The main characteristics and the potential advantages of generalized drift flux models are presented. In particular it is stressed that the issue on the propagation properties and on the mathematical nature (hyperbolic or not) of the model and the problem of closure are easier to tackle than in two fluid models. The problem of identifying the differential void-drift closure law inherent to generalized drift flux models is then addressed. Such a void-drift closure, based on wave properties, is proposed for bubbly flows. It involves a drift relaxation time which is of the order of 0.25 s. It is observed that, although wave properties provide essential closure validity tests, they do not represent an easily usable source of quantitative information on the closure laws

Boure, J. A.

English

NASA Technical Reports Server

N90-10386
THE GENERALIZED DRIFT FLUX APPROACH:
IDENTIFICATION OF THE VOID-DRIFT CLOSURE LAW
J. A. Bour_
Commissariat _ l'Energie Atomique
Centre d'Etudes Nucl_aires de Grenoble
Service d'Etudes Thermohydrauliques
85 X, F 38041, Grenoble Cedex, France
https://ntrs.nasa.gov/search.jsp?R=19900001070 2020-03-20T01:05:55+00:00Z
ABSTRACT
The main characteristics and the potential advantages of
generalized drift flux models are recalled. In particular it is
stressed that the issue on the propagation properties and on the
mathematical nature (hyperbolic or not) of the model and the
problem of closure are easier to tackle than in two-fluid models.
The problem of identifying the differential void-drift closure law
inherent to generalized drift flux models is then addressed. Such
a void-drift closure, based on wave properties, is proposed for
bubbly flows. It involves a drift relaxation time which is of the
order of 0.25 s.
It is observed that, although wave properties provide essential
closure validity tests, they do not represent an easily usable
source of quantitative information on the closure laws.
I. INTRODUCTION
Generalized drift flux models were recently shown (Bour4, 1988b)
to be attractive alternatives for current I-D two. fluid models.
Drift flux models are characterized by the uses of a single
momentum balance (the mixture balance instead of two phasic
balances) and of a void-drift closure law. In classical drift flux
models the void, drift closure law is expressed through an
algebraic equation, which amounts to ignoring nonequilibrium drift
effects. In generalized drift flux models, the void-drift closure
equation is a partial differential equation.
Generalized drift flux models and two-fluid models are compared in
the next section and in table i. The comparison brings out the
drawbacks of two-fluid models. However, both kinds of model must
be complemented by closure laws. In particular, generalized drift
flux models need a void, drift closure law which remains to be
specified.
Since generalized drift flux models were introduced to account for
the properties of kinematic waves (Bourn, 1988a), it seems
logical to use these properties to identify (i.e. to evaluate the
coefficients of) the void-drift closure equation. The purpose of
the present paper is to discuss the identification problem,
assuming that the void-drift closure equation may be approached by
a quasi linear differential equation of the first order.
II. A REMINDER ON GENERALIZED DRIFT FLUX MODELS AND TWO-FLUID
MODELS
The comparison between generalized drift flux models and two-fluid
models is summarized in table i which, like must of the substance
of this section, is taken from Bour_ (1988b). The mass and energy
balances are parts of both kinds of models, and they are not
discussed further hereafter. It is only noted for completeness
that they require a few closure equations, in particular for the
mass and energy transfers at the walls and at the interfaces.
The momentumbalances contain the two phasic averaged pressures PG
and PL' the subscripts G and L corresponding to the two phases.
For the following discussion, it is convenient to express PGand
PL in terms of the average pressure P and the pressure difference
PLG' with :
P _ _ PG + (I - _) PL ' PLG - PG - PL (2.1)
being the void fraction. Now, in the set of balance equations,
the two phasic momentum balances are equivalent to a subset of two
equations, namely the mixture momentum balance and the pressure
difference equation, obtained on eliminating P between the two
phasic balances.
The mixture momentum balance may be written :
OR O_r
+ bz + 8z + MLG (wG WL ) I_ + Fw + Pg - 0
(2.2)
the terms of the second line representing respectively :
a term accounting for fluctuation and transverse distribution
effects
a term accounting for longitudinal stress variations
an interfacial mass transfer term
a surface tension term (interfacial)
- a friction term (walls)
- a gravity term.
t and z are respectively the time and the space variables, OG and
@t are the phase densities, W G and W t are the phase average
velocities, g is the gravity acceleration, and :
p _ o_ PG + (I - o_) PL (2.3)
w being the local instantaneous velocity along Oz and Y the local
instantaneous deviatoric stress tensor, indicating the
conditional time or ensemble averaging operator and < > the
space averaging operator, and D being the unit vector of the Oz
Z
axis, R and T are defined as •
G L
R _ <_ or,(wG - %)2 + (1- =) _, (wL _ WL)2> (2.4)
=- _ + (l-u) r
-G -L _nz> • _nz (2.5)
Independently of the mass transfer, already present in the mass
balances, eq. 2.2 requires four closure laws for R, T, I_ and Fw.
Turbulence effects are present through R.
The pressure difference equation may be written :
[ (_ WG _ WG 1 _ WL _ WL I _ PLG
+ I_ + L_ + MLG [(I - _) (W G - WGI ) + _ (W L - WLI) ] (2.6)
+ L I + LFW + LFI - O_ (1 - (Z) PG L g - 0
the terms of the second and third lines representing respectively •
a term accounting for fluctuation and transverse distribution
effects
a term accounting for longitudinal stress variations
an interfacial mass transfer term (WG; and WL] are interfacial
averages of the phasic velocities)
an induced inertia ("added mass") term
two friction terms (respectively wall and interfaces)
- a gravity term, with
QGL _ PL - PG (2.7)
The above terms are defined in Bour_ (1988b). For instance :
5
a G a L
(2.8)
L_ _ - (l-<x) _z _ 9:G" nz n_z + C_ _z (i-o_) T:L" nz n_z (2.9)
Equation (2.6) requires seven closure laws for I_, L_, WGI,
WLI , Ll , LFW , LFI. Moreover, retaining eq. (2.6) implies, at least
in theory, retaining also the interfacial momentum balance, which
requires one supplementary closure law.
In two-fluid models, eqs (2.2) and (2.6) are both implicitly
written. A crucial point is that eq. (2.6) is stiff. This means
that, due to the relative orders of magnitude of its terms, small
variations on the closure laws (and in particular on the induced
inertia and interfacial friction closure laws) induce large
variations on PLG (when PLG is not merely ignored) and/or W G-
Hence the difficulty to adjust the very badly known closure laws
of eq. (2.6) to avoid unrealistic values of the light phase
velocity.
A second crucial point is that, whenever eq. (2.6) is closed by
algebraic laws only, a non-hyperbolic set results : two
characteristic velocities are complex conjugate, with the
consequence that the model is unconditionally unstable.
In drift flux models, eq. (2.6) is not written, which is
acceptable since the value of PLG does not really matter in
practice. The two foregoing difficulties are not encountered.
Besides the closure laws already mentioned, current two-fluid
models are closed through an assumption on PiG (the set of
6 phasic balance equations involves 7 dependent variables, namely
two pressures, two velocities, two enthalpies and the void
fraction). Such an assumption imposes an artificial constraint on
the pressures and pressure gradients. It disturbs the description
of the corresponding propagation phenomena.
Generalized drift flux models are closed through the direct
description of the void-drift dynamic dependency. Assuming a
quasi-linear, partial differential relationship of the first
order, and using the convenient variables :
W _ _ W G + (i - _) WL, & -_ _ (I - _) (W G - WL) = _ (W s - W) (2.10)
(center of volume velocity and drift), it may be approached by :
+ wz + (w + w4 - Z) S--i+(ww4 - H) a--_
86 86 I
-- --= -- (f - 6) (2.11)
+ 8t + W4 8z 8
Equation (2.11) requires seven closure laws, i.e. the same number
as eq. (2.6) but with simpler physical significances and more
straightforward consequences : f is the fully-developed drift
value, 8 a relaxation time, Z and _ are respectively the sum and
product of the two characteristic velocities corresponding to the
kinematic waves, _ expresses inertia effects, W 2 and W4 are
averaged velocities close to W.
It can be concluded that developing a closure set for eq. (2.11)
and using generalized drift flux models appear as less hazardous
and hopefully easier than developing a closure set for eq. (2.6)
and using current two-fluid models.
llI. INFLUENCE OF THE VOID-DRIFT CLOSURE ON THE PROPERTIES OF
KINEMATIC WAVES ("Direct problem")
As long as the kinematic wave velocities are small with respect to
the sonic velocity, the properties of small harmonic kinematic
waves of the form
x - x0 ei ([ot - kz) (3.1)
where x0 (constant) [o and k are real or complex quantities, result
from the approximate dispersion equation (Bourn, 1988b)
(0 - Cc_k + i O ([o2 _ Y- [ok + ]'[k 2) = 0 (3.2)
where •
= W + _, = (3.3)
A first consequence is that the kinematic wave properties, which
do not significantly depend on _, W2, W4, cannot be used to
evaluate these quantities (_ is related to the sonic velocity, W 2
and W 4 have only weak influences).
In the exploitable experiments (Tournaire, 1987, Bourn, 1988a) [o
is imposed and the kinematic wave velocities V and their spatial
amplification coefficients k i result from the data processing.
Eqs. (3.1) and (3.2) must therefore be used with [o real and :
k ---qk r + i k i (3.4)
from which
- - k i z i (0)t - krZ)
x - x 0 e e (3.5)
[o
V =- (3.6)
k r
kr, which does not depend on the frame of reference, can be used
instead of _, which does, to characterize a wave.
Introducing eqs. (3.4) and (3.6) in the dispersion equation (3.2)
and separating the real and imaginary parts lead to •
k r [V - Ca + e Z V k_ - 2 e II k i ] - 0 (3.7)
Ca k i + e V2 2 _ e _ v 2 2- kr k r + e II (k2r - ki) -- 0 (3.8)
Equations (3.7) and (3.8) provide the solutions to the direct
problem, viz. computing the properties of the kinematic waves of
wave number kr when C_, 8, _ and R are known. There are two
solutions corresponding to two modes (noted with the subscripts 3
and 4). In the experiments it was found that modes 3 and 4 are
respectively predominant at low void fractions (0 < _ < 0.25) and
at "large" void fractions (_ > 0.30)
In particular for kr = 0, the two solutions are •
k i e (Ca - C3) (C_ - C4)
k i = 0 with -- = , V = Ca (3.9)
k_ Ca
and
C_ I i I I
t
ki - e H r V = Ca with , _ + (3.10)
Ca C3 C4 Ca
C3 and C4 being defined by :
Z _ C3 + C4 II _ C3 C4 (3.11)
For k r _ _, the two solutions are :
C_ - C3
V = C3 k i = - (3.12)
e c3 (c4 - c3)
C4 - C_
V = C4 k i = - (3.13)
e c4 (c4 - c3)
9
In equations (3.7) and (3.8) the three quantities @ kr, k i are
present only through the two products (@ k r) and (@ ki). In actual
experimental runs, k r is never zero and the equations may be
written '
V - C_ + (C 3 + C 4) V (B k i) - 2 C 3 C 4 (Q k i) = 0 (3.14)
- C_
@ k i (@ ki )z
(@ k r)2 + V2 - (C3 + C4 ) %' + C3 C4 - C3 C4 (B k r)2
- 0 (3.15)
Since in the exploitable
significantly depend on k r
convenient to eliminate V
(3.14) yields :
experimental data, V does not
(no r significant dispersion), it is
between eqs. (3.14) and (3.15). Eq.
V --
C_ + 2 C3C 4 (B k i)
I + (C 3 + C 4) (8 k i)
(3.16)
and eq. (3.15) yields :
V --
C3 + C 4 J(C 4 - C3 )2 @ k i (e k i)2
__ _ + C_ )2 + C3 C42 4 (@ k r (8 k r )2
(3.17)
The solutions for @ k i are then the real solutions of the
equation
2 C_- (C 3 + C 4) - (C 4 - C3)2 (B k i)
i + (C 3 + C 4) @ k i
J @ ki (@ ki )2
+ (C 4 _ C3)2 + 4 Cc_ + 4 C 3 C4
(8 k r )Z (e k r )2
(3.18)
Computing directly @ k i as a function of @ k r from eq. (3.18) is
not straightforward. It is more convenient to transform eq. (3.18)
to express e k r as a function of 8 k_
10
2
(e k r )
D k_
[C_ + C3 C4 @ ki] [i + (C 3 + C 4) @ k i ]2
(Co- C3 ) (Co- C4 ) - (C 4- C3)28 k i [Ccx+ C3C 4 0 ki]
(3.19)
For a given value of ki, eq. (3.19) yields zero or one real
positive value of k r.
IV. IDENTIFICATION OF THE VOID-DRIFT CLOSURE EQUATION FROM
KINEMATIC WAVE PROPERTIES ("Inverse problem")
The problem posed in this paper is the determination of C_, 0, Z
and _, using the experimental data on kinematic waves. It is the
inverse of the problem of section 3.
The principle of the method is to write eqs. (3.7) and (3.8), for
instance, for several sets of experimental conditions for which
kr, V and k i are known and to use the resulting equations to
compute C_, 8, Z and R.
In the exploitable data, as already noted, V does not depend
significantly on k r . Accordingly, when a single mode is
predominant, V may be expected to be close to both C 3 (or C 4) and
C_ (or C_). This has two consequences :
i. Whenever a single mode is predominant, the experimental
correlation for V should be a correlation for C_ as well. This is
corroborated by the fact that is satisfies the definition (3.3).
In the experimental conditions of Tournaire (1987) (upward
vertical flow, low pressure), it leads to (Bourn, 1988a, f and
C_ - W in m/s) :
For 0 < _ < 0.2 (mode 3 predominant)
f = 0.22 _ (I - =) [i - 1.25 _ (i - _)]
J
C_ - W = 0.22 (I - 2_) [I - 2.5 _ (I - _)]
(4.1)
II
For 0.3 < _ < 0.41 (mode 4 predominant)
f = 0.22 _ - 0.028_
fC_ -W= 0.22
(4.2)
For 0.2 < _ < 0.3 (the two modes coexist) the values of f and
C_ - W may be interpolated between (4.1) and (4.2) with, from the
experimental data •
for _ : 0.25
C_ - W -_ 0.08 (4.3)
2. When mode 3 (respectively mode 4) is predominant, C_ - C3
(respectively C4 - C_) should be "small"
C_ being known and V eliminated, the problem may now be
reformulated, eqs. (3.7) and (3.8) being replaced by eq. (3.18) or
(3.19) to be solved for e, C3, C4. Only mode 3 results are
exploitable since mode 4 results for k i are too few in number. In
view of the forms of eqs. (3.18) and (3.19), the foregoing problem
is far from simple. This is confirmed by fig. i in which, as
suggested by eq. (3.19), the experimental results for - k_/k i are
plotted as a function of - ki, and which exhibits an important
scatter (in the representation of fig. i, the points corresponding
to Iki I < 0.i are subject to large errors and therefore
meaningless. They were not plotted in the figure).
In the domain in which mode 3 is predominant, the conditions :
#
C3 < C_ < C_ < C4 (4.4)
may be expected to hold (see Bour@, 1988a). They entail
C_ - C3 i C4 - Ca Ca
< < <
8 c3 (c4 - c3) 8 (C] + C4) 0 C4 (C4 - C3) e c3 c4
(4.5)
12
Then,
starts
which
kr2/ki
as k i decreases from zero, k_/k i resulting from eq. (3.19)
from the value corresponding to eq. (3.9) with a slope
may be of either sign. However, when Iki I is sufficient,
also decreases. It tends towards - _ for the value
corresponding to eq. (3.12). For the values of k i comprised
between those given by eqs. (3.12) and (3.13), there is no
physical solution (k2r _< 0). Finally, for the values of k i
comprised between those given by eqs. (3.13) and (3.10), there is
a solution again corresponding to mode 4.
By trial and error, 8, C 3 and C 4 may be adjusted to fit the curve
representing eq. (3.19) to mode 3 data. In view of the absence of
dispersion C 3 may be expected to be close to C a.
For 0 < _ _ 0.20 : the following set of values is acceptable
8=0.25 s 1Ca C3 = 0.02 m/s
C4 Ca = 0 08 m/sJ
(4.6)
Precise adjustement would need more accurate data for the wave
velocities and especially for the damping/amplification
coefficients. Such data does not exist and cannot be expected to
be obtained soon in view of the available instrumentation. For
> 0.25, no sufficiently accurate data is available.
Equations (4.1) and (4.6) confirm that, even at low pressure, the
relevant velocity differences corresponding to fully developed
conditions are fairly small in bubbly flows. They are negligible
as soon as W is large enough (say 3 m/s). Accordingly the
correlations for Ca - W, C 3 - W, C 4 - W are probably not crucial.
On the other hand, the drift relaxation time 8 is an essential
parameter of the generalized drift flux model.
13
V. CONCLUSIONS
After a reminder on the main characteristics and the potential
advantages of generalized drift flux models, the problem of the
identification of the differential void-drift closure law they
imply has been addressed.
Two advantages of generalized drift flux models versus complete
two-fluid models are :
i. The correct description of the kinematic wave phenomena and the
straightforward control and interpretation of the mathematical
nature (hyperbolic or not) of the model set of partial
differential equations that they enable.
2. The simplification of the closure problem, involving only
closure laws of simple physical significance and easy to assess.
On the other hand, it has been found that the available data on
kinematic wave properties is not quite adequate to enable the
identification of the void-drift closure law. Such an
identification would need a very good accuracy (difficult to reach
in practice) on the wave damping or amplification coefficients.
Accordingly kinematic wave properties seem to be more useful as a
closure validity test than as a source of quantitative information
on the closure laws.
However, a void-drift closure, suitable for bubbly flows, has been
adjusted on the available kinematic wave data. It involves several
velocities which differ only slightly from each other and from the
average fluid velocity but whichare necessary to the description
of the kinematic wave properties. It also involves a drift
relaxation time which is an essential parameter of generalized
drift flux models and which was tentatively found to be of the
order of 0.25 s for the upward flow of air-water mixtures at low
pressure.
14
Table '1
Simplified comparison of modeling strategies
D_ift flux approach Common features Two-fluid approach
I Void-drift closure eq.
(partial differential eq.)
÷
Closure laws :
fully.developed drift f
Drift relaxation time e
Kinematic wave
properties (C 3, C4)
Dynamic wave prop. (r1)
Balance equations •
I Mixture momentum
2 Phasic mass
2 Phasic energy
÷
Closure laws :
Bulk terms
Wall: Momentum, (Mass],
Energy transfers
Interface • Mass and
Energy transfers
t
Generalized drift flux models
(PLG not calculated) il
Pressure difference eq.
(Balance eq.)
÷
Closure laws •
Bulk terms
Wall terms (frictlonJ
Interfaclal terms, incl.
friction & Induced inertia
Problem : Stiff equation
J Assumption on PLG J
Optionally
I Pressure difference eq. J+ Clo laws (as above)
I
Complete two-fluid models JJ
(PLG calculated) Jl II Current two-fluid models(PLG assumed)
15
00
8
X
e
8
X
e
&
13
0
$ °e
0
e
O
e
II
A
A+
X
X
0
A
÷
X
+ = = 0.047
x = = 0.075
& = = 0.123
o = = 0.191
o = = 0.255
o = = 0.306
X
-ki (m -I)
Fig. I: Experimental data (Mode 3)
18
REFERENCES
Bourn, JoA., 1988a, Properties of kinematic waves in two-phase
pipe flows. Consequences on the modeling strategy, European
Two-Phase Flow Group Meeting, Brussels.
Bourn, J.Ao, 1988b, An alternative strategy for the development of
averaged two-phase flow models, Seminar on Theoretical Aspects of
Multiphase Flow Phenomena, Ithaca.
Tournaire, A., 1987, Detection et _tude des ondes de taux de vide
en _coulement diphasique _ bulles jusqu°_ la transition
bulles°bouchons, Th_se, Universit_ Scientifique, Technologique et
M_dicale et Institut National Polytechnique de Grenoble.
I?



The generalized drift flux approach: Identification of the void-drift closure law

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