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PART 1
Approach and General Equations
Part 1 of this book deals with the definition and description of the basic approach, which can be simply presented as the generalization to several phases of the usual approach of continuous media in the thermomechanics of fluids in turbulent flows. We thus often speak of âpiecewise continuous mediaâ. Its objective is to give a framework for the theoretical representation, its hypotheses, possibilities and also its limits, to identify the different types of equations that are necessary to make this approach useful, i.e. by taking the âpredictionsâ concerning specific flows. It should be noted that this approach is the only one that allows us to provide exact, instantaneous equations of the complete medium, from which we can then develop equations (which require closure assumptions, such as the more classical equations for turbulent flows with a single phase) of variances and probability density functions (PDF). The general description consists of four aspects, which are each the object of a chapter: the theoretical description of a âpiecewise continuous flow or mediumâ; the definition of an averaged or filtered description, usable in practice, and in particular the statistical averages that will be mainly used here; the writing of balance equations for averaged quantities that can be chosen to represent the medium in its evolution; and finally, the definition of the necessary constitutive laws, which include âequations of stateâ and laws of irreversible processes, and the problem of choosing them.
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Chapter 1
Towards a Unified Description of Multiphase Flows
1.1. Continuous approach and kinetic approach
In classic fluid mechanics theories, fluid is usually considered as a âcontinuous mediumâ, described locally and at each instant using a certain number of characteristic variables, and its evolution is represented by âbalance equationsâ. These balance equations are partial differential equations in three-dimensional space and time, whose original writing uses the Eulerian point of view: a geometric point in space is designated, and at this point, the characteristics of the fluid at each instant are observed, in particular its velocity. This leads to the so-called continuity equation and then to Euler and NavierâStokes equations and necessitates the definition of the Cauchy stress tensor, containing the pressure and tensor of viscous stresses. The description is, therefore, not complete until these new variables can be given by specific âconstitutive lawsâ, which represent the nature and small-scale properties of the fluid in question. These constitutive laws are not necessarily based solely on a theory; they can, with more generality, be of empirical origin, but in any case they must follow the principles of thermodynamics.
There is another way to find these same equations and laws by considering the fluid medium from the beginning as a set of atoms or molecules in steady motion and colliding frequently in the void space, under the action of the laws of classic mechanics for material points. This approach, often called the kinetic approach, was introduced by Boltzmann and Maxwell and has been used to recover classical Eulerian continuity equations and NavierâStokes equations since the work of Chapman and Enskog in the early 20th Century. In this context, the characteristic variables that we might call âmacroscopicâ variables are defined in a small volume around each point in space, and in a small interval of time around a given instant. When the volume and the interval of time considered are very small in relation to the spatial and temporal scales of variation expected for macroscopic variables, in the macroscopic experiments and situations with which we are concerned, these macroscopic variables no longer depend on the real size of the volume nor the time interval, and have continuity properties with respect to space and time. It is in these conditions that we can consider the medium as continuous, on the macroscopic scale and with our limited view.
The base equation of the kinetic approach is the Boltzmann equation, which concerns the distribution function of the velocities of molecules at a given point M(x) and a given instant t. To be exact, and for a gas medium containing only one kind of molecule, this function, f(u, x, t), is defined as the number of molecules that can be found, in the medium, in a small volume around point x approximately dx, at an instant t approximately dt, and which have velocity u approximately du (here, of course, the variables x and u are three-component vectors, as are dx and du). The number of molecules per unit of volume of the medium is, thus, defined as the integral in space of the values of velocities (from minus infinity to plus infinity) of this distribution function, and what we might call the macroscopic velocity of the medium is the integral, in this same space of velocities, of the product uf, divided by the integral of f itself. All the molecules of the gas being identical and of known mass, the volumetric mass of the medium is the product of the number of molecules per unit of volume and the mass of a molecule, and the velocity of the medium thus appears as the ratio between the momentum of the medium and its mass, both per unit of volume. The historical development of this approach is discussed in [CHA 60] and a very complete recitation of the developments in [HIR 54]. In the framework of the approximation of continuous media, this approach allows us to find the same forms of the balance equations of mass and momentum by using the first continuous approach discussed above. However, in addition, the fact that we can describe in more detail the microscopic structure of the material enables us to obtain the constitutive laws in a theoretical manner, that is expressions of all the variables that appear in the balance equations, namely the Cauchy stress tensor, pressure, viscous stress tensor, etc. These expressions are calculable only because of certain hypotheses on the microscopic characteristics of the gas being considered (for example the approximation of molecules as hard spheres of a certain mass). The energy balance equation of the continuous medium can also be found by defining the heat conduction flow in particular. When the approximation of the continuous medium is no longer valid, as is the case for rarefied gases, the Boltzmann equation remains valid and the balance equations of mass, momentum and energy still exist with the same overall form, but they contain additional terms, and the usual terms can no longer be calculated using the same approximations. In certain conditions, it is no longer useful to define a macroscopic velocity for the medium, and the calculation of the velocity distribution function itself, by using adequate approximations to represent collisions between molecules in the Boltzmann equation, is recommended.
When undertaking a theoretical study of multiphase fluid media, we might ask ourselves which of the two approaches is of more interest, especially when we wish to focus on a case in which one of the phases is dispersed into numerous parcels of different sizes.
In fact, it seems that the first studies involving clouds of drops in a gaseous medium used the kinetic approach to study the liquid phase composed of the set of drops by examining a distribution function where the variables were both the velocities and sizes of the drops, or f(u, v, x, t), where v is the volume of the drops. At a given point and at a given instant, the number per unit of volume (of the medium) of drops of volume v, approximately dv, is still the integral of f on the space of velocities, and the total number of drops of all sizes is the integral of f both on the space of velocities and that of volumes. The function f follows a type of Boltzmann equation here again, in which the most difficult term to approximate is the term due to the collisions of particles. We may refer, for example to, [WIL 58]. However, it has never been suggested to represent the gaseous phase in which these drops are dispersed using a Boltzmann equation that, when the quantity of drops is relatively large, might inclu...
