## 1. Introduction

Engineering and Nature very often are concerned with media that do contain some randomness. One particularly well known example is given by turbulent flows, where random fluctuations are the results of the growth of unstable motions from small perturbations in the initial or boundary conditions, when the velocity gradients or/and temperature gradients are too large in some place within the flow. These fluctuating motions result in effective additional friction, effective additional diffusion of species or heat, and additional energy dissipation (i.e. transfer from organized kinetic energy to turbulent kinetic energy, and finally to internal energy). Another example of random flows is seen in two-phase flows, either with particles dispersed into a liquid or gaseous continuous phase, or built with a bubbly liquid. Here, a first reason of randomness is given by the fact that the locations and velocities of the particles (or bubbles) at initial time and in entrance sections cannot be known or controlled, and the induced fluctuations are not rapidly damped with time. Generally, this randomness becomes rapidly uncorrelated with these initial conditions due to the complex interactions between the flow of the continuous phase and the moving inclusions, interactions that modify also the number and sizes of these inclusions. Consequently, the flow conditions in both phases are or become turbulent, with interconnected fluctuations of velocities. Again, the effective result of the fluctuations is additional friction, additional diffusion or dispersion, and additional energy dissipation. Two-phase or multiphase flows with a large amount of solid particles are used in the industrial devices called as “Fluidized Beds”, and are encountered naturally in “Granular Flows”, and in this case there are many lasting contacts between the particles. The so called “Granular Media”, even without actual flow but just with some deformations or slow motions, are also subjected to randomness, due to the preparation of the medium but also to the differences between the grains shapes, and even the global properties of these media are very difficult to predict.

The properties of these kinds of media and flows have been studied since more than hundred years, and very useful prediction methods have been proposed so far, with methodologies becoming more and more clearly similar. Although all these media are showing clearly irreversible and dissipative processes, the typical reasoning of irreversible thermodynamics has never been used, at least explicitly (it might be present, but not consciously, in the brain of the authors). Our purpose here is to show that the Irreversible Thermodynamics and the “Second Principle” can justify many features of the models that have been proposed until now for these three types of applications. It has to be emphasized that Irreversible Thermodynamics deals with the modelling of media in evolution, whatever can be this model. The usual model of classical continuous medium, where the randomness lies only at the infinitely small scale of moving molecules, is not the only one that can be addressed. The so called “Extended Irreversible Thermodynamics” (Jou et al., 2001) allows us indeed to build pertinent entropy functions adapted to each type of model, in particular depending of the basic variables of the model used, as we will see.

The first part of this paper deals with classical turbulent flows. The general problem of building a model for these flows, i.e. an approximate mathematical representation that could give predictions close to experiments, has in practice given rise to several solutions, which are valid in different domains, i.e. for different kinds of experimental situations. More precisely, the domains of efficiency of the different models appear to be larger and larger for models using an increasing number of variables. The basic variables that allow the building of these models are “averaged variables” (now precisely defined as statistical mean values), in a finite number, and then they do clearly constitute a truncated representation of the medium, but it is expected that these variables are sufficient for the purpose of the approximated knowledge needed. Some of these variables do satisfy balance equations, which are obtained averaging the primitive balance equations of the mechanics of continuous media, but these equations do need “closure assumptions” before to be useful in constituting a “Turbulence Model”. Once the basic variables are chosen, the “Extended Irreversible Thermodynamics” allow us indeed to define an entropy function for this representation of the medium, for each of the models proposed. We will see that this entropy function is not the statistical mean value of the classical entropy for the fluid used in this turbulent flow, but has to be built taking into account the averaged form of the state equations of the fluid. Then, the Second Principle may lead to conditions concerning the ingredients of the turbulence model, conditions that are different for the different models. We will show that here simply considering two very popular models, and we will see that the usual practice does satisfy these conditions.

The second part of the chapter considers the modelling of two-phase flows. This modelling can be attacked with a similar approach and similar tools as turbulent flows: a set of averaged variables are defined; balance equations are written from the primitive equations of a piecewise continuous medium and closure assumptions are needed, to be adapted to each kind of two-phase flows. Here the closure assumptions are needed for both additional dispersion fluxes and exchange terms between the phases. These two-phase flows are again associated with irreversible effects, and again the framework of extended irreversible thermodynamics helps the modelling. We will recover again assumptions that are used in practice and supported by experiments.

The third part is devoted to granular media, with the same point of view. The approach generalizes the case of two-phase flows, because the granular medium is considered as a multiphase medium where each grain is considered as one phase, and the fluid in between as the last phase (Borghi § Bonelli, 2007). Again, the variables of the model are defined as statistically averaged grains phase and fluid phase variables, and their balance equations can be found from the primitive equations of mechanics, to be completed with closure assumptions. The approach can handle the situations where the medium is “quasi-static” as well as the ones where the medium is flowing with large velocities, giving rise to the limiting case of “Granular Gases”. In both cases, Extended Irreversible Thermodynamics can be used for building entropies and giving closure assumptions, without linear laws in this case particularly. The classical models of “Granular Gases” are recovered in the limit of large velocities, with the difference that the grain size is not the single length scale to be considered. For quasi-static situations, it is necessary to include in the model the six components of the averaged “Contact Cauchy Stress Tensor” of the grains phase. Intermediate situations can be handled as well.

## 2. Irreversible thermodynamics for turbulent flows models

Turbulent flows at present time are studied with models of two different types, namely Reynolds averaged Navier-Stokes (RANS) models, or Large Eddies Simulations (LES) models. We consider here for simplicity the framework of RANS models, although the extension to LES models may be not difficult. Then, the object of the study is not one given flowing medium, but an infinite number of such media, flowing submitted to initial and boundary conditions randomly perturbed. Indeed, what we call “one turbulent flow” is this ensemble of similar individual flows, because only statistical mean values calculated from this ensemble (or, at best, probability density functions) can be possibly predicted by some model.

The “mean medium” that is the subject of modelling is then described by “mean velocities”, “mean densities”, “mean enthalpy per unit of mass”, etc.. For N realizations, one defines:

at each given location within the flow field, and similarly for pressure. But in the general case where the density of the fluid can vary noticeably, it has been found more interesting to consider for other variables the mean value weighted by density, and one defines:

i.e. the mean velocity is nothing but the mean momentum divided by the mean density. Similarly for the internal energy per unit mass (or the enthalpy per unit mass), the mean values are weighted with density.

Within a turbulent flow, these quantities are defined for each location and for each time, and vary with the location and eventually with time, but these variations are much smoother than the local and instantaneous quantities measured in one single realization of the flow, and by definition they are not sensitive to perturbations of initial and boundary conditions, that are inevitably present.

Is it possible to characterize the turbulent medium by

The first invented turbulence model, called “Prandtl mixing length model”, is of this type, and has been very useful in many practical cases. But a larger domain of validity can be covered taking into account additional fields variables pertinently chosen, described with their own partial differential evolution equations. There are different turbulence models of this improved type, the simplest one introduces only as additional field variable the kinetic energy of the random turbulent fluctuations, say

### 2.1. Basic equations

The physical basic equations corresponding to balance of mass, momentum and total energy give birth in any case to balance equations for the mean variables

The total energy here comprises the internal energy, the kinetic energy of the mean motion, and the kinetic energy of turbulent fluctuations, so

From (2) and (1), one obtains the equation for the kinetic energy of the mean motion:

Then, from (3) and (4) we have:

It is possible also to derive directly from the non-averaged momentum equation a balance equation for

From (5) and (6), finally the equation for the internal energy is written:

The last term

Equations (1) to (3) are able to calculate the mean variables

The simplest model, the “Prandtl mixing length model”, does not use a balance equation for

We will now show that the approach of irreversible thermodynamics can be of great help, once the type of model is chosen, to solve the problem of modelling. We will see that it justifies the usual choices done by researchers since 30 years, and suggests possibilities for unsolved problems.

### 2.2. Entropy for the Prandtl mixing length model

In this framework we expect to model the turbulent flow only with the variables

In addition, the rate of dissipation, which can be calculated through

The three basic variables

The two functions

The fluid itself, independently of the turbulent flow, has proper equations of state, and if we assume that it is an “ideal gas”,

It has to be remarked that the previous development cannot be extended to the case of fluid satisfying a non linear Joule law

Once we have defined the entropy and found the equations of state, we can use the classical approach of irreversible thermodynamics for deriving some laws for the additional fluxes appearing in the mean balance equations (2) and (7).

The Gibbs-Euler equation, now written as

The production terms are the two last terms, and in the last one the contribution due to the diagonal of the tensors is vanishing because the fluid is incompressible. It results that a simple extension of the linear classical irreversible thermodynamics is possible here, giving laws for the turbulent additional fluxes of heat and momentum.

Concerning the total friction term, a positive “total viscosity coefficient”

Considering that the fluid is Newtonian, one has already:

This is the well known “Boussinesq relation” proposed for turbulent flows at the end of the 19th century.

Similarly, for the total heat flux, the linear law of irreversible thermodynamics gives:

This introduces the classical « turbulent Prandtl number », constant, whose value (about 0.83) has been given by many experiments.

The eddy viscosity coefficient has now to be found. Just by dimensions, it can be written as

In this model, the turbulence length scale

The conclusion is then here that the classical approach of linear irreversible thermodynamics does justify perfectly the old empirical approach of Boussinesq and Prandtl.

### 2.3. Entropy for the K-epsilon model

We consider now the very popular “k-epsilon” model, where a balance equation for

In this case, the independent variables

The C’s are constants (numbers, without dimension) or algebraic functions of the variables in some cases (the two first being always positive), depending of the exact version of the model used. Within the most classical k-ε model,

Using now (1), (6), (7) and (14), we can derive as in § 2.2 the balance equation for the entropy as:

We have written as

Before discussing the implications of the second principle, it is of interest to discuss the physical meaning of each term and to precise the equations of state. The first remark that has to be done is that the pressure has not to appear in the entropy production term, because it is not a source of irreversibility. Then the terms where the pressure appears have to be zero, and that implies that

In addition, within the last term of (15), the directly scalar part of the entropy production, there is one contribution of the energy equation but also the contribution

These three relations do determine constraints on the equations of state, or on the C’s. It has to be remarked that the equation (14) has been built, and most often used, for cases where the turbulence is incompressible, then the divergence of the velocity fluctuations is vanishing, and the relation

Taking into account

where

We can now discuss in details the implications of the second principle, which prescribes that the entropy production rate, the right hand side of the following equation (16), has to be always positive, or zero at equilibrium.

We know already that

Secondly, the term

Here again, we have to write

The last point deals with the terms

As previously for the Prandtl mixing length model, we see that the well known k-epsilon model can be perfectly explained in the frame work of irreversible linear thermodynamics. The constraints we have found that

## 3. Irreversible thermodynamics for two phase flows models

The theoretical description of two-phase flows is now firmly based by describing at each instant this medium as piece-wise continuous, and then considering that the model has to deal with statistical averages, like for turbulent flows (Drew, 1983). The randomness here is due both to the poorly known initial and limiting conditions concerning the presence of the phases, and to the existence of local perturbations permanently created by unstable phenomena within the flow, similarly to classical turbulent flows. There is again the need of defining how many mean variables are necessary, and of finding their equations. We first give here the set of variables and equations for representing the flow, without details concerning the calculations but with the relevant physical interpretation, and then we show how the Extended Irreversible Thermodynamics helps the modelling.

### 3.1. Variables and equations for two-phase flows modelling

The first classical attempt for representing a two-phase flow uses statistically averaged variables for volumetric mass, momentum, internal energy for the two phases, and their probability of presence, also known as the “volume fraction” of each phase. It is defined a “phase indicator”

The averaged forms of mass, momentum and energy equations can be found from the primitive equations for the piecewise continuous medium (Borghi, 2008). For momentum and energy they display additional diffusion fluxes due to the presence of local perturbations around the mean values, similarly as for the turbulent flows. We will call these terms “pseudo-turbulent” fluxes. The averaged equation for the mass of phase i is:

The last term on the R.H.S is the possible exchange of mass between the two phases:

The pseudo turbulent momentum diffusion flux is

The total energy is defined as

It is possible also to write a balance equation for the volume of phases, simply obtained from the convection equation of the field of

The terms on the R.H.S. have to be approximated by models, the second one being only due to the exchange of mass between phases.

There are also important instantaneous interface conservation relations, linking at the interface the exchanges between the phases: first, the mass lost or gained by one phase is identical to the mass gained or lost by the other phase; second, if we neglect the surface tension phenomena, it follows that the momentum lost or gained by one phase is identical to the momentum gained or lost by the other one ; and third a similar relation holds for the total energy (Kataoka, 1986). That leads to:

At the interface, the tangential velocity and the temperature fields are continuous, but there are discontinuities of density, momentum and energy per unit mass. The equations (21) give “jump relations” relating the values of these quantities on both sides of the discontinuity. We will not consider further here the exchanges of mass between phases, and these relations are simplified as:

The third relation of (22) is deduced from the second one because both phases have the same velocity on the interface.

The instantaneous behaviour of each phase has to be given by the knowledge of

One sees the similarities of the problem of modelling these equations with the one of modelling turbulence in single phase flows. We will show now how the irreversible thermodynamics can help this modelling task. For simplicity, we will consider in the sequel that there is no exchange of mass between the phases, this could be taken into account later.

### 3.2. Entropy for a simple two-phase flows model

We will consider the simple case of a model similar to the Prandtl mixing length model, where simply the variables:

The kinetic energies of fluctuations, namely

We do not consider here, in the Prandtl mixing length framework, that each

The last term of (23) and (23’) is due to the exchange of energy from the other phase, by the contact force.

With the point of view of the Prandtl model applied to both phases, we have to build an entropy with the variables

Similarly, we take linear equations of state for phases, which can be averaged without needing additional variables. For a gaseous phases

We can first show that the convective time derivative for the mean entropy of the medium is such that:

The convective time derivative for the medium follows the averaged velocity, while for the phases they follow the averaged velocity of each phase. Of course, the Euler-Gibbs relation is written now as:

It is easy to verify that the equation (17) allows to write:

It is necessary then to obtain the balance equation for the mean internal energy of phases from (19), removing the kinetic energy of the mean motion by taking into account (18), and removing also the kinetic energy of the fluctuations with (23). It is found:

With (23’), it gives:

where

Now from (24), (25), (26), (27), we obtain the entropy balance equation for the medium:

We recognize in this equation the rate of production of entropy, which has to be positive or zero in any case.

The last sum deals with the exchange of heat between the phases. As

The second last term involves

The two first terms of the entropy production deal with the diffusion fluxes of heat and momentum. For momentum, when each phase is divergence free (which is originally the scope of Prandtl model), we find for each phase the same formulation as for a single phase turbulent flow in §2.2. Similarly, for heat diffusion, if this term is considered uncoupled with others, the classical model found in §2.2 is again obtained for each phase.

The term

All these discussions do justify the assumptions currently done for the modelling of two-phase flows, in a large range of applications.

## 4. Irreversible thermodynamics for modelling of granular media

Granular flows are nothing but two-phase flows of solid irregular particles within a gaseous or liquid phase. But the high number of particles per unit of volume, and the low velocity that may persist allow numerous and long duration contacts between the particles, unlikely to usual two-phase flows. The approach that has been presented in §3 for two phase flows can be used also, but the problem that arises immediately is that very often the solid phase, locally, is built with two or three particles (or more) in contact, and consequently the solid here is not a true solid for which behaviour laws are known. Actually, each grain is one solid, but a couple of grains does not. So, it is necessary to consider the medium as a multi-phase medium, where each grain represents one phase. This approach has been proposed recently by R.Borghi and S.Bonelli (Borghi § Bonelli, 2007), and we will develop a while its connection with Irreversible Thermodynamics.

### 4.1. Basic equations for granular media

For granular media, the equations (17) to (20) do hold, but we can simplify them because it is not needed to follow the motion of each grain. Considering that there are N solid grains, there are N+1 phases and N+1

Here the mean intrinsic density of the grains

We see that the effective Cauchy stress tensor for the grains has two parts:

The last term of (31) is the influence of the fluid phase. Indeed, the contact forces between grains do compensate themselves in the summation, and only the contact force with the fluid remains. More precisely, as in §2., we have

When the fluid is air, or any gas whose intrinsic density is very small with respect of the solid,

The averaged total energy of the grains can also be studied, following the same route, and from (19) a balance equation for the mean total energy of the grains phase can be written similarly. From this equation, by removing the kinetic energy of mean motion and the kinetic energy of fluctuating motions, we can get an equation for the mean internal energy of the grains phase, similar to (28). When the fluid is light, and has a temperature similar to the one of the grains, this equation is nothing but the mean internal energy equation for the entire medium:

We have neglected in (33) the viscous friction in the fluid phase.

The equations (32) and (33) need closure assumptions for the contact stress tensor and the Reynolds-like tensor, before being useful. The fluid pressure can be calculated with the momentum equation for the fluid phase, often reduced simply as

The modelling of the lasting contact stress tensor

### 4.2. The modelling of the mean Cauchy stress tensor of lasting contacts

The approach of Borghi and Bonelli gives an evolution equation for

The first term on the R.H.S. is a classical dispersion term due to the fluctuations of the velocity and appears for any averaged quantity within a “randomly moving” medium. The appearance of the Jaumann objective derivative is justified by the last term,

By definition

The last term groups all the irreversible effects of sliding contacts,

The granular medium is described by the variables

In this case, the entropy is simply:

The balance equation of this entropy can again be found using the balance equations of the corresponding variables, i.e. (30), (34), (35), (36). After some algebra, it is found:

The reversible terms have to be cancelled in the entropy production. That implies three relationships giving three partial derivatives of the entropy, which prescribe three of the needed equations of state:

Then, the Maxwell relations give the form of the forth needed equation of state:

The entropy production rate of (37) can be rewritten as:

In agreement with the Curie theorem, each term separately has to be positive. That gives, first, with a linear law, that the heat flux can be represented again with a Fourier law.

Concerning the effective contact Cauchy tensor, a first linear model could give:

The coefficients

But a very non linear model, with a threshold, could equally be compatible with the second principle, and may appear in better agreement with experiments for granular media. For instance, we could envisage (*H* being the Heaviside step function):

The physical meaning of the discontinuity involved in (39) is that when the medium is compressed (then

With (40), the medium will remain “quasi-elastic” below one plastic limit defined by k and the angle

If we consider the closures (39), (40) for the equations (35), (36), beyond the plastic limit, in the case where the last two terms are predominant and counterbalancing themselves (because

A similar law has been proposed in 1994 (Hutter & Rajagopal, 1994) and recently studied with numerical simulations (Frenette et al. 2002).

Before the plastic limit, the medium is purely elastic if the first terms on RHS of (34) and (35) are negligible. We can specify then that

In this case, (34) is reduced to

That means that the effective pressure due to the contacts is directly function of the mean volumetric mass of the medium, here simply

Similarly, (35) gives for elastic situations:

In any situation we can define an elastic part and an irreversible part for the stresses,

### 4.3. The modelling of the mean kinetic stress tensor

We consider now the contribution

In this case, it is clear that we can apply the irreversible thermodynamics approach developed in §3.2 for usual two-phase flows, without lasting contacts, similar to the framework of the Prandtl mixing length. The result with a linear approximation will be that again a Boussinesq relation can be proposed for the mean kinetic stress:

Of course, we have defined

It remains then to find a model for the “effective kinetic viscosity coefficient”

In addition, it is shown in §2.2 that the kinetic energy of fluctuations is:

In 1954, R.A.Bagnold has found in his pioneering experiment, for the highly sheared regime called “grain-inertia regime”, a quite similar law (even if the reliability of the experimental results is not so clear, see Hunt et al., 2002). He found the length

How to specify this length scale is a critical point of the modelling. The framework of the Prandtl model does not say anything about it. The experiments of Bagnold have given an empirical law linking it to the solid fraction

It is not conceptually difficult to obtain a balance equation for

The last term in this equation is nothing but the dissipation rate of kinetic energy, due to the irreversible effects of the contacts, that we can note

We see first, if we compare this equation to the one for turbulence kinetic energy, eq. (6), that applying the Prantdl mixing length approach for obtaining the Bagnold formulas (46) needs to neglect the terms

Second, this equation is very similar to the equation for the granular temperature in “granular gases”. For instance we can take the one proposed by S. B. Savage and co-workers (Lun et al, 1984).

The tensor

We notice the close analogy with our equation (48) for

The problem of finding the length scale that plays in these models, taking into account or not that there are “cooperative motions” of groups of grains, remains. We can remark that a model similar to the k-epsilon model is able to address this problem. We could follow the approach of §2.3 for the modelling of the equation for

## 5. Conclusion

We have applied the “Extended Irreversible Thermodynamics” approach in order to built models for three examples of “random media”, and we have found that this approach do justify the bases of classical models, which have been proposed without any reference to Thermodynamics. Of course, this approach gives only the shapes of the laws, and there are constants or non-dimensional functions remaining, to be determined from experiments. The approach can be applied to different models, more or less detailed, for the same kind of situation. We have shown only a few examples, and other models, in other situations can be studied with the same route.