A multi-temperature multiphase flow model.
In this paper we formulate a multiphase model with nonequilibrated temperatures but with equal velocities and pressures for each species. Turbulent mixing is driven by diffusion in these equations. The closure equations are defined in part by reference to a more exact chunk mix model developed by the authors and coworkers which has separate pressures, temperatures, and velocities for each species. There are two main results in this paper. The first is to identify a thermodynamic constraint, in the form of a process dependence, for pressure equilibrated models. The second is to determine one of the diffusion coefficients needed for the closure of the equilibrated pressure multiphase flow equations, in the incompressible case. The diffusion coefficients depend on entrainment times derived from the chunk mix model. These entrainment times are determined here first via general formulas and then explicitly for Rayleigh-Taylor and Richtmyer-Meshkov large time asymptotic flows. We also determine volume fractions for these flows, using the chunk mix model.
Introduction
Requirements of theromodynamic and mathematical consistency impose limits
on possible multiphase flow models. The length scales on which the mixing
occurs impose further restrictions, through limits on the range of physical
validity of these models. The models are distinguished by the dependent variables
allowed in the description of each fluid species. The thermodynamic,
mathematical and physical restrictions act to limit the freedom in the choices
of these dependent variables, or the flow regimes to which the models apply.
Two models for compressible mixing which meet the standards of thermodynamic
and mathematical consistency are (a) chunk mix and (b) molecular
mix. In the former, each fluid has distinct velocities and a full thermodynamic
description. In the latter, all fluid and thermodynamic variables are
shared, with a common value for all species. Only a volume or mass fraction
gives individual identity to the species.
We are mainly interested in models which describe fluid mixing layers. A
layer, by definition, has a small longitudinal dimension relative to its transverse
dimensions, and is thus quasi two dimensional in the large. For this
reason, there is an inverse cascade of enstrophy, with growth of large scale
structures. This growth is dominated by the merger of modes to form larger
modes. The number of mode doublings in a typical experiment [14, 17, 6]
is difficult to quantify precisely since the initial disturbance is not well char-
acterized, but a range of 3 to 5 doublings might be expected. Over this
range, i.e., within the experiments, a steady increase in structure size, and
adherence to resulting scaling growth laws for the mixing zone thinkness are
observed. To the extent that scaling laws and the regular doubling of structure
size persist, the chunk model appears to be an accurate description of
the mixing process.
Scaling laws and large scale structures go hand in hand. Among the events
which could disrupt these structures, we mention turbulence and shock waves.
If the large structures shatter, so that the structure size is fine scale and
somewhat uniformely dispersed, the flow regime will change in character to
one dominated by form drag and thermal conductivity. This regime will thus
have equilibrated pressures and temperatures, and will be well characterized
by the molecular mix models.
Data by Dimonte et al.[6] shows two dimensional slices through a mixing
layer, based on lasar illuminesence with indexed matched fluids. These
experiments show a wealth of small scale structures in a period for which
scaling laws still hold. However in this data, the structures at the edge of the
mixing zone have not broken up, an appearently important fact in explaining
the persistence of scaling laws.
The appeal of intermediate models, for example with common pressures
but distinct temperatures, lies in a hope for an optimal balance between
smplicity and fidelity to details of physical modeling. It has long been understood
that common pressures require common velocities or a high degree
of regularization of the velocities to ensure mathematical stabiliity (avoidance
of complex eigenvalues for the time propagation) [16].
The purpose of this paper is twofold. Our first main result is to identify
a thermodynamic constraint on equal pressure mix models. Equal pressure
models require the specification of a thermodynamic process or path along
which pressure equilibration is achieved or maintained. In practice, specification
of this path appears limited to certain relatively simple flow regimes.
One such case is that of incompressible flow and another is the case in which
all but one of the phases are restricted to be isentropic.
A consequence of the equal velocities required of equal pressure models
is that dispersive mixing must be modeled by a turbulent diffusion term.
Our second main result is to derive the required diffusion term, particularly
the diffusion coefficient, from the closed form solution of the more complete
chunk mix model, in the incompressible case.
The chunk mix model, with distinct pressures and velocities for each
phase, and with distinct temperatures in the compressible case, has been
studied in a series of papers [7, 9, 8, 10, 4]. Other authors have also considered
two pressure, two temperature two phase flow models [16, 11, 13]. Our
work goes beyond these references in several respects, including (a) a zero
parameter closure in agreement with Rayleigh-Taylor incompressible mixing
data and (b) closed form solutions, also for the incompressible case. Two
pressure closures based on drag are discussed in [18, 19].
Diffusion in the context of fluid mixing has been considered by many
authors [12]. Work of Youngs [18, 19] can be discussed in relation to the
acceleration driven mixing we consider here. The diffusion in [18, 19] derives
from a second order closure (k-ffl model) and requires new equations (k
and l equations) and parameters. Cranfill [5], in the same spirit, considers
a diffusion tensor, rather than scalar diffusion. This tensor is related to a
phenomenological drag frequency (defined by a k-ffl model) and the disordered
Reynolds and viscous stresses, and again requires additional equations
and parameters. Cranfill distinguishes ordered from disordered flow in his
approach. Shvarts et al. [1] propose diffusion as a model for velocity fluctuation
induced mixing. Our derivation of the dispersion closure differs from
[1] in several respects.
The chunk mix model and the diffusion model proposed here have as input
the velocities or trajectories of the edges of the mixing zone. Beyond this the
chunk mix model has zero (incompressible) or one (compressible) adjustable
parameter. We regard it as more accurate than the two temperature diffusive
mixing model proposed here and, for this reason, we use the chunk mix
model as a reference solution from which closure relations for the diffusive
two temperature model are derived.
Section 2 will develop a multi-temperature, multi-species thermodynamics
with equilibrated pressures, on the basis of an assumed EOS for each
species and no energy of mixing. The critical role of a thermodymanic process
for pressure equilibration, to complete the closure of these equations will
be stressed. Section 3 will develop model equations and closure expressions.
Section 4 will determine the coefficients in the diffusion closure relations.
Section 5 determines the Reynolds stress tensor. Conclusions are stated in
Section 6.
Thermodynamics
We assume that an equation of state ffl for the specific
internal energy of each species, We seek a composite EOS for
the n-species mixture which has achieved mechanical equilibrium (p k
for all k), but not thermal equilibrium. We assume no energy of mixing, so
that the total system specific internal energy density ffl satisfies
and
1.
A microscopic physical picture to describe this set of assumptions would
be a container consisting of n chambers separated by thermally
insulating frictionless moving partitions. Each chamber exerts pressure
forces on its two neighbors through the partition, and at equilibrium, each
chamber has expanded or contracted to achieve equal pressures. We argue
that this picture is an approximate description of the thermodynamics of
pressure equilibrated chunk mix.
Under the chunk mix assumption, we postulate chunks large enough that
the bulk energies and other thermodynamic quantities dominate surface ef-
fects, so that infinite volume thermodynamics applies within each chunk.
Thus we start with a system defined by 3n \Gamma 1 independent thermodynamic
variables, consisting of volume fractions fi k and two independent
thermodynamical variables per species. Within this space, we define a mechanical
equilibrium subspace, defined by equal pressures (n \Gamma 1 constraints),
and thus defined by volume fractions, a common pressure, and one additional
thermodynamic variable per species (e.g., S k
, or ae k
variables for the kth species are determined from the common pressure
and this one variable, using the k species EOS. The role of the fi k
is to give
relative species concentration. Other, equivalent, thermodynamic variables
with this role include the specific mass density - k
, the mass fraction
the species number density n k and the chemical potential
. Any of the above, in combination with the k species EOS and k species
thermodynamic state, determines fi k , and any can be used in place of fi k as
independent thermodynamic variables.
In summary, we describe the mixture at mechanical equilibrium with
concentration variables, n species dependent thermodynamic variables and 1
global thermodynamic variable, or in total 2n thermodynamic variables.
From the point of view of pressure equilibration, or of the definition of
pressure as a function of thermodynamically independent variables, in an
incomeplete EOS, the 2n thermodynamic variables are not all on an equal
footing. The variables fi k ae k represent mass of species k, and must be conserved
in any thermodynamic process. Likewise total energy, ffl is a conserved
quantity. The remaining are less fundamental. In effect, any specific
choice for these variables, to be held fixed during pressure equilibra-
tion, i.e., to serve as independent variables for the definition of the common
equilibrated pressure, is equivalent to the specification of a thermodynamic
process for the equilibration.
The n+ 1 variables which are conserved for all thermodynamic processes
and the process dependent choice of the remaining the
basic conserved dependent variables for the multiphase hydro equations of
x3. The domain of validity of these equations is thus limited to processes for
which pressure equilibration is maintained with preservation of these
variables.
As an example, consider One variable can be constrained
to vary adiabatically. Starkenberg [15] adopts this point of view in modeling
the initiation of a detonation wave, with the distinguished species being the
(unburned) reactants.
As a second example, consider the nearly incompressible case. Then the
volume fractions fi k
are (approximately) conserved. Choosing these as the
remaining process dependent variables completes the description of the
pressure equilibration process.
We outline an algorithm for the determination of pressure p as a function
of the 2n independent variables, based on the nearly incompressible closure
of the pressure equilibration process. Assume we are given fi k
and ffl. The
problem is to find the individual ffl k defined by these variables, subject to a
given total ffl, and then to apply the individual species EOS to determine
We start with an initial guess for the values of the ffl k .
The guess ffl k
together with the given ae k
determines a (non-equilibrated) p k
Each p k is monotone in ffl k . Thus we increase the specific internal energy of the
species with the smallest pressure, as based on this guess, and decrease the
specific internal energy of the species with the largest specific energy, while
preserving species mass fi k ae k and density ae k . In other words we heat the low
pressure species, and cool the high pressure species under constant volume,
total energy preservation conditions. This process terminates only when
all pressures have equilibrated,
, thus defining the common
pressure
The above algorithm determines a computational method for evaluation
of the incomplete EOS needed for solution of the multiphase fluid equations
(4), (7), (9), and (14), subject to an assumption of approximate incompress-
ibility. For other pressure equilibration processes, a different set of
process dependent conserved variables should be chosen.
3 Model Equations and Closure
As microphysical equations, we consider the multifluid compressible Euler
equations. Let X k (x; t) be the characteristic function of the region occupied
by the fluid species k at time t. Let h\Deltai denote an ensemble average, and let
be the volume fraction of fluid k.
Before considering the averaged equations in detail, we discuss some principles
which guide their derivation. As usual, we expand the variables in
terms of means and fluctuations. Specifically, for the velocity v,
The fluctuation v 0 plays an important role, because
the individual species velocities v k
which drive the mixing in the
chunk mix model [8] do not occur in the diffusion model proposed here. While
we expect
of the chunk mix model to correspond to the v of the
diffusion model, v 0 must play the role of v k \Gammav, for all
enters the diffusion model only indirectly, through the correlations which
define the diffusivity. Since the velocity diffusion of species volumes fi k
is the
important phenomena driving the mixing process, the fluctuations resulting
from the (v
correlations are important and must be retained in our
analysis of closure hypotheses.
We also consider mass weighted velocities e and the fluctuation
v. With ~ v defined as
the mass per unit volume available to species k, we have haee
We have previously used [2] the approximation v k - e v k , justified by the
smallness of the density fluctuations within a single species relative to those
between two species. Here we use the approximation v 0 - v 00 . Our basis for
this approximation is that difference between these quantities is not captured
within the phenomenological model of fluctuations via a diffusion process as
developed here.
Fluctuations are neglected for species dependent thermodynamic quantities
such as ae k and the species specific internal energy ffl k j hX k ffli=fi k or
e
These quantities enter into the conservation laws either
directly, as dependent variables (ae k ) or are available as thermodynamic functions
of the conserved dependent variables (ffl k ). For this reason fluctuations
in these quantities are of less importance and a strictly first order closure in
these quantities is sufficient. Any correlation which is linear in fluctuating
quantities will vanish. Fluctuation correlations arise when averaging non-linear
terms in the equations. In particular, the Reynolds stress R, defined
in terms of velocity-velocity fluctuation correlations, will play an important
role in the momentum equation. We assume here that the pressure has also
equilibrated, and thus the individual phase pressures p k
are also missing from
our model. On this basis, one might expect the pressure fluctuations p 0 to
play an important role also. But pressure, in contrast to velocity, enters
the momentum equation linearly, so that fluctuations do not arise. How-
ever, pressure fluctuations could appear in a pressure velocity fluctuation
correlation arising in the closure of the averaged energy equation.
The microphysical equation for X k is
@t
which upon averaging yields
@t
or
@t
Closure requires a model for the fluctuation term. As is common in models
of turbulent transport, we adapt a diffusion model, so that
@t
Following conventional ideas [12] (see Sect. 4), the diffusivity tensor D k is
defined as a velocity - velocity fluctuation correlation.
Next consider the microphysical continuity equation, multiplied by X k ,
@ae
@t
r
We multiply (1) by ae and add the result to (5), to obtain
@t
which implies
@t
including closure as in (4). Within the approximations of Section 4, the
diffusion constants in (4) and (7) coincide.
Next we consider the microphysical momentum equation,
@t
On averaging, this gives
@aee v
@t
ve
with R a Reynolds stress tensor, analyzed in Section 5, and
Finally we consider the microphysical energy equation,
@t
Our thermodynamic picture is that the dynamics will propagate within each
species with conservation of species energy, and then (on a slower time scale)
equilibrate adiabatically between species to a uniform pressure state with
conservation of total system energy. Thus we start by deriving conservation
equations for the energy of the individual species; for convenience these are
written nonconservatively in terms of the internal energy alone. Thus we
multiply (10) by X k and add aeffl times (1) to obtain
@t
Upon averaging, (11) becomes
e
@t
As above, this equation is closed with a diffusion term to give
e
@t
e
e
We use these equations to determine the total internal energy effl, which in
combination with the volume fractions fi k
and species densities ae k
, is a complete
thermodynamic description of the system constrained to equal pressure
in each species. Thus we sum (13) and obtain
@t
with neglect of (p; v) fluctuation correlations, where D \Delta raeeffl j
e
In the derivation of the model equations, we have applied the closure
With this closure, the model equations (4), (7), (9), and (14) together with
the evaluation of D given in x4, of R given in x5, and the EOS are expressed
in a closed form and provide a complete description of the system. Moreover,
the model equations are symmetric regarding different materials so that this
diffusion model can be applied to a system with any number of materials.
For 1-D flow dominated by velocity dispersion induced mixing associated
with velocity dispersion along a single coordinate direction (z), R
. The diffusivity tensor reduces to D . For radially dominated
dispersion, the 1-D diffusion model equations are simplified to
@t
@h
@
@h
(D k h -
@h
@t
@
@h
@
@h
(D k h -
@h
@t
@
@h
@
@h
@h
@t
@
@h
@h
@
@h
(D k h -
@h
where h is the spatial coordinate along the longitudinal axis,
geometry and cylindrical and spherical geometries. Accordingly,
are respectively for planar, cylindrical, and spherical geometries

If the pressure equilibration closure is based on an adiabatic process in
species, then the volume fraction equations are dropped from the finite
difference equations and equations for entropy advection in these
are added in their place.
4 Evaluation of Diffusion Coefficients
4.1 Lagrangian Formulas Evaluated within Chunk Mix
Model
Diffusive modeling of the velocity - volume fraction fluctuating correlation
(v
) is a consequence of the low order of closure in the model proposed
here. Such eddy diffusivity closure is justified on physical grounds by assuming
a mixing process dominated by concentration gradients, but it is also
required by a model in which the velocity fluctuations do not appear directly.
Thus we adopt the general form of the closed equations given in x3. It remains
to evaluate the diffusion constant, D. Here we appeal to a Lagrangian
framework. Since our problem is strongly time dependent, the conventional
definition of time averaged diffusivity,
where X(t) is the Lagrangian particle displacement, must be replaced by the
incrementally defined diffusivity,
dt
Normally the Lagrangian framework is difficult to use with Eulerian field
quantities. To justify use of this framework, we refer to the special feature,
or physical assumption of flow regimes the chunk mix model is intended
to describe, and then use those assumptions in the evaluation of (20). In
the chunk mix model, the major uncertainty, or fluctuation, is the choice of
distinct species, or label, to which a Lagrangian particle belongs.
We have neglected within species fluctuations, and closed the system at first
order in terms of mean flow quantities only. Thus any flow quantity with
a subscript k, such as ae k or v k , can be regarded as deterministic, and thus
a Lagrangian diffusive mix description as well as an Eulerian chunk mix
description of a species k fluid element. It follows that the expectation in
(20), when evaluated in terms of chunk mix flow fields is simply a weighted
sum over the index k.
We now restrict to the case of materials. As a fundamental modeling
or closure asumption, we use the known analytic solutions of the chunk
mix model for incompressible flow to generate expressions for these correla-
tions. Continuing with the evaluation of (20), for 1-D flow, we have
dt
and so
\Theta

For incompressible flow,
can be further simplified to
Here z(t) is the Lagrangian path of a fluid element of species k which enters
the mixing zone at the edge at some time s k - t and moves with the
chunk mix [8] velocity
to the point z at time t.
Here k 0 is the complementary index to k, and V k
is the velocity of the
edge of the mixing zone. Observe that D vanishes at both edges of
the mixing zone, as k and the k 0 the displacement z(t) \Gamma Z k 0 (s k )
also vanishes at Z k 0 .
We t). The function s k is the entrainment time for a
fluid parcel of species k located at the position z(fi k ; t) (volume fraction fi k
at time t. Since species k is entrained through the
the entrainment location is Z k 0 (s k
t, or in other words, a particle of species k at the z
is in the process of being entrained.
From the structure of (22), we see that separate evaluation of the species
dependent diffusion constants D k is possible. For Richtmyer-Meshkov mix-
ing, the individual D k have distinct asymptotics, which can be determined
by the methods developed here.
The main difficulty to be overcome in this section is the determination
of the entrainment time s k in large time asymptotic Rayleigh-Taylor (RT)
and Richtmyer-Meshkov (RM) flows. We believe this information is of independent
interest, beyond its contribution to the computation of D. For
RT mixing, the time of entrainment for a fluid parcel of species k, located
at volume fraction fi k and time t is proportional to t with a fi dependent
coefficient which we determine in closed form. For large Atwood number RT
mixing and especially for RM mixing, the entrainment times are significantly
nonlinear, in fact sublinear in fi indicating that more. For RM mixing, most
of the entrained material is deposited into the mixing zone early in the mixing
process. This difference between RT and RM mixing reflects a decrease
in the late of loss of memory of initial conditions for RM mixing as opposed
to RT mixing. In both the RT and RM mixing cases, we thus determine
residence times for entrained material in the mixing zone.
We determine s k as a function of fi k and t by integrating the evolution
equation
ds k
@z
ds k
@z
from together with condition Here we have used
expressions dz k given in the chunk mix
model, and
which leads to
The volume fraction fi k (z; t) is determined implicitly by
Z tv   (s; fi k )ds: (28)
Thus we have
@z
Z t@v
ds
Z tds
We consider two distinct flow regimes for the determination of D. The
first is self similar Rayleigh-Taylor (RT) mixing, and the second is the time
asymptotic Richtmyer-Meshkov (RM) mixing. We assume that the RM mixing
is governed by unequal power laws for the two edge velocities, Z k -
4.2 Diffusive Modeling of Self-Similar Rayleigh-Taylor
Mixing
Now we specialize to RT mixing. For self-similar flow,
Substituting
@z
Ags 2
Substitution into (25) and (27) yields
ds k
Integrating this separable equation finally generates the entrainment times
ff
and
ff
This is an exact consequence of chunk model closure. The fact that the
entrainment times grow linearly with t reflects, and makes precise, the idea
that RT mixing forgets its initial data, and thus is universal in the large
time asymptotic limit. The distribution of s k =t within the mixing layer for
shown in Fig. 1.
For
and s k
required by the definition of entrainment
time. For indicating that material in the minority
phase at the edge of the mixing zone was entrained at time
similar flow.
Incorporating the solutions (33) into (23) and noting that z(fi 1
we obtain an explicit expression for the diffusion coefficient
ff
ff
within the mixing zone or interval [Z 1 (t); Z 2 (t)] and
interval. Fig. 2a and Fig. 2b respectively display the distributions of D
across the mixing layer for both
0:1807 at various times. From Fig. 2b we see that for large A, the
diffusivity is more prominent on the bubble side than on the spike side.
similar RT mixing takes place in an expanding domain [Z 1 (t); Z 2 (t)],
which can be transformed into a fixed domain [Z 1 (t)=t
t. Then (4) becomes
(D
and mixing occurs on a log time
scale in this model, as it should. In the time asymptotic regime, the solution
asymptotically independent of t 0 . It is thus approximately
a solution of the time independent equation
@
4.3 Large Time Asymptotic RM Mixing in the Chunk
Mix Model
We obtain a closed form solution for RM mixing for z(fi) and the velocities.n
the large time. This is a complete solusion of the chunk model with the
exception of the pressure field. Our first objective is to determine z and
dz=dfi k as functions of fi k . For the RM case, we assume Z
We introduce the variables
and
We have
\Theta
\Theta
2:
Substituting (39) and (41) into (28) and (29) for
@z
ts ds
In terms of - , (43) and (44) can be rewritten as
Z --
@z
The integrals in (45) and (46) can be expressed through the hypergeometric
function F (ff; fi; fl; u) by use of the formula
Z ux
is called a generalized hyper-geometric
function. This leads to
and
@z
where we have used (37) and (38). The numerical solutions for z(fi; t) with
are respectively shown in Fig. 3a and Fig. 3b. From
the figures, we see that the nonlinearity of the mixing front in RM mainly
depends on the ratio of ' 2 =' 1 . In order to understand the dominant behaviour
of (48), we use the hypergeometric function transformation formulae
and rewrite the F as
Employing this expression, (48) is reduced to
@z
The transformed F has a series expansion
which converges uniformly in the unit circle
4.4 Diffusive Modeling of Large Time Asymptotic RM
Mixing
To obtain the diffusion coefficient D, we need to solve for s k . Substituting
(42), (38) and (50) into (25) gives the evolution equations
ds 1
@z
and
ds 2
@z
for fi k . We attain the solutions s k by integrating the resulting equation for
from fi k
(t). The numerical simulations for s k
at different
times for are displayed in Fig. 4a and Fig. 4b. The diffusion
coefficient D in RM mixing is evaluated as
In Fig. 5a and Fig. 5b, we show the distribution of D across the RM mixing
layer at different times for both 0:96. From the figures, we
see features also demonstrated in RT mixing. The diffusivity on the bubble
side is more prominent than on the spike side. With the exception of the Lagrangian
time of entrainment s k , the formula (54) is a closed form evaluation
of D, which is exact within the chunk model assumptions and the large time
RM asymptotic assumptions. To proceed, we require a deeper understanding
of the RM mixing zone asymptotics to evaluate s k approximately.
4.5 Scaling Law Behavior of the RM Mixing Zone
Since the relationship between z(fi k ; t) and fi k
is significantly nonlinear, we
now determine approximately the principal features of fi k (z; t). In units of
most of the mixing occurs at a speed V 1 , near the bubble interface Z 1 ,
while in units of z, most of the mixing zone is occupied nearly exclusively by
light material, and trace amounts of heavy material involved in the mixing
move with speed close to V 2 . We believe this heavy material motion has the
form of widely isolated jets, each containing a high concentration of heavy
material. Thus we do not expect that the trace amounts of heavy material
will be uniformly distributed in the ambient light material.
To substantiate these statements, observe that v - ff
a small range of fi 1 near the spike interface, i.e., for
the same reason, the coefficient of -
2dominates that of
more than compensates for the larger size of -
Z 2 . Thus except for
most of the transport is light material moving at a
speed
into the heavy material.
With two distinct power laws assumed for the two edges of a RM mixing
zone, the solution within the mixing zone displays a continuous range of
power law exponents. We introduce two asymptotic regions:
For large and fixed t, the relations - AE 1 and -=(1 are satisfied for
most of the mixing zones as measured in fi values except for the neighborhood
of the spike interface (fi 2 - 0).
In the case of region I,
which solely depends on - or ' k converges uniformly. Thus
the equations (52) and (53) are separable and have the form
ds k
\Theta
2:
In the case of region II, and the
equations (52) and (53) become
ds k
\Theta
2:
We introduce the curve boundary between the two asymptotic
regions. A more accurate analysis, which we forego here, would include a
third region, - 1 intermediate between regions I and II. In this intermediate
region, use of the variables fi k and - in place of fi k and s k still gives a separable
pair of equations, which after approximation of F by a finite power series,
can be integrated in closed form.
Let s
and fi
respectively represent the time and the fi k value
along the solutions to (57)-(58) at
s
2: (59)
Consider in region I. The particle path defined by the solution of the
equation remains in region I and so it can be integrated to yield s 2 . We
integrate fi 2 from 1 to fi 2 (t) and s 2 from s 2 to t in (57) and obtain
As t.
The path for the solution of the fi 1 equation must terminate in region II.
Thus its solution is attained by a matched asymptotic expansion. We first
integrate fi 1 from fi 1 (t) to fi  and s 1 from t to s  in region I, we have
Next consider a point in region II. Region II includes the mising
zone edge 1. Thus the particle path defined by the solution of the fi 1
equation remains in region II and so it can be integrated to yield s 1 . We
integrate the equation (58) for region II, and obtain the entrainment time s 1
for this region:
As
The path for the solution of the fi 2 equation must terminate in region I.
Thus its solution is attained by the same treatment as fi 1 in region I. We
integrate fi 2 from fi 2 (t) to fi
2 and s 2 from t to s
2 in region II, we obtain
s
To complete the integrations for paths which cross both regions, we need
to integrate fi 1 from fi
1 to 1 and s 1 from s
1 to s 1 in region II, and to integrate
fi 2 from fi
2 to 1 and s 2 from s
2 to s 2 in region I. The former gives
and the later generates
which are consistent respectively with (62) and (60) when these are evaluated
at
Combining (59) and (61), we obtain
and thus
s
Analogously, incorporating (59) and (63) yields
which leads to
Substituting the " " variables in (64) and (65), we finally obtain the
entrainment time s k
in RM mixing as
and
Incorporating (70) and (71) into (54), and considering the facts that - AE
1 in region I and - 1 in region II, we obtain the approximate expressions
for D in these two asymptotic regions:
ae
oe
and
ae
oe
As fi k
approaches zero as it should.
5 The Reynolds Stress Tensor
By definition the Reynolds stress tensor is
ve
For mixing induced by velocity dispersion in a single (z) coordinate direction,
and with only two materials, the Reynolds stress tensor becomes R
, where
R
This correlation is conventionally simplified to a two variable model, e.g.,
describing the total turbulent kinetic energy and the rate of dissipation of
turbulent kinetic energy (k-ffl models). The new variables acquire their own
dynamic equations. Such an approach is problematic for several reasons. The
new equations, which thus define a second order closure, contain additional
unknown parameters of uncertain value. The k-ffl models refer to isotropic
turbulence, and only model the trace of R. Setting parameter values for
nonisotropic turbulence closures in still less well understood. The expanded
equation set has two new equations in contrast to the one new equation for
the chunk mix model. The uncertain coefficients in the k-ffl model are in
contrast to the nearly parameter-free status of the chunk mix model. Thus
the use of a k-ffl or related model for turbulent diffusion will surely be more
complex than the more exact chunk mix model we are trying to simplify.
Here we derive an expression for the Reynolds stress from the chunk
model in the case of two materials. This expression has two advantages:
lack of unknown parameters and use of a first order closure, so that no new
dependent variables and equations are introduced. In the chunk model, the
Reynolds stress is represented as a sum of three terms. The first two are
the Reynolds stresses associated with the fluids strictly within each species,
and the third term is the Reynolds stress associated with the two species
mixture. We have assumed for the chunk model that the first two terms
can be neglected, and we retain that assumption here. Thus we evaluate the
Reynolds stress as equal to the third term alone following [3],
ae
The expression (75) can be evaluated in terms of fi k , ae k , and V k (t), i.e.,
variables defined within the diffusion model. We use the
Then
ae
Thus we see that R behaves like a potential well, and accelerates ev going into
the mixing layer, and decelerates it upon exit.
To determine the scaling behavior of the Reynolds stress term rR, we
specialize to RT and RM mixing, as in Section 4. In the RT case, velocities
are O(t) and r is O(z) In the RM case, we have
two scalings or a continuum range of scalings, but in all cases, rR scales like
6 Conclusions
For equilibrated pressure multiphase models,we have identified a thermodynamic
constraint or process dependence which relates the choice of
of the conserved variables of the model to the domain of physical validity
of the model. For incompressible multiphase flow, we have determined the
coefficient of turbulent diffusivity and the form of the Reynolds stress tensor.



--R


A renormalization group scaling analysis for compressible two-phase flow

Boundary conditions for a two pressure two phase flow model.
A new multifluid turbulent-mixing model
Turbulent Rayleigh-Taylor instability experiments with variable acceleration
Renormalization group solution of two-phase flow equations for Rayleigh-Taylor mixing
Statistical evolution of chaotic fluid mixing.


Multipressure regularization for multi-phase flow
The Physics of Fluid Turbulence.
Hyperbolic two-pressure models for two-phase flow
Experimental investigation of turbulent mixing by Rayleigh-Taylor instability
Private communication

Numerical simulation of turbulent mixing by Rayleigh-Taylor instability
Modeling turbulent mixing by Rayleigh-Taylor instability
Numerical simulation of mixing by Rayleigh-Taylor and Richtmyer-Meshkov instabilities
--TR
Renormalization group analysis of turbulence I. Basic theory
Modelling turbulent mixing by Rayleigh-Taylor instability
Transport by random stationary flows
turbulence;multiphase flow