scispace - formally typeset

Journal ArticleDOI

A fractional step method to compute a class of compressible gas–liquid flows

15 Feb 2012-Computers & Fluids (Pergamon)-Vol. 55, pp 57-69

AbstractWe present in this paper some algorithms dedicated to the computation of numerical approximations of a class of two-fluid two-phase flow models. Governing equations for the statistical void fraction, partial mass, momentum, energy are presented first, and meaningful closure laws are given. Then we may give the main properties of the class of two-fluid models. The whole algorithm that relies on the fractional step method and complies with the entropy inequality is presented afterwards. Emphasis is given on the computation of pressure–velocity–temperature relaxation source terms. Conditions pertaining to the existence and uniqueness of discrete solutions of the relaxation step are given. While focusing on some one-dimensional test cases, the true rates of convergence may be obtained within the evolution step and the relaxation step. Eventually, some two-dimensional numerical simulations of a heated wall are shown and are briefly discussed. Some advantages and weaknesses of algorithms are also discussed.

Topics: Relaxation (approximation) (56%), Uniqueness (51%), Computation (50%)

Summary (3 min read)

1 Introduction

  • Two distinct types of models are used in order to compute liquid-gas or watervapour two-phase flows in industrial codes: the homogeneous approach and the two-fluid approach.
  • The two-fluid approach is assumed to be more general, and it is also expected to predict more accurately flows for which the phasic desequilibrium plays a crucial role (see [9, 25, 26]).
  • One must be aware that at least two difficulties are hidden in these sets of PDE.
  • The second section gives emphasis on the approximation of velocity-pressure-temperature relaxation effects.

2.1 The two-fluid model

  • Throughout the paper, indexes l, g refer to the liquid and gas phases; the statistical void fractions of gas and liquid are noted classically αg and αl, which should agree with: αl + αg = 1.
  • The so-called conservative variable W will be defined as: W = (αl, αlρl, αlρlUl, αlEl, αgρg, αgρgUg, αgEg) Moreover, PI(W ) and VI(W ) respectively denote in this paper the interfacial pressure and velocity, and will precised afterwards.
  • The term S1,l will also be introduced later on.
  • External sources might be included but are not considered herein.
  • Standard viscous contributions may be included, which of course comply with the entropy inequality that will be detailed in the next subsection.

2.2 Closure laws for interfacial transfer terms

  • The same holds when tackling three-phase flows, as emphasized in [18].
  • Now, the second requirement (H1) implies that the field associated with λ = VI should be linearly degenerate.
  • As shown in [8, 11], few expressions guarantee this behaviour.

2.4 Main properties of the two-fluid model

  • The authors may now recall in brief the main properties of system (2) using the previous closure laws.
  • Apart from the field associated with the eigenvalue λ =.
  • Otherwise the computation of shock solutions would be meaningless, since multiple shock solutions may be obtained using various -stable- solvers (see for examples [17]) .
  • Obviously, an alternative formulation of jump conditions in genuinely non linear fields that is probably more convenient may be: σ = [ρφUφ]RL/[ρφ].

3.2 Computing the evolution step

  • Many solvers have been proposed in the literature for such a purpose.
  • Approximate solutions in the evolution step may also be obtained using either the non-conservative form of Rusanov scheme, or the non-conservative form of the approximate Godunov scheme VFRoe-ncv [6]; the authors refer to [11, 10] for such a description.
  • Obviously, the ultimate scheme has not been proposed yet.
  • The proof is classical but is briefly recalled.
  • The authors detail afterwards the relaxation step, with special focus on the pressure relaxation step which is rather tricky.

3.3 Computing the velocity relaxation step

  • It may be easily checked that internal energies remain positive through this step.
  • The proof is obvious considering formula (16) and is left to the reader.

3.4 Computing the temperature relaxation step

  • The proof provides a practical way to compute solutions of system (18), that is actually used in the code.
  • The authors emphasize that other -simpler- algorithms may be exhibited ; however, practical computations seem to show that the non-linear temperature relaxation scheme described here provides better results (see [24] and section IV).
  • The authors turn now to the most difficult part which corresponds to the pressure relaxation step.

3.5 Computing the pressure relaxation step

  • The first one is a semi-implicit scheme, that is such that the existence and uniqueness of the discrete solution is ensured, whatever the equations of state would be.
  • The authors focus here on the second one which is totally implicit with respect to the unknown (Pl, Pg, αl).
  • The main lines of the proof are given below, also known as Proof.
  • The implicit scheme (20) is exactly the same as the one introduced for dense granular gas-particle flows in [10], also known as Remark 2.

4 Numerical results

  • In the first subsection, the authors focus on the computation of the evolution step involving all convective terms.
  • The following unities are used : m for distances, kg/m3 for densities, m/s for velocities, Pa for pressures, K for temperatures, s for times and J/s/m2 for heating fluxes.

4.1 Verification of the evolution step

  • The authors use for this test case perfect gas EOS within each phase, setting γg = 1.4 and γl = 1.1.
  • The second test case is another Riemann problem taken from [13], where initial conditions are given in table 2.
  • This Riemann problem also contains a contact wave associated with Ug, and a right-going gas shock wave.

4.2.1 Velocity relaxation substep

  • Two different series of verification test cases have been considered in [24] for the velocity relaxation step.
  • The first one refers to a constant time scale τ3.
  • In that case, the scheme (16) is perfect, since it computes the exact value at time tn+1.
  • In that case, analytic solutions allow computing the true error occuring in the velocity relaxation step.
  • Several examples can be found in [24], which confirm that a first-order rate of convergence is achieved.

4.2.2 Temperature relaxation substep

  • The authors present below some tests corresponding to constant time scales τ4, when computing approximate solutions of (17) with the scheme (18).
  • Φ, unlike the non-linear scheme (18); actually, round-off errors are found for the linear scheme in that case.

4.2.3 Pressure relaxation substep

  • Eventually, the authors provide an example of measured convergence rates in the pressure relaxation step.
  • This one is crucial, and should be handled with great care.
  • Otherwise, both the present model (where the relaxation time scale τ2 is non-zero) and standard two-fluid models (corresponding to τ2 = 0) may be confused, if inadequate “rough” schemes are used to provide approximate solutions of (19).
  • The initial conditions of the test case are given in table 5. Figure 6 and Figure 7 show the behaviour of the scheme (20) and the comparison with another half-implicit scheme introduced in [22] and recalled in [24], focusing on void fractions and pressures within each phase at time t = 10−5.
  • This enables to retrieve the fact that the initial-value problem associated with τ2 = 0 is ill-posed: spurious oscillations arise when the mesh size is sufficiently small, and this is particularly spectacular for void fraction profiles, since the algorithm guarantees bounded variations owing to properties 4 and 5 (see [21] and [17] also for a similar study).

4.3 Two-dimensional numerical results

  • The authors consider now the two-dimensional unsteady computation of a heated wall in an almost square domain, where the wall contains a small cavity in the middle of the lower part .
  • On the contrary the two cells at the exit corners of the cavity do not receive any heat flux.
  • It also seems worth mentionning that almost similar results have been obtained while changing the pressure relaxation time step within the range τ2 ∈ [10−9, 10−6], keeping other relaxation time steps unchanged and using the same mesh.
  • Nonetheless, one must be aware that the 3D counterpart of the present 2D mesh would contain more than 150 millions of cells, which is of course far beyond what one can afford in an industrial situation.

Did you find this useful? Give us your feedback

...read more

Content maybe subject to copyright    Report

HAL Id: hal-01265315
https://hal.archives-ouvertes.fr/hal-01265315
Submitted on 1 Feb 2016
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entic research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diusion de documents
scientiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
A fractional step method to compute a class of
compressible gas–liquid ows
Jean-Marc Hérard, Olivier Hurisse
To cite this version:
Jean-Marc Hérard, Olivier Hurisse. A fractional step method to compute a class of compressible gas–
liquid ows. Computers and Fluids, Elsevier, 2012, 55, pp.57-69. �10.1016/j.compuid.2011.11.001�.
�hal-01265315�

A fractional step method to compute a class of
compressible gas-liquid flows
Jean-Marc H´erard
EDF, R&D
Olivier Hurisse
EDF, R&D
Abstract
We present in this paper some algorithms dedicated to the compu-
tation of numerical approximations of a class of two-fluid two-phase flow
models. Governing equations for the statistical void fraction, partial m ass,
momentum, energy are presented first, and meaningful closure laws are
given. Then we m ay gi ve the main properties of the class of two-fluid mod-
els. The whole algorithm that relies on the fractional step method and
complies with the entropy inequality is presented afterwards. Emphasis
is given on the computation of pressure-velocity-temperature relaxation
source terms. Conditions pertaining to the existence and uniqueness of
discrete solutions of the relaxation step are given. While focusing on some
one-dimensional test cases, the true rates of convergence may be obtained
within the evolution step and the relaxation step. Eventually, some two-
dimensional numerical simulations of a heated wall are shown and are
briefly discussed. Some advantages and weaknesses of algorithms are also
discussed.
Keywords:
Two-phase flows / Finite Volume schemes / Two-fluid model / Hyperbolic s ys-
tems / Closure laws / Entropy inequality / Relaxation effects.
1 Introduction
Two distinct types of models are used in order to compute liquid-gas or water-
vapour two-phase flows in industrial codes: the homogeneous approach and
the two-fluid approach. Within the framework of nuclear safety codes[16] ,
the homogeneous approach is generally adopted for codes dedicated to compo-
nents such as reactor cores and steam generators (THYC, FLICA and GENEPI
codes in France), whereas the two-fluid approach is prefered in system codes
Correspond ing author. EDF, R&D, Fluid Dynamics, Power Generation and Environ-
ment, 6 quai Watier, 78400, Chatou, France. Tel: (33)-1-30 87 70 37. Email: jean-
marc.herard@edf.fr
EDF, R&D, Fluid Dynamics, Power Generation and Environment, 6 quai Watier, 78400,
Chatou, France.
1

(CATHARE or RELAP codes for instance) and also in 3D commercial codes
(CFX, Star-CD, Fluent) and inhouse codes (NEPTUNE-CFD). The two-fluid
approach is assume d to be more general, and it is also expected to predict more
accurately flows for which the phasic desequilibrium plays a c rucial role (see
[9, 25, 26]).
Now, two distinct two-fluid approaches may be considered. The first one,
which is the most standard one, relies on an instantaneous pressure equilibrium
between both phases. The second one no longer assumes this hypothesis, which
means that the seven unknowns corresponding to the statistical fraction of liq-
uid, the two mean velocities, the two mean temperatures and the two me an
pressures are evaluated by searching approximations of solutions of a coupled
set of seven partial differential equations (PDE). These equations correspond to
the mass balance, the momentum balance and the energy balance within each
phase, and is supplemented by the governing equation for the statistical liquid
fraction.
This paper is devoted to the simulation of water-gas flow models belonging
to the second class. Though this approach was introduced approximately thirty
years ago (see [33]), few workers have been investigating this class until the
late 90’s. Within the framework of gas-particle flow s, and more precisely when
studying Deflagration to Detonation Transition, these models gained a consid-
erable interest within a small community. Among other studies, one should at
least point out the contributions [4, 28, 5, 27, 32] which are concerned with the
modeling aspects. Fewer papers tackle the problem of water-gas or water-vapour
flows, among which we must quote [33, 34] and more recently the article [12] ,
that examines a medium of small oscillating bubbles in a liquid medium, and
also provides a general formalism in order to derive meaningful governing equa-
tions. A classification of closure laws related to the interface pressure P
I
and
the interface velocity V
I
was proposed in [8, 11] , which provides a general
framework relying on two m ain ingredients:
(H1) the interface velocity V
I
, which gove rns the evolution of the sta-
tistical void fraction, should be such that the field associated with the
eigenvalue λ = V
I
were linearly degenerate;
(H2) a physically relevant entropy inequality should control smooth solu-
tions.
Based on these two keystones, it has been shown in [8, 11] that one may re-
trieve the well-known Baer-Nunziato model, that c orresponds to the particular
choice P
I
= P
l
, V
I
= U
g
, among other possibilities (where P
l
and U
g
respec-
tively refer to the liquid pressure and the gas velocity). We emphasize that this
approach was recently extended to the framework of two-phase flow in porous
media [19, 13] . T he same procedure provides some way to tackle the modeling
of three-phase flows[18] . It also gives a relevant approach for the modeling of
dense granular flows[10] . These extensions confirm the relevance of the whole
2

modeling approach. However, one must be aware that at least two difficulties
are hidden in these sets of PDE.
First, the convective part of the system of PDE is hyperbolic with no con-
straining condition on the physical states, but it contains two (or three, de-
pending on the closure law for V
I
) linearly degenerate fields. A straightforward
consequence is that the asymptotic rate of convergence of so-called first-order
(respectively second-order) Riemann solvers is 1/2 (respectively 2/3). This has
recently motivated great efforts in order to build accurate enough Riemann
solvers for the Baer-Nunziato system (see at least [1, 2, 7, 31, 35, 36] ). Sec-
ond, the system contains stiff source terms which are linked with the pressure-
velocity-temperature relaxation effects. These require specific algorithms, and
the main part of the present paper is actually dedicated to this work. This is
basically motivated by the fact that few available articles discuss these tricky
problems. In the sequel, the two-fluid two-pressure model accounts for velocity,
pressure and temperature relaxation, each one b eing associated with a non-zero
time scale. Actually, most of the papers in the literature focus on some specific
situations. Dealing with the velocity relaxation effects is not really challenging,
and a lot of schemes have been proposed in order to cope with it. On the con-
trary, there are few articles dealing with the numerical treatment of the pressure
relaxation. In the case of instantaneous relaxation (i.e. with “zero time scales”)
one can refer to [34] (which deals with instantaneous velocity and pressure
relaxation) or [21] (which only deals with instantaneous pressurerelaxation)
among others. For non-zero pressure relaxation time scales, a scheme has been
proposed in [11], with an important drawback : the total energy of the mix-
ture is not conserved by the scheme. Moreover, most of the references in the
literature are based on models where the temperature relaxation source terms
are neglected, and consequently there are no numerical schemes accounting for
them.
Hence the paper is organized as follows. We first recall the set of PDEs that
governs the two-fluid model, and recall its main properties. Next we present
approximate Riemann solvers and algorithms used to compute approximations
of solutions of the coupled ODEs arising when taking relaxation effects into
account. The most difficult task dwells in the building of suitable algorithms in
the pressure relaxation step. Of course this difficulty vanishes when the pres-
sure relaxation time scale is set to zero, but in that case convergence difficulties
-with respect to the mesh size- may be expected (see [21, 17, 10]) when focusing
on unsteady computations. The first section of numerical results focuses on the
practical estimation of the rate of convergence when looking for approximate
solutions of the convective subset. The second section gives emphasis on the
approximation of velocity-pressure-temperature relaxation effects. The last sec-
tion is devoted to the two-dimensional simulation of the flow close to a heate d
wall.
3

2 Governi ng equations and main properties of
the two -fluid m odel
2.1 The two-fluid model
Throughout the paper, indexes l, g refer to the liquid and gas phases; the sta-
tistical void fractions of gas and liquid are noted classically α
g
and α
l
, which
should agree with:
α
l
+ α
g
= 1
The mean pressures, mean velocities and mean densities of the two phases are
denoted P
φ
, U
φ
and ρ
φ
respectively, for φ = l, g. The total energy within each
phase is:
E
φ
= ρ
φ
e
φ
(P
φ
, ρ
φ
) + ρ
φ
U
2
φ
2
, φ = g, l (1)
Internal energy functions e
φ
are provided by users.
The so-called conservative variable W will be defined as:
W = (α
l
, α
l
ρ
l
, α
l
ρ
l
U
l
, α
l
E
l
, α
g
ρ
g
, α
g
ρ
g
U
g
, α
g
E
g
)
Moreover, P
I
(W ) and V
I
(W ) respectively denote in this paper the interfacial
pressure and velocity, and will precised afterwards. These interface terms V
I
and P
I
will be such that:
jump conditions are well defined within each isolated field;
a physically relevant entropy inequality holds for smooth solutions of (2).
Given these notations, the governing set of equations for first-order moments
may be written as follows in a multi-dimensional framework:
t
(α
l
) + V
I
α
l
= S
1,l
t
(α
l
ρ
l
) + .(α
l
ρ
l
U
l
) = S
2,l
t
(α
l
ρ
l
U
l
) + .(α
l
ρ
l
U
l
U
l
+ α
l
P
l
I) P
I
(α
l
) = S
3,l
t
(α
l
E
l
) + .(α
l
U
l
(E
l
+ P
l
)) + P
I
t
(α
l
) = S
4,l
t
(α
g
ρ
g
) + .(α
g
ρ
g
U
g
) = S
2,l
t
(α
g
ρ
g
U
g
) + .(α
g
ρ
g
U
g
U
g
+ α
g
P
g
I) P
I
(α
g
) = S
3,l
t
(α
g
E
g
) + .(α
g
U
g
(E
g
+ P
g
)) + P
I
t
(α
g
) = S
4,l
(2)
where right-hand side terms S
k,l
(W ) represent the source terms (for k = 2, 3, 4),
which enable to account for mass transfer, momentum and energy transfer
through the inte rface between the two phases. The term S
1,l
will also be in-
troduced later on. External sources might be included but are not considered
herein. A derivation of the first governing equation of the statistical liquid frac-
tion α
l
can be found in[20] . Standard viscous contributions may be included,
which of course comply with the entropy inequality that will be detailed in the
4

Figures (20)
Citations
More filters

Journal ArticleDOI
Abstract: Computation of compressible two-phase flows with single-pressure single-velocity two-phase models in conjunction with the moving grid approach is discussed in this paper. A HLLC-type scheme is presented and implemented in the context of Arbitrary Lagrangian–Eulerian formulation for solving the five-equation models. In addition, the extension to multicomponent cases is also examined. The method is first assessed on a variety of Riemann problems including both fixed and moving grids applications showing its simplicity and robustness. The method is also tested on 2-D moving mesh applications including fluid–structure interactions. The heat and mass transfer modeling is finally examined for two-phase mixtures. Computations using a fractional step approach of water hammer and fast depressurization with flashing are performed. Good agreement is obtained with available experimental data. All computations are performed with the Europlexus fast transient dynamics software.

38 citations


Journal ArticleDOI
Abstract: We construct an approximate Riemann solver for the isentropic Baer−Nunziato two-phase flow model, that is able to cope with arbitrarily small values of the statistical phase fractions The solver relies on a relaxation approximation of the model for which the Riemann problem is exactly solved for subsonic relative speeds In an original manner, the Riemann solutions to the linearly degenerate relaxation system are allowed to dissipate the total energy in the vanishing phase regimes, thereby enforcing the robustness and stability of the method in the limits of small phase fractions The scheme is proved to satisfy a discrete entropy inequality and to preserve positive values of the statistical fractions and densities The numerical simulations show a much higher precision and a more reduced computational cost (for comparable accuracy) than standard numerical schemes used in the nuclear industry Finally, two test-cases assess the good behavior of the scheme when approximating vanishing phase solutions

37 citations


Journal ArticleDOI
TL;DR: This is the only existing scheme for which the approximated phase fractions, phase densities and phase internal energies are proven to remain positive without any restrictive condition other than a classical fully computable CFL condition.
Abstract: We present a relaxation scheme for approximating the entropy dissipating weak solutions of the Baer-Nunziato two-phase flow model. This relaxation scheme is straightforwardly obtained as an extension of the relaxation scheme designed in 16 for the isentropic Baer-Nunziato model and consequently inherits its main properties. To our knowledge, this is the only existing scheme for which the approximated phase fractions, phase densities and phase internal energies are proven to remain positive without any restrictive condition other than a classical fully computable CFL condition. For ideal gas and stiffened gas equations of state, real values of the phasic speeds of sound are also proven to be maintained by the numerical scheme. It is also the only scheme for which a discrete entropy inequality is proven, under a CFL condition derived from the natural sub-characteristic condition associated with the relaxation approximation. This last property, which ensures the non-linear stability of the numerical method, is satisfied for any admissible equation of state. We provide a numerical study for the convergence of the approximate solutions towards some exact Riemann solutions. The numerical simulations show that the relaxation scheme compares well with two of the most popular existing schemes available for the Baer-Nunziato model, namely Schwendeman-Wahle-Kapila's Godunov-type scheme 39 and Tokareva-Toro's HLLC scheme 44. The relaxation scheme also shows a higher precision and a lower computational cost (for comparable accuracy) than a standard numerical scheme used in the nuclear industry, namely Rusanov's scheme. Finally, we assess the good behavior of the scheme when approximating vanishing phase solutions.

30 citations


Cites background from "A fractional step method to compute..."

  • ...Actually, comparing with Lax-Friedrichs type schemes is quite significant since for such stiff configurations as vanishing phase cases, these schemes are commonly used in the industrial context because of their known robustness [30]....

    [...]

  • ...This is actually the case when, in addition to the convective system (1), zero-th order source terms are added to the model in order to account for relaxation phenomena that tend to bring the two phases towards thermodynamical (T1 = T2 and u1 = u2) and mechanical (p1 = p2) equilibria (see [12, 21] for the models and [30] for adapted numerical methods)....

    [...]

  • ...However, this work is mainly concerned with the convective effects and these relaxation source terms are not considered here (see [12] for some modeling choices of these terms and [30] for their numerical treatment)....

    [...]


Journal ArticleDOI
Abstract: A class of non-equilibrium models for compressible multi-component fluids in multi-dimensions is investigated taking into account viscosity and heat conduction. These models are subject to the choice of interfacial pressures and interfacial velocity as well as relaxation terms for velocity, pressure, temperature and chemical potentials. Sufficient conditions are derived for these quantities that ensure meaningful physical properties such as a non-negative entropy production, thermodynamical stability, Galilean invariance and mathematical properties such as hyperbolicity, subcharacteristic property and existence of an entropy–entropy flux pair. For the relaxation of chemical potentials, a two-component and a three-component models for vapor–water and gas–water–vapor, respectively, are considered.

30 citations


Journal ArticleDOI
Abstract: This paper is devoted to the validation of a two-fluid two-phase flow model in some highly unsteady situations involving strong rarefaction waves and shocks in water-vapor flows. The two-fluid model and its associated numerical method that were introduced in a previous work are first recalled, and details on the computational scheme and the verification of interfacial mass transfer terms are provided. Consistency with experimental data is checked in three configurations. First, a comparison with the speed of sound in a two-phase mixture is detailed. Afterwards, numerical approximations obtained with the two-fluid approach are discussed and compared with some experimental data documented in the Simpson water-hammer experiment and the high depressurization with flashing associated with Canon experiment.

27 citations


References
More filters


01 Jan 1959
Abstract: The method of characteristics used for numerical computation of solutions of fluid dynamical equations is characterized by a large degree of non standardness and therefore is not suitable for automatic computation on electronic computing machines, especially for problems with a large number of shock waves and contact discontinuities. In 1950 v. Neumann and Richtmyer proposed to use, for the solution of fluid dynamics equations, difference equations into which viscosity was introduced artificially; this has the effect of smearing out the shock wave over several mesh points. Then, it was proposed to proceed with the computations across the shock waves in the ordinary manner. In 1954, Lax published the "triangle'' scheme suitable for computation across the shock" waves. A deficiency of this scheme is that it does not allow computation with arbitrarily fine time steps (as compared with the space steps divided by the sound speed) because it then transforms any initial data into linear functions. In addition, this scheme smears out contact discontinuities. The purpose of this paper is to choose a scheme which is in some sense best and which still allows computation across the shock waves. This choice is made for linear equations and then by analogy the scheme is applied to the general equations of fluid dynamics. Following this scheme we carried out a large number of computations on Soviet electronic computers. For a check, some of these computations were compared with the computations carried out by the method of characteristics. The agreement of results was fully satisfactory.

1,696 citations


Book
29 Nov 2005
Abstract: Part I Fundamental of two-phase flow.- Introduction.- Local Instant Formulation.- Part II Two-phase field equations based on time average.- Basic Relations in Time Average.- Time Averaged Balance Equation.- Connection to Other Statistical Averages.- Part III. Three-dimensional model based on time average.- Kinematics of Averaged Fields.- Interfacial Transport.- Two-fluid Model.- Interfacial Area Transport.- Constitutive Modeling of Interfacial Area Transport.- Hydrodynamic Constitutive Relations for Interfacial Transfer.- Drift Flux Model.- Part IV: One-dimensional model based on time average.- One-dimensional Drift-flux Model.- One-dimensional Two-fluid Model.- Two-Fluid Model Considering Structural Materials in a Control Volume.

1,197 citations


Journal ArticleDOI
Abstract: In this paper, a two-phase mixture theory is presented which describes the deflagration-to-detonation transition (DDT) in reactive granular materials. The theory is based on the continuum theory of mixtures formulated to include the compressibility of all phases and the compaction behavior of the granular material. By requiring the model to satisfy an entropy inequality, specific expressions for the exchange of mass, momentum and energy are proposed which are consistent with known empirical models. The model is applied to describe the combustion processes associated with DDT in a pressed column of HMX. Numerical results, using the method-of-lines, are obtained for a representative column of length 10 cm packed to a 70% density with an average grain size of 100 μm. The results are found to predict the transition to detonation in run distances commensurate with experimental observations. Additional calculations have been carried out to demonstrate the effect of particle size and porosity and to study bed compaction by varying the compaction viscosity of the granular explosive.

1,008 citations


"A fractional step method to compute..." refers background in this paper

  • ...2 Closure laws for interfacial transfer terms The interfacial transfer contributions have been studied in [4, 5, 17, 10, 28, 29], and a summary can be found in appendix A....

    [...]

  • ...Among other studies, one should at least point out the contributions [4, 28, 5, 27, 32] which are concerned with the modeling aspects....

    [...]


Journal ArticleDOI
TL;DR: A new model and a solution method for two-phase compressible flows is proposed that provides reliable results, is able to compute strong shock waves, and deals with complex equations of state.
Abstract: We propose a new model and a solution method for two-phase compressible flows. The model involves six equations obtained from conservation principles applied to each phase, completed by a seventh equation for the evolution of the volume fraction. This equation is necessary to close the overall system. The model is valid for fluid mixtures, as well as for pure fluids. The system of partial differential equations is hyperbolic. Hyperbolicity is obtained because each phase is considered to be compressible. Two difficulties arise for the solution: one of the equations is written in non-conservative form; non-conservative terms exist in the momentum and energy equations. We propose robust and accurate discretisation of these terms. The method solves the same system at each mesh point with the same algorithm. It allows the simulation of interface problems between pure fluids as well as multiphase mixtures. Several test cases where fluids have compressible behavior are shown as well as some other test problems where one of the phases is incompressible. The method provides reliable results, is able to compute strong shock waves, and deals with complex equations of state.

807 citations


"A fractional step method to compute..." refers background in this paper

  • ...with “zero time scales”) one can refer to [34] (which deals with instantaneous velocity and pressure relaxation) or [21] (which only deals with instantaneous pressurerelaxation) among others....

    [...]

  • ...Fewer papers tackle the problem of water-gas or water-vapour flows, among which we must quote [33, 34] and more recently the article [12] , that examines a medium of small oscillating bubbles in a liquid medium, and also provides a general formalism in order to derive meaningful governing equations....

    [...]