Open access peer-reviewed chapter

Irreversible Thermodynamics and Modelling of Random Media

By Roland Borghi

Submitted: November 21st 2010Reviewed: March 16th 2011Published: September 22nd 2011

DOI: 10.5772/22630

Downloaded: 1154

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:

ρ¯=lim(N)(1Nn=1Nρn)E1

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:

v˜α=lim(N)(n=1Nρnvαn/n=1Nρn)E2
,

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 ρ¯,e˜,v˜αonly, when a non turbulent flow of the same fluid is correctly characterized byρ,e,vα? Of course, the random fluctuations do have some influence. However, in some situations, it may be possible to take into account this influence by some algebraic formulas in terms of the mean variables and their derivatives. Of course, these formulas are only approximations, with a limited domain of validity assessed by experiments.

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, sayk˜, the “length scale” characterizing the two-point correlation of the fluctuations, lt, being given with algebraic formulas. More detailed models use k˜andε, the dissipation rate ofk˜, described by partial differential evolution equations, or k˜andω, a characteristic frequency of turbulence, or the Reynolds stress tensor components themselves andε(Schiestel, 2006).

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 variablesρ¯,e˜,v˜α. We write them with a tensor notation and the Einstein convention of repeated indices. We assume an orthogonal reference frame and then upper or lower indices are equivalent:

t(ρ¯)+xα(ρ¯v˜α)=0E3
t(ρ¯v˜β)+xα(ρ¯v˜αv˜β)=xα(p¯δαβ+τ¯αβρv'αv'β¯)+ρ¯gβE4
t(ρ¯e˜t)+xα(ρ¯v˜αe˜t)=xα(j¯αQpvα¯+ταβvβ¯ρv'αet¯)+ρ¯v˜βgβE5

The total energy here comprises the internal energy, the kinetic energy of the mean motion, and the kinetic energy of turbulent fluctuations, soρ¯e˜t=ρ¯e˜+12ρ¯v˜αv˜α+12ρv'αv'α¯. We have definedv'α=vαv˜α, and soρv'α¯=0. The well known « Reynolds stress tensor » is Rαβsuch that:ρ¯Rαβ=ρv'αv'β¯. Similarlye'=ee˜, butp'=pp¯. The “turbulent kinetic energy” introduced above is nothing but:p'=pp¯.

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

t(ρ¯v˜βv˜β/2)+xα(ρ¯v˜αv˜βv˜β/2)=xα(p¯v˜α+v˜βτ¯αβρv'αv'β¯v˜β)+(p¯δαβτ¯αβ+ρv'αv'β¯)v˜βxα+ρ¯v˜βgβE6

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

t(ρ¯(e˜+k˜))+xα(ρ¯v˜α(e˜+k˜))=xα(j¯αQpv'α¯+ταβv'β¯ρv'αe¯12ρv'αv'βv'β¯)p¯v˜αxα+(τ¯αβρv'αv'β¯)v˜βxαE7

It is possible also to derive directly from the non-averaged momentum equation a balance equation forρ¯k˜=12ρv'αv'α¯. That is done by scalar multiplying the instantaneous (non-averaged) momentum equation by the velocity in order to obtain an equation for the total kinetic energy, and then by subtracting the equation of the kinetic energy of the mean motion (4). One gets:

t(ρ¯k˜)+xα(ρ¯v˜αk˜)=xα(v'βταβ¯v'αp¯12ρv'αv'βv'β¯)ρv'αv'β¯v˜βxα+pv'αxα¯ταβv'βxα¯E8

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

t(ρ¯e˜)+xα(ρ¯v˜αe˜)=xα(j¯αQρv'αe¯)p¯v˜αxαpv'αxα¯+ταβv'βxα¯+τ¯αβv˜βxαE9

The last term ταβv'βxα¯=ρ¯εof (6) is called « volumetric rate of dissipation for turbulence », due to the viscosity (and εis the rate of dissipation by unit of mass). This term is a transfer of energy from k˜toe˜, while the last term of (7) is the dissipation rate of kinetic energy of the mean motion, directly from 12ρ¯v˜αv˜αtoe˜. There is in (6) a term corresponding to a transfer from12ρ¯v˜αv˜α: the term ρv'αv'β¯v˜βxαis called the production term fork˜. The terms p¯v˜αxαpv'αxα¯in (7) are also transfer terms from 12ρ¯v˜αv˜αandk˜, respectively, but these terms are reversible, unlike the last terms due to the viscosity. If the medium is incompressible (i.e.vαxα0), these pressure terms are vanishing. When the velocity of the flow is low, very often the two last terms can be neglected in (7) with respect to the heat flux term (Chassaing et al., 2002).

Equations (1) to (3) are able to calculate the mean variables ρ¯,e˜,v˜αif and only if the turbulent fluxes ρ¯,e˜,v˜αandpvα¯+ταβvβ¯ρv'αet¯, additional to the classical viscous stress and heat conduction flux, can be provided by additional closure expressions, based on theoretical or empirical grounds. Instead of (3), we can use (7), and the transfer terms from k˜to e˜are also to be modelled. As we have previously said, this needs to take into account additional variables that represent in some sense the turbulent fluctuations, and, at least, k˜is one of these new variables. It is seen here that one balance equation for k˜can be found, which can be used provided again that the correlations involving the fluctuations that appear in this equation can be given also by expressions involving onlyρ¯,e˜,v˜α, k˜and eventually other ones…

The simplest model, the “Prandtl mixing length model”, does not use a balance equation for k˜but expresses it directly form the mean velocity gradient. We will see how that is done, and in addition it is postulated that turbulence involves a single (mean) turbulence length scale, k˜, which is simply related to the geometry of the turbulent flow. The second simplest turbulence model, the “Kolmogorov model”, uses the balance equation (6), the unknown terms in it being conveniently modelled with closure assumptions Then, the model provides expressions for the correlations involved in (1) to (6), in particularRαβ, as functions ofρ¯,e˜,v˜α, k˜, ltand their spatial derivatives. The very popular “k-epsilon” model, instead of giving directlylt, assume that the dissipation rate εitself can be calculated by an additional balance equation, postulates such an equation, and writes the correlations as functions ofρ¯,e˜,v˜α, k˜, εand their spatial derivatives.

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 variablesρ¯,e˜,v˜α, given by their balance equations. It is assumed that the balance equation (6) for the kinetic energy of the fluctuating motion can be simplified in keeping only the production and destruction terms:

ρv'αv'β¯v˜βxα+pv'αxα¯=ταβv'βxα¯E10

In addition, the rate of dissipation, which can be calculated through k˜and ltonly, is necessarily written asρ¯ε=Cdk˜3/2/lt, where Cdis a non dimensional constant. This model has been initially proposed for incompressible flow, and the pressure term, with the divergence of velocity fluctuation, is totally negligible in this case, as well as the two pressure terms in (7). Then the turbulence kinetic energy is an algebraic function of the gradient of the mean velocity, and does not need to be calculated with the differential equation (6):

ρ¯Cdk˜3/2ll=ρv'αv'β¯v˜βxαE11

The three basic variables ρ¯,e˜,v˜αbeing given, the entropy function has to be searched as ς=ς(e˜,1/ρ¯)only (the velocity being a non-objective variable has not to appear), with the Gibbs and Euler-Gibbs equations classically written as:

e˜=θςπ1ρ¯+μ,
de˜=θdςπd(1ρ¯)E12

The two functionsθ, πare the partial derivatives of e˜with respect to ςand1/ρ¯, or 1/θand π/θare the partial derivatives of ςwith respect to e˜and1/ρ¯, and the relations 1/ρ¯et π=π(e˜,1/ρ¯)are the laws of state, to be found. Of course, the formula μ=μ(e˜,1/ρ¯)also can be deduced. In addition, these equations of state do satisfy necessarily the Maxwell relation:μ=μ(e˜,1/ρ¯).

The fluid itself, independently of the turbulent flow, has proper equations of state, and if we assume that it is an “ideal gas”, e=CvT+e0,p=ρRTdo hold, with three constants:Cv,e0,R. Then, we can derivee˜=CvT˜+e0,p¯=ρ¯RT˜, just taking the average (weighted average for the first one, classic average for the second one). It appears that if we identify θ=T˜andπ=p¯, we findθ=e˜e0Cv,π=ρ¯Re˜e0Cv, and these relations are convenient equations of state because they satisfy the Maxwell relation : here(1/θ)(1/ρ¯)=(π/θ)e˜=0. The thermodynamic description of the model is complete. From these equations of state we can derive the entropy function, similarly as in the classical case of an ideal gas:(1/θ)(1/ρ¯)=(π/θ)e˜=0. We emphasize that this entropy is not the statistical mean value of the entropy of the fluids=s(e,ρ). Indeed, this mean value cannot be expressed in terms of e˜,ρ¯only, because s=s(e,ρ)is non linear, and is not usable in our modelling framework.

It has to be remarked that the previous development cannot be extended to the case of fluid satisfying a non linear Joule lawe=e(T). In this case, when the fluctuations are not infinitely small, it comes thatρ¯e˜=f(ρ¯T˜,ρT'2¯,ρT'3¯,...), and it is no more possible to limit the model toς=ς(e˜,1/ρ¯). If the nonlinearity of e(T)is important, we have to consider, at least, the additional variableρT'2¯/ρ¯.

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 asde˜=T˜dςp¯d(1ρ¯), allows us to obtain the balance equation for the (new) entropy, showing the term of entropy production, which has to be always positive, and zero at equilibrium. Using (1), (2), (7), (8), one gets (with the “Lagrangian” derivative of the mean motion such thatDDt(.)=t(.)+v˜αxα(.)):

ρ¯DςDt=xα(j¯αQ+ρv'αe¯T˜)+(j¯αQ+ρv'αe¯)1/T˜xα+(τ¯αβρv'αv'β¯T˜)v˜βxαE13

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” μttcan be defined and:

τ¯αβρv'αv'β¯+23ρ¯k˜δαβ=μtt(v˜αxβ+v˜βxα)E14

Considering that the fluid is Newtonian, one has already: τ¯αβ=μ(v˜αxβ+v˜βxα)(here μ=cstandv˜αv¯α), and it appears for the pure turbulent contribution an “eddy viscosity coefficient” μt=μttμsuch as:

ρv'αv'β¯+23ρ¯k˜δαβ=μt(v˜αxβ+v˜βxα)E15

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:

j¯αQρv'αe'¯=(λ+λt)T˜xα=(λ+μtPrt)T˜xαE16

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 asμt=ρ¯Ck˜1/2lt, with (9) it comes:k˜=CCdlt2(v˜αxβ+v˜βxα)v˜βxα=C2Cdlt2(v˜αxβ+v˜βxα)(v˜αxβ+v˜βxα), and we get finally the well known formula of “Prandtl mixing length” :

μt=ρ¯CMLlt2((v˜αxβ+v˜βxα)(v˜αxβ+v˜βxα))1/2E17

In this model, the turbulence length scalelt, called “mixing length”, is given algebraically from the geometry of the flow. The constants CMLand Cdhave been found actual constants when the turbulent fluctuations are well developed, and thenμtμ, but when it is not the case they vary in function of the “turbulence Reynolds number”Ret=μt/μ. Experiments have given such corrective functions.

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 fork˜, based on (6), and a balance equation forε=ε2νk˜1/2xαk˜1/2xα, less firmly based on physics, are used. When turbulence is well established, εε, but the additional term is not negligible close to walls, where viscosity remains playing. There are other models that use equations forlt, or1/τt, the inverse of a time scale of turbulence, instead ofε, and similar discussions could be done for these models.

In this case, the independent variables ρ¯,e˜,k˜,εhave to be taken into account in order to build an entropy function, and we have to search ς=ς(e˜,1/ρ¯,k˜,ε)with:

e˜=θςπ1ρ¯ykk˜yεε+μ,
de˜=θdςπd(1ρ¯)ykdk˜yεdεE18
Hereθ=θ(e˜,1/ρ¯,k˜,ε), π=π(e˜,1/ρ¯,k˜,ε), yk=yk(e˜,1/ρ¯,k˜,ε), yε=yε(e˜,1/ρ¯,k˜,ε)are the new equations of state to be found, satisfying the six corresponding Maxwell relations. We will have to discuss the finding of these equations of state in connexion with the discussion of the second principle. Concerning the εequation, we use here the general following form, with a diffusion term and production and destruction terms:
t(ρ¯ε)+xα(ρ¯v˜αε)=xα(Jεα)ρ¯C32ε3v˜αxαρ¯C1εk˜Rdαβv˜βxα+C4εk˜pv'αxα¯ρ¯C2εεk˜E19

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,C1=1.44,C2=1.92,C3=1.44,C4=0.

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

ρ¯DςDt=xα(j¯αQ+ρv'αe¯θykJkα+yεJεαθ)+(j¯αQ+ρv'αe¯)1/θxα++Jkαyk/θxα+Jεαyε/θxα+1θ(πp¯23ρ¯k˜(yk+C3εyεk˜))v˜γxγ1ykC4εyε/k˜θpv'βxβ¯yk+C1εyε/k˜θρ¯Rdαβv˜βxα++1θτ¯dαβv˜βxα+13θτ¯γγv˜αxα+1ykC2yεε/k˜θρ¯εE20

We have written as τdαβand Rdαβthe deviators of the tensors ταβandRαβ, and Jkαis the diffusion term of (6). We recognize in this equation a diffusion term, the first on the right hand side, and all the following terms do constitute entropy production terms.

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 π=p¯+23ρ¯k˜yk+23ρ¯C3yεεand1ykC4εk˜yε=0.

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 (yk+C2εk˜yε)is due to the destruction terms of the equations for k˜andε. Globally, this term has to be positive, but the contribution of k˜and εmay be dangerous for that. This problem is cancelled by choosingyk+C2εk˜yε=0, and that corresponds to a “mutual equilibrium of destructions rates” for k˜andε. Assuming this mutual equilibrium ensures that no additional irreversibility is brought by this new kind of model.

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 1ykC4εk˜yε=0does not matter in this circumstance.

Taking into account yk+C2εk˜yε=0andπ=ρ¯RT˜+23ρ¯k˜yk+23ρ¯C3yεε, e˜=CvT˜+e0as well as the six Maxwell relations (saying that the second partial derivatives of ς=ς(e˜,1/ρ¯,k˜,ε)are identical when the derivation variables are commuted), it is possible with a few algebra to find a convenient set of four equations of state only in the case where C3/C2is a constant (possibly zero). The solution is:

π=ρ¯Rθ, e˜e0=Cvθ(12Rk3R+2RkC33RC2)=CvT˜, yk=Rkθk˜and
yε=RkθC2εE21
,

where yε=RkθC2εis a positive constant (with ykchosen positive, without loss of generality). These formulas are not the unique solution of the problem, but they constitute a solution that can be supported by measurements, as approximations, at least in a certain limited range of variation of of the variables. One can remark that for keeping θpositive, which is necessary in thermodynamics, it is necessary that12Rk3R(1C3C2).

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.

ρ¯DςDtxα(j¯αQ+ρv'αe¯θ+ykJkα+yεJεαθ)=(j¯αQ+ρv'αe¯)1/θxα+Jkαyk/θxα+Jεαyε/θxα1yk+ykC4/C2θpv'βxβ¯yk1C1/C2θρ¯Rdαβv˜βxα+1θτ¯dαβv˜βxα+13θτ¯γγv˜αxα+1θρ¯εE22
-

We know already that τγγ=μvvαxαand τdαβ=μ(vβxα+vβxαδαβ3vγxγ)with μvand μvpositive, in such a way that ταβvβxαis always positive, or zero if the velocity gradients are zero. We remember also thatρ¯ε=ταβv'βxα¯, and then the three last term can be grouped giving1θταβvβxα¯.Then, the group of the three last term of (16) is clearly always positive, by definition, even if the two first ones are not positive, depending on the fluctuations of density.

Secondly, the term ykθ(1C1/C2)ρ¯Rdαβv˜βxαconcerning the influence of turbulent friction has also to be positive always, or zero. At the same time, we remember that ρ¯Rdαβv˜βxαis a production term for k˜and has to be essentially positive, although not necessarily everywhere. With C2,C1constant, it is implied that 1C1/C2is necessarily positive and then the linear irreversible thermodynamics again supports the Boussinesq relation, here written for compressible flows:ρ¯Rdαβ=μt(v˜βxα+v˜βxα13v˜γxγδαβ), with μtpositive.

Here again, we have to writeμt=ρ¯Clk˜1/2lt=ρ¯Cμk˜2/ε, Cμbeing positive, constant or function of the Reynolds number of turbulence, defined here asRet=k˜2/εμ, if this number is sufficiently small. This is exactly what is used in all versions of the k-epsilon model. For versions able to consider small turbulence Reynolds number, 1C1/C2may become negative, and in this case this term has to be smaller than the three last terms corresponding to1θταβvβxα¯1θταβ¯v˜βxα.

The last point deals with the terms(j¯αQ+ρv'αe¯)1/θxα+Jkαyk/θxα+Jεαyε/θxα. Again, we see that the modelling of JkαandJεα, as well as (j¯αQ+ρv'αe¯)can be done in terms ofk˜xα, εxαandθxα, with possible coupling. No coupling has been attempted so far, however. The coefficients of diffusivity of turbulence kinetic energy, dissipation rate, and heat are all proportional to μtpreviously defined.

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 C2C1do correspond with the common practice for large turbulence. The value of C3is not well known, because the flows where v˜αxα0are more complex, but we have shown that we need v˜αxα0even if the C’s are varying. ConsideringC4, we have found that1Rkθk˜(1C4C2)=0. That implies then a constraint forC4, which isC4=C2k˜C2θRk. In case where the influence of k˜/θis poor, C4=C2simply. However, it remains to find an approximation for pv'αxα¯when it is not negligible. In practice, this quantity plays only in flows wherev'αxα0, which are less well studied turbulent flows, and the experiments up to now have only confirmed that C4has not to be put to zero...

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” Φi(x,t)which is unity within the phase i and zero outside, and the statistical average Φ¯i(x,t)is nothing but the probability of presence of this phase. Of course,i=1,2Φ¯i(x,t)=1. For each phase, we defineρ¯i=Φiρi¯=Φ¯iρ¯i, Φiρiviα¯=ρ¯iΦ¯iv˜iα,Φiρiviα¯=ρ¯iΦ¯iv˜iα.

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:

t(ρ¯iΦ¯i)+xα(ρ¯iΦ¯iv˜iα)=ρivriαniασs¯E23

The last term on the R.H.S is the possible exchange of mass between the two phases: vriαis the velocity of the phase i relative to the interface, non zero only when there is exchange of mass between phases (vaporization-condensation for instance) ; σsis the instantaneous interface area by unit of volume of the medium, mathematically defined byniασs=Φixα, where niαis the normal to the interface, oriented outward phase i. Of course, the gradient of the phase indicator has the structure of a Dirac peak, and σsis zero except on the interface (Kataoka, 1986). The averaged equation for the momentum of phase i is.

t(ρ¯iΦ¯iv˜iβ)+xα(ρ¯iΦ¯iv˜iαv˜iβ)=xα(Φ¯iσ¯iαβρiΦiv'iαv'iβ¯)+ρ¯iΦ¯igβρivriαniαviβσs¯+σiαβniασs¯E24

The pseudo turbulent momentum diffusion flux isρiΦv'iαv'iβ¯. The two last terms of (18) represent the momentum given to the phase i by the other one, by exchange of mass and by the contact force, respectively. The tensor σiαβis the Cauchy stress tensor within the phase i. The equation for the mean total energy of phase i is:

t(ρ¯iΦ¯ie˜ti)+xα(ρ¯iΦ¯iv˜iαe˜ti)=xα(Φ¯ijiQα¯+Φ¯iσiαβ¯v˜iβ+Φiσiαβv'iα¯ρiΦiv'iαeti¯)+ρ¯iΦ¯iv˜iβgβρivriαniαetiσs¯+σiαβniαviβσs¯jiQαniασs¯E25

The total energy is defined as, the pseudo-turbulent energy flux ise˜ti=e˜i+ρiΦiviαviα¯/2ρ¯i=e˜i+v˜iαv˜iα/2+ρiΦiv'iαv'iα¯/2ρ¯i, and the last three terms are the exchanges of energy from the other phase, due to exchange of mass, power of contact force, and heat flux, respectively.

It is possible also to write a balance equation for the volume of phases, simply obtained from the convection equation of the field ofρiΦv'iαeti¯:Φi(x,t). When averaged, this equation gives:

tΦi+vsαxαΦi=tΦi+(viαvriα)xαΦi=0E27

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:

t(Φ¯i)+v˜iαxα(Φ¯i)=vi'αΦixα¯+vriαΦixα¯E28

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:

i=1,2(ρivriαniα)s=0,i=1,2(ρivriαniαviβ+σiαβniα)s=0,i=1,2(ρivriαniαeti+σiαβniαviβjiQαniα)s=0E29

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 vr1α=vr2α=0,(σ1αβn1α)s+(σ2αβn2α)s=0,(σ1αβn1αv1β)s+(σ2αβn2αv2β)s=0,(j1Qαn1α)s+(j2Qαn2α)s=0and σiαβas function ofjiQα. We will notice here ρi,viα,Tifor any type of phase (but usually for a solid phase the pressure is defined by the third of the trace ofσiαβ=piδαβ+τiαβ, with an opposite sign). For a fluid phase, σiαβis the viscous stress tensor, which can be assumed Newtonian for instance, and an equation of state givesτiαβ. For solid particles, we are not concerned with the full elasticity properties and we can limit the description as a “quasi-liquid”, very viscous. For any type of phase the Fourier law for heat conduction can be adopted and a Joule law linking the temperature to the internal energy can be considered, neglecting the energetic aspects of compressibility for liquid or solid phases. Very often it suffices to takepi=pi(ρi,Ti).

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: ρi=csthave to be represented by their own balance equations. The development then follows the same path as § 2.2 above, but with two phases i=1, i=2.

The kinetic energies of fluctuations, namelyρ¯i,v˜iα,e˜i,Φ¯i,i=1,2, follow a balance equation that can be found from the primitive equations. It writes, without exchange of mass:

ki=+ρiΦiv'iαv'iα¯/2ρ¯i,i=1,2E30

We do not consider here, in the Prandtl mixing length framework, that each t(ρ¯iΦ¯iki)+xα(ρ¯iΦ¯ikiv˜iα)=xα(ρiΦiv'iβv'iβv'α/2¯+Φiv'iβσiαβ¯)ρiΦiv'iαv'iβ¯v˜iβxα+piΦiv'iαxα¯ρ¯iΦ¯iεi+σiαβniβv'iασs¯does follow this full equation, we assume that it can be reduced to a local balance between production terms and destructions terms, giving simply that :

kiE31

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 variablesρiΦiv'iαv'iβ¯v˜iβxα+piΦiv'iαxα¯+σiαβniβv'iασs¯=ρ¯iΦ¯iεi, such asρ¯i,v˜iα,e˜i,Φ¯i,i=1,2, because entropy is an extensive quantity. The entropies of each phase are functions only of the variables of this phase, unlike molecular mixtures:ρ¯ς=i=1,2ρ¯iΦ¯iςi. As in § 2.2, we can adopt

ςi=ςi(ρ¯i,e˜i),
e˜i=T˜iςip¯i1ρ¯i+μiE32

Similarly, we take linear equations of state for phases, which can be averaged without needing additional variables. For a gaseous phasesde˜i=T˜idςip¯id(1ρ¯i), and for liquid or solid phasese˜i=CviT˜i+ei0,p¯i=ρ¯iRiT˜i, neglecting the influence of compressibility on the internal energy.

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

e˜s=CsT˜s+es0,p¯s=Ks(ρ¯sρ0)E33

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:

ρ¯DςDt=iρ¯iΦ¯iDςiDtixα(iρ¯iΦ¯iςi(v˜iαv˜α))E34

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

DDtie˜i=T˜iDDtiςip¯iDDti(1ρ¯i), without exchange of mass between phases. By using (20), we obtain as well:
ρ¯iΦ¯iDDti(1ρ¯i)=Φ¯iρ¯iDρ¯iDti=Φ¯iv˜iαxα+DΦ¯iDtiE35

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:

ρ¯iΦ¯iDDti(1ρ¯i)=Φ¯iv˜iαxαv'iαΦixα¯E36

With (23’), it gives:

ρ¯iΦ¯iDe˜iDti=xα(Φ¯ijiQα¯ρiΦieiv'iα¯)Φ¯ip¯iv˜iαxα+Φ¯iτiαβ¯v˜iβxαpiΦiv'iβxβ¯jiQαniασs¯+ρ¯iΦ¯iεi
ρ¯iΦ¯iDe˜iDti=xα(Φ¯ijiQα¯ρiΦieiv'iα¯)Φ¯ip¯iv˜iαxα+Φ¯i(τiαβ¯ρ¯iRiαβ)v˜iβxαE37
,

whereρ¯iΦ¯iDe˜iDti=xα(Φ¯ijiQα¯ρiΦieiv'iα¯)Φ¯ip¯iv˜iαxα+Φ¯i(τiαβ¯ρ¯iRiαβ)v˜iβxαis the deviator of the Reynolds tensor for phase i. The two last terms take into account the exchange of heat by conduction Ridαβand the exchange of kinetic energy by contact forceφ¯iQ=jiQαniασs¯, respectively, from the other phase. In the momentum balance (18) the total contact force has to be split inσiαβniβv'iασs¯. The first part is simply the influence of the variation of the volume of the phase, while σiαβniβσs¯=+p¯Φ¯ixα+F¯si×βis really the force from the other phase, and the jump condition (22) givesF¯si×β=τi×αβniασs¯. It follows thati=1,2F¯si×β=0, giving finally:

σiαβniβv'iασs¯=+p¯v'iαΦixα¯+Fsi×αv'iα¯E38

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

σiαβniβv'iασs¯=+p¯v'iαΦixα¯+Fsi×αv'iα¯E39

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. Asρ¯DςDt=xα(i(ρ¯iΦ¯iςi(v˜iαv˜α)+Φ¯iT˜ijiQα¯+ρiΦieiv'iα¯T˜i))+i(Φ¯ijiQα¯+ρiΦieiv'iα¯)xα(1T˜i)+iΦ¯iT˜i(τiαβ¯ρ¯iRiαβ)v˜iβxα+iFsi×βv'iβ¯T˜iip¯ip¯T˜iv'iαΦixα¯+iφ¯iQT˜i, this term can be written as:ρ¯DςDt=xα(i(ρ¯iΦ¯iςi(v˜iαv˜α)+Φ¯iT˜ijiQα¯+ρiΦieiv'iα¯T˜i))+i(Φ¯ijiQα¯+ρiΦieiv'iα¯)xα(1T˜i)+iΦ¯iT˜i(τiαβ¯ρ¯iRiαβ)v˜iβxα+iFsi×βv'iβ¯T˜iip¯ip¯T˜iv'iαΦixα¯+iφ¯iQT˜i, and the linear classical irreversible thermodynamics proposesρ¯DςDt=xα(i(ρ¯iΦ¯iςi(v˜iαv˜α)+Φ¯iT˜ijiQα¯+ρiΦieiv'iα¯T˜i))+i(Φ¯ijiQα¯+ρiΦieiv'iα¯)xα(1T˜i)+iΦ¯iT˜i(τiαβ¯ρ¯iRiαβ)v˜iβxα+iFsi×βv'iβ¯T˜iip¯ip¯T˜iv'iαΦixα¯+iφ¯iQT˜i.

The second last term involvesφ¯1Q(1T˜11T˜2), expression implied in the closure of (20), and we see that assuming that v'iαΦixα¯eliminates this term and at the same time the need of (20). Instead of this assumption, the linear irreversible thermodynamics suggests the hypotheses that p¯i=p¯and gives a closure for (20). The phase volume fraction can be calculated then withv'iαΦixα¯(p¯ip¯). Such an equation is almost identical to the equation proposed by M. Baer and I. Nunziato for compressed granular medium (Baer & Nunziato, 1986).

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 t(Φ¯i)+v˜iαxα(Φ¯i)=v'iαΦixα¯=K(p¯ip¯)is connected with the exchange of momentum between phases. The “jump relations” (22), similarly asiFsi×βv'iβ¯T˜i=i1T˜i(Fsi×βviβ¯F¯si×βv˜iβ), gives i=1,2F¯si×β=0(and gavei=1,2Fsi×βviβ¯=0previously used). Because at the interface between the phases the velocities of phases are equal, an averaged interface velocity i=1,2φ¯iQ=0can be defined such thatv¯sβ, and the related source term of entropy can be writtenFsi×βviβ¯=F¯si×βv¯sβ. The classical linear irreversible thermodynamics leads toi1T˜iF¯si×β(v¯sβv˜iβ), and withi1T˜iF¯si×β(v¯sβv˜iβ), that gives an expression for the averaged interface velocity :F¯s1×β=F¯s2×β. Thev¯sβ=K1v˜1β+K2v˜2βK1+K2are positive constants, depending on the characteristics of the two phases in contact, to be found with experiments. If we consider the possibility of a coupling, in agreement with the so called Curie theorem, we find that it is possible that the contact force Kito the phase i is linked to the gradients of temperature within the phases. We recover here a “thermophoretic “effect.

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+1F¯si×β, but we are interested only in the knowledge ofρ¯i,v˜iα, andΦ¯g=i=1,NΦ¯i,ρ¯gΦ¯g=i=1,NΦiρi¯,ρ¯gv˜gαΦ¯g=i=1,NρiΦiviα¯. From (17), (18), we get the mass and momentum balance equations of the “grains phase” as:

ρ¯fρ¯N+1,v˜fαv˜N+1αE40

Here the mean intrinsic density of the grains t(ρ¯gΦ¯g)+xα(ρ¯gΦ¯gv˜gα)=0can be considered generally as known and constant. If all the grains are made with the same almost incompressible solid, ρ¯gis the density of this solidρ¯g.

ρ¯g=ρsE41

We see that the effective Cauchy stress tensor for the grains has two parts:t(ρ¯gΦ¯gv˜gβ)+xα(ρ¯gΦ¯gv˜gαv˜gβ)=xα(i=1,NΦ¯iσ¯iαβi=1,NρiΦiv''iαv''iβ¯)+ρ¯gΦ¯ggβ+i=1,Nσiαβniασsi¯. The first one takes into account the stresses within the grains but mainly also the influences of the lasting contacts, with friction between grains depending on pertinent laws, and we can call it “mean Cauchy stress of lasting contacts”; the second one is due to the velocity fluctuations, here defined with respect to the averaged velocity of all the grains:σ¯tgαβ=i=1,NΦ¯iσ¯iαβi=1,NρiΦiv''iαv''iβ¯, and is similar to the Reynolds tensor for turbulent flows, and we will call it “mean kinetic stress”. It is important only when the motion of the medium is well established, but even with slow mean motion there are some large and sudden fluctuations of velocity. The modelling of both terms has to be studied with care. Without models, eq.(31) is without interest.

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 havev''iα=viαv˜gα, where i=1,Nσiαβniασsi¯=σfαβnfασsf¯=p¯fΦ¯fxβ+Ffβ¯is the pure contact force of the fluid on the grains.

When the fluid is air, or any gas whose intrinsic density is very small with respect of the solid, Ffβ¯, we have ρ¯fρsandρ¯gΦ¯gρ¯. In addition, the contact force and the gradient of fluid pressure (which is of the order ofρ¯gΦ¯gv˜gαρ¯v˜α) are negligible in (31). Then equation (30) is nothing but the averaged continuity equation for the entire medium, and (31) gives the averaged momentum balance for the entire medium:

ρ¯fgβρ¯ggβE42

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:

t(ρ¯v˜β)+xα(ρ¯v˜αv˜β)=xα(i=1,NΦ¯iσ¯iαβΦ¯fp¯fδαβi=1,NρiΦiv''iαv''iβ¯)+ρ¯gβE43

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 astρ¯e˜+xγ(ρ¯v˜γe˜)=xγ(φ¯Qtγ)+(i=1,NΦ¯iσ¯iαβΦ¯fp¯fδαβi=1,NρiΦiv''iαv''iβ¯))v˜βxα.

The modelling of the lasting contact stress tensor p¯fxβ+ρ¯fgβ=0has been studied in details in (Borghi & Bonelli, 2007). We will only summarize their results in the next paragraph, but we will develop the question how the irreversible thermodynamics can help this modelling. The kinetic contribution, due to the velocities fluctuations, can be modelled with the principles that have been described in the first part of this chapter, and we will discuss this point in §4.3.

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

The approach of Borghi and Bonelli gives an evolution equation forσ¯cgαβ=i=1,NΦ¯iσ¯iαβ. The grains are assumed elastic, with small strains but large possible displacements, and the contacts are displaying partly a reversible behaviour, as proposed by B. Cambou (Cambou,1998, Emeriault & Cambou, 1996), and partly irreversible sliding. The obtained evolution equation displays the rotational objective derivative of Jaumann, without any additional assumption:

σ¯cgαβE44

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, tσ¯cgαβ+xγ(σ¯cgαβv˜gγ)=xγ(σcgαβv'gγ¯)+(c˜¯e)αβγδ:(Φ¯g(v¯gδxγ)symD¯pγδD¯fγδ)2(σgαγωcγβ¯)sym2(σgαγ.Q'gγβ¯)sym2(σ¯gαγ.Q¯gγβ)symbeing the averaged rotation rate tensor of the medium, mainly related toQ¯gγβ. The tensor Q¯gγβ(fourth order) is an effective elastic stiffness tensor of the medium, due to the elastic behaviour within the grains and also particularly of the lasting contacts. The tensor c˜¯eis due to the irreversible sliding in the contact zone between two grains, and D¯pis related to the change of shape of “void volume” between the grains. The two following terms are due to the correlations of the fluctuations of the Cauchy stress with the ones of the rotation rates of the contacts and of the medium, respectively. They are not expected to be important terms. All the details concerning the definition of these terms can be found in (Borghi & Bonelli, 2007). Similarly as for turbulent and two-phase flows previously discussed, most of these terms, even when their physical meaning is clear, are not under a closed form in this equation, because the influence of fluctuations are imbedded within them, and they have to be modelled. We can remark that both the unweighted and weighted averaged velocities appeared, but when D¯fthese two velocities are equal.

By definition ρ¯g=ρsis an averaged quantity for solid grains,σ¯cgαβ, we can define as well an intrinsic mean contact stress tensor asσ¯cgαβ=i=1,NΦ¯iσ¯iαβ. Equation (33), together with (30), can give an equation forΦ¯gσ¯cgαβ=σ¯cgαβ=i=1,NΦ¯iσ¯iαβ, and also equations for the trace and deviator of this tensor. The elastic stiffness tensor σ¯cgαβhas been formally defined, but it cannot be written explicitly, only numerical calculations have been done (Emeriault & Cambou, 1996). Even its dependence on c˜¯ecannot be theoretically written. For a statistically homogeneous and isotropic granular medium submitted to an isotropic stress, it will result that this elastic stiffness tensor can be represented with simply an effective compression modulus and an effective shear modulus, sayΦ¯g. These quantities are not necessarily constant, and in particular they depend onχe,Ge, in an unknown manner. We will consider in the sequel this case for simplicity, but the general case can be developed as well. Then, the equation for the trace and for the deviator of Φ¯gare simply written in terms ofσ¯cgαβ. If we take into account (30), remembering that we consider χe,Geas constant and definingρ¯g=ρs, it comes:

πg=trace(σ¯g)/3E45

The last term groups all the irreversible effects of sliding contacts,Φ¯gddtπ¯g=xα(πgΦgv'gα¯)3χeΦ¯gv˜gαxα+I¯.

I¯=tr(c¯e:(D¯p+D¯f))/3E46
Φ¯gddtσ¯cdαβ=xγ(σcdαβΦgv'gγ¯)+2GeΦ¯g(v¯gβxα)symdI¯dαβ2Φ¯g(σ¯gdαγQ¯gγβ)skewis a second order tensor, without trace, taking into account of all the irreversible effects resulting in production and destruction for the effective Cauchy tensor of contacts. Now the modelling of I¯dαβand I¯has to be discussed, and the extended irreversible thermodynamics will help us a lot.

The granular medium is described by the variablesI¯dαβ, for which we know balance equations, although some terms remain to be closed in it. First we will consider for simplicity only the case where the medium is “quasi-static”. It is possible then to neglect all terms related to the fluctuations of velocity, and we can concentrate our attention on the closure of equation for the mean Cauchy stress of contacts, again in the framework of the Extended Irreversible Thermodynamics.

In this case, the entropy is simply:e˜,ρ¯,π¯g,σ¯cdαβ,v˜α, and we can writes the Euler-Gibbs relation, introducing the pertinent partial derivatives:ς=ς(e˜,1/ρ¯,π¯g,σ¯gdαβ).

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:

dς=1Tde˜+pTd(1ρ¯)ypTdπ¯gydαβTdσ¯gdαβE47

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:

ρ¯ddtς=xγ(φ¯QγT)+φ¯Qγxγ(1T)++ρ¯Tσ¯cdαβρs(v˜gβxα)symdρ¯Tπ¯gρsv˜gαxα1T(Φ¯fp¯fp)v˜gαxαypT(3χeρ¯v˜gαxα+ρsI¯)ydαβT(2Geρ¯(v¯gβxα)symdρsI¯dαβ), p=Φ¯fp¯f,
p=Φ¯fp¯fE48

Then, the Maxwell relations give the form of the forth needed equation of state:3χeyp=π¯gρs.

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

e˜=(π¯g)26χeρs+σ¯cdαβσ¯cdβα4Geρs+fct(T)E49

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:

ρ¯Pς=φ¯Qγxγ(1T)π¯gI¯3Tχe+σ¯cdαβIdαβ2TGe,
I¯=3χeηvΦ¯gπ¯gE50
.

The coefficients I¯dαβ=2GeηΦ¯gσ¯cdαβare phenomenological viscosity coefficients, necessarily positive. The result would then be very similar to the old Maxwell model for visco-elastic materials.

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):

η,ηvE51
I¯=H(πcπ¯g)3χeηvΦ¯g|v¯gα/xα|πcπ¯gE52

The physical meaning of the discontinuity involved in (39) is that when the medium is compressed (then I¯dαβ=H[(σ¯cdαβσ¯cdαβ)1/2(k+sinϕ|π¯g|)]2GeηΦ¯g(v¯gβ/xα)symdk+sinϕ|π¯g|σ¯cdαβis positive) more than the positive valueπ¯g, the medium behaves very similarly to an elastic compressed solid, but when the medium is less compressed, or expanded, the normal contacts are lost in such a way that the "contact-pressure" decreases exponentially to zero due toπc.

With (40), the medium will remain “quasi-elastic” below one plastic limit defined by k and the angleI¯. The existence of this limit is strongly suggested from numerical experiments with Discrete Elements Method, and experimental finding give that the medium may experience a “solid-fluid transition”. The Mohr-Coulomb law displayed here is given just as example. The appearance of the solid fraction in ϕis expected because they involve the mean interface area between grains per unit of volume, and this quantity increases with the fraction of grains. The simple linear dependence in this quantity may be replaced by an empirically found formula.

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 I¯,I¯dαβis very large), (35) and (36) are reduced into algebraic (then local) equations, giving simply:

(v¯gβ/xα)symd, and
σ¯cdαβ=η(k+sinϕ|π¯g|)(v¯gβ/xα)symd(v¯gβ/xα)symdE53

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 π¯g=ηvπc(v¯gα/xα)|v¯gα/xα|and deduce the equations for these elastic stresses.

In this case, (34) is reduced toπ¯g=π¯e,σ¯cdαβ=σ¯edαβ. As (30) givesddtπ¯e=3χev˜gαxα, it follows that ddt(ρ¯)=ρ¯v˜gαxαand it gives nothing but that:

dπ¯edt=3χeρ¯dρ¯dtE54

That means that the effective pressure due to the contacts is directly function of the mean volumetric mass of the medium, here simplyπ¯g=3χeρ¯dρ¯. Of course, the effective compression modulus may be also function of the mean volumetric mass, and so this relationship is not simply logarithmic. Physically, the effective pressure goes to infinity when the compaction goes to the maximum possible (to be empirically known), and becomes zero when the grains do not experience lasting contacts. There is again in this case a pressure between grains due to short collisions of fluctuating grains, but this pressure is taken into account in the second part of the total mean Cauchy stress tensor, which we will consider in the following paragraph.

Similarly, (35) gives for elastic situations:

ρ¯=ρsΦ¯gE55

In any situation we can define an elastic part and an irreversible part for the stresses, ddtσ¯edαβ=+2Ge(v¯gβxα)symd2(σ¯edαγ(v¯gγxα)skewd)skew, where equations (41) and (42) give the elastic parts and the total is given by (35) and (36) with (39) and (40).

4.3. The modelling of the mean kinetic stress tensor

We consider now the contributionπ¯g=π¯e+π¯i,σ¯cdαβ=σ¯edαβ+σ¯idαβ, due to the fluctuations of the local velocities within the grains with respect to the mean velocity i=1,NρiΦiv''iαv''iβ¯of all the grains. This term is quite similar to the Reynolds stress for the turbulent flows, and that suggests that a model similar to turbulent models can be proposed. We will study now this modelling, considering that we have left the quasi static regime, and even that the motion of the granular medium is sufficiently fast in such a way that the total mean Cauchy stress tensor is dominated by the kinetic stress.

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:

vgαE56

Of course, we have definedi=1,NρiΦiv''iαv''iβ¯ρsΦ¯g23δαβkg=ρsΦ¯gνgeff(v˜gαxβ+v˜gβxα23δαβv˜gγxγ). This quantity is the kinetic energy of fluctuations for the grains, related to what is called “granular temperature” (with a relation likeρsΦ¯gkg=12i=1,NρiΦiv''iαv''iα¯) in the framework of granular gases. At the same time, this quantity is directly related to the normal mean kinetic stress:kg=RkTG.

It remains then to find a model for the “effective kinetic viscosity coefficient”P=+2ρsΦ¯gkg, and following again the same path a model similar to the Prandtl mixing length is necessarily found. Similarly as in §2.2, transposing formula (13), we have:

νgeffE57

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

νgeff=Cglt2((v˜gαxβ+v˜gβxα)(v˜gαxβ+v˜gβxα))1/2E58
kg=CBlt2(v˜αxβ+v˜βxα)(v˜αxβ+v˜βxα)are constants. In a simple case of sheared granular medium, that gives for the normal and the shear stressesCB,Cg:
P,TE59

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 P=12CBρ¯lt2(|Vy|)2,T/P=Cg/(2CB)was given proportionally to the mean diameter of the grains, aslt, (Bagnold, 1954).

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 fractionlt=d/((Φ¯gmax/Φ¯g)1/31), and to the grain diameter, but for turbulent flow, it has been found that this length scale is only related to the size of the experimental device. That is because the flow field imposes to the fluctuations its own scale. We can suspect that this can be true also for granular flows, meaning that the fluctuations do concern not simply each grain independently of the others, but some times, in some places, do concern groups of grains, whose size depends on the geometry of the flow. In this case, the law for the mixing length would necessarily take into account this geometry, as well to the grain diameter and the solid fraction…

It is not conceptually difficult to obtain a balance equation forΦ¯g, following the same route as for two phase flows. Here, we can continue to neglect the friction force of fluid phase, but we keep the lasting contact Cauchy stress, which is the only source of dissipation of kinetic energy. Transposing (6), and recognizing the elastic parts and irreversible parts of the mean contact stresseskg, we get:

σcgαβ=σegαβ+σigαβ,σcdαβ=σedαβ+σidαβ,πg=πe+πiE60

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 notet(ρ¯k˜g)+xα(ρ¯v˜gαk˜g)=xα(v"gβΦgσcdαβ¯v"gαΦgπg¯Φ¯fv"fαpf¯12ρsΦgv"gαv"gβv"gβ¯)i=1,NρiΦiv"iαv"iβ¯v˜gβxα+πeΦgv'gαxα¯+pfΦfvfxα¯Φ¯fp¯fv˜gαxασedαβΦgv'gβxα¯σigαβΦgv'gβxα¯. The first term on the R.H.S. is a classical dispersion term, the second one is a production term directly related to the gradient of the mean velocity, which appears classical also. The other terms are responsible for transfer of kinetic energy from the elasticity of the grains phase or the fluid phase. They are not in closed form, because they involve fluctuations, exceptρ¯εg, and their modelling is to be studied. One can already propose that the fluid motion between the grain, with very low velocity, is divergence free, thenΦ¯fp¯fv˜gαxα. We can then rewrite formally:

pfΦfvfxα¯0E61

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 t(ρ¯k˜g)+xα(ρ¯v˜gαk˜g)=xα(jkα)(i=1,NρiΦiv"iαv"iβ¯)v˜gβxα++πeΦgv'gαxα¯σedαβΦgv'gβxα¯Φ¯fp¯fv˜gαxαρ¯εgwithin (48). Indeed, the two first of these terms are due to lasting contacts between grains, and they are expected to be less and less efficient when the solid fraction decreases. The term due to the fluid phase is also expected to be small when the fluid is not submitted to compression or expansion.

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).

+πeΦgv'gαxα¯σedαβΦgv'gβxα¯Φ¯fp¯fv˜gαxαE62

The tensor +πeΦgv'gαxα¯σedαβΦgv'gβxα¯Φ¯fp¯fv˜gαxαis what we have called in §4.1 the kinetic stress, and the constitutive relation “formula” is given by the classical theory of granular gasesΣαβ. The first diagonal contribution in this stress tensor is simply proportional to the granular temperature itself, the two last ones are linked to the mean velocity gradient with effective viscosity coefficientsΣαβ=(ρ¯TG(1+4Φ¯gηφ0(Φ¯g))δαβ+ρ¯η(v˜gβxα+v˜gαxβ)ρ¯ηvδαβv˜gγxγ), which the theory relates to the variablesη,ηv. The first diagonal term involves a reversible contribution which is nothing butTG,d,Φ¯g, but involves also an irreversible contribution related toρ¯TG=ρsΦ¯gTG. The coefficients ηin (49) are also diffusion coefficients proportional toK,D. The last term of (49) is a destruction term due to inelastic collisions, and the classical theory givesη,ηv. The functionsΓ=φ5(Φ¯g)TG1/2dare non dimensional functions of the solid fraction only.

We notice the close analogy with our equation (48) forφ0,φ5. We recognize the dispersion flux, the production term due to mean velocity gradient, and the last term is the dissipation rate. Similarly to the Prandtl model, the kinetic stress is related to the gradient of mean velocity with the Boussinesq relation, and the dissipation rate is calculated proportional toρ¯k˜g, with the length scale TG3/2ltlinked to the mean diameter of the grains. There is also a dependence on the solid fraction, which is due to the occupation of the space when there are many grains. We get in (48) additional terms due to the elastic properties of the grain phase, which are not considered in granular gases because the lasting contacts are not considered, only instantaneous collisions are supposed to play. We can then consider that (48) agrees with the classical developments of the theory of granular gases and extend their validity for granular medium where the solid fraction is sufficiently high and the global velocity is low, in such a way that the enduring contacts can no more be neglected.

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 ltand of an equation fork˜g, from which we can extract a length scale. The irreversible thermodynamics will hep us, as shown in §2.3, for choosing the terms and the coefficients of these balance equations.

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.

© 2011 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike-3.0 License, which permits use, distribution and reproduction for non-commercial purposes, provided the original is properly cited and derivative works building on this content are distributed under the same license.

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Roland Borghi (September 22nd 2011). Irreversible Thermodynamics and Modelling of Random Media, Thermodynamics - Kinetics of Dynamic Systems, Juan Carlos Moreno Piraján, IntechOpen, DOI: 10.5772/22630. Available from:

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