Local instantaneous equations form the foundation for almost all two-phase modeling procedures. They may be used directly, in the study of bubble dynamics or film flows, for example. More commonly, however, they are used in averaged form as in the study of flow in pipes and conduits. Averaged forms of the local instantaneous equations will be considered in the section below on averaged equations.
The formulation of local instantaneous equations involves deriving the appropriate conservation equations and then closing the set, the latter problem discussed in the section below on closure and applications. Conservation equations for a two-phase flow can be derived by writing integral balances in much the same way as for the single-phase case (see Conservation Equations, Single-Phase). The problem is how to take into account behavior at the interface.
A phase interface may be considered as a three-dimensional region which separates the bulk portions of two phases and in which the constitutive equations may differ from those applicable in the bulk portions of each phase.
A surface is said to be singular with respect to a quantity at a point if the limiting value of this quantity at a point on the surface—obtained by approaching the point along a path restricted to one side of the surface—differs from that obtained by approaching the point from the other side of the surface. The phase interface may be envisaged as a singular surface.
The above allows integral balances to be written encompassing both phases and the interface. These result in phase equations which are identical to their single-phase counterparts and jump conditions which are unique to multiphase flow analysis. The derivation leads to the local instantaneous equation set.
The formulation of integral balances for single-phase flow under the continuum hypothesis has well-proven validity. Here, these balances must be generalized to the case of two-phase flow. This is done by considering two-phase flow as a field which is subdivided into single-phase regions with moving boundaries between the phases. The standard single-phase balances hold for each subregion and are matched by the interfacial jump conditions.
First, a general balance must be written for any quantity, Ψ, over a material volume V
where J is the influx of Ψ through A, while φ is the supply of Ψ within V.
Consider now a material volume V within which there occurs a surface A1(t) splitting V into V1 and V2, and splitting A into A1 and A2, (Figure 1). The surface is a persistent singular surface with respect to a quantity Ψ, and possibly also with respect to u. It is assumed to be smooth and may be in motion with any speed of displacement, u1.
The next step is to generalize the well-known transport theorem [Slattery (1972), Truesdell and Toupin (1960)] to regions containing such a singular surface. The transport theorem in its usual, single-phase formulation may be written as:
for any Ψ. In other words, this states that: the rate of change of the total Ψ over a material volume V equals the rate of change of Ψ over V, plus the flux of uΨ out of the bounding surface.
Turning back to the region divided by a singular surface, consider the case where the region is arbitrarily small and arbitrarily smooth. Thus it can be assumed that Ψ and u are continuously differentiable in the two regions, V1 and V2. In general, the regions and surfaces V1, V2, A1, A2 are not material. The fields u1 and u2 can now be defined as follows:
Since A is a persistent common boundary of V1 and V2, the expression may be written as:
where the operator du1/dt indicates that the time derivative of the integral over a region that instantaneously coincides with V1 and is material with respect to the field u1, must be taken while du2/dt is analogously defined. To each of the integrals on the right—since Ψ and u are continuously differentiable in V1 and V2 and approach continuous limits on the entire boundaries A1 + AI and A2 + AI — the basic transport theorem, Eq. (2), may be applied to obtain
These results can then be substituted into Eq. (4) to give
where [Ψ] is the jump of quantity Ψ across the singular surface denoted by
where Ψ+ is the limiting value of Ψ− as the interface is approached from one side and , the limiting value from the other side. The sign of the jump is a matter of convention. Equation (7) is the transport theorem written over a region containing a singular surface.
A general balance at a surface of discontinuity must be written next. It is assumed that a general balance equation of the form of Eq. (1) holds for an arbitrary material volume, irrespective of whether it contains a singular surface or not. Then, combining Eq. (1) and Eq. (7) for a sufficiently small material volume V containing a singular surface AI, will yield
It can now be assumed that in the neighbourhood of AI, the quantities and ρφ are bounded, while on each side of AI the quantities ρΨ, u·n, and n·J approach limits that are continuous functions of position. Under these conditions, let A1 and A2 shrink down to A1 (Figure 2) so that the volume of V1 + V2 vanishes while the area of AI remains finite in the limit. The volume integrals then vanish in the limit, and Eq. (9) becomes
This equation holds for an arbitrary area on AI, thus the integrand must be zero. The general balance for a singular surface may then be written as:
where is the mass transfer per unit area of interface and per unit time. By choosing appropriate values for the quantities Ψ and J, Eq. (11) may be used to express the balances of mass, momentum and energy across the phase interface for a two-phase flow, i.e., the interfacial jump conditions.
The balances of mass, momentum and energy for a single-phase flow (see Conservation Equations, Single-Phase) may be expressed by a single general balance equation, thus
for a single-phase, k. As previously noted, phase equations of two-phase flow are identical to their single-phase flow counterparts; the former may be extracted from the general balance equation by using the values of Ψk, Jk and φk given in Table 1. In this table, u is the internal energy per unit mass; is the heat flux vector; τ is the total stress tensor; and g is the gravity vector (all body forces have been assumed to be of gravitational origin).
The interfacial jump conditions follow from the general balance for a singular surface, Eq. (11), by using the appropriate values for the quantities Ψk and Jk given in Table 1.
Closure laws may be considered as those necessary and sufficient relations which must be added to conservation equations to allow calculation.
For each of the phases, three constitutive equations are sufficient to effect closure. These are the same as the closure laws for single-phase flow, and one set might be: One scalar equation of state for ρ, one vectorial constitutive law for and one tensorial constitutive law for τ. However, the problem of achieving a correct and consistent description of the interfaces is much less simple.
The practical applications of the local instantaneous equation set are rather limited. This is because although the set is closed, the resulting initial moving boundary-value problem is intractable except in the simplest of cases. The difficulties stem from the existence of deformable moving interfaces, with their motions unknown, and fluctuations of variables due to turbulence and to the motion of the interfaces. The former leads to complicated coupling between the field equations of each phase and the interfacial conditions, whilst the latter inevitably introduces statistical characteristics. In addition, the resulting set of partial differential equations is generally singular when the void fraction is identically one or zero since many coefficients vanish in the equations of the corresponding phase. They are also numerically ill-conditioned for small and large values of the void fraction. They are of use, however, in theoretical studies since they do not involve any further algebra. Typical examples are found in situations where the interfacial geometry is particularly simple, such as the study of a single bubble or of a laminar separated flow.
For most practical purposes, for example modeling flow in a pipe, the local instantaneous formulation is not very useful largely due to the intractability of the set as discussed in the preceding section. Moreover, the local instantaneous behavior of the various flow variables is often not required; the prediction of averaged quantities appears sufficient. Thus, the almost universal approach is to average the local instantaneous equations using one or more of the following averaging operators: ensemble, time and distance. The resulting averaged equation set is much simplified, but information is inevitably lost during the averaging procedure. This information must be resupplied in the form of auxiliary relationships, i.e., additional closure laws. The problem is exactly analogous to that encountered in single-phase turbulent flow. The closure problem is a considerable one and is discussed in a separate section below.
Here, focus is only on Eulerian averaging since it is closely related to human observations and instrumentations. Eulerian averaging is based on time-space description of physical phenomena. Since in the Eulerian description time and space coordinates are taken as independent variables, it is natural to consider averaging with respect to these independent variables, i.e., time and spatial averaging.
The local instantaneous equation set is spatially averaged in the following section. Then in the succeeding section, the spaced-averaged equations are averaged over time to give a more useful composite-averaged equation set. It is also possible (although it is not given here) to time average the local instantaneous equation set and then volume average the resultant time-averaged, leading to the same result.
The local instantaneous equations may be averaged over either area or volume. In practice, volume-averaged equations are generally more useful than their area-averaged counterparts. One reason for this is that discontinuities may arise in area averages. Discontinuities do not arise if averaging is performed over a thin slice with a finite volume, rather than over an area. Consider a fixed tube with axis Oz (unit vector nz) in which a volume is cut by two cross-sectional planes located a distance Z apart over area Ak1 and Ak2 (see Figure 3). Let Vk be the volume limited by Ak1, Ak2, and the portions AI and AkW of interface and wall enclosed between the two cross-sectional planes. The unit vector normal to the interface and directed away from phase k is denoted by nk. The cross-sectional planes limiting the volume Vk are not necessarily fixed and their speeds of displacement are denoted by −uA11· nz and uA12· nz.
The transport theorem, [Eq. (2)], may be generalized to give
Applying to volume Vk leads to
where uI·nk is the speed of displacement of the interface AI and the integral over the wall vanishes since the velocity of the wall is assumed to be zero.
The Gauss divergence theorem applied to volume Vk for any vector v gives
The volume average operator is defined as follows:
Integrating the general balance. Eq. (13), over the volume Vk leads to
Applying Eqs. (15) and (16) leads to
By using the definition of the volume averaging operator, Eq. (17) and also Eq. (12), this result can be written as
Use of the appropriate values for Ψk, ρk and φk (see Table 1) give the instantaneous volume-averaged equations for the balances of mass, momentum and total energy.
Composite-averaged equations are those which, for example, have been averaged over both space and time. The vast majority of practical situations are analyzed with a composite-averaged equation set.
Here, the volume-averaged equations will be time-averaged to give a composite-averaged equation set. A single time-averaging operator will be used, which is defined as follows:
where [T] is the time interval over which averaging occurs. If fk has no jump discontinuities, as is the case here since volume averaging has been performed, then
i.e., the partial differentiation of a time-averaged function and the time-average of a partial derivative commute if the function is smooth.
Consider the volume-averaged general balance, Eq. (20), for the case when the cross-sectional planes are fixed such that the velocities uAk1 and uAk2 vanish. Integrating the resultant expression over the time interval [T] gives
which, using the above definition of the time averaging operator along with Eq. (22) and also the commutativity properties of integrals, gives
The two terms in the above expression involving integration over the cross-sectional planes Ak1 and Ak2 may be combined into a differential term since the distance Δz is arbitrarily small. This results in
In practice, it is generally more useful to have a composite-averaged equation set expressed in terms of average over the control volume, V, rather than the phasic volume, Vk. Fortunately, the transformation is a simple one which, defining
where εk is the time fraction of phase k and . This is the space/time-averaged equation expressed in a more useful form.
Integrating Eq. (11) over space and time results in
This is the composite-averaged general interfacial jump condition which may be used in conjunction with the composite-averaged general balance equation [Eq. (27)].
A brief indication of how averaged equation sets may be closed in order to allow calculation is provided below, but restricted to the composite (volume/time)-averaged equation set derived above.
The closure laws of the composite-averaged equation set may be subdivided into three distinct types:
Constitutive laws, these are so called due to their similarity with the constitutive laws of single-phase flow. They describe the constitution or state of a material.
Transfer laws which express the boundary terms in the balance equations.
Topological laws which restore necessary information on flow structure lost during the averaging process.
The term topological law is not a standard one and the motivation for its introduction merits further discussion. It has been popularized by Bouré (1986) who used the term to highlight the difference in the closure problem between local instantaneous and averaged equation sets. The local instantaneous equation set is closed very classically by true constitutive laws, as discussed in the section on closure and applications. Averaging the local instantaneous equation set to obtain equations of practical usefulness inevitably entails a loss of information. In particular, details of flow structure are smoothed out, i.e., fluctuations whose characteristic times are shorter than the averaging operator characteristic time and distributions whose characteristic lengths are smaller than the averaging operator characteristic length. This information must be resupplied in the form of auxiliary relationships since some of it plays an essential role in the thermohydraulic behavior of the flow (e.g., geometry of the interfaces, velocity, and temperature gradients within each phase). Closure laws that are primarily responsible for resupplying interfacial structure information are the topological laws.
The phase equations and jump conditions can be made dimensionless, using, for example, characteristic values as units for lengths, times, densities, velocities, and enthalpies, where ukz is the projection of the velocity vector along the z axis (ukz = uk · nz). The only difference between dimensional and dimensionless balance equations lies in the appearance of the ratio
in energy equations. The equations presented here may be regarded as dimensionless, although setting η = 1 in energy equations results in the dimensional form again.
The first difficulty concerns the averages of products of the dependent variables in the composite-averaged general balance equation [Eq. (27)]. What is needed is the products of the averages of the dependent variables since it is the averaged variables that are generally measured experimentally and that are related to physically. In fact, it is very difficult to relate the averages of products to the product of averages and the almost universal approach is to assume that they are equal. This is equivalent to making a statement something like the following:
Whenever fluctuations and/or transverse distributions are present, this practice is obviously questionable. In this discussion, the above convention is followed and henceforth, the averaging symbols are omitted. Thus, the classical phase fraction, <<εk>>3 is denoted by εk.
Composite (space/time)-averaged phase equations are obtained by substituting the appropriate values of Ψk, Jk, and φk from Table 1 into Eq. (27). The composite (space/time)-averaged interfacial jump conditions are obtained by substituting the appropriate values of Ψk, Jk, and φkk from Table 1 into Eq. (28). No mass transfer is assumed to take place between the fluid and the walls of the pipe, although this could easily be included if required.
(AI is the instantaneous interfacial area present in volume V.)
Here, the phasic total stress tensor τk is decomposed in the classical manner in the bulk phase (i.e., not at the walls or interface), thus
where I is the unit tensor. This decomposition holds for any scalar field, pk and any tensor field τk; pk is identified with the pressure and , with the deviatoric stress tensor field. The momentum balance is thus:
Again, the stress tensor is decomposed. Thus,
where velocity of the wall has been assumed to be zero.
where, once again, the stress tensor has been decomposed and surface tension effects have been neglected.
Possibly the most common, simplified two-fluid model for pipe flows is the single-pressure model. In single-pressure models, it is recognized that the averaged phasic pressures are not very different in practical pipe flows (except for surface tension effects). Thus it has become customary to assume
which enables elimination of one dependent variable and serves as a substitute for the void fraction (topological) closure law.
This is particularly convenient since a satisfactory void fraction closure law has yet to be found, as discussed by Bouré (1986). Unfortunately, assuming that the phasic pressures are equal dictates that their partial derivatives must also be equal. Thus, under this assumption, pressure disturbances have the same averaged effect on the two phases; in particular, they propagate at the same velocity within the two phases. This is clearly unrealistic in many cases and very restrictive where propagation phenomena are important. Another assumption commonly made is that the field and viscous conduction terms
may be neglected. This assumption is motivated by the fact that these terms are generally very small compared to the other terms of momentum and energy balances. Care must be exercised, however, since in some cases they are not negligible; the sizeable conduction term for liquid metals is one such example.
With these simplifications, the following closure laws are required: two equations of state (one per phase), one interfacial mass transfer law, three interfacial momentum transfer laws, seven interfacial energy transfer laws, two wall momentum transfer laws and four wall energy transfer laws. They are usually obtained in a rather ad hoc manner, and are often empirical. Further simplifications may be possible for a given situation.
A surface area
f arbitrary quantity
f+ limiting value of f as the interface is approached from one side
f− limiting value of f as the interface is approached from other side
g gravity vector
h specific enthalpy
I unit tensor
Jk flux term
mass transfer per unit of interface area and per unit time
n unit normal to a surface, principle normal of a curve, etc.
q heat flux vector
T time interval
u velocity vector
uI velocity vector of surface point
uk velocity vector of phase k
ux component of u in x direction
uz component of u in z direction
u internal energy per unit masse
v arbitrary vector
x spatial displacement vector
x,y,z spatial coordinates
z spatial displacement in the z direction
ε time fraction or classical phase fraction
η ratio appearing in dimensionless form of energy equation
Ψ arbitrary scalar, vector or tensor
τ total stress tensor
τD deviatoric stress tensor
φ source term
[Ψ] jump of Ψ across a surface
time average operator defined by Eq. (21)
phasic volume average of fk defined by Eq. (17)
phasic volume average of fk defined by Eq. (26)
K phase index
c characteristic value
Bouré, J. A. (1986) Two-phase flow models: The closure issue. In European Two-Phase Flow Group Meeting, Munchen, Germany, June.
Bouré, J. A. and Delhaye, J. M. (1982) General equations and two-phase flow modeling. In G. Hetsroni, editor. Handbook of Multiphase Systems. Hemisphere.
Ishii, M. (1975) Thermo-Fluid Dynamic Theory of Two-Phase Flow. Eyrol- les, Paris.
Slattery, J. C. (1972) Momentum, Energy, and Mass Transfer Continua. McGraw-Hill.
Truesdell, C. and Toupin, R. A. (1960) The classical field theories. In S. Flugge, editor, Handbuch der Physik. Springer-Verlag.