT2
2 dy
d [wul]2
9y
d [ai i]2
Dy
(54)
(55)
dTT1 2 dPT1 d [TUi]2 1 T (56)
+ (56)
Dt 5 + n Dt Dy 2 Dy2
DU1T3 dWJT2 d [aU1]3
dy dt Dy
DUlT2 1 OPT3 D [WUl]2 d [UlUa]3 + [WU1U1]
Dt 2 0y Dt Dy
Pr (4 Pb 02 1T2 1 37 T1 (58)
2 (3 po) y2 2 Dy3 '
3 + n dPT2 5 + n dlT3
2 Dt 2 Dy
d [iui] 5 +n [PU1]3
dt 2 dy
9 [aiii] 5 + n 92 (
S[ululul] + 5+n 2 T2 +
Dy 4 y2
The notation [ ] represents
m1
[ab]m= (asrbTmr + aTrbsmr
r=l
2 [TT) (59)
+ aTrbTmr),
(60)
[abc] = (ast + aTl)(bsi + bTl)(cs1 + CT1)
asibsicsi.
Finally, we consider the Knudsen layer correction.
In the Knudsen layer, where the characteristic length is
of the order of mean free path, the order of the time
derivative terms in the GaussianBGK Boltzmann equa
tion is smaller than that of the other terms and hence for
the leading order of the approximation, we can treat the
vapor flow in the Knudsen layer as a steady one. The
Knudsen layer analysis can also be carried out in the
same way as that in Sone et al. (1978) and thereby the
Knudsen layer corrections and the slip boundary con
dition for the fluiddynamictype equations can be ob
tained. For the case of a 0(1), the results of the
leading order of the approximation are given as follows:
On the sound source
usi(0, t) 0, (62)
TsI (0, t) + TTI(0, t) 0. (63)
On the vaporliquid interface
Psi(L, t) P = C^4lsl(L,t), (64)
Tsl(L,t) + TT1(0, t) 7, = d4Ulsi(L,t), (65)
and the value of slip coefficients 04 and d4 for the case
of methanol has been obtained by Yano et al. (2005),
C4 2.0723 and d = 0.2185.
For the case of a O(c), the results of the leading
order of the approximation are given as follows:
On the sound source
si(0, t) = 0,
Tsl(0, t) + TT1(0, t) 0.
On the vaporliquid interface
uisl(L,t) = 0,
Ts1(L,t) + TT1(L,t)
(69)
7 = 0. (70)
Results
In the present section, we shall derive a solution of lead
ing order of approximation. Especially the high fre
quency waves and a 0(1) is considered. For the case
of high frequency waves, the variation of liquid temper
ature is almost negligible, because A in Eq. (34) become
large. Therefore the boundary conditions for ulsi and
Psi can reduce to
lsi(0,t) = 0,
Psi(L,t) = C4 si(L,t).
The general solution of Euler set of equations is given
by
ulsa (x, t) = f(t xa/co) + g(t + xl/co), (73)
Psi(x1, t) = V [f(t xl/co) g(t + xl/co)],
(74)
where f and g are arbitrary real functions, and co
S/2. Substituting Eqs. (73) and (74) into the boundary
conditions (71) and (72), we then get
f(t + L/co) f(t L/co). (75)
Let N be a positive integer and f be a bounded function,
we finally get
f(t) 4 f(t 2L/co)
04 +27
S04 f (t 2NL/co)
\04 + v)
0 (N oo) .
The only trivial solution exists, i.e., no shock waves is
formed in the system. Therefore, the order of nondi
mensional distribution function need to assume to be of
the same order of Mp for the case of a 0(1). Such
flow is almost equivalent to that for the case of the lin
ear analysis (Inaba et al. 2008) except for the thermal
boundary layer near vaporliquid interface.
Conclusions
Vapor flows in a onedimensional space between an os
cillating plate and a vaporliquid interface is theoret
ically analyzed by applying the asymptotic theory of
GaussianBGK Boltzmann equation for Kn < M < 1.
The result shows that the gas region consists of the fol
lowing three regions, i.e., the Knudsen layer, the ther
mal boundary layer and the isentropic sound region. For
the case where the evaporation coefficient a is of the or
der of unity and the high frequency oscillation, no shock
waves is formed in the system. Details of the solution for
a O(c) and the low frequency waves will be shown at
the conference.
Acknowledgements
This work is supported in part by the Japan Society for
the Promotion of Science, GrantinAid for Scientific
Research (A) 21246031. The authors would like to ex
press their deepest gratitude to this grant.
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