An assessment of centered finite volume methods for applications to one dimensional multiphase flows

Contents

2.4 Disperse phase number continuity . . . 10

2.5 Conservation of momentum . . . 11

2.6 Conservation of energy . . . 13

2.7 Mixed fluid . . . 15

2.8 Equations of state for liquid phase . . . 16

3 Centred methods for hyperbolic conservation laws 17 3.1 Introduction . . . 17

3.2 The second order scheme . . . 19

3.3 Slope limiters . . . 23

3.4 Stability conditions . . . 24

3.5 Hyperbolic equations and source terms . . . 27

4 Numerical tests 29 4.1 Burgers equation . . . 29

4.1.1 Periodical initial condition . . . 30

4.1.2 Riemann initial condition . . . 32

4.2 Euler equations . . . 33

4.3 Conduit with variable section . . . 40

5 Applications to multiphase fluids 45 5.1 Two phase flow with six governing equations . . . 45

5.1.1 Toumi’s shock tube test . . . 46

5.1.2 Water faucet . . . 47

Summary and Keywords

Summary

In questa tesi si studia una famiglia di metodi ai volumi finiti centrati, che sembrano essere di particolare interesse per applicazioni a problemi di flu-idodinamica multifase. Dopo una prima validazione dell’implementazione di tali metodi realizzata nell’ambito della tesi, i metodi sono stati applicati a problemi multifase monodimensionali che si presentano in applicazioni su scala di laboratorio e su scala geofisica.

In this thesis, we study a family of centered finite volume methods, that appear especially suited for applications to multiphase fluid dynamics problems. After a first validation of the implementation of such methods developed within the thesis work, the methods have been applied to one-dimensional multiphase problems at the laboratory and geophysical scale.

Keywords

Keywords: centered methods, hyperbolic laws, volcanic eruptions, Euler equa-tions

Parole chiave: metodi centrati, equazioni iperboliche, eruzioni vulcaniche, equazioni di Eulero

Chapter 1

Introduction

This thesis is focused on the study of centered finite volume methods for applications to multiphase fluid dynamics problems. In particular, the methods developed by H. Nessyahu and E. Tadmor in [15], [9], [13] and further improvements in [12], [14] have been studied and implemented for simulations of one dimensional problems at the laboratory and geophysi-cal sgeophysi-cale.

Centered finite volume methods are attractive for problems with com-plex equations of motion, since they do not require the solution of Rie-mann problems to determine the numerical fluxes. Centered methods are usually based on the Lax - Friedrichs scheme. Other important centered methods have been proposed in [11], [10], [24] and [25]. The methods by H. Nessyahu and E. Tadmor are especially interesting because the great semplicity of implementation and their versatility to treat different types of problems with or without the presence of source terms.

Multiphase problems arise in a number of differents contexts ranging from aereospace engineering to geophysical applications. The model equa-tions are usually derived by an averaging approach and the interacequa-tions among the different phases results in complex coupling terms that make the solution of the associated Riemann problem a difficult task, as we can see in [18], [22] where the cell interface Mach numbers or velocities have to be calculated for each time step in order to know the ”direction of the wind”, or in [2] and [4] where the eigenstructure of the problem has to be derived to implemented the methods used, which is really complicate in the case of multiphase problems.

The methods by H. Nessyahu and E. Tadmor have been implemented in a MATLAB code, both for one phase and multiphase one-dimensional

models. The properties of the methods described in the literature were accurately reproduced in a first validation of their implementation to one phase flows. Finally, the methods have been applied to two examples of multiphase fluid dynamics problems, such as in sec: 5.1 and sec: 5.2.

The numerical results obtained in sec: 5.1 and sec :5.2 are qualitative in accordance with those in literature, espacially for volcanic eruptions a better accordance is not possible because of the high sensitivity of the problem on the input data which, also, are not entirely reported in [16]. On the other hand the applications of the numerical schemes to the scalar and system tests in chapter 4 show very good accordance.

In chapter 2 we derive the equations which governs the behaviour of a multiphase fluid either for a constant cross section area and for a not constant cross section area.

In chapter 3 we derive the second order numerical schemes with stag-gered and non stagstag-gered grids and its TVD property. We report also the various limiters we implemented in the different types of problems anal-ysed.

In chapter 4 we carry out different numerical examples: scalar problem, Euler equations and variable cross section conduit for a single-phase flow. Chapter 5 is focused on the resolution of multiphase problems. Firstly we deal with two-fluid model with six governing equations and coupling terms, whereas in the last section we model a volcanic eruption using a mixed fluid model.

In chapter 6 we report the results obtained in all the problems faced and further improvements both for the mathematical and numerical model are discussed.

Chapter 2

Multiphase fluids

The term multiphase flow is used to refer to any fluid flow consisting of more than one phase or component. A large number of multiphase flows exist, such as: gas-solid, liquid-solid, gas-liquid, gas-particle, bubbly flows.Almost all industrial processes deal with some occurence of multi-phase flow. There are several aerospace applications, such as hot jets ejected from rocket nozzles, and geophysical ones, such as simulation of volcanic eruptions, avalanches, mud slides and sediment transport. Mul-tiphase phenomena are also a fundamental feature in nature: avalanches, mud slides, sediment transport and others.

In this chapter we introduce the models and equations which deal with description of the behaviour of multiphase flows. Three general typologies of multiphase flow can be identified: disperse flows, separated flows and flows in which components are well mixed at molecular level. We will treat the latter at the end of this chapter. By disperse flows we mean those consisting in finite particles, drops, or bubbles distribuited in a connected volume of the continuos phase. On the other hand separated flows consist of two or more continuos streams of different fluid separated by interfaces. Due to very high computational cost of resolving the full Navier Stokes equations for each of the phases or components and of the computation of every detail of a multiphase flow, the motion of all the fluid around every particle, the position of every interface, we must introduce some simplifications.

In disperse flows, two types of models are prevalent, trajectory models and two-fluid models.

In trajectory models the motion of the disperse phase is assessed by fol-lowing the motion of either the actual particles or larger representative particles. In two-fluid models, the disperse phase is treated as a second continuos phase intermingled and interacting with the continuos phase.

In the following, we will introduce the continuous equations for two-fluid models along the lines of the presentation in [3].

2.1

Averaging

We make the fundamental assumption that an infinitesimal volume must exist such that its dimension is much smaller than the typical distance over which the flow properties vary significantly, at the same time it has to be much larger than the size of the individual phase elements (drops, dis-perse, bubbles). The first condition allows to write differential equations, while the latter is important because the infinitesimal volume needs to be a representative sample of each components or phases. We will proceed to write the conservation equations taking into account these two conditions.

2.2

Notations

The lowercase subscripts ij refer to the cartesian component. The upper-case subscript (N) refers to the property of a specific component or phase, for example N = D means that we are refering to the disperse phase. Vol-umetric fluxes are indicated with jAi, jBi.

The total volume flux is then given as follows:

ji = jAi+ jBi+ ... = ∑

N

jNi (2.1)

The volume fraction of each component is given by :

αN =

VN

V (2.2)

and it is related with the volumetric flux:

MULTIPHASE FLUIDS

It is clear that a multiphase mixture has a certain mixture properties, one of the most important is the mixture density given by the following:

ρ =∑

N

αNρN (2.4)

In the following sections we developed the tools to construct the gov-erning equations either for trajectory model and two-fluid model.

2.3

Conservation of mass

We now want to construct the differential equation governing the con-servation of mass for disperse multiphase flow if we chose the two-fluid model.

As usual we define the ratio of incresing density, the flux through the faces of our infinitesimal volume and the source term due to phase transitions (IN). These give the mass equation for each component:

∂ ∂t(ρNαN)+ ∂ ∂xi (ρN jNi)= IN (2.5) Using jNi= αNuNi: ∂ ∂t(ρNαN)+ ∂ ∂xi (ρNαNuNi)= IN (2.6)

We notice that for conservation one must have: ∑

N

IN = 0 (2.7)

∂ ∂t    ∑ N ρNαN    + _∂x∂ i    ∑ N ρN jNi    = ∑ N IN ∂ρ ∂t + ∂ ∂xi    ∑ N ρN jNi    = 0 ∂ρ ∂t + ∂ ∂xi    ∑ N ρNαNuNi    = 0 (2.8)

If we have a problem in which a fluid has to flow inside a duct with a cross section area which varies with A(x) the equation of mass becomes:

∂ ∂t(ρNαN)+ 1 A ∂ ∂x(AρNαNuN)= IN (2.9)

The sum over each phases yields the following combined phase conti-nuity equation: ∂ ∂t(ρ αN)+ 1 A ∂ ∂x(A ∑ N ρNαNuN)= IN (2.10)

2.4

Disperse phase number continuity

Along with the equations of conservation of mass are the equations gov-erning the conservation of the number of bubbles, drops, particles, and so on that constitute the disperse phase. If we denote the number of bubbles per unit of total volume by

nD =

total number of particles of N total volume = ND V we obtain: ∂nD ∂t + ∂ ∂x (nDuDi)= ID (2.11)

MULTIPHASE FLUIDS

If the volume of a particle is denoted by vDit follows that:

αD = nDvD (2.12)

and substituing we obtain the following:

∂

∂t(nDρDvD)+_∂x∂ i

(nDuDi, ρDvD)= ID (2.13)

2.5

Conservation of momentum

We now continue our treatment by introducing the terms which act into Newton’s law, but prior to doing so we make some modifications to the control volume to skip some difficulties arising cutting through the dis-perse phase. So we modify the control volume in such way that the boundaries remain everywhere within the continuos phase. This allow us to overcome the issue of knowing the stresses acting within the disperse phase or to deal with the complication of determing the interaction forces between the two phases with boundaries which intersect the dispersed phase.

Then using the conservation principle the net force acting in the k di-rection on the N component must be given by:

∂

∂t(ρNαNuNi)+ ∂

∂xi(ρNαN

uNiuNk)= FNk (2.14)

Now we must identify the forces that act on FNk. We will consider the following:

1. body forces acting within the control volume; only the gravity force will be considered here: α ρN gk.

2. stress forces acting on the surface of the control volume: σCki = −pδki+ σ′_Cki

. We consider the deformation of the disperse phase negligible. 3. interaction forces acting on the N component by the other

compo-nents: FN k. Often is useful for the dispersed phase to separate the

contribution of pressure field inside the control volume from other effects and write:

FDk = −αD∂P ∂xk

+ FD′k = −FCk .

Put together all the above contributes we obtaing the following:

∂ ∂t(ρNαNuNi)+ ∂ ∂xi (ρNαNuNiuNk)= (2.15) = − α ρNgk + Fk+ δN { −_∂x∂P k − ∂σ′ Cki ∂xi }

We setδN equal to zero if we are considering the desperse phase, instead we haveδC= 1 if we are considering the continuos phase.

Furthermore, if we sum up the equations for the various phases we arrive at this equation:

∂ ∂t    ∑ N ρNαNuNi    +_∂x∂ i    ∑ N ρNαNuNiuNk    = (2.16) = − ρ gk− ∂P ∂xk + ∂σ′Cki ∂xi

MULTIPHASE FLUIDS Considering also the case of monodimensional flow inside a variable area duct we can derive the conservation equation for the single phase:

∂ ∂t(ρNαNuN)+ 1 A ∂ ∂x(AρNαNu2N)= (2.17) = − α ρNgx+ Fx+ δN { −∂P_∂x + pτw A }

Summing over all components:

∂ ∂t    ∑ N ρNαNuN    + _A1_∂x∂   A ∑ N ρNαNu2N    = (2.18) = − ρ gx− ∂P ∂x + pτw A

2.6

Conservation of energy

Even in a single phase flow some complications arise by treating terms such as heat conduction and viscous dissipation. For that reason and be-cause their effects are often negligible in gasdynamic problems we don’t consider those contributions in our equations. These limitations are often minor compared with other dificulties that arise in constructing an energy equation for multiphase flows. Now we face the tasks of derive an equa-tion for this kind of flow firstly by categorizing the various terms which are considered. We start with total energy of the single phase which is defined as follows:

e∗_N = eN + 1

2uNiuNi+ gz (2.19)

Then the terms involved in changing the total amount per unit time in the control volume are:

1. rate of heat transfer to N from outside of control volume, QN. 2. rate of heat transfer to N from within control volume, QIN.

3. rate of work done to N by the exterior sourrandings of control

vol-ume,WAN.

4. rate of work done to N by the interior of control volume,WIN. The second term contains two contributions: one takes into account the work made by the viscous forces, the other considers the rate of external

shaft work denotated byWN. The individual phase energy equation may

be then write as follows:

∂ ∂t(ρNαNe∗_N)+ ∂ ∂xi (ρNαNe∗_NuNi)= (2.20) =δN_∂x∂ j (uCiσCij)+ QN+ WN+ QIN+ WIN

As with the continuity and momentum equation we sum all the equa-tions for each phase and obtain:

∂ ∂t( ∑ N ρNαNe∗_N)+ ∂ ∂xi ( ∑ N ρNαNe∗_NuNi)= ∂ ∂xj (uCiσCij)+ Q + W (2.21)

Finally, we note that the one-dimensional duct flow version for one phase and the combining one are as follows:

∂ ∂t(ρNαNe∗_N)+ 1 A ∂ ∂xi (AρNαNe∗_NuN)= (2.22) = − δN_∂x∂(uCP)+ QN + WN + QIN + WIN

MULTIPHASE FLUIDS ∂ ∂t( ∑ N ρNαNe∗_N)+ 1 A ∂ ∂x( ∑ N AρNαNe∗_NuN)= (2.23) =_∂x∂ (uCP)+ Q + W

2.7

Mixed fluid

In this chapter we are going to see which are the equations that govern a mixed fluid flow. In this kind of flow the physical properties are calculated in order to construct a single averaged flow. Retracing the approach used in [16] for the volcanic eruption where the mixed fluid is composed by melt, water and vapour we set the governing equations as follow:

∂ρ ∂t + ∂ρv ∂x = j (2.24) ∂χ ∂t + ∂(χ v) ∂x = 0 (2.25) ∂ρv ∂t + ∂(ρv2_{+ P)} ∂x = −ρg − fµ (2.26) ∂E ∂t + ∂((E + P)v) ∂x = −ρgv − fµv− q (2.27) E= (1 − α)clρT + (1 − α)cgρT + ρv2 2 (2.28) ρ = α P RT + (1 − α)ρl (2.29)

whereα is the gas void fraction as defined in eq.(2.2), while χ is defined as

χ = ρm(1− α)(1 − c)

and represents the pure melt without either water nor vapour. We can see how density and energy are wighted with the respective void fractions and we have two equations of conservation of mass: one for the total mass

eq.(2.24) and one for the pure melt eq.(2.25). Obviously in this case we are refering to a single averaged fluid so we have single temperature, velocity and pressure. In sec: 5.2.1 this set of equations are used to compute a volcanic eruption.

2.8

Equations of state for liquid phase

In this section we deal with various examples of state equations found in literature:

1. Incompressible liquid: we consider a liquid which as P(x, t) = const. 2. Stiffened gas equation: in this case we use a modified equation of state

of perfect gas:

ρ = γ(P + P_{(γ − 1)h}∞) (2.30)

whereγ is the ratio of the specific heats and h is the specific enthalpy. Using this simple equation of state liquid behaves as though it is an ideal gas that is already under P∞ = const which represents the

molecular attractions within the fluid. In chapter 5 we use it to model the behaviour of water-vapour bi-phase flow.

Chapter 3

Centred methods for hyperbolic

conservation laws

3.1

Introduction

In this chapter we deal with the numerical solution of hyperbolic conser-vation laws. An example of this equations is a generic system of equation as:

∂

∂tq(x, t) + ∂

∂xF(q(x, t)) = 0 (3.1)

where q(x, t) is the vector of states and f(q(x, t)) is the flux vector, func-tion of the state vector. This type of equafunc-tions may contain discontinuities in the solution and these could lead lo computational difficulties, for this reason classical finite difference methods, in which derivatives are approx-imated by finite differences, can be expected to fail near discontinuities in the solution where the differential equation equation does not hold. This work concerns with finite volume methods, which are based on the inte-gral form of eq.(3.1):

∫ L ∂ ∂tq(x, t) + ∫ L ∂ ∂xf(q(x, t)) = 0 (3.2)

Rather than pointwise approximations at grid points of the solution we brake the domain into grid cells and approximate the total integral of q

over each grid cell, or actually the cell average of q, which is the integral of q diveded for the volume of the cell.

Building a numerical volume method one principal requirement is that it is Total Variation Diminuishing or TVD, because we want to avoid spu-rious oscillations near discontinuities. To achieve this we need a way to measure this oscillations in the solution. This is provided by the notion of the total variation of a function:

Definition 1 For an arbitrary function we can define:

TV(q) = sup

N ∑

i=1

|q(ξi)− q(ξi−1)| (3.3)

where the supremum is taken over all subdivision of the real line.

For a grid function Q we define the TVD property:

Definition 2 A numerical method is TVD if, for any set of data Qn_{, the valus}

Qn+1_{computed by the method satisfy:}

TV(Qn+1)≤ TV(Qn) (3.4)

If the numerical method introduces oscillations we would expect the total variation of Qn _{to increase with time. We can thus attempt to avoid} oscillations by requiring that the method not increase the total variation.

As we see in sec: 3.3 we will deal with slope limiters in order to com-pute numerical derivatives which enter in the definition of our numerical method and we want that the scheme will preserve the TVD property. In simple limiters like MinMod we can easly check from its definition (3.23) that it is TVD, but if we introduce a more complex limiter we would like to have an algebraic proof that the resulting method is TVD. A foundamental tool in this direction is the following theorem of Harten [1]:

CENTRED METHODS FOR HYPERBOLIC CONSERVATION LAWS

Qn_i+1 = Qn_i − Cn_i−1(Qn_i − Qn_i−1)+ Dn_i(Qn_i+1− Qn_i) (3.5)

over the time step, where the coefficients Cn

i−1and D n

i are arbitrary values. Then

TV(Qn+1)≤ TV(Qn) (3.6)

provided the following conditions are satisfied:

Cn_i−1≥ 0 ∀ i

Dn_i ≥ 0 ∀ i (3.7)

Cn_i + Dn_i ≤ 1 ∀ i

This theorem can be used to derive algebric condition for the limiters required for the TVD method.

Many numerical centered methods are presented in literature such as: [9], [24], [25], the numerical methods presented in this chapter follows [15]. It is a non-oscillatory central differencing scheme for hyperbolic conserva-tion laws. The main advantage of this family of methods is to integrate the conservation equations on the entire Riemann fan. This approach avoids to compute the numerical fluxes, which is typical of Godunov type methods and hence field-by-field decomposition are not needed. This method is based on the alghoritm of Lax-Friedrichs (LxF) where the second order of accurancy is obtained by a picewise linear reconstruction of the numerical solution. The building block is the LxF scheme which is a prototype of a central difference approximation, which offers a great semplicity over the upwind Godunov method.

3.2

The second order scheme

We describe the procedures to derive a second order scheme on a staggered grid for the scalar conservation law:

∂q ∂t +

∂ f

∂x = 0 (3.8)

where the flux is a function of the state q (x, t) : f = ˜f(q (x, t)).

Firstly we integrate the scalar equation over the rectangular space-time domain [tn_{, t}n+1_{] × [x} i, xi+1] and obtain: ∫ xi+1 xi ∫ tn+1 tn [qt+ fx]= 0 (3.9) ∫ xi+1 xi q (tn+1, x) = (3.10) = ∫ xi+1 xi q (tn, x) − [∫ tn+1 tn f (q (τ, xi+1))− ∫ tn+1 tn f (q (τ, xi)) ]

Than dividing by _∆x1 we obtain:

1 ∆x ∫ xi+1 xi q (tn+1, x) = (3.11) = _∆x1 ∫ xi+1 xi q (tn, x) − 1 ∆x [∫ tn+1 tn f (q (τ, xi+1))− ∫ tn+1 tn f (q (τ, xi)) ]

The first member rappresents the average value of the function at

t = tn+1 in the cell of width [xi, xi+1], and we associate it with the cen-tral value of the cell:

Qn_i+1/2+1 = 1

∆x ∫ xi+1

xi

CENTRED METHODS FOR HYPERBOLIC CONSERVATION LAWS now reconstruct the solution at time tnwith a picewise linear approxi-mation utilizing the central values Qn

i. The linear approximation is:

Li(x, t) = ¯qi+ σi

∆x(x− xi), with xi−1/2≤ x ≤ xi+1/2 (3.13) where ¯qicorresponding to the average Qn_i.

The numerical derivatives

σi = Qn_i+1/2− Qn_i−1/2 are defined as:

σi

∆x =

∂q (tn_{, x} i) ∂x

If we now use the linear approximation into the integral we obtain:

Qn_i+1/2+1 = 1 ∆x [∫ xi+1/2 xi q (tn, x) + ∫ xi+1 xi+1/2 q (tn, x) ] (3.14) −_∆x1 [∫ tn+1 tn f (q (τ, xi+1))− ∫ tn+1 tn f (q (τ, xi)) ] = =_∆x1 [∫ xi+1/2 xi Li(x, t) + ∫ xi+1 xi+1/2 Li+1(x, t) ] −_∆x1 [∫ tn+1 tn f (q (τ, xi+1))− ∫ tn+1 tn f (q (τ, xi)) ]

The first two integrals of the second member are integrated exactly. The two last integrands f (xi, τ) and f (xi+1, τ) are smooth functions of τ and so may be approximated with the midpoint rule which has a truncation error

O(∆t3_): 1 ∆t ∫ tn+1 tn f ( q (τ, xi)) F ( q (tn+1/2, xi) ) + O(∆t2_{) (3.15)}

Now we gather all the above steps and arrive at the final formula for the staggered scheme:

Qn_i+1/2+1 =1 2(Q n i + Qni+1)+ 1 8(σ n i − σni+1) (3.16) − λ[ f (q (tn+1/2, xi+1))− f ( q(tn+1/2, xi)) ]

The family of centered staggered schemes of eq:(3.16) could be rewrit-ten as: Qn_i+1/2+1 = 1 2(Q n i + Q n i+1)− λ[gi+1− gi] (3.17)

where the modified numerical flux, gi, is given by:

gi = f (Qi(t+ 0.5∆t)) + 1

8λσi. (3.18)

After evolving the function to obtain the new cell averages Qn+1 it is projected back into the space of costant-piecewise grid functions:

1 ∆x

∫ xi+1

xi

q (x, tn+1)= ¯q (x1+1/2, t + ∆t) (3.19)

In order to compute the state value q (t+ ∆t₂, xi) we use the Taylor ex-pansion: q (t+ ∆t 2 , xi)= qi− 1 2λ f ′ i (3.20)

CENTRED METHODS FOR HYPERBOLIC CONSERVATION LAWS To complete the treatment we report the non-staggerd version of the scheme: Qn_i+1 =1 2(Q n i−1+ Q n i+1)+ 1 4(σ n i−1− σ n i+1) (3.21) − λ 2 [ f(q (tn+1/2, xi+1) ) − f(q (tn+1/2, xi) ) ]

We can easly check the second order accuracy of the scheme by calcu-lating the truncation error in space and time:

ϵ (∆x) = ∫ xi+1/2 xi ˜q′′(xi) 2 (∆x) 2₊ ∫ xi+1 xi+1/2 ˜q′′(xi+1) 2 (∆x) 2_{+ O(∆t)}3 = ˜q′′(xi) 12 (∆x) 3₊ ˜q′′(xi+1) 12 (∆x) 3_{+ O(∆t)}3 = O(∆x3_{)+ O(∆t}3_{) (3.22)}

3.3

Slope limiters

To guarantee the nonoscillatory property we have to carefully choose the numerical derivatives σi and f_i′ so we use slope limiters. Following [21] we present the definitions of the principal limiters used in this work

• Standard MinMod limiter:

MM(x, y) = MinMod(x, y) = 1

2[sgn(x)+ sgn(y)] · Min(|x|, |y|) (3.23)

• MinMod limiter with α parameter: σi = MM (α∆Qi+1,1

• Superbee limiter, proposed by Roe [20]: σn i = MaxMod(σ1i, σ2i) (3.25) where: σ1 i = MinMod (( Qn i+1− Q n i ∆x ) , 2 ( Qn i − Q n i−1 ∆x )) σ2 i = MinMod ( 2 ( Qn i+1− Qni ∆x ) , ( Qn i − Qni−1 ∆x ))

• Monotonized central-difference limiter (MC limiter), which was pro-posed by van Leer [26]:

σn i = (3.26) = MinMod (( Qn i+1− Qni−1 2∆x ) , 2 ( Qn i − Q n i−1 ∆x ) , 2 ( Qn i+1− Qni ∆x ))

Many other choices of non-linear limiter are possible and could be found in literature.

3.4

Stability conditions

In order to ensure that this scheme is non-oscillatory in the sense described below, our numerical derivatives _∆x1 w′_i, should satisfy for every grid func-tion w= {wi}:

CENTRED METHODS FOR HYPERBOLIC CONSERVATION LAWS The constraint in eq.(3.27) is required in order to guarantee the Total Variation Diminishing (TVD) property for the family of central differenc-ing schemes. We recall that TVD is a desiderable property in the current setup, for it implies no spurious oscillations in our approximate solution.

Theorem 2 Let the numerical derivatives 1

∆xσi and _∆x1 f_i′ be chosen such that the

TVD requirement eq.(3.27) holds, say,

0≤ σisgn (∆σ±1/2)≤ Constσ|MM{∆σi+1/2, ∆σi−1/2}|,

Constσ≡ α, (3.28)

0≤ f_i′sign (∆σ±1/2)≤ Constσ|MM{∆σi+1/2, ∆σi−1/2}|. (3.29)

Assume that the following CFL condition is satisfied

λ maxi|a(σi)| ≤ β, β ≡ λ Constf Const_σ ≤ 1 2α ( √ (4+ 4α − α2)− 2). (3.30)

Then the family of high-resolution central differencing schemes eq.(3.16)is TVD.

Proof 1 By eq.(3.18) we have:

λ|δgi+1/2 δQi+1/2 | ≤ λ|f (Qi+1(t+ 0.5∆t)) − f (Qi(t+ 0.5∆t)) ∆Qi+1/2 | + 1 8| ∆σi+1/2 ∆σi−1/2 | ≤ ≤ λ|f (Qi+1(t+ 0.5∆t)) − f (Qi(t+ 0.5∆t)) Qi+1(t+ 0.5∆t) − Qi(t+ 0.5∆t) | |Qi+1(t+ 0.5∆t) − Qi(t+ 0.5∆t) ∆Qi+1/2 | + 1 8| ∆σi+1/2 ∆σi−1/2| (3.31)

Our CFL condition eq.(3.30) implies taht the first on the right of eq.(3.31) does not exceed:

λ|f (Qi+1(t+ 0.5∆t)) − f (Qi(t+ 0.5∆t))

Qi+1(t+ 0.5∆t) − Qi(t+ 0.5∆t) | ≤ β

(3.32)

Using the midvalue Qi(t+ 0.5∆t), we can estimate the second term on the

right of eq:(3.31), |Qi+1(t+ 0.5∆t) − Qi(t+ 0.5∆t) ∆Qi+1/2 | ≤ 1 + 0.5λ| ∆ f′ i+1/2 ∆Qi+1/2|, (3.33) ∆ f′ i+1/2≡ fi′+1− fi′

where in view of eq.(3.28) and eq.(3.31)

|∆ f ′ i+1/2 ∆Qi+1/2| ≤ max ( | fi′ ∆Qi+1/2|, | f_i′₊₁ ∆Qi+1/2| ) ≤ Constf ≤ 1 λαβ. (3.34)

Finally, the TVD requirement, eq.(3.29) gives us an upper bound for the third term on the right of eq.(3.31),

|∆σi+1/2 ∆Qi+1/2 | ≤ max(| σi ∆Qi+1/2 |, | σi+1 ∆Qi+1/2 |) ≤ α, (3.35)

Using eq.(3.32). eq.(3.33), and eq.(3.34), we find that eq.(3.35) boils down to the quadratic inequality:

β(1 + 0.5αβ) + 1 8α ≤

1

2, (3.36)

CENTRED METHODS FOR HYPERBOLIC CONSERVATION LAWS Follow eq.(3.30) the CFL conditions to achieve the TVD property of the methods are: • Non-staggered version: β ≤ 1 α( √ 4+ 4α − α2− 2) _(3.37) • Staggered version: β ≤ _2α1 (√4+ 4α − α2− 2) _(3.38)

Setting, as example, α = 1 we obtain: CFLstag = 0.32 and CFLnon−stag = 0.65

3.5

Hyperbolic equations and source terms

In this section we deal with hyperbolic equations with the presence of non-conservative terms, namely source terms. There are more than one way to compute them, two most important are: fractional-step method and well-balanced schemes. The latter deal with the inclusion of the source terms in the definition of the numerical flux, like in [6]. The first approach is followed in this work. As reported in [21] it is applied by first splitting the equation into two subproblems that can be solved independently. As example we consider a generic hyperbolic scalar equation as follow:

∂

∂t(q (x, t)) + ∂

∂xf (q (x, t)) = s (3.39)

where, as usual, q(x, t) is the state and f (q(x, t)) is the flux function of the state q(x, t). In this case we have a source term s, and the two subproblems are:

Problem A ∂ ∂t(q (x, t)) + ∂ ∂xf (q (x, t)) = 0 (3.40) Problem B ∂ ∂t(q(x, t)) = s (3.41)

If we split eq.(3.39) into homogeneous conservation law (Problem A) and a simple ODE (Problem B), then we can use a centered method for the conservative part and an ODE solver like Crank-Nicholson for the non-homogeneous part, as we do in sec: 5.2. For complicated problems this approach has a great advantage over derive an unsplit method.

Chapter 4

Numerical tests

In this section we present some numerical results for non-linear scalar and system of conservation laws. Different combination of limiters are anal-ysed in order to evaluate the best combination considering time consuming and accurancy. Crucial for our decision will be the quality of the resolution of the contact wave presented in the density because this discontinuity is linearly degenerate so it is over-smeared by the numerical method. Central methods don’t need the derivation of the eigenstructure of the system but their great versatility allows to include some characteristic informations in order to better tackle the problem of the contact wave.

4.1

Burgers equation

We present the first numerical applications of this centered method solving Burger equation:

ut+ fx = 0 (4.1)

f = 1

2(u)

2 _(4.2)

with different initial condition: u(x, 0) = sin(πx) and a Riemann prob-lem defined as follows:

u(x, 0) =

 

4.1.1

Periodical initial condition

NUMERICAL TESTS 0 0.5 1 1.5 2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Figure 4.2: Scalar: staggered T= 0.4 s, N=400

0 0.5 1 1.5 2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

4.1.2

Riemann initial condition

NUMERICAL TESTS 0 0.5 1 1.5 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 4.5: Scalar: non-stg T= 0.4 s, N=400

4.2

Euler equations

In this subsection we show the convergence property of the method ap-plied to the Euler equations for an ideal gas:

  ρuρ E    t +   (ρuρu2_{+ P)} (E+ P)u    x = 0 P= (γ − 1) · (E − 1 2ρu 2_{) (4.4)}

With the Riemann problem denominated RIM1:

u(x, 0) =

 

uulr for xfor x≤ 0, u> 0, ur = (0.125, 0, 0.25)l = (1, 0, 2.5)T T.

(4.5)

For the flux derivatives we can calculate the Jacobian matrix:

or we may use the two Minmod limiters: f_i′_,k = MM{∆ f_i+1 2,k, ∆ fi−12,k} (4.7) f_i′_,k = MM{α∆ f_i+1 2,k, 1 2( fi+1,k + fi−1,k), α∆ fi−12,k} (4.8)

Another possible choice which include the characteristic information into the definition of the numerical derivatives is to implement the Roe matrix: ˆA_i+1

2(qi, qi+12):

We summarized the different cases analysed:

• The non-staggered version of the method using eq. (3.23) and eq. (4.6) for the flux is referred to as ORD with CFL: 0.95.

• The non-staggered version using eq. (3.24) with α = 2 and eq. (4.6) is referred to as ORD2, CFL= 0.475.

• The central method in his staggered form with eq. (3.23) and eq. (4.6) is referred to as STG. CFL= 0.475.

• The central method in his staggered form with eq. (3.24) α = 2 and eq. (4.6) is referred to as STG2. CFL= 0.475.

• The central method in his staggered form using a Roe matrix is re-ferred to as STGR. CFL= 0.475.

We want to verify the order of accurancy of the central method and evaluate the best variant. In the table below we report for density the order of convergence for different interval with which the grid is divided:

N = 50,100,200. We can see how the staggered versions are pretty more

precise than the non-staggered one, with a sensible improvement with α = 2 for all the methods. We would expect a better resolution for the STGR because its integration of characteristic information of the linearly degenerate field but the best one in terms of accurancy is the STG2 version.

NUMERICAL TESTS

Table 4.1: L1 NORM for RIM1 Density 50 100 200 ORD 0.0231 0.0125 0.0064 order 0.8886 0.9647 ORD2 0.0245 0.0122 0.0066 order 1.0146 0.8767 STG 0.0179 0.0095 0.0052 order 0.9089 0.8684 STG2 0.0165 0.0085 0.0045 order 0.9516 0.9228 STGR 0.0170 0.0170 0.0094 order 0.8535 0.8678

We show all the graphs computed in the tests carried out. Firstly we report the scalar case and then the system case with particular attention on density. In the last part the results for pressure and velocity are displayed:

0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 4.6: Density T= 0.1644 s, ORD, N=200

0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

NUMERICAL TESTS 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 4.8: Density T= 0.1644 s, STG, N=200 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 4.9: Density T= 0.1644 s, STG2, N=200

0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 4.10: Density T= 0.1644 s, STGR, N=200

NUMERICAL TESTS 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 4.11: Velocity T= 0.1644 s, STG1, N=200 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 4.12: Pressure T= 0.1644 s, STG1, N=200

4.3

Conduit with variable section

In this section we test our scheme on the computation of a flow inside a convergent-divergent nozzle. We follow the mathematical model and compare the results with ones find in the article [4].

Wt+ Fx+ B′ = C (4.9) where: W(x, t) =    ρA ρuA A(ρu₂2 + _γ−1P )    F(W)=    ρuA (ρu2_{+ p)A} Au(ρu₂2 + P_γ−1γ )    (4.10) B′(P, x) =    0 PdA_dx 0    C(W)=    −gρA0 l qρAl    (4.11)

with u=velocity, P=pressure, ρ=density, A=cross-section, γ=ratio of specific heats, q=term of heat transferred to the walls, g=friction term,

t=time, x=distance, Alcorresponds to the wall surface per unit of length. The method implemented in [4] requires the evaluation in each time step of the approximated right and left eigenvector matrices (Roe matrix [19]) which complicate the writing and the computation of the numerical fluxes. With a centered method we avoid it and using a flux splitting technique we can easily take into account of the source terms of this problem.

The nozzle has a very simple symmetric geometry with Dinlet = Doutlet = 0.05 m and a throat diameter of Dt = 0.038 m. The overall length is L = 1 m. Initial conditions are discontinuos and are set like:

   PuLL T    =    1.8 bar0 m/s 300 K       PuRR T    =    1.5 bar0 m/s 300 K    (4.12)

NUMERICAL TESTS

Istead only pressure at the boundaries of both side of the domain are set constant and equal to their initial value.

For this test we set the parameters [p, q] to zero, so we assume no wall friction and an adiabitic expansion. The condition for stability is CFL equal to 0.95 and we discretize the length of the nozzle with 137 cells in order to well represent steep gradients.

We present the numerical solution of the Riemann problem eq:(4.12) when the solution reaches a stationary point at about T= 0.15 s.

0 0.2 0.4 0.6 0.8 1 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 10 5 _{Pressure at Time=0.15005} Length [m] P [Pa] Figure 4.13: Pressure at T= 0.15 s

0 0.2 0.4 0.6 0.8 1 40 60 80 100 120 140 160 180 Velocity at Time=0.15005 Length [m] u [m/s] Figure 4.14: Velocity at T= 0.15 s 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Mach number at Time=0.15005

Length [m]

M [−]

NUMERICAL TESTS 0 0.2 0.4 0.6 0.8 1 6 8 10 12 14 16 18 20 22 Density at Time=0.15005 Length [m] ρ [Kg/m 3] Figure 4.16: Density at T= 0.15 s

We can clearly note from fig.4.13, fig.4.14, fig.4.15, fig.4.16, the subsonic expansion in the convergent, the reaching of sonic point exactly in the throat and then the supersonic expansion in part of the divergent because

at about x = 0.8 m there is a stationary shock which causes the sudden

Chapter 5

Applications to multiphase fluids

5.1

Two phase flow with six governing equations

In this section we present the model and the results of the simulation of a two-phase two-fluid flow in which each phase is represented with its own equations of conservation. The model used is the same of [2]: we consider two fluid (water and vapour) coupled with interaction terms. Both fluids have been characterized using simple equations of state, namely stiffened gas for the liquid phase and perfect gas for the gas phase. The set of equa-tions used are as follows:

∂ ∂t(αgρg)+ ∂ ∂x(αgρgug)= 0 (5.1) ∂ ∂t(αlρl)+ ∂ ∂x(αlρlul)= 0 (5.2) ∂ ∂t(αgρgug)+ _∂x∂ (αgρgu2g+ αgP)= Pi ∂αg ∂x + αgρgg (5.3) ∂ ∂t(αlρlul)+ _∂x∂ (αlρlu2_l + αlP)= Pi∂α_∂xl + αlρlg (5.4) ∂ ∂t(αgρgEg)+ _∂x∂ (αgρgHgug)= −P ∂αg ∂t + αgρggug (5.5) ∂ ∂t(αlρlEl)+ _∂x∂ (αlρlHlul)= −P∂α_∂tl + αlρlgul (5.6)

whereαk is the void fraction of the phase k, either liquid (l) or vapour (g),ρk the density, uk the velocity, P the pressure, Ek = ek+ u

2

total energy, with ek its specific internal energy, Hk = hk + u

2

2 is the specific

total entalpy of phase k with hk its specific enthalpy, g is the acceleration of gravity (9.81 m/s), Pi_{stands for the difference of pressure between each} phase and the interface, this term is included to make the system hyper-bolic:

Pi = P − Pi = σ

αgαlρgρl αgρl+ αlρg

(ug− ul)2 (5.7)

We have also considered this two expressions for the density:

ρg= γgP (γg− 1)hg (5.8) ρl = γg(P+ P∞) (γl− 1)hl (5.9) with γg = 1.4, cpg = 1008 (J/kgK), γl = 2.8, cpg = 4186 (J/kgK) and

P∞ = 8.5 108 Pa. Enthalpy and speed of sound of phase k are given by

hk = cpkTk and ck = ((γk− 1)cpkTk)1/2.

For the resolution of this system of equations we use our second order accurate central scheme. Some numerical test are presented: Water faucet and Toumi’s shock tube.

5.1.1

Toumi’s shock tube test

This test is proposed by Toumy [8]. In a 10 m long horizontal tube we place two mixtures of water and air separated by a membrane which are characterized by the following initial conditions: left state,αLg = 0.25 and

PL _{= 20Mpa and right state α}L

g = 0.1 and PL = 10Mpa. Fluid temperatures are constant in both sides and equal to 35 C. This test differs from the original one by setting the initial velocities to zero.

APPLICATIONS TO MULTIPHASE FLUIDS 0 2 4 6 8 10 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26

Gas void fraction

length [m]

Void fraction []

Figure 5.1:

5.1.2

Water faucet

Water faucet consists of a homogeneous mixture of water and air flowing ownwards through a vertical pipe, (12 meters long) under the action of gravity. This test allow us to analyse the stability of the scheme, its accu-rancy and the numerical diffusion it may produce. This test also shows the performance of the scheme when simple gravity effects are included in the system of equations and its ability to deal with a situation in which phases mechanically decoupled.

Initial conditions are given by a gas void fraction αg = 0.2, the phase ve-locities, ug = 0 m/s and ul = 10 m/s, pressure P = 105bar and the phase

temperatures Tg = 323 K and Tl = 323 K. On the other hand, boundary

conditions are characterized by the following inlet values: αl = 0.8, the

phase velocites ug = 0 m/s and ul = 10 m/s and the phase temperatures

Tg = Tl = 323 K. Outlet is characterized by a discharge pressure of 105Pa. For this problem we have considered a value of CFL= 0.9 and an arti-ficial pressure correction parameter ofσ = 3.

0 2 4 6 8 10 12 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38

Gas void fraction

length [m]

Void fraction []

Figure 5.2: Void fraction at T=0.2,0.4,0.8,1.2

5.2

Volcanic eruptions

In this section we present a transient model for explosive and phreatomag-matic eruptions. The eruption is started by plug disruption at the top of the conduit. The model takes into account of the injection of water from an unconfined acquifer and we study how this affects the eruption. The magma flow is modeled as a two-phase homogenous fluid using the model sec: (2.7): before fragmentation we have bubbly fluid afterwards we have gas with liquid suspansion. The injection of water causes a lowering of temperature and a deeper fragmentation. There are several different mech-anisms by which water interacts with the magma: seawater, water from a crater lake, ground water; also many studies confirm the great importance of magma-water interaction in phreatomagmatic events.

5.2.1

Governing equation

We repropose the equation of section sec. 2.7 in which we add some costi-tutive relations:

APPLICATIONS TO MULTIPHASE FLUIDS ∂ρ ∂t + ∂ρv ∂x = j ∂χ ∂t + ∂(χ v) ∂x = 0 ∂ρv ∂t + ∂(ρv2_{+ P)} ∂x = −ρg − fµ ∂E ∂t + ∂((E + P)v) ∂x = −ρgv − fµv− q Costitutive equations: E= cmρT + ρv2 2 ρ = α P RT + (1 − α)ρm P= P(α, T) c= kp √ (P) (5.10)

The system written include: three hyperbolic conservation laws with source terms: j costant water injection from the acquifer, ρg gravitational force, fµ viscous friction, q loss of energy to heating water injected. For

costitutive equations we define:

α = Vgas

V

which represents the volume fraction of exolved gas, cmis the specific heat of melt, ρm is the density of melt, c is the concentration of water in melt. The total water influx J is the product of the volume of the influx zone and the water mass flux

J= π 4D 2_h aj and j= 4J πD2_h a .

Many models where studied to well approximate the effects of the vis-cosity on the flow of magma inside the volcanic conduit, see for example:

[17], [5], [23]. We chose for semplicty the simple criterion presented in [23] which sets the fragmentation level when the concentration of void fraction α reaches the threshold value of αcritic = 0.75. After that value the viscous forces are neglected. Fragmentation level is the hight at which magma fragments and its consistency drammatically change: from a viscous melt filled with bubbles it turns out to be a hot low-viscous gas with liquid droplets sospension.

Viscosity of the melt is modelled with the regression formula derived in [7], which is function of temperature and percentual concentration of wa-ter (w):

logη = [−3.545 + 0.833ln(w)] + [9601− 2368ln(w)]

T− [195.7 + 32.25ln(w)] (5.11)

We consider a cylindrical conduit with constant diameter. We consider the response of our model to disruption of the volcanic plug. Initially the pressure under the plug is magmastatic, and the temperature is constant; above the plug, pressure and temperature are atmospheric:

t= 0

x≤ xp P(x)= Pmst(x), T = Tch (5.12)

x≥ xp P= PatmT = Tatm (5.13)

The initial magmastatic pressure profile below the plug Pmst(x) is calcu-lated form the balance equation of momentum with velocity equal to zero:

dPmst dx = − ( αPmst RT + (1 − α)ρm ) g (5.14)

And the void fractionα:

α = c0− ρm(kp

√

Pmst)

APPLICATIONS TO MULTIPHASE FLUIDS

Eq.(5.14) and eq.(5.15) are solved with the boundary condition Pmst(0, Tch)=

Pch for 0 < x < xp where Pmst(xp, Tch) determines the pressure drop DP at the plug:

DP= Pmst(xp, Tch)− Patm .

For most explosive eruptions, the erupted volume is much smaller than the chamber volume; hence, we assume that at the bottom of the conduit the chamber pressure and temperature remain constant. At the top of the conduit the pressure is equal to atmospheric when the exit velocty is sub-sonic, otherwise no conditions can be specified.

The set of governing equations of sec: 5.2.1 is solved by means of time splitting method as described in sec: 3.5 conserved equations are solved with our centered scheme insted where source terms are calculate by Crank-Nicholson method for ordinary differential equations.

We report the results obtaibed with a discretization of 40 cells of the conduit. CASE1 and CASE2 are carried out with the parameters reported below:

Table 5.1: eruption parameters CASE1 CASE 1

chamber pressures (MPa) 150,175,200

conduit length L (km) 5 conduit diameter D (m) 50 melt density (kg/m3) 2400 melt temperature (K) 1150 acquifer position (km) 3 J (kg/m3_{) 6*10}6 computed time (m) 3

Table 5.2: eruption parameters CASE2 CASE 2

chamber pressures (MPa) 175,190

conduit length L (km) 10 conduit diameter D (m) 40 melt density (kg/m3_{) 2600} melt temperature (K) 1200 acquifer position (km) 5 J (kg/m3_{) 6*10}6 computed time T (m) 4

In the figure below we report the results for CASE1 and CASE2. For CASE1 the mass fluxes for three different pressure show that an increase in chamber pressure leads to an increase of the mass flow either for the case without injection of water (solid line), and for the phreatomagmatic eruption ( blue dashed line). The influence of water is strong and causes a reduction of the discharge peak and stationary mass flow. This behaviour can be explained by the absence for test CASE1 of the fragmentation front, so the leading process of strong mass flow erupted cease to exist making the eruption less explosive due to the diminished exit velocity caused by the lower temperature and higher viscosity. The order of magnitude is in agreement with that found in [16], the temperature variations range from few dagrees to 40 degrees in agreement. In CASE1 we do not observe a fragmentation point due to the small length of the conduit (5 km).

In CASE2 we can see a futher effect of the presence of the acquifer, the lowering of the fragmentation level: fig. 5.9 and fig. 5.10.

APPLICATIONS TO MULTIPHASE FLUIDS 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 108 Mass flux Time [min] Mass flow [Kg/s]

Figure 5.3: CASE1: Mass flux with and without water influx. Chamber pressure 150 MPa 11340 1136 1138 1140 1142 1144 1146 1148 1150 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Temperature at t=90.0517 Temperature [°K] Depth [m]

Figure 5.4: CASE1: Temperature with and withou water influx. Chamber pressure 150 MPa

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 108 Mass flux Time [min] Mass flow [Kg/s]

Figure 5.5: CASE1: Mass flux with and without water influx. Chamber pressure 175 MPa 11340 1136 1138 1140 1142 1144 1146 1148 1150 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Temperature at t=90.0136 Temperature [°K] Depth [m]

Figure 5.6: CASE1: Temperature with and withou water influx. Chamber pressure 175 MPa

APPLICATIONS TO MULTIPHASE FLUIDS 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 108 Mass flux Time [min] Mass flow [Kg/s]

Figure 5.7: CASE1: Mass flux with and without water influx. Chamber pressure 200 MPa 11250 1130 1135 1140 1145 1150 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Temperature at t=90.0125 Temperature [°K] Depth [m]

Figure 5.8: CASE1: Temperature with and withou water influx. Chamber pressure 200 MPa

0 50 100 150 200 250 6000 6500 7000 7500 8000 8500 9000 Fragmentation front Time [s] xf f [m]

Figure 5.9: CASE2: fragmentation level, pressure 175 MPa

0 50 100 150 200 250 7000 7500 8000 8500 9000 9500 Fragmentation front Time [s] xf f [m]

Chapter 6

Conclusions

In this thesis the applications of centered finite volume methods to mul-tiphase fluid dynamics problems have been assessed. In particular, the methods developed by H. Nessyahu and E. Tadmor in [15], [9], [13] and further improvements in [12], [14] have been studied and implemented for simulations of one dimensional problems at the laboratory and geophysi-cal sgeophysi-cale.

The methods by H. Nessyahu and E. Tadmor have been implemented in a MATLAB code, both for one phase and multiphase one-dimensional models. The properties of the methods described in the literature were accurately reproduced in a first validation of their implementation to one phase flows.

Finally, the methods have been applied to two examples of multiphase fluid dynamics problems.

The application to water faucet problem and Tuomi tests shows a qual-itative accordance with the results in [2] and [18], this may be explained by the difficulty of the centered methods to well approximate this types of problems, resulting in a not so good and high dissipative resolution, caused by the absence of any characteristic informations.

Application to conduit model presents a good qualitative agreement than the results in [16], a better accordance is not possible because of the high sensitivity of the problem on the input data which, also, are not entirely reported.

The applications of the numerical schemes to the scalar and system tests in chapter 4 show a very good agreement. Espacially in section 4.3 the problem is well solved with the centered scheme which avoids the calculation of the Jacobian matrix based on Roe’s linearization as in [4].

Perspectives for future work are to implement a biphase two-fluid model for the problem in section 4.3, this could lead, for example, to

the numerical resolution of hot gas-particle flow inside a rocket nozzle. Regarding the modelling of volcanic eruption a important improvement is to adapt the numerical scheme in order to calculate magma flow inside a variable cross section conduit. For a better understanding into volcanic process an accurate study of the input data is needed.

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