European Congress on Computational Methods in Applied Sciences and Engineering
ECCOMAS 2000
Barcelona, 11-14 September 2000
c
ECCOMAS
NUMERICAL STUDY OF NONEQUILIBRIUM
CONDENSATION IN UNSTEADY TRANSONIC VISCOUS
FLOW PROBLEM
Satoru Yamamoto
Dept. of Aeronautical and Space Engineering, Tohoku University,
Sendai 980-8579, Japan
e-mail: [email protected],
web page: http://www.caero.mech.tohoku.ac.jp/
Key words: Nonequilibrium Condensation, Transonic Flow, Unsteady Flow
Abstract. The purpose of the present paper is to propose a higher-resolution numerical
method for simulating unsteady subsonic and transonic viscous flows in wet or steam
condition. The fundamental equations contain the compressible Navier-Stokes equations
and the model equations for vapor and liquid phases. The classical condensation theory is
implemented to model the nonequilibrium condensation. These equations are solved using
the fourth-order accurate compact MUSCL TVD scheme and the maximum second-order
implicit scheme.
As numerical examples, unsteady transonic viscous flows around the 2-D airfoil in
atmospheric wind tunnel conditions are first calculated. The present method is also applied
to the calculation of unsteady turbine stator-rotor interactions in wet-steam condition.
1
S.Yamamoto
1
INTRODUCTION
A number of numerical studies of 2-D transonic wet-steam flows through a steam
turbine channel have already been reported[1]-[3]. In these studies, the Euler equations
are solved using the time-marching method, and the approximation of nonequilibrium wetsteam gas integrates the growth of droplets along each stream line. Unsteady transonic
inviscid and viscous flows of moist air through the nozzle have been also calculated[4]-[7].
Recently, Ishizaka et al. calculated transonic turbulent flows through the steam turbine cascade using a high-resolution finite-difference method[8]. This method was applied to 2-D and 3-D transonic viscous flows around the airfoil under the atmospheric
wind tunnel conditions[9]-[10] coupled with the shock-vortex capturing method[11]. This
method contains the fourth-order compact MUSCL TVD(FCMT) scheme and the maximum second-order implicit diagonal approximate-factorization scheme. In the present
study, the condensation model[8] and the shock-vortex capturing method is extended
to a higher-resolution method for solving unsteady subsonic and transonic viscous flow
problems.
As numerical examples, unsteady transonic viscous flows around the NACA0012 airfoil at high angle of attack in atmospheric wind tunnel conditions are calculated. Also
unsteady turbine stator-rotor interactions in wet-steam condition are calculated changing
the degree of superheat.
2
NUMERICAL METHODS
The fundamental equations are composed of the 2-D compressible Navier-Stokes equations and the mass conservation equations of vapor, liquid, and number density of droplet,
in general curvilinear coordinates. The homogeneous multi-phase fluid without velocity
slip is assumed. These equations are written by
∂Q/∂t + F (Q) =
∂Q ∂Fi
+
+S+H =0
∂t
∂ξi
(1)
where

Q = J












ρ
ρu1
ρu2
e
ρv
ρβ
ρn








,

















Fi = J
ρUi
ρu1 Ui + ∂ξi /∂x1 p
ρu2 Ui + ∂ξi /∂x2 p
(e + p)Ui
ρv Ui
ρβUi
ρnUi
2







,






S=




∂ξi ∂ 


−J
∂xj ∂ξi 




0
τ1j
τ2j
τkj uk + (κ + κt )∂T /∂xj
0
0
0













S.Yamamoto

H = −J












0
0
0
0
−Γ
Γ
ρI







,





(i = 1, 2)
t, xi , ξi , ρ, ui , and e are the time, the components of the cartesian coordinate, the components of the general curvilinear coordinate, the density, the cartesian components of
the physical velocity, and the total internal energy per unit volume, respectively. Ui ,
T , κ, κt , and J are the components of the contravariant velocity, the temperature, the
heat conductivity, the eddy heat diffusivity, and the Jacobian, respectively. ρv , β, n, Γ,
and I are the density of vapor, the mass fraction of liquid phase, the number density of
droplet, the mass generation rate due to condensation, and the homogeneous nucleation
rate, respectively. The components of the stress tensor, τij , are defined by
τij = (µ + µt )[(
∂ui ∂uj
2 ∂uk
+
) − δij
]
∂xj
∂xi
3 ∂xk
where µ, µt and δij are the molecular viscosity, the eddy viscosity, and the Krónecker delta,
respectively. The molecular viscosity µ is derived from the linear combination between
that of gas phase and that of liquid phase using the mass fraction β. The eddy viscosity
is evaluated by the modified Baldwin-Lomax turbulence model[12] or the low Reynolds
number k − ε model[13].
The equation of state and the speed of sound derived by Ishizaka et al.[8] are also used
herein. These equations are written by
p = ρRT (1 − β)
Cpm
p 1/2
]
c = [
Cpm − (1 − β)R ρ
(2)
(3)
where
R = (
ρa Ru
ρv Ru
+
)
ρg Ma ρg Mv
ρa and Ma are the density of dry air and its mass per molecule, ρv and Mv are those of
vapor, and ρg is the density of gas phase. Ru is the universal gas constant. Cpm is defined
by the linear combination of the specific heat at constant pressure between the gas phase
and the liquid phase using the mass fraction β.
The mass generation rate Γ assuming nonequilibrium condensation is modeled using
the classical condensation theory. It is written in the sum of the homogeneous nucleation
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and the mass increment according to the growth of the droplet[14] as
dr
4
Γ = πρ Ir∗3 + 4πρ nr 2
3
dt
(4)
r and r∗ are the droplet radius and the critical droplet radius defined by Kelvin-Helmholtz,
respectively. The nucleation rate I is referred from Frankel[15] and the surface tension is
evaluated by Young[16]. The growth of the droplet is calculated from the Hertz-Knudsen
model[17]. In the second term of Eq.(4), the simple form of the growth of the droplet
dr/dt defined by Young[18] is used.
3
3.1
NUMERICAL RESULTS
NACA0012 airfoil at high angle of attack
The unsteady 2-D transonic viscous flows around the NACA0012 airfoil at 15◦ angle
of attack in atmospheric wind tunnel conditions are calculated. The computational grid
forms the overset grid composed of a C-type grid around the airfoil(121 × 41 grid points)
and a rectangular grid for the flow field(121 × 101 grid points). As flow conditions, the
uniform flow Mach number and the Reynolds number(dry air) are specified to 0.8 and
1.377 × 106 .
The relative humidities, Φ = 0%(dry air) and Φ = 60%, are considered herein. Figures
1(a)(b) show the calculated instantaneous vorticity contours. Vortices generated periodically behind the airfoil are clearly seen in both figures. These vortices are first generated
by the shock/boundary layer interaction on the airfoil upper side, and are being merged
according to the movement. In the case of 0% humidity, another larger structure of vortex
is being constructed far behind the airfoil, while vortices are being broken and merged in
the case of 60% humidity.
Figure 2 shows the contours of the calculated instantaneous condensation mass fraction
of the liquid phase in the case of 60% humidity. The result indicates that droplets are
first generated near the airfoil. Additional droplets are also produced in the wake region.
Finally both droplets are mixed and streaming downward as liquid phase. The generation
of droplets results in the disturbance at the wake region.
3.2
Turbine stator-rotor cascade
Unsteady 2-D transonic flows through the turbine stator-rotor cascade channels are
calculated. The stator and the rotor blades are developed in Toshiba corporation for a
steam turbine cascade. The pitch ratio is fixed to 3:2 herein. Two passages in stator
and three passages in rotor are simultaneously solved. The grid points of 151 × 121 and
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201 × 121 are generated in each stator and rotor passage. A modified H-type grid system
is introduced to improve the grid orthogonality in each inlet and outlet region.
Wet-steam flows are calculated. The flow conditions specify that the inlet total pressure
is 7.15 × 104[Pa], the inlet Mach number is 0.181, the Reynolds number is 1.0 × 105[1/m],
and the outlet static pressure is 5.56 × 104 [Pa]. Four different degrees of superheat, 8◦ C
(CASE 1), 4◦ C (CASE 2),0◦ C (CASE 3), and −4◦ C (CASE 4) at the inlet, are considered.
At first, the space-averaged boundary conditions are adopted at the connecting boundary
between the stator and the rotor. Figures 3(a)-(d) show the calculated instantaneous mass
fraction contours of liquid phase. An expected difference is observed in these figures. The
liquid phase appears at the downstream regions both in the stator and the rotor passages.
In accordance with the decrement in the degree of superheat, the region with the liquid
phase is obviously spreading. Especially in Fig.3(d), the liquid phase are widely observed
in the stator passages. It maybe due to the flow expansion near the suction side of the
stator blade. In the rotor passages, the wet flow is observed not only at wake region but
at the region close to the rear side of the rotor blade.
Next, the time-dependent boundary conditions are adopted at the connecting boundary
between the stator and the rotor. Only the result of the degree of superheat −4◦ C(CASE5)
is shown herein. Figure 4 shows the calculated instantaneous mass fraction contours of
liquid phase. The pattern of the contours is different in each passage due to the timedependent calculation. In the stator passages, these patterns are qualitatively in good
agreement with the CASE4. However, the amount and the location of the liquid in the
rotor passages is quite different with the CASE4. It might be due to the unsteady wakes
from the stator passages.
4
CONCLUDING REMARKS
A higher-resolution numerical method for simulating unsteady transonic viscous flows
in wet or steam condition were applied to the steam turbine stator-rotor cascade flows.
The calculated results suggest that the nonequilibrium condensation occurs not only
around the airfoil or the cascade blade but in those wake regions. The condensation
in the wake region could be captured only by the unsteady flow calculation, because
the nucleation and the condensation are associated only with the low pressure in each
unsteady vortex. In the turbine stator-rotor cascade case, the onset of the condensation
is successfully demonstrated at different degrees of superheat.
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S.Yamamoto
References
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14. Moses, C.A. and Stein, G.D., 1978, “On the Growth of Steam Droplets Formed in a
Laval Nozzle Using Both Pressure and Light Scattering Measurement,” Transaction
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(a) 0% humidity
(b) 60% humidity
Fig.1 Instantaneous vorticity contours
Fig.2 Instantaneous condensation mass fraction contours of liquid phase(60% humidity)
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(a) CASE 1
(b) CASE 2
(c) CASE 3
(d) CASE 4
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Fig.4 Instantaneous mass fraction contours of liquid phase(CASE5)
9