Multiphase Flows: Compressible Multi-Hydrodynamics - Springer Link

 Multiphase Flows: Compressible Multi-Hydrodynamics Part I: Effective Field Formulation of Multiphase Flows Daniel Lhuillier ⋅ Theo G. Theofanous Part II: Computation with Effective-Field Models of Multiphase Flows Meng-Sing Liou ⋅ Theo G. Theofanous 

Institut Jean le Rond d’Alembert, CNRS and University Paris, Paris, France [email protected]  Department of Chemical Engineering, Department of Mechanical Engineering, Center for Risk Studies and Safety, University of California, Santa Barbara, CA, USA [email protected]; [email protected]  NASA Glenn Research Center, Cleveland, OH, USA [email protected]

PART I: EFFECTIVE FIELD FORMULATION OF MULTIPHASE FLOWS . . . .. . . .. . .





Introduction and Scope I . . .. . . .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . .



 . . .

Basics of Coarse-Graining. .. . . .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . The Two-Fluid Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . The Kinetic Theory Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . The Hybrid (Symmetry-Breaking) Model . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

   



A General Formulation . . . .. . . .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . .



 . .. .. .

Non-Dissipative Model .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . .  Rigid Particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .  Rigid Particles, No Velocity Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .  Rigid Particles with Added-Mass Velocity Fluctuations . . . . . . . .. . . . . . . . . . . . . . .  Compressible Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 

 . .. .. . .. .. .

Dissipative Model.. .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . Noncompressible Particles: Solid Grains or Drops . . . . . . . . . . . . . .. . . . . . . . . . . . . . . The Dissipation Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . The Constitutive Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Compressible Particles: Bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . The Dissipation Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . The Constitutive Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Final Form of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .

Dan Gabriel Cacuci (ed.), Handbook of Nuclear Engineering, DOI ./----_, © Springer Science+Business Media LLC 

       



  . . .. .. .. .. . .

Multiphase Flows: Compressible Multi-Hydrodynamics

Summary of Key Results. . . . .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . .. .  Hybrid Approach for Dispersed Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Supplementary Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Pseudo-Turbulent Kinetic Energies .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Volume Fraction Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Interfacial Energy Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Particle Deformation and Dynamics of Interfaces .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Hyperbolicity .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Nuclear Reactor (Design) Systems Codes.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

APPENDIX A: Rigid Spheres in a Nonviscous Fluid . . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . .. .  APPENDIX B: Hyperbolicity Aspects of the Effective Field Model.. . . .. . . .. . . .. . . .. . . .. .  APPENDIX C: Including Surface Tension. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . .. .  PART II: COMPUTATION WITH EFFECTIVE-FIELD MODELS OF MULTIPHASE FLOWS. .. . . .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . .. .  

Introduction and Scope II . . .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . .. . 



Strategy for Computing Compressible Multi-Hydrodynamics . . .. . . .. . . .. . 

 . .

Basics: The Riemann Problem and the Godunov Method . . . .. . . .. . . .. . . .. .  The Riemann Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  The Godunov Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

 . . .

Approximate Flux “Splitting” Schemes for Single Phase Flows . .. . . .. . . .. .  Characteristics-Based Flux Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Direct Flux Splitting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Advection Upstream Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

 . . .

Advection Upstream Splitting for the Effective Field Model ... . . .. . . .. . . .. .  Recasting the System of Equations in Quasi-Conservative Form . . . . . . . . . . . .  Numerical Discretization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Time Integration.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

 . . . . .

Numerical Testing in the ARMS Code . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . .. .  Uniformly Translating Body-and-Fluid System .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  The Faucet Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Fitt’s Problem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Shock Tube Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Particle Cloud Dynamics in Gaseous Shocks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 



Conclusions and Outlook . . .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . .. . 

APPENDIX D: Sample Computational Results . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . .. .  References . . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . .. . 




Abstract: Effective field modeling of two-phase flow has provided a critical part of the foundation upon which light water (power) reactor technology was made to rest some - years ago. We can envision a similarly significant role in the future as simulation capabilities are poised to meet new kinds of practical demands at the interplay between economics, safety assurance, and regulatory needs. These new demands will require better predictive reliability for larger departures from past practices, and this in turn will require strengthening of the scientific component along with translating past empiricism into more and more fundamental terms. In this perspective, the mathematical formulation of the effective field model, as well as the numerical implementation of this formulation needs to be revisited and reassessed. Helping respond to this need is the purpose of this chapter. We delineate a conceptual framework for addressing prediction of multiphase flows at the three-dimensional, phase distribution level. This is in terms of a local, disperse system description (bubbles/drops in a continuous liquid/vapor phase). The requirement that follows is a well-posed formulation and a high-fidelity numerical treatment that allows capturing of shocks and contact discontinuities over all (relative) flow speeds, consistently with what is physically allowable according to the density ratios involved — in particular, high relative Mach numbers for droplet/particle flows. The importance of inviscid interactions (between the phases) in this context is highlighted. The scope is for disperse-phase volume fractions up to about % as pertinent to fluid-fluid systems. This provides the basis for addressing phase transitions through coalescence as this process becomes significant at still higher volume fractions. The theoretical framework provides also the basis for extensions (outlined only in general terms here) to the high-volume fractions pertinent to dense solid-particle systems. The computational approach is readily applicable to both these extensions. A general disperse system formulation is derived by means of a new, “hybrid” method that incorporates features of a statistical approach and reveals more clearly the nature of phase interactions at the individual particle scale. Moreover in this manner the formulation lends itself to elaboration of the constitutive treatment by means of numerical simulations (based on the direct solution of the Navier-Stokes equations) resolved at the particle scale. The formulation is exemplified by successive applications to various increasingly complex situations, starting with non-dissipative systems, where one or the other phase may be incompressible. At each step we examine the hyperbolic character of the system of equations, and we include consideration of high (relative) Mach numbers. The basic constitutive treatment concerns pseudo-turbulent fluctuations of the continuous phase, and the resulting systems of equations are fully closed and hyperbolic even in their non-dissipative form (ready for computation), except for a nonhyperbolic corridor around the transonic region. Results obtained are discussed in relation to formulations that form the basis of current numerical tools (codes) employed in nuclear reactor design and safety analyses (mostly addressing bubbly flows), as well as formulations found in other contexts. This mathematical formulation is pursued further to its numerical implementation. With an emphasis on flow compressibility, we focus on capturing shocks and contact discontinuities robustly for all flow speeds and at arbitrarily high spatial resolutions. As we learn from relatively recent progress in single-phase flows, the key role is that of “up-winding” applied on the basis of a scheme that emphasizes conservative discretization. This background is briefly reviewed, culminating with a rather detailed exposition of the most recent advance in this line of development: the Advection Upstream Splitting Method (AUSM). The essential and new step here is to extend the basic ideas of the AUSM to the compressible multi-hydrodynamics








problems of interest here, and the above-mentioned EFM in particular. We also include calculations illustrative of numerical performance. The presentation is arranged into two autonomous parts: Part I addresses the formulation of the EFM and Part II deals with the numerical implementation and testing. An overall summarization of where we stand at the completion of this work and what we see as needed future developments is provided in the following. One basic objective was to address inviscid interactions by means of a coherent theoretical formulation and through to computational testing. In particular, we wanted to access highfidelity simulations of the type needed to address flow regimes at all flow speeds; especially at the high relative Mach numbers pertinent to disperse particle/droplet flows. This requires that the system of equations be hyperbolic, and we wanted to achieve a solid foundation rather than adopting any of the several ad hoc constitutive models as post facto “remedying” the problem. On theory, we begin with entropy rather than energy transport equations and we derive, consistently with thermodynamics and the momentum equations, a condition for satisfying conservation of total energy. This condition is of utmost importance showing the tight link between the conservation laws employed, and the transport equations of volume fraction and of pseudo-turbulent kinetic energies of the continuous (included) and disperse (not yet included in the derivation) phases. On this basis we demonstrate a systematic way to deduce closed systems of equations for non-dilute disperse flows, and thusly we arrive at an EFM that is hyperbolic except for a “corridor” around the transonic region. The key is a function of the disperse phase volume fraction E(α d ). It enters as a coefficient of the disperse phase pseudo-turbulent kinetic energy. Awaiting further definition as a function of the Mach number, by means of the type of direct simulations noted above, it is employed here throughout in its zero-Mach form. A much needed extension would also involve the pseudo-turbulent kinetic energy of the disperse phase, along with physics of dense dispersions (collisions etc.). While terms such as those proposed previously for “interfacial pressure” and “added mass” phenomena can be identified, the complete formulation is not reducible to any of those ad hoc models. Notably, the disperse phase pressure appears nowhere in the momentum equations. Also we find that the claimed as hyperbolic, Baer-Nunziato model involves a volume fraction transport equation, which is not physically tenable for dispersions, or is it an appropriate means to dealing with ill-posedness. On the other hand, we find perfect agreement with the formulations obtained at the incompressible limit by Geurst (), employing a complex variational approach, and by Wallis (), employing a rather involved development based on potential flow theory. On computations our objective is to capture shocks and contact discontinuities, for conditions that are within the hyperbolic regions in the Mach number space, and to explore () behaviors within the non-hyperbolic corridor, and () means of stabilization as necessary. Given the EFM development needs expressed above, it is understood that this testing in the Mach number space is strictly provisional. We begin with an adaptive mesh refinement infrastructure, and the Advection Upstream Splitting Method (AUSM), currently the method of choice for single-phase compressible flows. A key point of adaptation to our EFM is treating the pseudo-turbulent stresses within the pressure flux splitting, and ensuring that the discretization of the nonconservative terms is done in a way that satisfies propagation of contact discontinuities in uniform steady flow without disturbing the pressure field. Our approach is readily extendable to any equation of state and to adding any number of equations (volume fraction transport, multiple equations for the disperse phase for tracking multiple length scales




as may be found when the disperse phase is subject to fragmentation). The testing performed for this work was done on D problems only. Extending this testing to D and D problems is underway. Testing was carried out independently with two computer codes: ARMS (all-regime multiphase simulation) and MuSiC-ARMS (Multi-scale Simulation Code-ARMS). The ARMS was built on an open access platform, the structured adaptive mesh refinement infrastructure (SAMRAI) developed at Lawrence Livermore National Laboratory. The MuSiC-ARMS was built, more recently, on the MuSiC platform, our own specialized code, using irregular grids to “fit” areas of highest refinement (shocks, interfaces, etc.), which are embedded in a multilevel (adaptive) Cartesian mesh. This platform is also used for a DNS code, the MuSiC-SIM, and a pseudo-compressible (incompressible) code, the MuSiC-ISIM. We focus on dispersed being the heavy phase (droplet/particle flows) so as to access realistically high Mach numbers, and significant inviscid interactions. The test cases were selected to include various kinds of Riemann problems with discontinuities in (a) Mach number only (Fitt’s problems) and (b) pressure, or pressure and disperse phase volume fraction (shock tube problems). In the Fitt’s problem case, we include parametric studies on the value of C that appears in function E(α d ). In addition, we consider shock wave “impact” problems on particle clouds that are either with sharp or smooth (in particle volume fraction) outer boundaries, and as part of this class also the case of dilute clouds for which we have the analytic solution for comparison. Finally, we considered the capturing of contact discontinuities in “mild” situations such as the so-called Faucet problem and the simple convection of a coherent second phase by uniform flow. The Faucet problem is well known to be failed under grid refinement in all published tests to date. The convection problem is important check of the pressure non-disturbing condition, a requirement that is hard to meet due to the non-conservative terms found in all effective field models. The emphasis being on stability and convergence under grid refinement, all problems were carried out in the inviscid limit (no interfacial drag), and all cases passed the test except for the high pressure ratio shock tube problems where instabilities developed within the expansion wave. However, these cases were stabilized with a minimal amount of dissipation effected by adding a small amount of interfacial drag (roughly one tenth of the normal amount). These numerical results render support to the idea that, notwithstanding the “mild” non-hyperbolic corridor found in the analysis of Part I, the present effective field model is hyperbolic, and along with the numerical treatment employed they provide access to rather extreme two-phase flow conditions in a robust and accurate manner. In an overall perspective of computational fluid dynamics, the presently offered capability is complementary to that already available through the “standard”, non-hyperbolic twofluid model as already found in the computational frameworks of the ICE (Harlow and Amsden ) and SIMPLE (Patankar and Spalding ) methods. The special purposes aimed here are to overcome limitations in grid refinement and to approach flows where the phasic-relative velocities are high enough to introduce significant compressibility effects. Rapid advancement in hardware makes computational analysis of complex multiphase flows, even direct numerical simulations, increasingly more practical and reliable. High-fidelity/resolution techniques such as those employed here can address problems of varying time and length scales and this paves the way for actual simulations of multiphase physics at the effective field level, and even allowing a seamless analysis transitioning across regimes of multiphase flows.





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PART I: EFFECTIVE FIELD FORMULATION OF MULTIPHASE FLOWS 

Introduction and Scope I

The essence of multiphase flows is in the space-time distribution of phases and their length scales. Hence the need for an elaborated description of the Flow Regime or Flow Pattern, a topic of inquiry that began from the very first engineering efforts on the subject nearly a century ago, yet a topic of inquiry that remains largely unfulfilled (Theofanous and Hanraty ). Once the flow pattern is known everything else follows in a basic way. In fact all key behaviors of multiphase systems depend on the relevant flow regimes at the appropriate scale(s); for example, the limits to coolability (critical heat flux) are to be found in the microscopic flow pattern in the immediate vicinity of the heating surface, at a scale much smaller than realized previously (Theofanous et al. ). The fact that the design of safety systems in light water reactors has reached maturity, notwithstanding the absence of such a foundation, is owed to a large extend to a huge, comprehensively defined set of experiments over several scales up to prototypic that served to (empirically) anchor the numerical simulation tools (so-called systems codes) developed in this era (RELAP, TRAC-BWR, and the CATHARE code that followed). Conversely, none of this would have been possible if not for the organizing principle provided by the mathematical framework (the so-called two-fluid model) at the basis of these computational tools. Indeed, it was the synergism thus created and fed by an international, enormous in scope and resources R&D effort under the leadership (and the major financial support) of the US Nuclear Regulatory Commission (Office of Nuclear Reactor Regulatory Research) that made possible to meeting the great practical need: assuring and demonstrating the safety of nuclear power reactors. Moreover, it was this juncture in the s and s that afforded the first quantum leap in the scientific development of the subject, and provided the basis for the further developments that were to follow in the intervening  years; notably in addressing steam explosions as part of severe accident management in the s (Fletcher and Theofanous ) and in spinoffs driven by other practical needs such as the chemical process, oil and gas, and manufacturing industries (catalytic crackers, internal combustion engines, etc). In these intervening  years, at the other extreme of the detailed local level, major strides have been made in the direct solution of the Navier-Stokes equations (DNS), addressing both turbulence and interfacial dynamics. The scope of such simulations is ever enlarging, in pace with ingredients of high-performance computing: machine performance, adaptive mesh platforms, and massively parallelizable schemes. On this basis and with a renewed interest in nuclear power, the “old” systems tools are poised to meet and leverage with these new developments and capabilities toward multi-scale treatments that gain in generality and predictive power as the next generation nuclear power design and safety analysis tools (the CATHARE-NEPTUNE program in France, and the RELAP program in the USA). If successful these efforts, both in very early stages of development, are likely to provide the next quantum leap in numerical simulation of multiphase flows. It is in this context that this chapter has been written. The focus being, as noted above, on the numerical simulation of Flow Regimes, the basic requirements are a D representation and a demarcation between regions of space with topologically similar character. For example, in a gas-liquid flow we seek the demarcation between domains of bubbles dispersed in liquid, and domains of drops dispersed in gas. We call these demarcations, as large-scale discontinuities or LSDs (Theofanous and Dinh ). In this way




all macroscopic two-phase flows may be seen as interacting assemblages (collective effects of bubbly or droplet flow regions) of disperse flows, and the flow regime identification depends upon the spatial-temporal positioning and multiplicity of the LSDs. As a consequence, coarsegraining or the derivation of effective field (interpenetrating continua) transport equations need only address disperse media, and this focusing is important in eliminating ambiguities inherent with the coarse-graining of internal scales much larger than those of the underlying disperse flows. Moreover, based on typical regime stability requirements the region of interest is over disperse phase volume fractions of less than about %. As another key consequence, the numerical solution of the effective field equations must be amenable to grid refinement, so as to capture shocks, contact discontinuities, and the LSDs at arbitrarily sharp focus, and this means, besides the numerical scheme being nondiffusive, that the system of equations be well posed. A further aim in the formulation is to define the phase interaction terms clearly enough as to support the development of closures (a task of renewed potential under DNS) on a physically sound basis. Finally, emerging areas of application require the consideration of supersonic and transonic flows, areas with their own particular demands (Theofanous et al.). At this basic level of treatment the major attention is given to the dynamics of (inertial or) inviscid interactions between the dispersed and continuous phases (with due account for the so-called pseudo-turbulent fluctuations). These underlie the mathematical character of the system of conservation laws. The incorporation of turbulent stresses and of breakup/coalescence phenomena can be added according to need, for example, in the manner developed by Lahey’s and Ishii’s teams respectively. Some remarks on the formulation and implications of current systems codes are integrated in > Sect. . Accordingly, in Part I we are particularly interested in a continuum-mechanical description of mixtures made of many particles dispersed in a carrier fluid. We will present the two coarsegrained models relevant to the description of dispersed mixtures in > Sect. . We will insist on the difficulties inherent to the modeling of the interphase force of the two-fluid model before presenting in > Sect.  a special form of the two-fluid model dedicated to the description of dispersions. The non-dissipative version of that model is presented in > Sect.  with special emphasis on the role of pseudo-turbulent velocity fluctuations. The dissipative counterpart is presented in > Sect. , while some extensions (concerning the role of surface tension in particular) and the main results are recapitulated in > Sect. . Emphasis is placed in delivering complete systems of equations that can be taken directly to numerical solution, and as such Part I provides the starting point for Part II. We also aim for clarity on the interfacial momentum transfers — a subject of prolonged, and often confusing debates. Their meaning is not readily open to intuition, while their importance is critical to the well-posedness of the mathematical system that constitutes the effective field model.



Basics of Coarse-Graining

.

The Two-Fluid Model

The most well-known coarse-grained model is certainly the two-fluid or two-phase model (Nigmatulin ; Drew and Passman ; Ishii and Hibiki ) whose exclusive specificity








to disperse systems has been acknowledged only implicitly. In its standard form, the two-fluid model is obtained as a combination of three ingredients: (a) The (microscale) conservation equations for mass, momentum, and energy in each of the two phases (b) The function of presence χ k of phase k in the mixture with χ c + χ d =  where the indices c and d stand for the carrier fluid and the dispersed particles respectively (c) A statistical (e.g., ensemble, time, or volume) averaging denoted by < ⋯ > For example, the microscale momentum balance of phase k is ρ k [

∂v k + (v k ⋅ ∇)v k ] = ∇ ⋅ σ k + ρ k g ∂t

where ρ k , v k and σ k are the microscale mass density, velocity, and stress-tensor. From that microscale conservation equation one deduces the macroscale momentum balances (with no mass exchange between phases, for simplicity here) (Nigmatulin ; Drew and Passman ; Ishii and Hibiki ) dc uc + ∇⋅ < χ c ρ c v ′c v ′c >= ∇⋅ < χ c σ c > − < σ c ⋅ n d δ I > +α c ρ c g , dt d u  ′ ′   α d ρ d d d + ∇⋅ < χ d ρ d v d v d >= ∇⋅ < χ d σd > + < σ c ⋅ n d δ I > +α d ρ d g . dt αc ρc

() ()

Here δ I is the function of presence of the interfaces between the two phases and n d is the normal to those interfaces pointing outwards phase d. Moreover, v ′k = v k − u k is the velocity fluctuation relative to the mean velocity u k , α k =< χ k > is the volume fraction or probability of presence of phase k, and d k /dt = ∂/∂t + u k ⋅ ∇ is a convected time-derivative. These macroscale equations are perfectly symmetric relative to the two phases, i.e., there is no apparent distinction between the continuous and the dispersed phase.

.

The Kinetic Theory Model

Concerning the dispersed (particulate) phase there exists a second coarse-grained model, the kinetic theory model which is built on the basis of three ingredients: (a) The balance equations for the mass momentum and energy of a single particle (b) The microscale number density δ d of the particles (c) A statistical averaging based on a probability distribution function and denoted by the same symbol < ⋯ > as in the two-fluid approach For example, the equation of motion for a single particle is m

 dw

dt







= ∮ σ c ⋅ n d ds + m g ,

where m  and w  are the mass and velocity of a particle. From this equation one deduces the momentum balance of the particulate phase (Buyevich et al. ; Lhuillier ; Zhang and




Prosperetti ; Jackson ) nm [

∂w + (w ⋅ ∇)w] + ∇ ⋅ [nm < w ′ ⊗ w ′ >] = n < ∮ σ c ⋅ n d ds > +nmg , ∂t

()

where w is the mean translational velocity of the particles, w ′ = w  −w is the fluctuation relative to that mean value, m is the mean mass per particle, and n =< δ d > is the mean number density of the particles. This momentum balance is rather similar to that written in the two-phase model except for two remarkable differences: (a) the particle stress which is present in () is absent from (), and (b) the force exerted by the fluid is simply the mean force per particle times the particle number density. The particle stress is not present in the momentum balance but it has not completely disappeared. In fact it is taken into account in a different equation, the equation for the moment of momentum of the particles which writes (Lhuillier ) < δd

d ρ  r ⊗ u dτ > −n < ∫ ρ d u ⊗ u dτ > dt ∫ d 



= n < ∮ r ⊗ (σc ⋅ n d ) ds > −n < ∫ σd dτ > ,

()

where u is the particle-internal motion (a pure rotation for a rigid particle) and r is the vector joining the center of a particle to a point of its surface. The antisymmetric part of the above equation is nothing but the angular momentum balance of the particulate phase. And the symmetric part can be inverted so as to give an explicit expression for the particle stress. When the role of inertia in internal motion can be neglected one recovers Batchelor’s expression (Batchelor ) < ∫ σd dτ >=

.

 < [r ⊗ (σ c ⋅ n d ) + (σ c ⋅ n d ) ⊗ r] ds > .  ∮

()

The Hybrid (Symmetry-Breaking) Model

The main advantage of the kinetic theory model is that it introduces quantities (e.g., the mean force n < ∮ σ c ⋅ n d ds > acting on the particulate phase), which have a far more intuitive meaning than the corresponding quantity for the two-fluid model (where the interphase force is written as < σ c ⋅ n d δ I >). It would be nice therefore if we could express some two-phase quantities like the mean particle stress or the mean interphase force in terms of quantities pertaining to the kinetic theory approach. This transformation between two-phase and kinetic quantities is in fact well known in electrodynamics of continuous media. One makes such a transformation when writing the mean charge per unit volume as the density of true particle charges minus the divergence of the mean density of particle dipoles, etc. For two-phase flows the corresponding transformations were derived in Buyevich et al.(), Lhuillier (), Zhang and Prosperetti (), and Jackson () with the following general results that apply to any quantity A defined on the interfaces and B defined inside the particles < A δ I >= n < ∮ A ds > −∇ ⋅ [n < ∮ rA ds >] + ⋯

()

< B χ d >= n < ∫ B dτ > −∇ ⋅ [n < ∫ rB dτ >] + ⋯ ,

()








where the integrals span over the particle surface and the particle volume respectively, and the dots stand for a development in series of higher and higher moments of A and B. When B stands for the small-scale charge density one verifies the well-known result mentioned above and the dots stand for the contributions of particle quadrupoles and higher multipoles. For the modeling of suspensions of particles, A will be the local force exerted by the continuous phase and B will be the local stress inside a particle for which the above transformation rules will be written as 





< σ c ⋅ n d δ I > ≈ n < ∮ σ c ⋅ n d ds > −∇ ⋅ [n < ∮ r ⊗ (σ c ⋅ n d ) ds >] 



< χ d σd > ≈ n < ∫ σd dτ >

() ()

One can be surprised that the simplest transformation rule is applied to the particle stress while a more complete one is necessary for the interphase force. The reason is that the particle stress appears with its divergence in the momentum balance. If we limit the interphase force to the first term of the transformation rule we miss a stress term, while if we limit the particle stress to the first term, we only miss higher-order terms comparable to the terms we neglected in the interphase force. The next step is to introduce the mean pressure p c of the continuous phase defined as < χ c σ c >= −α c p c I+ < χ c τ c >

()

where τ c is the microscale fluid viscous stress. With the continuous phase pressure and the transformation rules () and () the two-phase momentum balances, () and (), now appear in a special (hybrid) form dc uc + ∇ ⋅ σ c + α c ∇p c = −F + α c ρ c g dt d u α d ρ d d d + ∇ ⋅ σd + α d ∇p c = F + α d ρ d g dt αc ρc

() ()

with the definitions F = n < ∮ (σ c + p c I) ⋅ n d ds >  ′

′



() 

σ c = < χ c ρ c v c ⊗ v c > − < χ c τ c > −n < ∮ r ⊗ (σ c + p c I) ⋅ n d ds > σd = < χ d ρ d v d′ ⊗ v d′ > −n < ∫ ρ d u ⊗ u dτ > + < δ d

d ρ  u ⊗ r dτ > dt ∫ d

() ()

The expression of the dispersed phase stress σd was obtained with the help of () and (). The carrier fluid does not play any role in that stress, which vanishes for massless particles. Whenever collisions (or direct inter-particle forces) must be taken into account a collision stress (or an inter-particle stress) is to be added to the right-hand side of (). In what follows we will use () and () as the starting point for the two-phase description of suspensions of particles. It is remarkable that the dispersed phase pressure nowhere appears in the two momentum equations, a feature which is specific to dispersed mixtures. We will see however that the pressure difference p c − p d is hidden in the stress σd and also that it plays a role in the energy equations as




well as in the transport equation for the volume fraction; so that the models we have to present are two-pressure models actually. The entropy balance equations of the hybrid model are obtained in a similar way. If one neglects the interfacial entropy (linked to surface tension) as well as any entropy production at the interfaces, then the mean specific entropies s k evolve in time as dd sd + ∇ ⋅ hd = Δd + Σ dt dc sc αc ρc + ∇ ⋅ hc = Δc − Σ dt

αd ρd

() ()

where Δ k and h k are the entropy production rate and entropy flux in phase k while Σ is the interphase entropy exchange. These quantities are defined as Σ = −n < ∮

q c ⋅ n d ds > , Tc

q q c > +n < ∮ r ( c ⋅ n d ) ds > ,  Tc Tc d  ′ ′   h d = < χ d ρ d s d v d > −n < ∫ ρ d s d u dτ > + < δ d ∫ ρ d s d r dτ > , dt  ′ ′

hc = < χc ρc sc vc > + < χc

() () ()

where q c /Tc is the microscale entropy flux of the continuous phase. There is an evident similarity with the corresponding definitions for the momentum balances.



A General Formulation

The main equations describing a two-phase mixture are the six balance equations for mass, energy, and momentum. The way these equations are obtained is well documented in many textbooks (Nigmatulin ; Drew and Passman ; Ishii and Hibiki ). Here we prefer an entropy balance instead of the energy balance and we will use the hybrid form of the momentum equations as stressed in the previous section. Thus accounting also for phase change, the six main balance equations are written as ∂ (α d ρ d ) + ∇ ⋅ (α d ρ d u d ) = Γ ∂t ∂ (α c ρ c ) + ∇ ⋅ (α c ρ c u c ) = −Γ ∂t d s ⋆ α d ρ d d d + ∇ ⋅ h d = Δ d + Σ + Γ(s − s d ) dt dc sc αc ρc + ∇ ⋅ h c = Δ c − Σ − Γ(s ⋆ − s c ) dt d u ⋆ α d ρ d d d + ∇ ⋅ σd + α d ∇p c = F + Γ(u − u d ) + α d ρ d g dt dc uc αc ρc + ∇ ⋅ σ c + α c ∇p c = −F − Γ(u ⋆ − u c ) + α c ρ c g . dt

() () () () () ()

In these equations appear the mass density ρ k , the entropy density s k , the volume fraction α k , and the velocity u k of phase k (k = c, d) together with the mean pressure p c of the continuous








phase. Besides the entropy production rate Δ k ≥  inside phase k, are defined the intra-phase entropy flux h k and the stress tensors σ k together with the exchanges between phases, Γ for the mass exchange, Σ + Γs ⋆ for the entropy exchange and F + Γu ⋆ for the momentum exchange. Our aim is to close these equations, i.e., to find by all means (DNS, exact analytical results, phenomenology, etc.) explicit expressions of all averaged quantities (like the stresses σ c , σd and the inter-phase force F but also s ⋆ and u ⋆ ) in terms of the basic variables such as the mean velocities and the volume fractions. To achieve that goal, an important step will be to assume that the thermodynamic properties of each phase in the mixture are the same as those at work for a pure phase. This means that the equations of state, which hold when written with local quantities are assumed to hold when written with the mean values of these quantities. In other words, one denies any role to the fluctuations around the mean pressure p k and the mean temperature Tk inside phase k. But one acknowledges a possible difference between Tc and Td as well as between p c and p d . For example the Gibbs relation of phase k in the mixture will be supposed to be d e k = Tk ds k − p k d(/ρ k ) ,

()

where e k is the mean internal energy per unit mass of phase k. For convenience that Gibbs relation is rewritten as αk ρk

p k d k (α k ρ k ) d k ek d s d α = Tk α k ρ k k k + − pk k k , dt dt ρk dt dt

where d k /dt = ∂/∂t+u k ⋅∇ is the convected time-derivative of phase k. The evolution equations of the internal energies are then deduced from the above evolution equations of entropy and mass per unit volume ∂ (α d ρ d ed ) + ∇ ⋅ (α d ρ d ed u d + Td h d ) = Td Δ d + Td Σ + h d ⋅ ∇Td ∂t ∂α Γ + Γ[ed + Td (s ⋆ − s d )] − p d ( d + ∇ ⋅ (α d u d ) − ) ∂t ρd ∂ (α c ρ c e c ) + ∇ ⋅ (α c ρ c e c u c + Tc h c ) = Tc Δ c − Tc Σ + h c ⋅ ∇Tc ∂t ∂α c Γ ⋆ − Γ[ec + Tc (s − s c )] − p c ( + ∇ ⋅ (α c u c ) + ) . ∂t ρc

()

()

Note the first appearance of the dispersed phase pressure and the work of pressure forces in the volume exchange between phases. The balances of the kinetic energies are deduced from the momentum balances u u ∂ (α d ρ d d ) + ∇ ⋅ (α d ρ d d u d + u d ⋅ σd + α d p c u d ) ∂t   = u d ⋅ F + σd : ∇u d + Γ (u d ⋅ u ⋆ −

u d ) + p c ∇ ⋅ (α d u d ) + α d ρ d u d ⋅ g 

()

u u ∂ (α c ρ c c ) + ∇ ⋅ (α c ρ c c u c + u c ⋅ σ c + α c p c u c ) ∂t   = −u c ⋅ F + σ c : ∇u c − Γ (u c ⋅ u ⋆ −

u c ) + p c ∇ ⋅ (α c u c ) + α c ρ c u c ⋅ g . 

()




The total phasic energy per unit mass is written as E k = e k (ρ k , s k ) +

u k + Kk . 

()

where K k is the pseudo-turbulent kinetic energy related to the velocity fluctuations around u k . It is to be stressed that in a fluid-particles mixture those velocity fluctuations are created by the shear motion of the particles, by the relative motion between the fluid and the particles or by a change in volume or shape of the particles. They disappear as soon as the shear motion, the relative motion, or the particle deformation stops. They are thus quite different from the velocity fluctuations generated by Brownian motion or by turbulence. After adding the balance of kinetic energy to the balances of internal energy one obtains the following evolution equations for the total energy ∂α ∂ (α d ρ d E d ) + ∇ ⋅ (α d ρ d E d u d + α d p c u d + u d ⋅ σd + Td h d ) = −p c d + Td Δ d ∂t ∂t + σd : ∇u d + h d ⋅ ∇Td + Td Σ + u d ⋅ F + Γ(μ d + Td s ⋆ + u d ⋅ u ⋆ − u d /) ∂α ∂ + (p c − p d ) [ d + ∇ ⋅ (α d u d )] + (α d ρ d K d ) + ∇ ⋅ (α d ρ d K d u d ) + α d ρ d u d ⋅ g ∂t ∂t ∂α c ∂ (α c ρ c E c ) + ∇ ⋅ (α c ρ c E c u c + α c p c u c + u c ⋅ σ c + Tc h c ) = −p c + Tc Δ c ∂t ∂t ⋆ ⋆  + σ c : ∇u c + h c ⋅ ∇Tc − Tc Σ − u c ⋅ F − Γ(μ c + Tc s + u c ⋅ u − u c /) ∂ + (α c ρ c K c ) + ∇ ⋅ (α c ρ c K c u c ) + α c ρ c u c ⋅ g . ∂t

()

()

In these equations μ k = є k + p k /ρ k − Tk s k is the chemical potential or Gibbs free-energy per unit mass of phase k. Note the asymmetry concerning the work of the pressure difference p d − p c , which appears in the equation for the particulate phase only. We will neglect surface tension and any interfacial energy so that the right-hand sides of the two above equations (apart from the work of external gravity forces) represent the exchange of energy between the two phases and their sum must vanish or must be equal to the divergence of some Galilean-invariant energy flux Q. Hence the necessary condition to be fulfilled for the total energy E = α d ρ d E d + α c ρ c E c to be conserved is ∂α d + ∇ ⋅ (α d u d )] ∂t + Γ[μ d − μ c + (Td − Tc )s ⋆ + (u c − u ⋆ ) / − (u d − u ⋆ ) /] ∂ + ∑ [ (α k ρ k K k ) + ∇ ⋅ (α k ρ k K k u k ) + σ k : ∇u k + h k ⋅ ∇Tk ] + ∇ ⋅ Q =  . ∂t k=c,d

Td Δ d + Tc Δ c + (Td − Tc )Σ + (u d − u c ) ⋅ F − (p d − p c ) [

()

The above equality will play a role of utmost importance for dispersed mixtures and it holds in all cases, whether the carrier fluid is compressible or not, whether the particles are rigid or not. It must be considered as a necessary condition to be satisfied between the (yet unknown) transport equations for α d , K d , and K c and the six main balance equations ()–(). Its significance will appear more clearly if we deal first with non-dissipative mixtures and then with dissipative ones.





 


Non-Dissipative Model

When all dissipative phenomena are discarded, condition () simplifies to R

(u d − u c ) ⋅ F − (p d − p c ) ( + ∑ [α k ρ k k=c,d

∂α d + ∇ ⋅ (α d u d )) ∂t

dk Kk R R + σ k : ∇u k ] + ∇ ⋅ Q =  . dt

()

stating the link to be verified between the non-dissipative form of the transport equations for α d , K d , and K c and the non-dissipative parts (with superscript R) of the phasic stresses and the interphase force. We consider the consequences of that equality for cases of increasing complexity.

. ..

Rigid Particles Rigid Particles, No Velocity Fluctuations

When the particles are rigid and the role of velocity fluctuations is neglected, condition () simplifies to (u d − u c ) ⋅ F R + σ cR : ∇u c + σdR : ∇u d + ∇ ⋅ Q R =  and there is apparently no other solution than F R = σ cR = σdR = Q R = . This result holds for compressible as well as incompressible fluid surrounding the (compressible or incompressible) particles. The equations describing that simple non-dissipative case are ∂α d + ∇ ⋅ (α d u d ) =  ∂t ∂ (α c ρ c ) + ∇ ⋅ (α c ρ c u c ) =  ∂t dk sk = dt d u α k ρ k k k + α k ∇p c = α k ρ k g . dt

() () () ()

When the carrier fluid is incompressible () is replaced by ∇⋅u =  where u = α c u c +α d u d is the volume-weighted mean velocity of the mixture and the remaining equations are left unchanged. In all cases (rigid particles in a compressible or incompressible carrier fluid but with negligible velocity fluctuations) the balances of total energy are given by

αk ρk

dk Ek ∂α + ∇ ⋅ (α k p c u k ) + p c k = α k ρ k u k ⋅ g . dt ∂t

()

It is known since long (Gidaspow ; Stuhmiller ; Jones and Prosperetti ) that the above set of non-dissipative equations is not hyperbolic.


..



Rigid Particles with Added-Mass Velocity Fluctuations

When the carrier fluid is incompressible and its flow is nonviscous and potential the two momentum equations () and () can be written as (see Appendix A): dc uc + ∇ ⋅ σ cK + α c ∇p c = −F K + α c ρ c g dt d u K K α d ρ d d d + ∇ ⋅ σd + α d ∇p c = F + α d ρ d g , dt αc ρc

() ()

with ∂J − ∇ ⋅ (u d ⊗ J) − (J ⋅ ∇)u c − α c ρ c ∇K c − J × (∇ × u c ) ∂t K σ c = (u d − u c ) ⊗ J K

F =−

() ()

where J is the (Kelvin) fluid impulse (defined in (A-)) and K c is the fluctuational kinetic energy of the fluid. Note that so far no expression is given for the particle stress σdK . Some general definition exists for this stress tensor (Sangani and Didwania ; Bulthuis et al. ) but a closed expression in terms of the fluid impulse and other main variables has been difficult to obtain. However, some insight concerning σdK can be provided by condition () as developed below. When particles move relative to the fluid they drag part of the fluid mass with them and they confer the fluid a supplementary kinetic energy. Neglecting the particles velocity fluctuations (K d = ) results in a well-known simplified expression for the fluid pseudo-turbulent (particleinduced) kinetic energy (Geurst ; Wallis )   K c = E(α d )(u d − u c ) , 

()

where E is a scalar depending on the particle volume fraction, which behaves like α d / for dilute suspensions. There is no real transport equation for K c but it is possible to deduce from () αc ρc

dc Kc dc (u − u c ) d c E = α c ρ c [E(u d − u c ) ⋅ (u d − u c ) + d ] dt dt  dt = (u d − u c ) ⋅ [

(u d − u c ) d c E ∂J + ∇ ⋅ (u c ⊗ J)] − α c ρ c ∂t  dt

()

where J = α c ρ c E(α d )(u d − u c ) .

()

With F K and σ cK given by the exact results () and (), and noticing that d d α d /dt = −α d ∇⋅u d for rigid particles, the above (pseudo-)transport equation for K c is finally written as αc ρc

dc Kc = −F K ⋅ (u d − u c ) − σ cK : ∇u c − α d P K ∇ ⋅ u d − ∇ ⋅ Q K . dt

()








with the pressure P K and the energy flux Q K defined as P K = −α c ρ c

∂K c , ∂α d

()

Q K = α c ρ c K c (u d − u c ) .

()

When compared to the transport equation for a turbulent kinetic energy, it is worthy to note that () exhibits three production terms and one flux (but no dissipation term). Taking () into account, relation () is satisfied with R

F =F

K

R

K

R

K

R

K

, σ c = σ c , σd = α d P I , Q = Q .

()

We have thus obtained explicit expressions for the fluid impulse and for the particle stress, which are consistent with assumption (). It is remarkable that the same scalar E(α d ) appears in the pseudo-turbulent kinetic energy () and in the Kelvin impulse (). This is not true in general as was emphasized by Biesheuvel and Spoelstra () but is a consequence of our neglect of the velocity fluctuations of the particles, as is clear when comparing with the general result (A-) in Appendix A. All these results hold for rigid spheres moving in an incompressible carrier fluid. Their extension to a compressible fluid is far from obvious. The velocity disturbances created in the fluid by the particle motion do not propagate instantaneously but with the speed of sound and they are accompanied by mass density disturbances. The simplest approximation is to consider that expression () is approximately correct in a compressible carrier fluid. Then it can be shown that identity () still holds and consequently, to describe the flow of rigid particles moving in a fluid with added-mass velocity fluctuations described by () one needs the following set of six non-dissipative equations ∂α d + ∇ ⋅ (α d u d ) =  ∂t ∂ (α c ρ c ) + ∇ ⋅ (α c ρ c u c ) =  ∂t dd sd dc sc = , = dt dt dc uc K K αc ρc + ∇ ⋅ σ c + α c ∇p c = −F + α c ρ c g dt d u α d ρ d d d + ∇(α d P K ) + α d ∇p c = F K + α d ρ d g . dt

() () () () ()

The corresponding balances of total energies are deduced from () and () as αc ρc

dc Ec ∂α c + ∇ ⋅ (α c p c u c + u c ⋅ σ cK + Q K ) + p c = −W K + α c ρ c u c ⋅ g dt ∂t d E ∂α K K α d ρ d d d + ∇ ⋅ [α d (p c + P )u d ] + p c d = W + α d ρ d u d ⋅ g dt ∂t

() ()

with the inter-phase energy exchange defined as W K = ud ⋅ F K + αd P K ∇ ⋅ ud .

()




When all terms associated with added-mass kinetic energy vanish we find back the results of K > Sect. ... Note that from () the sum p c + P can be considered as the particle pressure. The mathematical character of the above set of mass and momentum equations has been investigated in Sushchikh and Chang () (summarized in Appendix B) and, in case of an incompressible fluid, it was found that its hyperbolicity requires the function E(α d )/α d and its first and second derivatives with respect to α d (denoted by a prime and a double prime) to satisfy the inequality 

[α d − α c

ρd ρd E E ′ E + ( − α c ) + α c α d ( ) ] ≥ [α d + α c + ] ρc αd αd ρ c αd

× [α d + α c

ρd E E ′′ + + α c α d ( ) ] . ρ c αd αd

()

It is interesting to note that this inequality is independent of the relative velocity and cannot be satisfied when E = , that is to say when one neglects the added-mass kinetic energy of the carrier phase. We looked for expressions of E(α d ) for which the above inequality is satisfied whatever the value of ρ d /ρ c . We found that the quadratic expression E = (α d /)(+Cα d ) (hence (E/α d )′′ = ) assures hyperbolicity for any value of the density ratio provided C ≤ −. Since E(α d ) must be positive, the choice of a particular value for C defines the range in volume fraction over which hyperbolicity can be assured on physical grounds. For example, with C = − we have E(α d ) =

αd ( − α d ) 

()

and hyperbolicity is assured up to α d = /. That special value of C was already pointed out by Geurst () who remarked that it is a sound choice in case of bubbly fluids for which a flow regime transition is expected close to α d ≈ .. This choice of C is also supported by theoretical arguments (Wallis , ). On the other hand, it must be noted that the physical validity of this second-order expansion, beyond the value of α d that corresponds to the maximum in E, would appear to require closer examination. While in bubbly flows the relative velocity is always small and the simplified incompressible analysis is quite adequate, this is not the case for solid particles or drops dispersed in a compressible phase. A complete analysis for this case (Sushchikh and Chang ) (Appendix B) shows that there is a non-hyperbolic corridor around the transonic region that depends weakly on the volume fraction. Clearly, if one demands hyperbolicity for any volume fraction and any Mach number, then new physics must be introduced, involving in particular a description of particle velocity fluctuations, either fluid-induced or resulting from collisions. As a consequence () would be replaced by a more elaborated expression like  K c = E(α d , M ⋆)(u d − u c ) + F(α d )K d . 

()

Although the two scalar functions E and F behave as α d / in the dilute limit (Zhang and Prosperetti ) we cannot take for sure that they are equal for all concentrations. The point is that no closed expression exists for K d itself or for its transport equation. However if () is accepted it is clear that the density α d ρ d + α c ρ c F(α d ) will play a main role in the transport equation for K d as well as in the modified form of inequality (). This is left for later studies.





 .


Compressible Particles

In dispersed liquid-vapour mixtures the bubbles are highly compressible and the carrier fluid is generally considered as incompressible. To simplify the issue the bubble shape will be assumed to stay spherical and only monodisperse collection of bubbles with a radius a(x, t) will be considered. The bubble radius changes with time and position and a relevant quantity is the convective time-derivative d d a/dt. When mass exchange occurs between the bubbles and the carrier fluid d d a/dt is different from w c the mean radial velocity of the fluid at the boundary with the bubbles. In the present section we neglect mass exchanges (a dissipative process) and take d d a/dt = w c but to be consistent with the next section dealing with dissipative phenomena, we will consider w c as the fundamental velocity to describe the non-dissipative radial motion of the bubbles. The velocity w c is responsible for a supplementary kinetic energy of the liquid, which adds to the virtual mass kinetic energy. The expression of the pseudo-turbulent kinetic energy K c becomes     K c = E(α d )(u d − u c ) + Q(α d ) w c  

()

where Q is a function of the volume fraction, which behaves like α d in the dilute limit. Because of the small mass density of the bubbles it is usual to assume that their kinetic energy of pulsation vanishes and we let K d =  in what follows. To describe the motion of the bubbly fluid the momentum equations are completed by two more (non-dissipative) equations (Nigmatulin ), the first one being the transport equation for the volume fraction with due account for the bubble expansion rate ∂α d wc + ∇ ⋅ (α d u d ) = α d ∂t a

()

and the second one being a generalization to finite volume fractions of the single particle Rayleigh-Lamb equation a d d w c /dt + (/)w c − (/)(u d − u c ) = (p d − p c )/ρ c . There is also a second quantity that must be properly generalized to finite volume fractions, the total (non-dissipative) stress, which is well known (Nigmatulin ) for dilute mixtures σ cR + σdR =

αd ρ c (u d − u c ) ⊗ (u d − u c ) + α d (p d − p c )I + α d ρ c w c I 

The two generalizations we are looking for can be deduced from constraint (). We insist again that there is no true transport equation for K c but, because of its dependence on the bubble volume fraction, on the relative velocity and on the bubble expansion rate, one can derive from () the identity αc ρc

dc Kc RK wc RK wc ∇α d ) + + α d P K − ≡ −(u d − u c ) ⋅ (F K + ( M) dt α c a αc a − (σ cK +

RK RK ) ∇ ⋅ ud − ∇ ⋅ Q K . I) : ∇u c − α d (P K +  α c

()




The Kelvin impulse J is still given by () and the quantities F K , σ cK , P K , and Q K are still defined by (), (), (), and () respectively, but with K c now given by (). The only new quantity is R K which is defined as K

R ≡

∂M + ∇ ⋅ (Mu d ) , ∂t

()

where M is the scalar Kelvin impulse of the liquid defined as M = α c ρ c Q(α d ) aw c .

()

The scalar M is playing for the radial motion the same role as the vector J for the relative motion. Taking () and () into account, relation () is satisfied with Q R = Q K and R

K

F =F +

RK RK RK R K R K ∇α d , σ c = σ c + )I . I , σd = α d (P + α c  α c

The generalized Rayleigh-Lamb equation simply appears as RK wc + α d P K − M = α d (p d − p c ) . αc a while the generalized total stress is σ cR + σdR = σ cK + α d (p d − p c )I + w c

M I. a

We now gather the above results to obtain the non-dissipative equations for a suspension of spherical bubbles moving in a non-compressible fluid with a pseudo-turbulent kinetic energy given by (), wc ∂α d + ∇ ⋅ (α d u d ) = α d ∂t a ∂α c + ∇ ⋅ (α c u c ) =  ∂t ∂α d ρ d + ∇ ⋅ (α d ρ d u d ) =  ∂t dc sc dd sd = , = dt dt dc uc RK K K ) + α c ∇p c = −F + α c ρ c g αc ρc + ∇ ⋅ σc + α c ∇ ( dt α c dd ud RK ) + α d ∇p c = F K + α d ρ d g + ∇(α d P K ) + α d ∇ ( dt α c ∂M wc K + ∇ ⋅ (Mu d ) − α c M = α d α c (p d − p c − P ) . ∂t a

αd ρd

() () () () () () ()

These eight equations reduce to six (see > Sect. ..) when the particles are incompressible : this is because () is redundant with (), and () reduces to p d = p c + P K . Note that this








justifies the statement made in > Sect. .. that p c + P K represents the particle pressure in case of rigid particles. It is sometimes interesting to have the transport equations for the total energy. The result is an extension of () and () dc Ec RK K K +Q ) + ∇ ⋅ (α c p c u c + u c ⋅ σ c + (α d u d + α c u c ) dt α c ∂α c = −W K + α c ρ c u c ⋅ g + pc ∂t d E ∂α α d ρ d d d + ∇ ⋅ [α d (p c + P K )u d ] + p c d = W K + α d ρ d u d ⋅ g . dt ∂t αc ρc

() ()

with the inter-phase energy exchange given by

W K = u d ⋅ [F K − α d ∇ (

RK wc )] + α d P K ∇ ⋅ u d − α d (p d − p c ) . α c a

()

The transport equation () for the particle volume fraction holds whether the fluid around the bubbles is compressible or not. In case of a compressible fluid the Rayleigh-Lamb equation () is modified (Keller and Miksis ). We showed with () that the Rayleigh-Lamb equation is intimately related to the fluid fluctuational kinetic energy and a possible way to represent the fluid compressibility is to start from a modified expression like   wc ⋆   K c = E(α d , M )(u d − u c ) + Q (α d , ) w c   cc

()

where c c is the isentropic sound speed of the carrier fluid. Deducing α c ρ c d c K c /dt and following the same route as above one can derive the new Rayleigh-Lamb equation in a compressible fluid. The main advantage of the method is that a lot of complex results can be deduced from rather simple expressions for the pseudo-turbulent kinetic energies. It is clear however that we supposed the particles to be spheres of variable radius. This is tenable in so far as the bubbles are not distorted by the mean flow or by the presence of neighbor particles. This simplistic description of the particle internal motion is certainly not the most general one but is the one that prevails at not too high particle volume fraction and for not too large bubbles.



Dissipative Model

This section will consider dissipative processes in dispersed mixtures. For condition () to be satisfied the first issue is to propose complete transport equations for α d , K d , and K c . These transport equations are very different depending upon whether the particles are compressible or not and we will consider the two cases separately.


. ..



Noncompressible Particles: Solid Grains or Drops The Dissipation Rate

Because the issue of particle velocity fluctuations has not yet been solved we assume for simplicity that K d =  while K c is given by the added-mass expression (). Since we now deal with dissipative phenomena we must take mass exchanges (phase transitions) into account and the transport equationl for the fluid pseudo-turbulent energy writes (compare with the former expression ()) ∂ (α c ρ c K c ) + ∇ ⋅ (α c ρ c K c u c ) = −F K ⋅ (u d − u c ) − σ cK : ∇u c ∂t K K K ∂α − α d P ∇ ⋅ u d − ∇ ⋅ Q + P ( d + ∇ ⋅ (α d u d )) + ΓK c . ∂t

()

As seen in > Sect. .., the pressure difference for rigid particles results from a Bernoulli effect expressed as pd − p c = P K .

()

Hence the transport equation for the volume fraction disappears from the condition for energy conservation (), which is transformed into an expression for the total dissipation rate Td Δ d + Tc Δ c = −(Td − Tc )Σ − (u d − u c ) ⋅ (F − F K ) − h c ⋅ ∇Tc − h d ⋅ ∇Td − Γ[μ d − μ c + (Td − Tc )s ⋆ + (u c − u ⋆ ) / − (u d − u ⋆ ) / + K c ] − (σ c − σ cK ) : ∇u c − (σd − α d P K I) : ∇u d − ∇ ⋅ (Q − Q K ) . This expression is not yet fully satisfactory: For a dispersed mixture a dissipation rate involving ∇u d or ∇Td holds when the particles are in permanent contact or display frequent collisions. At variance the particulate stress originating from the viscosity of the carrier fluid and the entropy flux originating from its heat conductivity must be described differently. These fluid-induced stresses and heat fluxes will be represented in what follows by τ and h and one expects them to give rise to a dissipation rate τ⋅∇u+h⋅∇T(Batchelor ; Prosperetti et al. ; Lhuillier ) where u = α d u d + α c u c , T = α d Td + α c Tc ,

()

are the mean velocity and mean temperature of the whole suspension. The simplest way to obtain the desired result is to adopt the definitions F = F K + f − αd ∇ ⋅ τ , K

σc = σc + τ , σd = α d P K I , K

Q = Q + (T − Tc )h + (u − u c ) ⋅ τ , hd =  , hc = h , Σ = H − αd ∇ ⋅ h .








With these definitions the initial equations ()–() are transformed into Γ ∂α d + ∇ ⋅ (α d u d ) = ∂t ρd ∂ (α c ρ c ) + ∇ ⋅ (α c ρ c u c ) = −Γ ∂t d s ⋆ α d ρ d d d + α d ∇ ⋅ h = Δ d + H + Γ(s − s d ) dt dc sc αc ρc + α c ∇ ⋅ h = Δ c − H − Γ(s ⋆ − s c ) dt d u α d ρ d d d + ∇(α d P K ) + α d ∇ ⋅ τ + α d ∇p c = F K + f + Γ(u ⋆ − u d ) + α d ρ d g dt dc uc αc ρc + ∇ ⋅ σ cK + α c ∇ ⋅ τ + α c ∇p c = −F K − f − Γ(u ⋆ − u c ) + α c ρ c g . dt

() () () () () ()

Note that, at variance with most two-fluid models, there is a single viscous stress and a single heat flux and that ∇ ⋅ τ and ∇ ⋅ h are shared in proportion to the volume fractions. With the above definitions the total dissipation rate becomes Td Δ d + Tc Δ c = −(Td − Tc )H − (u d − u c ) ⋅ f − h ⋅ ∇T − τ ⋅ ∇u − Γ[μ d − μ c + (Td − Tc )s ⋆ + (u c − u ⋆ ) / − (u d − u ⋆ ) / + K c ] .

()

Expression () plays a very important role for dissipative phenomena, similar to the role played by () for non-dissipative ones. The quantity Td Δ d + Tc Δ c is slightly different from the total entropy production rate Δ d + Δ c but it represents a closely related quantity, the total energy dissipation rate in the mixture. And instead of the usual thermodynamic inequalities Δ d ≥  and Δ c ≥  we will rely on the related inequality Td Δ d + Tc Δ c ≥ . Hence the right-hand side of () must be positive in all circumstances.

..

The Constitutive Laws

According to () there are five different sources of dissipation corresponding to the the heat, momentum, and mass exchanges at interfaces together with the heat and momentum transport through the bulk of the mixture. The thermodynamic forces associated with these five kinds of dissipation have the intuitively expected form except for the one linked to mass exchange, which is associated with a disequilibrium involving u ⋆ and s ⋆. Since the mass exchange (phase transition rate) is expected to be driven by thermodynamic quantities mainly, with a limited influence of the relative velocity, the best choice for u ⋆ is apparently u⋆ =

u c + ud . 

()

That special value for u ⋆ was adopted by several authors (Baer and Nunziato ; Young ) but a different one (u ⋆ = u d ) was preferred by Marble () who supposed that no thrust force could act on the particles due to mass exchange. At variance () implies that the same thrust force acts on the two phases of the mixture. In other words, with choice () there is no “rocket




effect” due to mass exchange. Concerning the specific entropy s ⋆ Marble () suggested (for liquid droplets surrounded by their vapor) to choose s⋆ = sc

()

because all the latent heat of the phase transitions are provided by (or given to) the droplets. If we adopt Marble’s choice, the driving force for mass exchange between a noncompressible particle and its vapor is μ c − μ d + s c (Tc − Td ) ≃ (s c − s d )(Ts at (p c ) − Td ), with Ts at (p c ) the saturation temperature at the continuous phase pressure. The dissipation rate is now rewritten in the final form Td Δ d + Tc Δ c = (Tc − Td )H + (u c − u d ) ⋅ f − τ ⋅ ∇u − h ⋅ ∇T + Γ[(s c − s d )(Ts at (p c ) − Td ) − K c ] ≥  .

()

Concerning the exchanges at interfaces a positive dissipation is guaranteed with the constitutive laws H = κ(Tc − Td ),

()

f = ζ(u c − u d ),

()

Γ = γ[(s c − s d )(Ts at (p c ) − Td ) − K c ]

()

where ζ, κ, and γ are three positive transport coefficients. The above expressions are the simplest ones with no coupling at all between the dissipative exchanges. Concerning the viscous and heat transport through the bulk of the mixture the constitutive laws are in their simplest form σ = −η∇u

()

h = −λ∇T

()

where η and λ are the effective viscosity and effective heat conductivity of the mixture and are positive transport coefficients that depend on the particle volume fraction. More complicated non-Newtonian constitutive relations are possibly obeyed by σ and non-Fourier ones by h. Whatever the chosen expressions they must satisfy the thermodynamic constraint expressed by inequality (). Once the expression of the total dissipation rate Td Δ d + Tc Δ c has been obtained, one must decide the way it is shared between its phasic components Td Δ d and Tc Δ c . Baer and Nunziato () proposed that the dissipation from compaction is apportioned to the dispersed phase and the dissipation from drag to the continuous phase (nothing is said concerning the thermal dissipation and mass exchange dissipation). Then it was shown in Bdzil et al. () that many other choices are possible, involving up to four sharing coefficients, one for each type of interfacial dissipation. Marble () proposed to give the whole dissipation to the continuous phase and this is obviously the simplest solution that we here adopt because the internal velocity field and internal temperature field inside a particle is not likely to give rise to a heat source that can compete with the heat exchanged through interfaces. We thus assume Δd =  .

()








We acknowledge that this is a crude assumption but the use of sharing coefficients would be a source of complication for the model without any guaranty of improvement.

. ..

Compressible Particles: Bubbles The Dissipation Rate

We neglect the pseudo-turbulent kinetic energy of the particles (K d = ) and assume that the pseudo-turbulent kinetic energy of the carrier fluid is given by () from which is deduced the pseudo-transport equation ∂ (α c ρ c K c ) + ∇ ⋅ (α c ρ c K c u c ) = −(u d − u c ) ⋅ F K − σ cK : ∇u c − α d P K ∇ ⋅ u d ∂t wc a˙ K K ∂α K + (R − M) + P ( d + ∇ ⋅ (α d u d )) − ∇ ⋅ Q + ΓK c , a a ∂t

()

where a˙ = d d a/dt and all other quantities are defined in > Sect. .. Concerning the transport of the particle volume fraction we now take mass exchange into account and () is transformed into its dissipative counterpart ∂α d wc Γ . + ∇ ⋅ (α d u d ) = α d + ∂t a ρc

()

a result obtained from the boundary condition for radial velocities at interfaces Γ α Γ α d = d wd + . wc + a ρc a ρd

()

With the two above transport equations, condition () for energy conservation now turns into Td Δ d + Tc Δ c + (Td − Tc )Σ + (u d − u c ) ⋅ (F − F K − ΔP∇α d ) + Γ[μ d − μ c + (Td − Tc )s ⋆ − (p d − p c − P K )/ρ c + (u c − u ⋆ ) / − (u d − u ⋆ ) / + K c ] K

K

+ (σ c − σ c − α c ΔP I) : ∇u c + (σd − α d P I − α d ΔP I) : ∇u d + (w c /a)(R K − α c ΔP) + h c ⋅ ∇Tc + h d ⋅ ∇Td + ∇ ⋅ (Q − Q K ) = 

()

where the generalized pressure difference ΔP is defined by

ΔP = α d (p d − p c − P K ) +

a˙ M . a 

()




We now follow the same strategy as for incompressible particles and introduce the definitions F = F K + ΔP∇α d + f − α d ∇ ⋅ τ , σ c = σ cK + α c ΔP I + τ , K

σd = α d P I + α d ΔP I , Q = Q K + (T − Tc )h + (u − u c ) ⋅ τ , hd =  , hc = h , Σ = H − αd ∇ ⋅ h . To these definitions we add the generalized Rayleigh-Lamb equation written as K

R + α d α c η c

wc = α c ΔP , a

()

where η c is the viscosity of the carrier fluid. Hence the final set of equations can be written as ∂α d wc Γ + ∇ ⋅ (α d u d ) = α d + ∂t a ρc ∂α d ρ d + ∇ ⋅ (α d ρ d u d ) = Γ ∂t ∂α c ρ c + ∇ ⋅ (α c ρ c u c ) = −Γ ∂t d s α d ρ d d d + α d ∇ ⋅ h = Δ d + H + Γ(s ⋆ − s d ) dt dc sc ⋆ αc ρc + α c ∇ ⋅ h = Δ c − H − Γ(s − s c ) dt dc uc αc ρc + ∇ ⋅ σ cK + α c ∇ ⋅ τ + α c ∇(p c + ΔP) dt K ⋆ = −F − f − Γ(u − u c ) + α c ρ c g d u α d ρ d d d + ∇(α d P K ) + α d ∇ ⋅ τ + α d ∇(p c + ΔP) dt K ⋆ = F + f + Γ(u − u d ) + α d ρ d g

() () () () ()

()

()

K

R wc + α d η c = ΔP , α c a

()

while the total dissipation rate becomes Td Δ d + Tc Δ c = −(Td − Tc )H − (u d − u c ) ⋅ f − h ⋅ ∇T − τ ⋅ ∇u + α d α c η c ( − Γ[μ d − μ c + (Td − Tc )s ⋆ − (p d − p c − P K )/ρ c + K c ] ,

wc  ) a ()

where the special value of u ⋆ given in () was used. The main difference with the case of incompressible particles is the existence of a new source of dissipation linked to the radial motion and a new term involving the pressure difference in the thermodynamic force driving the mass exchange.





 ..


The Constitutive Laws

Most of the constitutive laws (those for H, f , h, τ, and u ⋆ ), as well as the assumption Δ d = , are unchanged compared to the case of incompressible particles. There are however two remarkable differences, the value of s ⋆ and the constitutive law for Γ. The gas bubbles can hardly provide or receive the latent heat associated with mass exchange while the liquid around them is a good candidate for that role. Hence it is likely that s⋆ = sd

()

so that the entropy balance of the bubbles () is independent of the mass exchange. The driving force for mass exchange between a bubble and the surrounding liquid is then μ c − μ d + s d (Tc − Td ) −

 (p c − p d ) ≃ (s d − s c )(Tc − Ts at (p d )), ρc

and the mass exchange at interfaces is given by Γ = γ[(s d − s c )(Tc − Ts at (p d )) − K c − P K /ρ c ]

()

The role of the two terms related to velocity fluctuations is presumably negligible.

.

Final Form of the Model

The above analysis of two special cases (incompressible particles and highly compressible ones) suggests that Δ d =  and u ⋆ is given by (). As a consequence the initial equations ()-() are conveniently replaced for any dispersed mixture by ∂ (α d ρ d ) + ∇ ⋅ (α d ρ d u d ) = Γ ∂t ∂ (α c ρ c ) + ∇ ⋅ (α c ρ c u c ) = −Γ ∂t d s α d ρ d d d + α d ∇ ⋅ h = H + Γ(s ⋆ − s d ) dt dc sc αc ρc + α c ∇ ⋅ h = Δ c − H − Γ(s ⋆ − s c ) dt d u α d ρ d d d + ∇ ⋅ σdR + α d ∇ ⋅ σ + α d ∇p c dt = F R + f + Γ(u c − u d )/ + α d ρ d g dc uc R αc ρc + ∇ ⋅ σ c + α c ∇ ⋅ σ + α c ∇p c dt = −F R − f + Γ(u c − u d )/ + α c ρ c g ,

() () () ()

()

()




The resulting balance equations for internal energy are ∂ (α d ρ d ed ) + ∇ ⋅ (α d ρ d ed u d ) = Td (H − α d ∇ ⋅ h) + Γ[ed + Td (s ⋆ − s d )] ∂t ∂α Γ − p d ( d + ∇ ⋅ (α d u d ) − ) ∂t ρd ∂ (α c ρ c e c ) + ∇ ⋅ (α c ρ c e c u c + T h) = Tc Δ c + (Td − Tc )H + h ⋅ ∇T ∂t ∂α c Γ + ∇ ⋅ (α c u c ) + ) . − Td (H − α d ∇ ⋅ h) − Γ[e c + Tc (s ⋆ − s c )] − p c ( ∂t ρc

()

()

while the conservation of total energy requires Tc Δ c = −(Td − Tc )H − (u d − u c ) ⋅ ( f + F R ) − h⋅ ∇T − σ : ∇u ⋆

− Γ[μ d − μ c + (Td − Tc )s ] + (p d − p c )[∂α d /∂t + ∇ ⋅ (α d u d )] ∂ R R − ∑ [ (α k ρ k K k ) + ∇ ⋅ (α k ρ k K k u k ) + σ k : ∇u k ] − ∇ ⋅ (Q ) . ∂t k=c,d Hence transport equations are needed for α d , K d , and K c in order to deduce the final form of the entropy production rate Δ c . In principle, if one of the two phases is incompressible the transport equation for the volume fraction is no longer independent since it stems from the mass balance. The transport equations for the pseudo-turbulent kinetic energies will provide explicit expressions for the non-dissipative stresses σ cR , σdR and the non-dissipative force F R . Conversely, these three quantities are likely to disappear in case one denies any role to the velocity fluctuations. The entropy flux h is fully dissipative while σ is the sum of a viscous stress (noted τ in > Sect. . and > .) and a second part representing all phenomena (depicted by ΔP in > Sect. .), which play a role in the pressure difference. When collisions between particles are relevant they are taken into account with a stress tensor σ col l and an entropy flux h col l . A force ∇ ⋅ σ col l is to be added on the left-hand side of the momentum balance () of the particulate phase and similarly ∇ ⋅ h col l is to be added to the left-hand side of (). Any physically admissible closure for σ col l and h col l must verify the positivity of the energy production rate −(σ col l : ∇u d + h col l ⋅ ∇Td ).



Summary of Key Results

.

Hybrid Approach for Dispersed Mixtures

The two-phase model (built on the characteristic function of presence χ k ) is capable, formally at least, to describe any kind of mixture. In case of dispersed mixtures with particles suspended in a carrier fluid its main drawback is the lack of reference to what happens to one particle and this has important implications on the closure problem. This is an old issue and an early example was given by Batchelor () who expressed (see () and ()) the dispersed stress < χ d σd > of the two-phase model in terms of the so-called particle stresslet, which is the first moment of the fluid force exerted on the particle surface n < ∮ r ⊗ (σ c ⋅ n d ) ds >.








The presence of the small-scale number density δ d is typical of a kinetic theory approach of the particulate phase and it happens that we know the general rules that allow to transform two-phase quantities into quantities pertaining to the kinetic theory approach. With these transformation rules elaborated in Buyevich and Schelchkova (), Lhuillier (), Zhang and Prosperetti (), and Jackson () not only we can generalize Batchelor’s result for the dispersed phase stress (see results () and ()), but we also obtain a relation between the inter-phase forces of the two models (see result ()). The later relation is remarkable and shows that the inter-phase force of the two-phase model is not only the mean force per particle times the particle number density, but that it also involves the particle stresslet. A first consequence is that the particle stresslet appears in the momentum balance of the fluid phase exclusively. A second consequence is that the two-phase momentum balances are to be written as () and () for a dispersed mixture. In these two equations nowhere appears the pressure of the dispersed phase, but only the continuous phase pressure involved in some Archimedes force apportioned to the volume fractions. This does not mean that the dispersed phase pressure can be discarded. The models that were described above are clearly “two-pressure” models, which show that in case of compressible particles the dispersed phase pressure is involved in the compression work appearing in the energy balance equations, not to mention its influence in the transport equation for the volume fraction. Conversely, whenever the energy equations decouple from the other equations (a widely spread but debatable assumption), the only pressure that remains is that of the continuous phase and a one-pressure model is enough to describe the flow. Note also that we refrained introducing a “mean interfacial pressure” p I as is done currently in many two-phase models and the reason is clear : we decided at the outset to work with the two mean pressures only and, to be consistent, we had no possibility for introducing a third pressure. Some forces however were deduced that look like p I ∇α d or (p I − p c )∇α d . But such forces never occur alone and are part of a more general force such as ∇(α d P K ) in the case of added-mass phenomena. Focusing the efforts on a p I ∇α d or (p I − p c )∇α d type of force is thus too restrictive, it is part of the truth only. Both in this philosophy of approaching inviscid interaction and in the final results we find agreement with Wallis () and Geurst (). Having been derived independently and with completely different methodologies (let alone nomenclatures) these three formulations may appear completely different, however, we have verified that they are completely equivalent. In the case of Geurst, one must understand that what he calls the hydrodynamic pressure is nothing but the mean pressure p = α c p c + α d p d . In the case of Wallis, one must understand that he adopts a slightly different definition of the interphase force, which amounts to F K + ∇ ⋅ σ cK and consequently the kinetic stress (u d − u c ) ⊗ J is assigned to the particulate phase instead of the fluid phase.

.

Supplementary Equations

The standard two-fluid model is made of six equations, representing the balances of mass, momentum, and energy for each of the two phases. Do we need more equations? The answer depends on a delicate balance between precision and efficiency. Using more equations means needing more closure relations and facing more difficulties with the numerical simulations. New equations are needed for mainly two reasons: A better description of the




physics and a better description of the geometry of the interfaces (and sometimes a mixture of the two).

..

Pseudo-Turbulent Kinetic Energies

Flow-induced velocity fluctuations are at the origin of a rather formidable problem sometimes coined as “pseudo-turbulence.” Much like what happens for the true turbulence one is led to introduce new equations giving the evolution in time (and coupling with the mean flow) of quantities like the pseudo-turbulent kinetic energies or the pseudo-turbulent kinetic stresses. Such supplementary equations are of common use for the description of collisioninduced velocity fluctuations of particles moving in a gas but they are still in their infancy for particles moving in a liquid of almost matching density. In the above presentation of the two-phase model we have not taken collisions into account (the divergence of a collision stress must, in some cases, be added to the right-hand side of ()) and we were not able to propose a general form for the time evolution of the pseudo-turbulent kinetic energy or kinetic stresses. But we have insisted on two particular kinds of flow-induced velocity fluctuations, those associated with the relative fluid-particle motion and those associated with the pulsating motion of particles. These two special cases are relatively simple in so far as the related pseudoturbulent kinetic energies are given by analytical expressions like () or () and instead of new equations we have just derived the pseudo-equations () and (). But these pseudoequations give a flavor of what the two extra equations for K d and K c may look like, while the constraint () details the way these new equations should be coupled to the six two-phase equations.

..

Volume Fraction Transport

The volume fraction is a parameter that is of crucial significance to the two-phase model. Is it necessary to write an extra equation for one of the two volume fractions? Obviously not when the particles are assumed to be incompressible because the transport equation for volume is then a mere consequence of the transport equation for mass, and this is independent of the compressibility of the continuous phase. For compressible particles new physics must be introduced, and the answer is readily obtained provided one makes simplifying assumptions concerning the particle deformation. The simplest answer is obtained with () after supposing that all the particles are monodispersed spheres. This introduces an extra geometric variable, the radial velocity w c at interfaces, which is coupled to the mean flow by the Rayleigh-Lamb equation (). Hence this is not one but two coupled extra equations that are needed in case of compressible particles. The only exception is the case of a slow radius change because the inertia of the liquid in the radial motion can be neglected, the Rayleigh-Lamb equation then reduces to η c (w c /a) = p d − p c and the transport equation for the volume fraction appears as ∂α d α Γ . + ∇ ⋅ (α d u d ) = d (p d − p c ) + ∂t η c ρc

()

We thus need an eight-equation model or a seven-equation model depending whether the inertia of the continuous phase is important or not for the radial motion of the particles. Moreover,








when phase changes occur at sufficiently slow rates to assure thermodynamic equilibrium, then () is replaced by the fluid equation of state, thus reverting back to the six-equation model. A seven-equation model was proposed by Baer and Nunziato () but with a transport equation for the volume fraction that markedly differs from () since the left-hand side of () is replaced by a convective time-derivative ∂α d /∂t + VI ⋅ ∇α d where VI is sometimes called the mean interfacial velocity. That convective time-derivative was taken for granted in many subsequent works (Bdzil et al. ; Kapila et al. ; Abgrall and Saurel ). However, while a transport equation with a convective time-derivative is correct for separated flows averaged over a cross section (Ransom /), its use cannot be extended to the dispersed mixtures we have considered here and for which (), or more generally (), is the correct answer.

..

Interfacial Energy Transport

Besides the volume fraction the density of interfacial area a I defined as a I =< δ I >

()

is also an important parameter for the inter-phase exchanges. The volume fraction and the interfacial area density are connected to each other via the mean particle radius α d /a I . For a monodisperse suspension of spheres one then obtains the transport equation ∂a I a I ∂α d ( + ∇ ⋅ (a I u d ) = + ∇ ⋅ (α d u d )) . ∂t α d ∂t

()

The interfacial energy of the mixture is γa I where γ is the surface tension. Requiring the total energy α d ρ d E d + α c ρ c E c + γa I to be conserved leads to a dissipation rate similar to () but with p d replaced everywhere by p d − γa I /α d with the consequence that p d = p c + γa I /α d at equilibrium. The above transport equation is very simple but it requires the particles to keep a spherical shape and it neglects the coalescence, break-up, and nucleation phenomena. When these phenomena occur, new terms appear in the right-hand side and an example is given below concerning the particle deformation. Concerning coalescence and break-up the main issue is to select the relevant physical phenomena (turbulence of the carrier phase, relative motion of the two phases, etc.) and the way the polydispersity of the particles modifies the intensity of the processes.

..

Particle Deformation and Dynamics of Interfaces

When the particles are deformable and depart from a spherical shape, another quantity (a tensor in general) will be needed to depict the deformations of the interfaces and their coupling to the ambient flow. Introducing a tensor as a new variable brings a lot of difficulties but it provides a way to explain part (and only part) of the lateral drift experienced by deformable particles (drops or bubbles) in a pipe flow. Usually deformable particles move toward places where they are deformed less. In a pipe flow the region of smallest deformation is the pipe axis where the




shear vanishes. Since the particle deformability depends on their size (and more generally on the capillary number) the larger particles will drift and accumulate close to the pipe axis while the smaller ones will be less sensitive to that drift and will have a more homogeneous distribution over the pipe cross section. Other effects due to the particle deformability are known to exist such as non-Newtonian behavior of the mixture with difference of normal stresses and shear-dependent viscosity. Doi and Ohta () proposed to describe the interfacial microstructure with two quantities, the interfacial area density and the anisotropy tensor q i j =< (n i n j − δ i j /) δ I >, where n is the unit vector normal to the interfaces, and they derived two coupled evolution equations for q i j and a I . Their model, however was restricted to a very special flow with α d ≈ . and equal viscosities of the two phases. Many subsequent works tried to go beyond that special case and are presented in Tucker and Moldenaers (). It is possible to use the above-defined anisotropy tensor for a dispersed mixture with α d < .. But it is much more convenient to use C i j = q i j /a I as the anisotropy tensor with δ

Ci j =

< (n i n j − i j ) δ I > . < δI >

()

As an example we give below the two coupled equations for an emulsion of two immiscible and incompressible phases where the interfaces are those of (slightly) deformed spheres and viscous effects are dominant. With p = η d /η c the ratio of the droplet viscosity to the carrier viscosity, α d the droplet volume fraction, and σ + σ I the emulsion viscoelastic stress (see Appendix C for the definition (C) of the interfacial stress σ I ) one obtains (Lhuillier ) ∂a I F  γa I C) + ∇ ⋅ (a I u d ) = −a I C : ( D + ∂t H H α d η c

()

γa I  F ∂C   C] + (u d ⋅ ∇)C + C ⋅ Ω − Ω ⋅ C = − [ D + ( + ) ∂t  H H p +  α d η c

()

σ + σ I = −η c [ + α d (p − )

F F ]D+ γa I C H H

()

where D and Ω are the strain rate and rotation rate of the suspension mean velocity u = α d u d + α c u c . Moreover, H(p, α d ) = p −  + F(α d ) while F(α d ) is related to the effective viscosity of a suspension of hard spheres, with F(α d ) = /( − α d ) as one of the simplest possible expressions (Palierne ). In the two above transport equations for C i j and a I one clearly distinguishes the two main phenomena: the shear-induced deformation and the retraction due to surface tension. But the breakup and coalescence phenomena are not represented and must be taken into account by extra terms. The above three equations were written for incompressible particles. The compressibility of particles has a role in the transport equation for a I only, the right-hand side of which is then the sum of the two right-hand sides of () and ().

.

Hyperbolicity

Ill-posedness has been a major issue in regards to the formulation of the two-phase model since the very early efforts on behalf of nuclear reactor safety in the early s (Gidaspow ),








and all along the way to the most recent renditions (Drew and Passman ; Prosperetti and Satrape ; RELAP/MOD. Code Manual ). Inviscid interactions between the phases were anticipated as the cause, and a number of empirical or ad hoc additions have been made as constitutive descriptions mainly under the names of “added mass” e.g., (Park et al. ) and “interfacial pressure” e.g., (Stuhmiller ), but the overall subject seems to still lack in clarity and definiteness. On the other hand, the unified treatments of these interactions in Wallis () and independently in Geurst () seemed to have gone unnoticed. These results are recovered completely independently in the present development, and so is a generalization of the requirement for hyperbolicity found in Geurst () for incompressible flows (low Mach number limit in the present work) equation (). This result is completely adequate for disperse systems as its applicability, which coincides with the natural range of disperse phase volume fractions being under about %. At still higher concentrations particle collisions have to be taken into account, as indeed the role of particle fluctuation adding to the kinetic energy as explained by (). On the other hand, for highly compressible flows the function E in () must be redefined to include the effect of flow Mach number, and this further (and hopefully final) development of the subject will be an important task for the future. For the time being, in practical terms we have found that the present formulation is sufficient to obtain robust numerical solutions at all flow speeds (Part II). The approach of Baer and Nunziato () must be mentioned in this context. These authors proposed, for dense particulate systems, one of the rare models to be hyperbolic. Why are we skeptical concerning the potentialities of that model for dispersed mixtures? We pointed out above a strong discrepancy concerning the transport equation for the volume fraction. Another important discrepancy concerns the two momentum balances which, in the BN model, appear as dc uc + ∇(α c p c ) = − f − Γ(u ⋆ − u c ) + α c ρ c g dt d u ⋆ α d ρ d d d + ∇(α d p d ) = f + Γ(u − u d ) + α d ρ d g dt αc ρc

() ()

These equations are different from () and (). The gradient of the dispersed phase pressure is assumed to drive the motion of the dispersed phase while we claimed that the gradient of p c is the only driving force for both the carrier fluid and the dispersed phase. It is true that the special way the momentum and volume fraction equations were written was enough to ensure hyperbolicity. But the paradox is that hyperbolicity is obtained with a set of equations that is definitely not suitable for particulate systems.

.

Nuclear Reactor (Design) Systems Codes

Here we specifically refer to the RELAP developed in the USA during the s and the French code CATHARE that followed; they are the main representatives of the genre. They are based on the premise that flow and transport of heat/mass can be captured generally for all flow regimes by one and the same set of effective field equations (a system of six equations plus the equations of state of the fluids involved). These equations were written in D, as (channel) cross-sectional averages, and augmented by volume sources and/or sinks that represent the wall transfers. The later are determined by a logic that reflects the estimated flow pattern on the basis of vapor and liquid flow rates and other pertinent quantities. The RELAP code is well documented in a




comprehensive set of volumes that is openly available (RELAP/MOD. Code Manual ). For CATHARE the main reference is a journal paper by Bestion (). The comments below are based on these sources, and we focus on the momentum equations as they are the critical link to hyperbolicity, and thus to the numerical performance of these computer codes. For RELAP, the afore-mentioned equations are formally the same as the D version of () and (), except for the following: (a) The phases are assumed to be at the same pressure, so in the gradient term instead of p c write p; (b) both phasic stresses are absent; (c) from the added-mass force only the component ∂(u d − u c )/∂t is retained; and (d) there are source terms to express wall transfers as noted above. For CATHARE, the spatial derivatives of the added mass term are retained and the Stuhmiller () treatment of interfacial pressure is added to “render the system hyperbolic.” It is known that with an appropriate selection of the coefficient in this expression the system becomes hyperbolic for incompressible flows, but there is no consideration of the effect of Mach number. For sensitive problems the results can depend on the value of this coefficient. The RELAP system of equations is ill-posed and solutions are obtained by controlling the growth of oscillations. In particular, damping is affected by means of added artificial viscosity terms, and by keeping the size of the (numerical grid) nodes sufficiently large. An interesting part of the RELAP numerical stability analysis (RELAP/MOD. Code Manual ) is that the drag terms stabilize the solution down to sufficiently small wavenumbers as to render solutions for nuclear reactor scale systems practicable. Much of the utility of these codes rests upon the dominance of phase-change effects present in applications, which these codes were intended for to begin with (nuclear reactor loss of coolant accidents). In a way this, along with conservation, provides an attractor strong enough to be a basis for tuning by means of experiments.

Acknowledgments This work was supported by the Joint Science and Technology Office, Defense Threat Reduction Agency (JSTO/DTRA), and the National Ground Intelligence Center (NGIC) of the US Army (Dr. Richard Babarsky). Dr. S. Sushchikh and Dr. C.-H. Chang (CRSS / UCSB) participated with many helpful discussions and derivations concerning the hyperbolicity issue and comparison to other formulations. Lively discussions with (and useful comments from) Dr. C. Morel (CEA Grenoble) and Professor R. Saurel (IUSTI Marseille) are gratefully acknowledged.

Appendix A: Rigid Spheres in a Nonviscous Fluid Our aim is to give a brief derivation of the fluid momentum balance () with definitions () and (). We consider many rigid spheres dispersed in a nonviscous and noncompressible fluid. The momentum balances of the two phases are expressed by the general equations () and () but definitions ()–() simplify to F = < δ d ∮ (p c − p)n d ds >

(A-)

σ c = ρ c < χ c v ′c ⊗ v ′c > − < δ d ∮ (p c − p)r ⊗ n d ds >

(A-)

σd = ρ d < χ d v d′ ⊗ v d′ >

(A-)








where p is the microscale fluid pressure and p c is its mean value. Note that we neglected the kinetic contribution to σd because the internal motion u is reduced to a rotation in case of rigid particles and the angular velocity is a constant when the rigid particles have a spherical shape. Hence the inter-phase force is linked to the pressure fluctuations on the surfaces of the particles while the phasic stresses depend on both the pressure and velocity fluctuations. To make some progress we must be able to calculate the various terms on the right-hand sides of definitions (A-) to (A-). This can be done if we further suppose that the nonviscous fluid has a microscale velocity v deriving from the potential ϕ. In that case the microscale equations of motion are v = ∇ϕ, ∇ ϕ = ,

(A-)

∂ϕ v  p −g⋅x =. + + ∂t  ρc

(A-)

The fluid velocity is split into a mean value u c and a fluctuation v ′ ′

′

v = u c + v , < χ c v >= α c u c , < χ c v >=  .

(A-)

With Wallis () we split the velocity potential into an averaged value Φ and a fluctuation ϕ′ defined by ′

′

ϕ = Φ + ϕ , < χ c ϕ >= α c Φ, < χ c ϕ >=  .

(A-)

A quantity that will prove of utmost importance is the mean fluid impulse J defined by J = ρ c < χ c ∇ϕ′ >= −ρ c < ϕ′ n d δ I > .

(A-)

J is a Galilean-invariant momentum, which plays a role in both the mean velocity and the velocity fluctuations v ′ = ∇ϕ′ −

J J J , u c = ∇Φ + , ∇ × (u c − )=. αc ρc αc ρc αc ρc

(A-)

It is now rather easy to deduce the momentum balance of the fluid phase. The microscale Bernoulli equation (A-) is multiplied by the fluid characteristic function χ c and one performs the statistical average noted by brackets < ⋯ >. The result is the averaged Bernoulli equation αc

∂ϕ′ αc pc ∂Φ v − αc g ⋅ x =  . + < χc > + < χc > + ∂t ∂t  ρc

(A-)

Taking the gradient of that equation and eliminating ∇Φ with the help of (A-) one obtains 

∇p c ∂ J  J ⋆ ) + ∇ (u c − ) − ∇K + = g, (u c − ∂t αc ρc  αc ρc ρc

(A-)

where K ⋆ is a specific kinetic energy related to the fluid impulse J and to the pseudo-turbulent fluid kinetic energy K c =< χ c v ′ > /α c by α c K ⋆ = α c K c + α c (



J ) − ∇⋅ < χ c ϕ′ ∇ϕ′ > . αc ρc

(A-)




Moreover, the pseudo-turbulent kinetic energy of the fluid is itself related to the Kelvin impulse by

α c K c = (u d − u c ) ⋅

J − < ϕ′ (v − u d ) ⋅ n d δ I > +∇⋅ < χ c ϕ′ ∇ϕ′ > . ρc

(A-)

Equation (A-) can be transformed into a momentum balance similar to () with

− F − ∇ ⋅ σc =

∂J J ⋆ ) + α c ρ c ∇K + ∇ ⋅ (u c ⊗ J) + (J ⋅ ∇)(u c − ∂t αc ρc

(A-)

The main conclusion at this point is that the dynamics of the fluid phase depends on two quantities only, the mean fluid impulse J and the kinetic energy K ⋆. We can simplify the above result (A-) if we assume the last term of (A-) to be negligible. Then, the non-dissipative force acting on the fluid phase becomes

− F − ∇ ⋅ σc =

∂J + ∇ ⋅ (u c ⊗ J) + (J ⋅ ∇)u c + α c ρ c ∇K c + J × (∇ × u c ) ∂t

(A-)

where, according to (A-), ∇ × u c in the last term can be replaced by ∇ × (J/α c ρ c ) at will. It is clear that the above equation of motion is equivalent to () with definitions () and (). Note however that our splitting into a force F K and a stress σ cK is somehow arbitrary in so far as only the sum F K + ∇ ⋅ σ cK can be deduced unambiguously from expression (A-).

Appendix B: Hyperbolicity Aspects of the Effective Field Model with S. Sushchikh and C.-H. Chang Here our aim is to study the mathematical character of the system ()–(), and as is common practice we do so in one space dimension. While the procedures are standard, the derivations are rather laborious and results for the fully compressible case (high Mach numbers) are presented here for the first time. These results are of essential use in Part II. The system of equations is: ∂ ∂ (α c ρ c ) + (α c ρ c u c ) =  ∂t ∂x ∂ ∂ (α d ρ d ) + (α d ρ d u d ) =  ∂t ∂x ∂p c ∂ ∂ [α c ρ c u c + α c ρ c E (u d − u c )  ] + α c (α c ρ c u c ) + = −F K ∂t ∂x ∂x ∂p c ∂ ∂  K K (α d ρ d u d ) + [α d ρ d u d + α d P ] + α d =F ∂t ∂x ∂x

(B-a) (B-b) (B-c) (B-d)








With ∂K c ∂J ∂u c ∂ − (u d J) − J − αc ρc ∂ t ∂x ∂x ∂x J = α c ρ c E(α d ) (u d − u c )  K c = E(α d )(u d − u c )  ∂K c K P = −α c ρ c ∂α d = pd − p c

FK = −

Equation of state information is introduced by defining the isentropic speeds of sound for each field: ∂p c ∂p d and a d = a c = ∂ρ c ∂ρ d We will cast this system in characteristic form and will determine the eigenvalues, first analytically, in the incompressible approximation, and then numerically for any Mach number flow. In the standard wave equation form the above system becomes:

A

∂Q ∂Q +B = ∂t ∂x

(B-)

where Q = (α d , u d , u c , p c )T is the vector of primitive variables, and matrixes A and B are: A  ⎛ −ρ c ⎜ A=⎜ ⎜ α c ρ c E ′ (u d − u c ) ⎝ −α c ρ c E ′ (u − u c ) d

⎛ u d A  ⎜ −ρ c u c B=⎜ ⎜ B  ⎝ B 

A   αd ρd + α c ρ c E −α c ρ c E

α d ρ d + u d A   B  −α c ρ c u d E

A   −α c ρ c E αc ρc + αc ρc E

u d A  αc ρc B  α c ρ c ( + E)u c

A  α c /a c  

u d A  α c u c /a c B  αc

⎞ ⎟ ⎟ ⎟ ⎠

⎞ ⎟ ⎟ ⎟ ⎠

(B-)

(B-)

with: A  = ρ d − A  =

α d ρ c (α c E ′′ − E ′ )(u c − u d ) a d

α d α c α d E ′ (u c − u d ) − a d a d a c

′



A  = −A  = −α c α d ρ c E (u d − u c )/a d




And: B  = / ρ c (u d − u c ) (α d E ′ − E − α c α d E ′′ ) + α c ρ c E ′ u d (u d − u c ) B  = −/ α c ρ c E ′ (u d − u c )(u d + u c ) B  = α d ρ d u d − α c α d ρ c E ′ (u d − u c ) + α c ρ c E(u d − u c ) + α c ρ c u d E B  = −α c ρ c Eu c + α c ρ c (u d − u c )(α d E ′ − E) ′ B  = α d + / α c (E − α d E )

(u d − u c ) a c

The eigenvalues of (B-) can be calculated by solving: ∣B − λA∣ =  or letting D = B − λA, we have to solve: &&& (u − λ)A  d &&& && −ρ c (u c − λ) ∣D∣ = &&& &&& D  &&& D  &&

α d ρ d + (u d − λ)A   D  −ρ c E(u d − λ)

(u d − λ)A  αc ρc D  ρ c ( + E)(u c − λ)

(u d − λ)A  α c (u c − λ)/a c D  

&&& &&& && &&& =  &&& &&& && (B-)

Where: ′ ′ ′′  D  = α c ρ c E (u d − u c )(u d − λ) + ρ c/(α d E − E − α c α d E )(u d − u c )

D  = (α d ρ d + α c ρ c E)(u d − λ) + α c ρ c (E − α d E ′ )(u d − u c )

D  = α c ρ c E(λ − u c ) − α c ρ c (E − α d E ′ )(u d − u c ) α c (E − α d E ′ ) D  = α d + (u c − u d ) a c ′ D  = ρ c E (u d − u c ) (λ − /(u d + u c )) Equation (B-) will lead to a fourth-order polynomial equation (quartic equation) whose roots (eigenvalues) will be examined analytically; first in the low Mach number approximation (B-), and then numerically for the general case (B-). We seek conditions for which all four roots are real for this is a necessary and sufficient requirement for our system to be hyperbolic.

B. Low Mach Number Approximation Under the conditions (Hancox et al. ): (u c − λ) .

and

(B-a) (B-b)

Here Z k are obtained by √ 

ϕ ω cos ( ) ;  √ ϕ Z  =   ω cos ( +  √ ϕ Z  =   ω cos ( +  Z = 

π );  π ). 

√  −p −q and ϕ = cos− ( ω ). where ω =  Once functional form of E(α d ) is provided, the hyperbolicity boundaries can be found by ˆ α d , ρ. ¯ For the particular example numerically scanning the domain of parameter space M, considered here, we remain with the form taken for the low Mach number analysis above, with C  = / so that (B-) E(α d ) =  α d ( + Cα d ) where C is now introduced as an additional parameter. As long as ρ¯ >>  the results are insensitive to the actual value ( is the value for which results are shown here). Also, as expected, and in contradistinction to bubbly flows, the hyperbolicity boundaries are insensitive to the presence of the added mass term, although the term is included for the results shown here. Representative results of this type of analysis are summarized in > Fig. B, and we can see that they correctly recover the low Mach number analysis presented above. Notable is also the increase of accessible Mach numbers within the hyperbolic domain (so that E(α d ) remains positive) with decreasing disperse phase volume fraction. Perhaps more important is the finding that in all cases the non-hyperbolic corridor is characterized by exceedingly small but positive values (typically of order − ) of the discriminant



Multiphase Flows: Compressible Multi-Hydrodynamics αd = 0.3

0

αd = 0.2

0 Non-hyperbolic

Non-hyperbolic

–5

–5 Hyperbolic

C

C

Hyperbolic

E(αd) < 0 Hyperbolic

–15

E(αd) < 0

–10

–10

0

0.5

1

–15

1.5

Hyperbolic

0

0.5

1

1.5

M

M αd = 0.1

0

αd = 0.01

0

Non-hyperbolic

Non-hyperbolic

–5

–5 Hyperbolic

Hyperbolic

Hyperbolic

C

C

Hyperbolic

–10

–10

E(αd) < 0 –15

–15 0

0.5

1

1.5

0

0.5

M

1

1.5

M

⊡ Figure B Representative sample of hyperbolicity maps of our effective field model employed for droplet (or particles in gas) flows. The Mach number(M) here is without the factor of / found in (B-)

D, which can be thought of as a symptom of “mild” non-hyperbolicity. Support for such a view derives from the implied roots with very small imaginary parts, as discussed further below.

B. Discussion To further explore the potential significance of the finding that failure to meet condition (B-a) was consistently by an exceedingly small margin, we computed directly the eigenvalues (using MATHEMATICA ()), for a number of cases, an example of which is provided in > Table B. Such results allow us to generalize as follows:

• • • •

ˆ for M ˆ < .; Re(λ∗ ) ≈ M ∗ − M ˆ +  for M ˆ > . Re(λ∗ ) ≈ M ∗ + M ∗ ∗ ˆ ˆ Re(λ  ) ≈ M + M for all M. ˆ −  for all M. ˆ λ∗ ≈ M ∗ − M ∗ ∗ ˆ ˆ ˆ for M ˆ > . λ  ≈ M − M +  for M < .; λ∗ ≈ M ∗ + M








⊡ Table B A set of sample solutions of (B-) with : C = −., αd = ., ρ¯ = ,  ˆ M

M∗

λ∗

.

.

.

.

.

.

λ∗

λ∗

λ∗

.

–.

.

. + .i

. – .i

−.

.

.

. + .i

. – .i

−.

.

.

.

. + .i

. – .i

−.

.

.

.

. + .i

.–.i

−.

.

.

.

.

.

−.

.

.

.

.

.

−.

.

• The imaginary parts are always at least two orders of magnitude smaller than the real parts. Moreover, using the definitions, we have the corresponding relations to the real speeds as follows: ˆ −  = uc − ac , M∗ − M ac

ˆ = u d , and M ∗ − M ˆ +  = uc + ac . M∗ + M ac ac

ˆ Sect. B.. From the last bullet above, it is clear that the exceedingly small value of the discriminant can be associated with eigenvalues whose imaginary parts are very small in relation to their real parts. This in turn can be related to the stability character of our wave system (B-), by recognizing that the unstable modes (those associated with the negative imaginary parts of the eigenvalues) grow with a characteristic time of τ G = χ/π λ i while the characteristic time for propagation is τ P = χ/πλ r , where λ i /λ r are the imaginary/real parts of the corresponding eigenvalue, and χ is the wave length of the disturbance. Thus we have a measure of the “mild” non-hyperbolicity mentioned above as instability growth rates being slower than onehundredth of disturbance propagation rates through our system. This, along with the fact in actual simulations the non-hyperbolic corridor would be traversed rather rapidly, suggest that our two-fluid model is effectively hyperbolic.

Appendix C: Including Surface Tension Taking surface tension into account means upgrading the interfaces to the status of a third phase. As a consequence the interfaces have their own balance equations and they exchange mass, momentum, and energy with the two neighbor phases. Usually that third phase is supposed to have no mass and the surface tension γ is then a free energy depending on the mean temperature TI of the interfaces. As a consequence the interfaces are endowed with an entropy




and an internal energy which, per unit volume of the mixture, are written as S I = −a I

dγ dγ , U I = a I (γ − TI ) dTI dTI

(C-)

where a I is the density of interfacial area defined in (). The thermodynamics of interfaces is made of the above definitions for energy and entropy together with the Gibbs relation dU I = TI dS I + γ d a I .

(C-)

It is also usual to introduce a mean transport velocity VI of the interfaces. That mean velocity is somehow arbitrary but whatever it is the above Gibbs relation can be transformed into a relation between transport equations that writes ∂U I ∂S I ∂a I + ∇ ⋅ (U I VI ) = TI [ + ∇ ⋅ (S I VI )] + γ [ + ∇ ⋅ (a I VI )] . ∂t ∂t ∂t

(C-)

To simplify the issue the interfaces are supposed to have no momentum and no kinetic energy. Moreover, heat transport as well as viscous phenomena are neglected in that third phase. But the forces associated with surface tension are taken into account with an interfacial stress tensor  σ I = − < γ(I − nn)δ I > = a I γ (C i j − I) 

(C-)

where the anisotropy tensor C i j is defined in (). The main modification brought by surface tension to the two-fluid equations concerns the exchanges between the two phases. Except for mass exchanges they no longer mutually cancel. The sum of the two entropy exchanges is transformed into interfacial entropy and the sum of the two momentum exchanges is transformed into interfacial forces (interfacial stress in fact). But how the interfacial entropy and the interfacial stress are to be shared between the two phases? Let us solve that issue with the following trick: Instead of writing the balance equations for a quantity X in the form d Xd = xd , dt

d Xc = xc , dt

d XI = −x d − x c dt

(C-)

we incorporate the third equation into the first two with a “sharing” coefficient a and write d Xd d XI =x−a , dt dt

d Xc d XI = −x − ( − a) dt dt

(C-)

Doing so we can forget the third equation and obtain a set of equations very close to the one in the main text. In fact with two sharing coefficients a s and a u the two-fluid equations for entropy and momentum are modified into dd sd ∂S I + α d ∇ ⋅ h = Δ d + Σ + Γ(s ⋆ − s d ) − a s [ + ∇ ⋅ (S I VI )] dt ∂t dc sc ∂S I αc ρc + α c ∇ ⋅ h = Δ c − Σ − Γ(s ⋆ − s c ) − ( − a s ) [ + ∇ ⋅ (S I VI )] dt ∂t d u α d ρ d d d + α d ∇ ⋅ σ + α d ∇p c = F + Γ(u ⋆ − u d ) − a u ∇ ⋅ σ I + α d ρ d g dt dc uc αc ρc + α c ∇ ⋅ σ + α c ∇p c = −F − Γ(u ⋆ − u c ) − ( − a u )∇ ⋅ σ I + α c ρ c g. dt αd ρd

(C-) (C-) (C-) (C-)








Note that we neglected the interfacial entropy production like we did in the main text. Moreover, because they have no role in surface tension phenomena we simplified the issue by discarding all forces and stresses bound to velocity fluctuations but they can be restored without any problem. And we did not write the mass balances because they are not modified by massless interfaces. Repeating the procedure detailed in > Sect.  we deduce from the above equations the transport equation for the total energy and we find ∂ (α d ρ d E d + α c ρ c E c ) + ∇ ⋅ (α d ρ d E d u d + α c ρ c E c u c + p c u + u ⋅ σ + T h) ∂t = (α d ρ d u d + α c ρ c u c ) ⋅ g + Td Δ d + Tc Δ c + (Td − Tc )Σ + (u d − u c ) ⋅ F + σ : ∇u + h ⋅ ∇T + Γ[μ d − μ c + (Td − Tc )s ⋆ + (u ⋆ − u c ) / − (u ⋆ − u d ) /] + (p c − p d )(∂α d /∂t + ∇ ⋅ (α d u d )) − TI [∂S I /∂t + ∇ ⋅ (S I VI )] − WI ⋅ (∇ ⋅ σ I )

(C-)

where TI = a s Td + ( − a s )Tc

(C-)

WI = a u u d + ( − a u )u c .

(C-)

Hence the mean interfacial temperature is connected to the sharing coefficient a s and a mean interfacial velocity WI can be defined, which is connected to the sharing coefficient a u . That second interfacial velocity is bound to the power developed by interfacial forces and there is no reason for that velocity to be equal to the mean transport velocity VI . The total energy is conserved and this imposes a very specific form for its evolution equation, which can be obtained with due account for (C-) and which writes ∂ (α d ρ d E d + α c ρ c E c + U I ) + ∇ ⋅ (α d ρ d E d u d + α c ρ c E c u c + U I VI ) ∂t + ∇ ⋅ (p c u + u ⋅ σ + WI ⋅ σ I + T h) = (α d ρ d u d + α c ρ c u c ) ⋅ g .

(C-)

As a consequence the total dissipation rate is Td Δ d + Tc Δ c = (Tc − Td )Σ + (u c − u d ) ⋅ F − h ⋅ ∇T + Γ[μ c − μ d + (Tc − Td )s ⋆ − (u ⋆ − u c ) / + (u ⋆ − u d ) /] − σ : ∇u − σ I : ∇WI + (p d − p c )[∂α d /∂t + ∇ ⋅ (α d u d )] − γ [∂a I /∂t + ∇ ⋅ (a I VI )] .

(C-)

For dispersed mixtures plausible values for VI and WI are VI = u d , WI = u,

(C-)

meaning that the interfacial area is transported with the dispersed phase velocity and that the work of interfacial forces involves the volume-weighted velocity defined in (), with the consequence that a u = α d . The two closures (C-) have important consequences. The total stress of the suspension is σ + σ I + p c I. While σ I is entirely proportional to the surface tension, a part of σ is also proportional to γ while the remaining part is a viscous stress τ D . One is thus inclined to write σ + σ I = α d (p d − p c −

γa I D γ )I −τ −τ . α d

(C-)




Closures for τ D and τ γ are given in () for the case of an emulsion. It is clear that τ γ is different from the anisotropic part a I γC i j of σ I . The last four terms of the dissipation rate (C-) are then transformed into ∂α d ∂a I + ∇ ⋅ (α d u d )] − γ [ + ∇ ⋅ (a I u d )] ∂t ∂t γa I ∂α = τ D : D + (p d − p c − ) [ d + ∇ ⋅ (α d u d ) − α d ∇ ⋅ u] α d ∂t ∂a I a I ∂α d τγ −γ[ ( + ∇ ⋅ (a I u d ) − + ∇ ⋅ (α d u d )) − : D] ∂t α d ∂t γ

− (σ + σ I ) : ∇u + (p d − p c ) [

(C-)

where D = (/)(∇u +∇u T ) is the suspension strain rate. The total dissipation rate is thus connected to the way one writes the transport equations for the volume fraction and the transport equation for the density of interfacial area. Let us begin with the equation for α d and the special case of bubbles moving in a noncompressible fluid. The starting point is w ∂α d Γ + ∇ ⋅ (α d u d ) = α d d + ∂t a ρd ∂α c Γ + ∇ ⋅ (α c u c ) = − , ∂t ρc

(C-)

where w d is the radial velocity inside the bubble and close to its surface. w d is related to the radial velocity w c in the liquid close to the interface by the boundary condition () and we deduce from the two above equations ∇ ⋅ u = α d (w c /a). Consequently wc ∂α d Γ , + ∇ ⋅ (α d u d ) − α d ∇ ⋅ u = α d α c + ∂t a ρc

(C-)

a result equivalent to (). That result also confirms that w c is a much more convenient quantity to handle than w d and that the transport equation for the volume fraction is more conveniently expressed as in () rather than in its more classical counterpart (C-). Concerning the density of interfacial area the above expression of the dissipation rate suggests to write its transport equation as ∂a I a I ∂α d τγ ( + ∇ ⋅ (a I u d ) = + ∇ ⋅ (α d u d )) + : D − Φγ , ∂t α d ∂t γ

(C-)

where Φ γ is a positive scalar reflecting the tendency of surface tension to decrease the total amount of interface. In () the C : C term is a contribution to Φ γ depicting the return to isotropic shapes. The coalescence phenomenon is represented by another contribution to Φ γ . And the final expression of the dissipation rate is Td Δ d + Tc Δ c = (Tc − Td )Σ + (u c − u d ) ⋅ F − h ⋅ ∇T + τ D : D + Γ[μ c − μ d ⋆

⋆



+ (Tc − Td )s + (p d − p c − γa I /α d )/ρ c − (u − u c ) / + (u ⋆ − u d ) /] + α d α c (w c /a)(p d − p c − γa I /α d ) + γΦ γ a result to be compared with ().

(C-)








PART II: COMPUTATION WITH EFFECTIVE-FIELD MODELS OF MULTIPHASE FLOWS 

Introduction and Scope II

In Part II we pursue the EFM developed in Part I to its numerical implementation. To recapitulate, the principal desiderata are: robust and high-fidelity simulations of disperse systems at all flow speeds, and this implies, sharp capturing of pressure and flow discontinuities (shock waves), as well as highly resolved material (or contact) “interfaces.” Besides a well-founded numerical scheme, these aims require an ability to refine the grid (thereby requiring stability under reduced numerical dissipation), as well as the means to do so for problems involving large spatial dimensions, which in turn translates into a requirement for adaptive mesh refinement. The cornerstone to all these is a hyperbolic system of equations, which was shown to have been effectively achieved for all conditions of interest here (Part I). This allows for a focused treatment that is based on the extensive theoretical foundation of hyperbolic conservation laws (single-phase gas dynamics, Euler’s equations), as well as ample practical experience in aerospace science and engineering. Our main challenge then will be to extend these concepts and practice to the case where the flow field is occupied simultaneously (in the sense of interpenetrating continua) by more than one phase. As we have seen in Part I, inviscid interactions (between the phases) play a critical role in securing the hyperbolic character of our system, and accordingly we can expect that the numerical treatment of these terms will be of central importance in adapting from the single-phase methods just mentioned. To further define the scope, it is sufficient for purposes of illustration (> Sect. ) to consider disperse systems with continuous to disperse density ratios much less than unity (i.e., droplet or particle flows). This is because such flows: (a) make accessible large slip between the phases, and thusly supersonic regimes that are of great interest to our aims here, and (b) can exhibit inviscid interactions that are realistically and quantitatively significant. Notably, slip is rather small in bubbly flows, even under forced accelerations; for example, in converging/diverging channels breakup due to interfacial instabilities leads to decreased length scales and thus closely followed near-equilibrium. On the other hand, recall (Part I) that for bubbly flows we need to also include (to the system considered here) equations that describe radial inertia and volume fraction transport (()-() in Part I). The extension of the present treatment to such situations is rather straightforward. Other terms in the equations that describe dissipative phenomena due to viscosity, heat transfer between phases, phase change, and breakup/coalescence can be readily added as source terms in the equations – they are indicated in Part I at the basic level, and they do not affect the numerical treatment discussed here. In other words, they concern fidelity of a simulation in a particular venue rather than robustness of the numerical scheme. Beyond this short introduction and the statement of our problem in > Sect. , the presentation is arranged in three parts. The first part provides the foundations from single-phase flow as mentioned already – it is separated into two section: > Sect.  deals with theory and exact results, and > Sect.  that leverages these essentially exact results to the most important approximate methods that address practical utility. The second part (> Sect. ) provides the extension of these methods and ideas to the treatment of the EFM; that is to the development of the AUSM-ARMS scheme specifically tailored to our present needs. The third part (> Sect. ) is on sample calculations that illustrate numerical performance. Throughout, we focus on




high-speed flows where the timescales are already limited by phenomena at the acoustic timescales and accordingly we limit our consideration to explicit schemes. Our present focus provides the complement to moderate speed and incompressible flows already well known via the methods belonging to the ICE and SIMPLE families. Indeed, they are at the genesis of the CFD era, the Implicit Continuous-Fluid Eulerian (ICE) method by Harlow and Amsden (), at Los Alamos Scientific Laboratory, and the Semi-implicit Method for Pressure-Linked Equations (SIMPLE) by Patankar and Spalding () at the Imperial College, UK (see also Harlow and Welch ). The former was the first method to overcome the Courant-Friedrichs-Lax (CFL) stability criteria, thus allowing practical simulations of large systems at all flow speeds (it formed the basis for all nuclear systems codes in existence to date – RELAP, TRAC, CATHARE, etc.). The later is arguably one of the most popular methods for low/moderate Mach number flows, especially in commercial software packages. Other well-known multiphase flow codes that are based on the ICE method include KACHINA (Amsden and Harlow ), K-FIX (Rivard and Torrey ), and MF-ICE/MAC (Kashiwa and Rauenzahn ). The first CFD type codes used to assess steam explosion energetics in severe nuclear reactor accidents, PM-ALPHA-D and ESPROSE-D (Theofanous et al. a,b), were based on the K-FIX solver. Commercial code packages that are based on SIMPLE include PHOENICS (Concentration Heat and Momentum Ltd.), FLUENT (Fluent Inc. joins ANSYS Inc.), and CFX (ANSYS Inc.).



Strategy for Computing Compressible Multi-Hydrodynamics

On the impetus of abounding gas dynamics problems, most notably from aerospace science and engineering, the numerical solution of hyperbolic conservation laws is well founded in theory (Lax ; Smoller ), and fully endowed in practice (Roe ; LeVeque ; Toro ), yet the supposedly similar path with multiphase systems has been filled with confusing retours and tangents. Starting with the first serious attempt at numerical simulation of nuclear reactor (loss of coolant) accidents, it was realized that coarse-graining of two-phase flows can be much more detrimental than that of turbulent flows – while they both share the issue of closure, reincorporation of fine-scale physics lost in averaging, the mathematical character of the two-phase system was found to have lost the hyperbolic property. First pointed out by Gidaspow (), this “ill-posedness” implies that numerical solutions are inherently susceptible to catastrophic instability and much of the efforts since have been focused on: (a) Finding numerical schemes that improve resiliency – numerical dissipation being a leading method to such an end (b) Aiming to show that such instabilities, moderated by nonlinear effects and by dissipative (algebraic) “source terms” would not significantly contaminate the numerical results (c) Adding “by hand” closure terms that would restore hyperbolicity On the other hand, and somewhat more severely, as a result of ill-posedness, there has been left some doubt on the basic correctness of the effective field model in any of the many shapes or forms that it has appeared in the literature (Drew and Passman ; Prosperetti and Satrape ; Prosperetti ).








On approaches (a) and (b) the major developments were made long ago and on behalf of the RELAP code – they have been invaluable for nuclear system simulations and are well documented in the open literature (RELAP-D Manuals ). The main ideas are that spatial discretizations normally employed in reactor system simulations introduce a sufficient amount of numerical dissipation to stabilize the solution, and that there is a “natural” cutoff on the unstable wave-numbers that is consistent with practical system discretization requirements (at least in terms of computer hardware available at the time). Furthermore, this dissipation role was shown to be aided by the drag terms in the momentum equations. On item (c) the efforts have been of more widespread and diverse origins. The idea was to “constitute” the phase interaction terms that appear in the averaged momentum equations in a manner that includes the “added mass” and “interfacial pressure” mechanisms (collectively called inviscid interactions herein and in Part I). More specifically, these are to address the two pairs of terms involving stresses in () and () – the first and second terms on the RHS, where the χ and δ are the phase-indicator and interface-indicator functions respectively (see () and () of Part I). d c uc + ∇⋅ < χ c ρ c υ′ c ⊗ υ′ c > = ∇⋅ < χ c σ c > − < σ c ⋅ nd δ I > +α c ρ c g dt d u  ′ ′   α d ρ d d d + ∇⋅ < χ d ρ d υ d ⊗ υ d > = ∇⋅ < χ d σ d > + < σ c ⋅ nd δ I > +α d ρ d g dt

αc ρc

() ()

In these equations subscripts c/d indicate continuous/disperse phases, respectively, σ’s are phasic stresses, u’s are velocities, ρ’s are densities, α’s are volume fractions, υ′ ’s are the pseudo-turbulent velocity fluctuations, and nd is the unit vector normal to interface. The added mass, first introduced by Drew et al. (), is to account for accelerations of one phase relative to the other. The interfacial pressure, first introduced by Stuhmiller (), was visualized as a Bernoulli effect due to the continuous phase flowing “around” the individual elements of the dispersion. Stuhmiller’s approach was found to be effective in rendering the system hyperbolic in incompressible, low-speed flows, and in fact it has been incorporated in the French nuclear code CATHARE (Bestion et al. ) (a descendant of RELAP). Many followed with combinations and/or variations of these two “fixes,” however, this practical path did not appear to intersect with theory to an adequate degree, therefore lacking in definitiveness of treatment (Toumi et al. ). In particular, no definitive treatment can be found that is fully integrated with rigorous and complete developments of the effective field model, even though two such developments have become available long ago (Geurst ; Wallis ). These two works (Geurst ; Wallis ) share with Part I the philosophy that rather than ad hoc and separate closures for added mass and interfacial pressure, inviscid interaction terms arise naturally and together with only minimal assumptions about the role of (pseudoturbulent) velocity fluctuations. Perhaps more importantly, the results, which can be shown (Part I, > Sect. .) to be in complete agreement among these three derivations, ()–() below, include terms other than those that have come to be recognized as added mass and interfacial pressure. The impact of these terms on hyperbolicity is provided in Appendix B of Part I. Notably, the methodologies employed in these three independent derivations are completely different, and thus we believe that there is a high degree of reliability in the result. Accordingly, this system is the starting point of our numerical treatment; ()–() of Part I, with the gravity terms omitted without loss of generality, and allowing for compressibility of the




disperse phase: ∂α c ρ c + ∇ ⋅ (α c ρ c uc ) =  ∂t ∂α d ρ d + ∇ ⋅ (α d ρ d ud ) =  ∂t d c uc αc ρc + ∇ ⋅ σ Kc + α c ∇p c = −FK dt d u α d ρ d d d + ∇(α d P K ) + α d ∇p c = FK dt dc Ec ∂α c αc ρc + ∇ ⋅ (α c p c uc + ud ⋅ σ Kc ) + p c = −W K dt ∂t d E ∂α α d ρ d d d + ∇ ⋅ (α d (p c + P K ) ud ) + p c d = W K dt ∂t

() () () () () ()

Where:     E c = e c + uc + K c ; K c = E(α d )(ud − uc ) ;   ∂K c K ; p c = f (ec , ρ c ); J = α c ρ c E(α d ) (ud − uc ); P = −α c ρ c ∂α d ∂J FK = − − ∇ ⋅ (ud ⊗ J) − (J ⋅ ∇)uc − α c ρ c ∇K c − J × (∇ × uc ) ∂t  K K K K E(α d ) = α d ( + C α d ); W = ud ⋅ F + α d P ∇ ⋅ ud ; σ c = (ud − uc ) ⊗ J  where p c is the continuous phase pressure along with the definitions of other quantities as given above. The hyperbolicity analysis for this system will be used (> Sect. ) in setting the finite volume rendition of this system and in the construction of our numerical scheme. As established by the landmark papers of Lax () and Courant et al. () and summarized in an excellent review article by Roe (), “shock-capturing methods have attained mathematical respectability, partly through reinterpretation of the numerical equations as expressions of integral rather than differential laws, and partly through the incorporation of ideas drawn from the theory of characteristics.” The former refers to expressing spatial derivatives in complete divergence form, and by use of Gauss’ theorem to manifest these terms as flux differences at the boundaries of finite volumes (used to discretize our flow domain) – an overall effect of conservation is quite obvious. The later refers to estimating these fluxes so that “state information” is propagated correctly from the upstream and/or the downstream directions, according to the characteristic speeds of our flowing two-phase mixture (as we will see below, these involve both the flow speeds of the phases and their sound speeds) – this we call generically “upwinding.” In particular this opens up access to “weak solutions”(Lax, ), that is, it allows the existence of discontinuities such as shock waves or contact discontinuities, and the numerical challenge is to accomplish this robustly and without oscillations in the solution. This is expressed as the monotonicity preserving constraint, which however according to a theorem due to Godunov () it is bound to be violated for any second-or higher-order accurate linear scheme. On the other hand, Godunov showed that first-order upwinding (see next section) is monotonicity preserving with the least truncation error, and this formed the








basis of a general strategy, namely to adding (carefully) EOS complexity and efficiency in computation. This strategy is also the foundation of our efforts in addressing the EFM. Accordingly, > Sects.  and >  are dedicated to setting up this foundation. Now in the same vein we note that our system of equations cannot be set in a fully conservative form ( > Sect. ). The “offending” terms are those involving the pressure gradient in the momentum equations as well as some of the terms representing inviscid interactions. More revealingly, the pressure gradient terms can be rewritten so that what is left outside the divergence sum are terms involving gradients in volume fractions – this is the real effect of “flow area” change for each of the phases, and in the extreme, it would represent a contact discontinuity (e.g., a gas flow entering a particle bed).



Basics: The Riemann Problem and the Godunov Method

Consider the motion of an (ideal) inviscid, compressible fluid in one space dimension. The “state” of fluid/motion at any point and time instant is described by the triplet [density, velocity, internal energy] = [ρ, u, e], and the evolution from any initial state (a prescribed spatial distribution of these quantities) is governed by the conservation laws of mass, momentum, and energy – the latter follows from the requirement that, in the absence of shocks (these require special treatment), the entropy of such an ideal flow must remain constant. These are the Euler equations, which along with the fluid (thermodynamic) equation of state (EOS) write: ∂U ∂F(U) + = , ∂t ∂x

t > ,

−∞ < x < ∞,

E = e + / u  ,

e = e(ρ, p)

()

U = (ρ, ρu, ρE)T and F(U) = [ρu, ρu  + p, (ρE + p)u]T It can be seen that our EFM in the previous section reduces to the Euler equations when we let the disperse phase vanish. We will refer to the original triplet as “primitive” variables, while U expresses a vector of “conserved” variables. The idea of conservation derives from the recognition that by use of Gauss’ theorem the divergence of the fluxes, F(U), can be written as the net (signed) flux crossing the boundaries of our system (by convection and pressure forces). In numerical implementation this crucial property can be assured at the discrete level by simply defining quantities U as averages over finite (cell) volumes, and the quantities F(U) as some appropriately averaged fluxes over the cell boundaries. For example, in one space dimension we can have our domain, say from x = a to x = b, discretized by setting cell boundaries at x j+/ ( j = ......N) a = x / < x / < ⋯ < x N−/ < x N+/ = b

()

Accordingly we have cells (C j ), cell centers (x j ), and cell “volumes” (Δx j ), respectively as: C j ≡ [x j−/ , x j+/ ] ,

xj =

 (x j−/ + x j+/ ) , 

Δx j = x j+/ − x j−/ .

()

On the time domain, we define Tn ≡ [t n , t n+ ] ,

Δt ≡ t n+ − t n

()




Integration of () over C j × Tn gives ∫

t n+ tn

x j+/ t n+ x j+/ ∂F(U(x, t)) ∂ U(x, t)d x) d t + ∫ (∫ dxdt =  ∫ ∂t ∂x x j−/ tn x j−/

()

which along with the definition for space and time averages: x j+/  U(x, t n )d x Δx j ∫x j−/ t n+  f(x j+/ ; t n , t n+ ) ≡ F(U(x j+/ , t) d t ∫ Δt t n n

uj ≡

() ()

yields: Δx j ∫

t n+ tn

(

t n+ ∂ n (F(U(x j+/ , t) − F(U(x j−/ , t)) d t =  uj ) d t + ∫ ∂t tn

()

If the fluxes in the above integral are evaluated at time t n+ we have an implicit scheme. For our purposes it is sufficient to focus on explicit treatment, so then by dropping the t n as well we finally have the discrete conservation law in terms of the approximate “numerical fluxes” at cell boundaries: Δt (f(x j+/ ) − f(x j−/ )) un+ = unj − () j Δx j At this point it should be clear that the whole question is on how to deal with these approximate numerical fluxes so as to preserve the basic natural behavior of our system with maximum efficiency and fidelity. In particular, since the existence of weak (non-differentiable) solutions has been established theoretically, of special interest is to obtain such fluxes in the presence of discontinuities in state variables.

.

The Riemann Problem

Focusing on the problem of dealing with discontinuities, as introduced above, let us then consider the simplest case first, that of an infinite flow domain initially separated into two uniform states – we denote them by superscripts (or subscripts when it is more convenient) L/R (for the state on the left/right). Moreover, for illustration let us focus on a particular example of this general class of Riemann problems; the one exemplified by a shock tube allowing by release of a diaphragm to instantaneously contact two initially stationary states of a fluid found at two different pressure levels. Our object is to find the structure of the solution, and more specifically to determine the flux vector at the position of the diaphragm; taken at the origin, x = . With the notation introduced above, this flux vector is denoted as f(x = ). As illustrated in > Fig. , the solution consists of three principal wave fronts – they correspond to the three eigenvalues and propagate at constant rates that are related to the three characteristic speeds of our system, so in the x-t plane the front positions of these waves can be found for any future time as shown in the figure. The central wave indicated as “contact,” provides the material displacement from the original position of the diaphragm. Both pressures and velocities across this wave are continuous, but there is a density jump on account








Wave 2: Rarefaction

Wave front 1: Contact

Wave front 3: Shock

pL *

pL

ρL

*

pR pR

* ρL

*

ρR ρR

*

uL uL

*

uR uR

t

x=0

⊡ Figure  Illustration of the solution structure to a Riemann (shock tube) problem. The solution is a constant on any “ray” (characteristics) emanating from the origin with a constant slope of x/t. The slopes of the characteristics that trace the wave fronts correspond to the eigenvalues

that on the left the fluid expands, while on the right it is compressed. This “material” characteristic speed is written as λ  = u ∗R = u ∗L ≡ u ∗ . The wave front on the left, marked as “rarefaction” propagates with the speed λ  = u ∗L − a ∗, where a ∗ is the speed of sound in the compressed state. The situation of the wave front on the right marked as “shock,” is a little more complicated. If its amplitude is sufficiently small, it propagates with the characteristic speed λ  = u ∗R + a ∗ , otherwise such acoustic waves (as their speed increases with amplitude) would “pile up” into a shock, a singularity in pressure across which energy/entropy conservation cease to be observed – accordingly the speed of such a shock needs to be determined by simultaneous consideration of the mass, momentum, and energy conservation equations (the so-called Rankine-Hugoniot (R-H) relations). The details are readily available and beyond our specific purposes here, but suffice to note that: . The characteristic speeds introduced above, the λ’s, are called the eigenvalues. For their explicit connection to () and > Fig. , and their further utility in our task it is best to wait until > .. . The nonlinearity (increasing/decreasing sound speed with state of compression/expansion) that steepens the compression front to a shock is responsible for “flattening” (with time)




the rarefaction (at the other end) through a fan of wave “fronts” all emanating from the origin. . The R-H conditions on the shock and an isentropic expansion on the rarefaction allow starred quantities to be related to the initial L/R states, UL ; UR , and these relationships along with the requirement that pressures and velocities are continuous across the contact discontinuity (note also the possibility of two different fluids involved in the use of EOS), yield an algebraic system to be solved (for the starred quantities) as the actual quantitative solution. This is the exact solution to the Riemann problem under consideration. . Once this solution is known, the sought-after flux at the origin is obtained simply as: f(x = , t) = {

f(U∗L ; U∗R ) = f(U∗L ) f(U∗L ; U∗R ) = f(U∗R )

if u ∗ ≥  if u ∗ < 

()

where the notation f(U∗L ; U∗R ) is to express the flux at the origin (the position of the diaphragm). As is to be seen in > ., this is a major result in motivating the concept of “upwinding.” It is important to note that in general the initial state of a Riemann problem will involve nonzero velocities – they would enter in the determination of the starred quantities, and in particular in the upwinding criterion () – in the particular shock tube example given above u ∗ >  obviously, but this is not to be so in general. Also note that since there is no characteristic length or time scales imposed, the solution to our problem depends solely on the combination x/t. Lines of constant x/t are called characteristics and the solution remains constant along a characteristic. Accordingly, the space between wave fronts is described by uniform states, except in the case of an expansion, where the “front” broadens out with time (> Fig. ). Godunov () capitalized on these simple ideas to propose a general numerical scheme capable of predicting evolutions from any initial state U (x), even when this initial distribution contains singularities. Moreover, even though the exact Riemann solution is not available for flows in higher dimensions, this numerical concept could be extended to provide robust and accurate results as summarized in > . and > .. In turn these provide the cornerstone for our own developments in regards to the EFM (> Sect. ).

.

The Godunov Method

Godunov’s brilliant and path-breaking idea was to provide the exact solution to an approximation of the initial fluid state “decomposed” in such a manner as to conform to a series of Riemann problems as illustrated in > Fig.  – in the simplest, first-order approximation, the decomposition is in terms of piece-wise constant values. Now according to notation introduced already, the unj ’s represent cell-average values of the “initial” state function in positions j at time cycle n, and the fluxes based on the Riemann problem across any two cells say [ j, j + ] are then denoted by f(x j+/ ) = f(uLj+/ , uRj+/ ). Thus, the average values at each cell in the next time increment Δt can be simply written out according to () as: n+

uj

Δt L R L R (f(u j+/ , u j+/ ) − f(u j−/ , u j−/ )) Δx j Δt n (f(u j , u j+ ) − f(u j− , u j )) ≡ uj − Δx j n

= uj −

()








L

uj+1/2

uj+1

uj

uj–1 xj–3/2

R

uj+1/2

xj–1/2

xj+1/2

uj+2

xj+3/2

xj+5/2

⊡ Figure  Illustration of the reduction of a general Euler governed problem to the solution of a series of Riemann problems – first-order approximation

L

uj+1/2

R

uj+1/2 uj+2

uj–1 xj–3/2

uj+1

uj

xj–1/2

xj+1/2

xj+3/2

xj+5/2

⊡ Figure  Illustration of the Godunov reduction in terms of higher-order approximations to a series of Riemann problems

For higher-order approximation the Riemann problems are defined in terms of states extrapolated from “sided” interpolations as illustrated in > Fig. . In two or three dimensions, the procedures are very similar, as boundary fluxes are composed from appropriately defined, one-dimensional Riemann problems at each of the cell boundaries. Details on practical implementation of the Godunov method can be found in van Leer (), Colella and Glaz (), LeVeque (), and Toro (). A recent example of the fundamental and broad reach of the Riemann solution is a particular implementation at the interface between two different fluids with high acoustic impedance mismatch – it brings the accuracy and robustness of front-tracking approach into the fast local level set front-capturing implementation of the characteristics-based matching (CBM) method (Nourgaliev et al. ), and further on to ab initio capturing of interfacial instabilities by the Sharp Interface Method (SIM) (Nourgaliev and Theofanous ; Nourgaliev et al. ; Chang et al. b). What remains to be done is to overcome the issues of inefficiency inherent in the method: the solution of a nonlinear set of algebraic equations by iteration methods, which is further




aggravated in the case of complex EOS (nonideal, stiffened gas, etc.). However, the Godunov method remains the “gold standard” in testing such other approaches.



Approximate Flux “Splitting” Schemes for Single Phase Flows

Three critical approaches in “translating” Godunov’s idea to practicality are outlined here. All three involve some sort of flux splitting aimed to “mimic” the characteristics-inspired “upwinding” discussed above but without solving the exact Riemann problem. In the first approach (flux difference splitting, or FDS), due to Roe (), this is accomplished by recasting the exact mathematical statement of the Riemann (eigenvalue) problem involved into an approximate (linearized) one whose solution can be obtained in a form convenient for numerical implementation. In the second approach (flux vector splitting, or FVS), due to van Leer (), the eigenvalue problem is bypassed altogether by means of a Mach-based flux splitting that provides exact upwinding for supersonic conditions and an approximate interpolation for the intermediate subsonic region. The third approach (advection upstream splitting method, or AUSM), due to Liou and Steffen () and Liou (), is a hybrid that combines the following specific features inherent to the first two: () property of FVS that selects only entropy satisfying approximate solutions (it can distinguish between expansions and shocks), and () decisive ability of some FDS methods to exactly capture stationary contact discontinuities and shocks. Indeed, the role of the convection cannot be overstated – it is the process that embodies the nonlinear effects of Euler equations, which can turn initially smooth profiles sharper and eventually into discontinuities (shocks). Along with a summary description of each of these approaches below we provide the successive improvements achieved. The figures of merit in this evaluation can be listed as follows:

• Discontinuity (shocks, contacts, expansions) capturing, stationary or moving, at all flow speeds

• Monotonicity preserving (not prone to spurious oscillations near discontinuities) • Positivity preserving (thermodynamic and material parameters remaining nonnegative) Clearly the aim is to achieve simultaneously all of the above in the most efficient computational manner and with maximum scope of application (generality). Maintaining conservation and satisfying appropriate entropy criteria are not to be sacrificed while trying to meet these ends. At a more philosophical level, the success of Godunov’s idea of using an exact solution to an approximation of the actual problem compels to further approximations in solving this Riemann problem itself.

.

Characteristics-Based Flux Splitting

Consider a Riemann problem, as the one in > Fig. , but now let us suppose that we “resolve” the discontinuities in the manner illustrated in > Fig. ; that is, the fluxes associated with each of the three waves are thought to be decomposed into a series of infinitesimal fluxes, δf k , k = ,  or  each associated with an incremental (infinitesimal) wave amplitude. In terms of our








f(UL, UR) δf2

δf1

Δf1 Δf2 δf3 F(UL)

Δf3

F(UR)

xj+1/2

⊡ Figure  Illustration of Roe’s flux difference-based splitting

Jacobian matrix (A), the three eigenvalues (λ  , λ  , λ  ) arranged along the diagonal of Λ, and the eigenmatrix R and its inverse matrix L ≡ R− , the decomposition can be written as: ∂F δU = A δU = R Λ L δU ∂U ⎛  ⎛ λ   ⎞ = R ⎜    ⎟ L δU + R ⎜  ⎝  ⎝    ⎠

δF =

 λ 

 ⎞ ⎛   ⎟ L δU + R ⎜  ⎝   ⎠

  

 ⎞  ⎟ L δU λ ⎠

= R Λ  L δU + R Λ  L δU + R Λ  L δU ≡ δf  + δf + δf

()

So the fluxes evaluated at the right and left states differ by:

F(UR ) − F(UL ) = ∫ =∫

UR UL UR UL

A dU R Λ  L dU + ∫

UR UL

R Λ  L dU + ∫

UR UL

≡ Δf + Δf + Δf

R Λ  L dU ()

which also shows that the sign of each λ k carries over to the sign of Δf k : (Δf k )± =

UR  R (Λ k ± ∣Λ k ∣) L dU  ∫U L

()

Moreover, () provides the path to smoothly connect the fluxes across the two neighboring cells by means of the three incremental characteristic fluxes. In particular, using upwinding the Riemann flux at the cell boundary can be expressed by any of the three forms


(see also

>



Fig. ): F(UL , UR ) = F(UL ) + ∑ (Δf k )− k +

= F(UR ) − ∑ (Δf k )

()

k

=

 − + [F(UL ) + F(UR ) + ∑ (Δf k ) − ∑ (Δf k ) ]  k k

So far these formal results are exact and not of much help in computation since the integrations in () are over a continuously changing eigensystem. Roe’s idea was to replace the whole series of A’s seen in the above derivation as one traverses from state UL or UR toward the common cell boundary, by one effective A. This he accomplished by evaluating A at some appropriately ˜ = U(U ˜ L , UR ), so that A(U ˜ L , UR ) = A (U(U ˜ L , UR )) satisfies the chosen “intermediate” state, U following: . . .

˜ has real eigenvalues and a complete set of eigenvectors (hyperbolicity) Matrix A ˜ (UL , UR ) (UR − UL ) so that conservation can be ensured F(UR ) − F(UL ) = A ˜ A (U, U) = A(U) for consistency

So now in place of () and () we have, respectively: ˜ ΔU F(UR ) − F(UL ) = A ˜Λ ˜  L˜ ΔU + R ˜Λ ˜  L˜ ΔU ˜ ˜  L˜ ΔU + R = RΛ

()

= Δf + Δf + Δf ˜ − (UR − UL ) F(UL , UR ) = F(UL ) + A ˜ + (UR − UL ) = F(UR ) − A

()

 ˜ (UR − UL )] = [F(UL ) + F(UR ) − ∣A∣  ˜ =A ˜+ −A ˜ − is a quantity related to the numerical dissipation. Due to the form of these where ∣A∣ equations Roe’s method is referred as Flux Difference Splitting (FDS). To illustrate, for an ideal gas, the flux Jacobian matrix is (with γ being the ratio of specific heats):  ⎛  A=⎜ −( − γ) u ⎜  ⎝ (γ − )u − γuE

 ( − γ)u γE −  (γ − )u 

 γ− γu

⎞ ⎟ ⎟ ⎠

()

The eigenvalues and eigenvectors can be obtained analytically as: ⎛ u Λ=⎜  ⎝ 

 u−a 

 ⎞  ⎟ u+a ⎠

()








⎛ ⎜ ⎜ ⎜ R=⎜ ⎜ ⎜ ⎜ ⎜ ⎝

−



ρ a

ρ a

ρ (u − a) a ρ u a − ( − ua + ) a  γ− −

u 

u 

γ −  u ⎛ − ⎜  a ⎜ ⎜  γ−  ⎜ L=⎜ − ( u + ua) ⎜ ρa  ⎜ ⎜ ⎜  γ−  ⎝ ρa (  u − ua)

⎞ ⎟ ⎟ ρ ⎟ (u + a) ⎟ ⎟ a ⎟   ⎟ ρ u a ⎟ ( + ua + ) ⎠ a  γ−

(γ − )

u a

γ− a γ− − ρa γ− ρa −

 (a + (γ − )u) ρa  (a − (γ − )u) ρa

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

()

()

˜ can be found analytically: Then from the condition  the Roe-“average” state U ρ˜ =

√

ρ L ρ R , u˜ =

uL

√ √ √ √ HL ρL + HR ρR ρL + uR ρR p ˜ ,H = ,H = E + √ √ √ √ ρL + ρR ρL + ρR ρ

()

Finally, the numerical flux for Roe’s approximate Riemann solver can be calculated by (). Remarks on performance: . Condition () also ensures that the Rankine-Hugoniot conditions, relating the speed s of a single discontinuity (contact or shock) to the jump across it, is satisfied, namely: F(UR ) − F(UL ) = s (UR − UL ) This together with condition () gives: ˜ (UR − UL ) = s (UR − UL ) A

Or

˜ −sI= A

˜ namely that s = u˜ + a˜ (u˜ > ) or s = which shows that s is an eigenvalue of the Roe matrix A; ˜ Clearly therefore, any two states that are “connected” by the Rankine-Hugoniot condition u. can be resolved by Roe’s approximate Riemann solver exactly (provided the grid moves along with the discontinuity), and stationary discontinuities can be captured in principle exactly. In practice however, the scheme is known to fail for super strong shocks (both in D and D) even if stationary (Kitamura et al. ), and also performance deteriorates (oscillations) in slow-moving shocks. Moving contact discontinuities can in principle be captured exactly at the CFL =  limit (> .), but this is not useful for practice, and the scheme is diffusive. . The downside of this exact property of Roe splitting is that it cannot distinguish between a shock and an expansion wave – that is, the splitting does not satisfy the entropy condition, a difficulty that leads to nonphysical jump within the expansion wave, known as “expansion shock.” This occurs because one of the eigenvalues decreases toward a zero value and concomitantly the numerical dissipation gradually disappears too. To remedy the problem, one needs to impose a lower limitation on the absolute magnitude of the eigenvalue. This procedure generally known as “entropy fix” was first suggested by Harten and Hyman () and has several variants now.




. For nonideal gas the eigenvalues are the same (with the speed of sound obtained from the applicable EOS), but the eigenvectors are not, and the Roe-“average” is not unique (Liou et al. ; Shuen et al. ). . Finally the method is susceptible to the so-called carbuncle instability – a transverse instability in D or D shocks that will be illustrated in > ..

.

Direct Flux Splitting

Direct flux splitting began with Steger and Warming’s () idea that, for the Euler system, the flux vector F(U) is a linearly homogenous function of degree one in U and accordingly: F=

∂F(U) U ≡ AU ∂U

()

Thus following the same diagonalization procedure as applied previously to δF (Roe), we can now split F directly into components associated respectively with the positive and negative eigenvalues: + − + − + − () F = R Λ L U ≡ R (Λ + Λ ) L U ≡ (A + A ) U ≡ F + F Thus, the numerical flux at the cell boundary can be expressed as: F(UL , UR ) = F+ (UL ) + F− (UR )

()

which automatically gives the exact flux at supersonic conditions: . If M L >  and M R > , then F+ (UL ) = F(UL ) and F− (UR ) = ; If M L < − and M R < −, then F− (UR ) = F(UR ) and F+ (UL ) = . + − . All the eigenvalues of matrices ∂F∂U(U) / ∂F∂U(U) are nonnegative/nonpositive. . F(U, U) = F+ (U) + F− (U) for consistency. The problem with this procedure is that the eigenvalues of A do not carry over individually to the splitted Jacobians; that is: A≡

∂F ∂F+ ∂F− ∂F+ ∂F− ≡ + ≡ A+ + A− does not imply = A+ and = A− ∂U ∂U ∂U ∂U ∂U

and the consequence in practical terms is that the zero eigenvalues expected in the splitting of A may now become small positive/negative values, thereby “contaminating” upwinding, which can become particularly problematic in subsonic flow. Moreover, while the splitting in () yields automatically proper fully one-sided upwinding at the sonic points ∣u∣ = a, the switching is not differentiable, thereby resulting in glitches there, even in a smooth acceleration from subsonic to supersonic speeds. The discontinuous switching also adds more dissipation at shock points than other flux schemes such as presented in this article. Independent work by van Leer () anticipated this switching difficulty and specifically required that the Jacobians of split fluxes be differentiable at sonic point. In addition, he sought to have one eigenvalue vanish identically so that the shock is captured in two cells. To achieve this he proposed to interpolate the splitting between the two extremes (upwinding) specified






1 M(2)(M)

+

–

M(2)(M) 0 –

+

M(2)(M),



+

M(2)(M)

–

M = M(2)(M) +M(2)(M)

–1

–1

0

1

M

⊡ Figure  The interpolation of Mach split function used in van Leer’s approach to subsonic flow

above for supersonic flow. More specifically, as indicated in > Fig. , in the intermediate subsonic region one operates with a combination of both nonlinear fields characterized by u + a >  and u − a < , namely M = M+ (M) + M− (M). Formally, the additional conditions imposed are: ±

. ∂F must be continuous and have one zero eigenvalue if ∣M∣ < . ∂U . The F+ / F− must be symmetric in M; F+ (M) = F− (−M); F− (M) = F+ (−M). We will build, according to van Leer, the vector f for this subsonic region from its components ( f  , f  , f  ) corresponding to the conserved variables (mass, momentum, energy), considered one by one separately. Moreover, we will use this as a way to anticipate and to bridge with AUSM – accordingly the derivation acknowledges AUSM’s key physics-based idea of separating out (and treating accordingly) convective from pressure contributions to the flux vector. From the continuity equation we have: f  =ρu = ρaM +

−

≡ρa (M (M) + M (M)) . + ≡ f

+

()

− f

When M > , we have f + = f  and f − = , giving that M+ (M) = M and M− (M) = . Similarly, when M < −, we have M+ (M) =  and M− (M) = M. For the interpolation van Leer used:  M±() (M) = ± (M ± ) 

()

where subscript “()” is to indicate the order of the polynomial employed. The numerical mass flux can then be written as: ( f  )/ = ρ L a L M+() (M L ) + ρ R a R M−() (M R )

()




For the momentum equation similarly, with the incorporation of splitting, we have: 

f  =ρu + p  =ρa  (M  + ) γ         = ρa (M + ) (M + ( − M)) + ρa (M − ) (−M + ( + M))  γ  γ

()

≡ f + + f − Which accordingly can be seen in more familiar terms to be:   ±   f  = ± (ρ u) a ⋅ (M ± ) + p ⋅ (M ± ) ( ∓ M)   ± ± =(ρ u) a ⋅ M() (M) + p ⋅ P() (M) with the interpolated pressure split function (illustrated in

>

()

Fig. ):

 ± P() (M) = (M ± ) ( ∓ M) 

()

Lastly, by combination of the above we obtain the splitting for energy flux: f ± =

γ ( f ± )  (γ − ) f ±

=±

 ρa γ− a  (M ± ) ⋅  ( ± M)  γ − 

±

= f  (H − m (u ∓ a))

1.5 +

1

–

P(3)(M), P(3)(M)

P(3)(M)

+

0.5

0 –

P(3)(M) –0.5 –1

0

1

M

⊡ Figure  The interpolation of pressure split function used in van Leer’s approach to subsonic flow

()








with

h () + h so that the square bracket in the energy flux can be factored into a perfect square for an ideal gas. This results in a nice property that one eigenvalue of the split-flux Jacobians vanishes, thus leading to reduced numerical dissipation and sharper resolution of shocks (van Leer ). This property however is hard to maintain for a general gas. To summarize, the splitted flux vector can be written so as to visualize not only the upwinding, but also the splitting between the convection and pressure components: m=

a

 ρ ⎞ ⎞ ⎛ ⎛ ± ± ± (M) ⎟ ⎟ + ⎜ p P() ρu f = a M()(M) ⎜ ⎠ ⎝ ρH − ρ m (u ∓ a) ⎠ ⎝ 

()

In this form van Leer’s method is not able to maintain constant total enthalpy along a streamline. One simple “fix” derived independently in Hänel et al. () and Shuen et al. (), is to drop the term ρ m (u ∓ a). Thus in final form the recommended van Leer splitting of the Riemann flux is: +

−

f/ (UL , UR ) = a L M() (M L ) ΦL + a R M() (M R ) Φ R  ⎞ ⎛ + − (M L ) + p R P() (M R )⎟ + ⎜ p L P() ⎠ ⎝  with

()

Φ = (ρ, ρ u, ρH)T .

It is observed that the whole splitting hinges on the split Mach numbers M± , which defines the mass and pressure fluxes, which in turn affects the momentum and energy fluxes.

Remarks on Performance

. It turns out that this flux vector splitting is significantly easier to implement in comparison to Roe’s scheme of building numerical fluxes, and surprisingly, it possesses a good capability of capturing shocks. . In addition to the advantage of being smooth at the transition points where eigenvalues change signs, this splitting does not involve differentiation of flux functions with respect to U (it does not involve the Jacobian). Instead, the imprint of the EOS appears covertly (via the speed of sound), the key parameter is the Mach number, and the form of split fluxes remains the same for all types of EOS. This point becomes a tremendous advantage in computing multiphase flows. Moreover, this splitting, being in terms of mass fluxes only, allows any additional conservation laws to be easily added making extensions to other kinds of systems rather straightforward. . As pointed out by van Leer himself, the dispensing with the eigensystem undermines the scheme’s performance in regards to contact discontinuities as can be seen in the following: Consider a moving contact discontinuity defined as: ρj = {

ρ ρ

for j < k for j ≥ k + 

∗

u j = u ≥ , p j = p

∗


From

>



., (), we can readily see that Roe’s numerical flux is:  ˜ (UR − UL )] F/ (UL , UR ) = [F(UL ) + F(UR ) − ∣A∣   ∗ = [F(UL ) + F(UR ) − ∣u ∣ (UR − UL )]  ⎛  ⎞ =u ∗ UL/R + ⎜ p∗ ⎟ ⎝u ∗ p∗ ⎠

()

Here the relevant velocity for upwinding is u ∗ , so UL/R = {

UL UR

if u ∗ ≥ ; if u ∗ < .

()

and by substitution into the discretized equations, we have: ∂U + F j+/ (U j , U j+ ) − F j−/ (U j− , U j ) =  ∂t ()

or ∂U  ∗ ∗ + [(u U) j − (u U) j− ] =  ∂ t Δx ∗

We can see that when uΔ Δx t = , Roe’s scheme is able to capture the contact continuity exactly. On the other hand, for van Leer’s scheme, (), we get: ∂U  + [(a M−() Φ) j+ − (a (M+() − M−() ) Φ) j + (a M+() Φ) j− ] =  ∂ t Δx

()

and it is quite clear that when ∣M j+ ∣ <  we have M−() ≠ , and a flux representing downwind properties enters the calculation – destroying upwinding and leading to false updating in the presence of stationary or moving contact discontinuities.

.

Advection Upstream Splitting

As we have seen above the basic ideas of Roe and van Leer have much to complement each other relative to mutual shortcomings in addressing the Riemann problem (and thereby the implementation of the Godunov idea). In the Advection Upstream Splitting Method (AUSM) (Liou and Steffen ; Liou ) a synthesis (of these two approaches) is made that retains the positive attributes while canceling the negative ones. The bases are to recognize and observe the physical correctness of the pressure-convection separating of van Leer (). As we will see ( > Sect. ) this is a crucial point in the implementation of the method to the effective field model of two-phase flow. Moreover, on this basis, it retains the advantages of bypassing the need for an eigensystem. This, besides efficiency in computation offers the means to addressing complex equations of state as well as problems that may be mildly non-hyperbolic – as is the








case in certain regions of the (relative) Mach number space in the multiphase flow model recommended and utilized herein. On the other hand, the method needs to incorporate the means of resolving the contact discontinuity issue explained above. Recalling the convective component of (): f (c) = a L M+() (M L ) Φ L + a R M−() (M R ) ΦR

()

and that the contact discontinuity issue arises from using two separate convective speeds, here we use only one convective speed comprising contributions from both “L” and “R” states and write instead: (c) f = u / ΦL/R = a / M / ΦL/R () where M / is split according to van Leer: +

−

M / = M (M L ) + M (M R )

()

and the sound speed a / is designed to satisfy certain discontinuity capturing properties as explained further below. It turns out that this has Roe’s (FDS) contact capturing quality we were also aiming for. The pressure splitting of () is maintained and likewise for the energy equation we use the total enthalpy H, while the interpolations involved are extended to still higher order. We thus have the AUSM+ (Liou ) as: ⎛  ⎞ F/ = a / M / ΦL/R + ⎜ p / ⎟ ⎝  ⎠

()

where M / and p / are defined as: +

−

M / ≡ M() (M L ) + M() (M R ), p / ≡

+ P() (M L )

− p L + P() (M R )

M L/R = u L/R /a / pR

() ()

With

M±() (M)

⎧  ⎪ ⎪ (M ± ∣M∣) ⎪ ⎪ ⎪  =⎨ ⎪   ⎪ ⎪ ± (M ± ) ± (M  − ) ⎪ ⎪  ⎩ 

if ∣M∣ ≥ , () Otherwise.

and ⎧ ∣M∣  ⎪ ⎪ ( ± ) ⎪ ⎪ ⎪  M ± (M) = ⎨ P() ⎪   ⎪ ⎪ ⎪ (M ± ) ( ∓ M) ± M(M  − ) ⎪ ⎩ 

if ∣M∣ ≥ , () Otherwise.

The a / is a “numerical” speed of sound, and its specification is done so as to assure that a stationary shock is captured by satisfying the Rankine-Hugoniot condition exactly. Once the




average state has been so fixed there is no more control; we will use this result even for moving shocks as experience shows that performance is still very good. Consider the two fluid states UL and UR separated by a stationary shock and such that u L > u R > . According to Rankine-Hugoniot condition, numerical fluxes F/ must satisfy FL = F/ = FR , which with the incorporation of the above convection specification yields: u / ρ L ⎞ ⎛ ρLuL ⎞ ⎛ FL = ⎜ ρ L u L + p L ⎟ = ⎜u / ρ L u L + p / ⎟ = F/ ⎝ ρ L u L H L ⎠ ⎝ u / ρ L H L ⎠ Now from (), if M L =

uL a /

()

> , we get M+() (M L ) = M L and

     u / = a / (M L − (M R − ) − (M R − ) )     = u L − (u R − a / ) ( (M R − ) + (M R + ) )  

()

Thus we can choose a / = u R , and it follows u / = u L , which leads to p / = p L and, hence, satisfy the Rankine-Hugoniot condition. The sound speed can be written in a more general form, which involves only “upwinding” states. From Prandtl’s relation across a stationary shock wave, we have (a ∗L ) = (a ∗R ) = u L u R , 



()

Thus we get a / =

(a ∗L )



uL

()

where a ∗ is the critical (sonic point) speed of sound. For ideal gas, it is expressed as: (a ∗) = 

(γ − ) H γ+

()

By extending the above analysis to cover other conditions, as u R < −a R or in subsonic flows, we have:  (a ∗) a / = min( a˜ L , a˜ R ) with a˜ = () max(a ∗, ∣u∣) By virtue of its derivation, () is only applicable to the capturing stationary shocks – it does not satisfy the Rankine-Hugoniot across a moving shock exactly. Still, based on a wide range of numerical tests we can state that this drawback is only in effecting a slight amount of numerical dissipation resulting only to a minimal smearing of the shock. In fact, even the rather simple choice () a / = /(a L + a R ) is quite satisfactory, again at the cost of smearing in the shock profile. In all cases both amplitudes and positions are captured accurately, a manifestation of satisfying conservation properties.








In a further development, the AUSM+ scheme was extended to addressing all-speed situations (Liou ). This is done by adding properly-scaled dissipation terms to the convection and pressure fluxes to enhance the coupling between pressure and velocity fields, which is especially important for multiphase flow (Chang and Liou , ) as we may be facing great disparities in fluid properties and flow speeds. The AUSM+ -up scheme is written as  ⎞ ⎛ F/ = (a / M / ρ L/R + D (p)) Ψ L/R + ⎜ p / + D (u) ⎟ ⎠ ⎝ 

()

With Ψ = ( , u , H)

T

()

¯  , ) (p L − p R ) ΔM max ( − M (p) D = κp a¯ (u) + − ¯ P() ( M) ¯ ρ¯ a¯ (u L − u R ) D = κ u P() ( M)

()

¯ L ) − M+() ( M ¯ L ) − M−() ( M ¯ R ) + M−() ( M ¯ R) ΔM = M+() ( M

()

()

And

¯ are obtained as the arithmetic means of “L” and “R” states, and ¯ a, ¯ and M The parameters ρ, usually the coefficients κ p = κ u =  are sufficient to suppress numerical oscillation in both the gas and liquid fields. Remarks on Performance . Discontinuity Capturing. This of course is the most crucial issue in respect to robustness

as well as accuracy. For both contacts and shocks the greatest the jumps in flow and state variables involved the greatest the “stress” in the numerical scheme. Another stress is due to unsteady/nonstationary discontinuities, and unfortunately, theoretical results are available only for the stationary case: one is looking for sharp (nondiffusive) capturing. While this provides some sense of what could be expected in general for unsteady cases, as for example the development of a shock wave in an explosion (e.g., starting with ignition and run up to detonation), much of the performance characterization has to rely on empirical evidence; namely analysis of actual calculations and comparison to known solutions (like the Chapman-Jouget self-similar limit in D detonations) and/or experiments. On stationary discontinuities the situation can be summarized as follows: (a) For a contact AUSM+ automatically captures exactly the discontinuity since the mass flux is zero. In other words ρu = , M L = M R = , p L = p R . As shown already the FDS also satisfies this requirement, while the FVS fails it. (b) For a stationary shock, AUSM+ with the choice of numerical speed of sound in () captures the discontinuity exactly (Liou ). Due to the design for possessing the Rankine-Hugoniot property, AUSM+ , like Roe, cannot distinguish whether the discontinuity is a shock or an expansion – this can be remedied (Liou ) by the following redefinitions in (): 



a˜ L = a ∗ /max(a ∗, u L ), a˜ R = a ∗ /max(a ∗, −u R ).

()


Roe



AUSM+-up

⊡ Figure  Supersonic flow past a sudden expansion (corner). The Roe method with entropy fix produces instability. The AUSM+ calculation (CFL=. on a  x  grid) employed a MUSCL-type linear interpolation of primitive variables along with the van Albada limiter (van Albada et al. ) to control monotonicity

In > Sect. , and in more detail in the Appendix, we illustrate these trends also for the case of multiphase flow, and also for the case of nonstationary discontinuities. . Positivity Preserving. Sudden expansions can lead to negative values of certain variables such

as pressure, density, or volume fraction, which would terminate the calculation, thus, positivity preserving is an essential attribute of robustness of a numerical scheme. In an D situation violation of positivity can arise in the case of a Riemann problem with the two states set to move in opposite directions with velocities that exceed some critical value. In D the same can arise in a supersonic flow moving past a sharp, ○ expansion (corner) as illustrated in > Fig. . In contrast to documented failings of several prominent flux schemes (Einfeldt ), AUSM+ exhibits a robust shock capturing capability typical of what is seen in > Fig. . As another example, for the D Riemann problem {(−u, p, ρ)L ; (u, p, ρ)R } it has been shown (Liou and Edwards ) that AUSM+ guarantees positivity as long as: u Δt/Δx <  and u Δt/Δx < /γ for pressure and density, respectively. For the linear case this condition coincides with the total variation diminish (TVD) condition of Harten ().

. Transverse Shock Instability (Carbuncle Phenomenon). This sort of problem can arise in multidimensional situations, such as the one illustrated in > Fig. , and it is due to spurious transverse waves that can amplify to a point that destroy the calculation. More specifically it has been suggested (Wada and Liou ) that this transverse action is originated by the few (unphysical) intermediate points of the profile approximating the shock, and their lateral influences, which in turn affect the shock in a multidimensional manner. A “shock fix” has been proposed, and it is effective but at the additional complication of the “if-then” check involved.





Multiphase Flows: Compressible Multi-Hydrodynamics AUSM+

Roe

Mach = 6.00 p/p ∞ x 0.25 T/T ∞

10.0

10.0

5.0

0.0 –2.0

T / T∞, p / p ∞ x 0.25

T/T ∞, p/p ∞ x 0.25

Mach = 6.00

–1.5 x/R

6.0

6.0

4.0

4.0

r 2.0 0.0 –2.0

p / p ∞ x 0.25 T/T∞

5.0

0.0 –2.0

–1.0

r



–1.5 x/R

–1.0

–1.5 x/R

–1.0

2.0

–1.5 x/R

–1.0

0.0 –2.0

⊡ Figure  Density contours in steady supersonic flow past a blunt body positioned at right angles. Inset profiles are along the stagnation streamlines

As in the case for Roe () (shown) also the Osher and Solomon () method, as do most others fail. On the other hand, as is the case of AUSM+ (shown) so is the van Leer VFS scheme not subject to this deficiency. . Efficiency, Generality. Unlike Roe’s FDS, the numerical dissipation in AUSM+ is merely a

scalar, not of matrix type. As a result, the system is decoupled and hence requires only O(n) operations, n being the number of unknowns (and equations). Moreover, the same formula is easily extendable to include other conservation laws, or to fluids with general EOS, as is the case in multiphase flow. As in van Leer’s FVS, the AUSM+ does not require differentiation, or the flux Jacobian matrix; and the flux splitting always involves only the common mass-flux term for any additional conservation laws.



Advection Upstream Splitting for the Effective Field Model

.

Recasting the System of Equations in Quasi-Conservative Form

As made clear in the previous section, we need to begin the task of extending AUSM to the effective field model by recasting the system of equations, ()-(), in conservation form. Unfortunately, this cannot be done in a complete manner – terms like p∇α and other similar terms remain, and as we shall see shortly these “nonconservative” terms create a new type of numerical challenge of significant proportions. The recasting then also aims to consolidate and




simplify these terms to the extent possible. The result is: ∂ (α c ρ c ) + ∇ ⋅ (α c ρ c uc ) =  ∂t ∂ (α d ρ d ) + ∇ ⋅ (α d ρ d ud ) =  ∂t ∂ (α c ρ c uc ) + ∇ ⋅ [α c (ρ c uc ⊗ uc + p c I + ρ c E w ⊗ w)] = p d ∇α c + ΦK ∂t ∂ K (α d ρ d ud ) + ∇ ⋅ [α d (ρ d ud ⊗ ud + p d I)] = p d ∇α d − Φ ∂t ∂ ∂α c (α c ρ c E c ) + p d + ∇ ⋅ [α c (ρ c E c uc + p c uc + ρ c E (ud ⋅ w)w)] = ud ⋅ Φ K ∂t ∂t ∂α ∂ K (α d ρ d E d ) + p d d + ∇ ⋅ [α d (ρ d E d ud + p d ud )] = −ud ⋅ Φ ∂t ∂t

() () () () () ()

+ ∇ ⋅ (ud ⊗ J) + (J ⋅ ∇)ud + J × (∇ × ud ). with w = ud − uc and Φ K = ∂J ∂t Note: In one-dimensional flow the last term of ΦK is identically zero, while in general it is convenient to combine the last two terms and use as: (J ⋅ ∇)ud + J × (∇ × ud ) ∂u d ∂v ∂w ⇀ + (v d − v c ) d + (w d − w c ) d ) e  ∂x ∂x ∂x ∂u d ∂v d ∂w ⇀ + α c ρ c E(α d ) ((u d − u c ) + (v d − v c ) + (w d − w c ) d ) e  ∂y ∂y ∂y ∂u ∂v ∂w ⇀ + α c ρ c E(α d ) ((u d − u c ) d + (v d − v c ) d + (w d − w c ) d ) e  ∂z ∂z ∂z

= α c ρ c E(α d ) ((u d − u c )

()

We will demonstrate the numerics in one space dimension, and based on the above our system is: ∂ ∂ () (α c ρ c ) + (α c ρ c u c ) =  ∂t ∂x ∂ ∂ () (α d ρ d ) + (α d ρ d u d ) =  ∂t ∂x ∂ ∂ ∂ (α c ρ c u c ) (α c ρ c ( + E(α d )) u c ) − (α c ρ c E(α d ) u d ) + ∂t ∂t ∂x ∂α c ∂u ∂ ∂ [α c (p c + ρ c E(α d ) (u d − u c ) )] = p d + + (α c ρ c E(α d )u d (u d − u c )) + J d ∂x ∂x ∂x ∂x () ∂ ∂ ∂  (α d ρ d u d ) − (α c ρ c E(α d ) u c ) + ((α d ρ d + α c ρ c E(α d )) u d ) + ∂t ∂t ∂x ∂ ∂u ∂ ∂α c + (α d p d ) = −p d − (α c ρ c E(α d )u d (u d − u c )) − J d () ∂x ∂x ∂x ∂x ∂α c ∂ ∂ [α c (ρ c H c u c + ρ c E(α d ) u d (u d − u c ) )] = u d Φ K () (α c ρ c E c ) + p d + ∂t ∂t ∂x ∂ ∂α ∂ () (α d ρ d E d ) + p d d + [α d ρ d H d u d ] = −u d Φ K ∂t ∂t ∂x








With pd pd  = ed + u d + ; ρd  ρd ∂J ∂u ∂ K Φ = + (u d J) + J d ∂t ∂ x ∂x Hd = Ed +

Hc = Ec +

pc pc   = e c + u c + E(α d )(u d − u c ) + ; ρc   ρc

In final vector form with the convection terms collected in the manner suggested by AUSM is: ˜ ∂T ∂G ∂H ∂Q ∂ ∂ ∂ + pd + (F(u c ) ) + (F(u d ) ) + (F(p) ) = p d + +S ∂t ∂t ∂x ∂x ∂x ∂x ∂x

()

Where: αc ρc ⎞ ⎛ ⎛⎞ αd ρd ⎟ ⎜ ⎜⎟ ⎟ ⎜ ⎜ ⎟ ⎜ α c ρ c ( + E(α d )) u c − α c ρ c E(α d ) u d ⎟ ⎜⎟ ⎟ ⎜ ⎟ ˜ =⎜ Q ⎜−α c ρ c E(α ) u c + (α ρ + α c ρ c E(α )) u ⎟ ; T = ⎜  ⎟ ⎜ ⎜ ⎟ d d d d d⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜αc ⎟ αc ρc Ec ⎠ ⎝ ⎝α d ⎠ αd ρd Ed ⎛ αc ρc uc  ⎜ ⎜ ⎜ α c ρ c u c (u c ) =⎜ F ⎜  ⎜ ⎜ ⎜ αc ρc Hc uc ⎝ 

 ⎞ ⎞ ⎛ αd ρd ud ⎟ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ (u d ) ⎜ ⎟  ⎟;F ⎟; =⎜  ⎟ ⎟ ⎜ αd ρd ud ⎟ ⎟ ⎜ ⎟ ⎟ ⎜  ⎟ ⎜α c ρ c E(α d ) (u c − u d ) u d ⎟ ⎠ ⎠ ⎝ αd ρd Hd ud

()

()

⎛⎞ ⎜⎟ ⎜ ⎟ ⎜ αc ⎟ ⎜ ⎟, ⎜α ⎟ ⎜ d⎟ ⎜ ⎟ ⎜⎟ ⎝⎠

()

⎛  ⎞ ⎜  ⎟  ⎜ ⎟ ⎛ ⎞ ⎜ ∂u d ⎟  ⎜ ⎟ ⎜ J ⎟ ⎜ ⎟ ⎜ ∂x ⎟ ⎜ α c ρ c E(α d )u d (u d − u c ) ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ H= ⎜ ⎜−α c ρ c E(α )u (u − u c )⎟ ; S = ⎜ ∂u d ⎟ . ⎜ ⎟ ⎜ −J ⎟ d d d ⎟ ⎜ ⎜ ∂x ⎟ ⎟ ⎜ ⎟ ⎜  ⎜ ⎟ ⎠ ⎝ ⎜ u ΦK ⎟  ⎟ ⎜ d ⎝−u d Φ K ⎠

()

 ⎞ ⎛  ⎟ ⎜ ⎟ ⎜ ⎜α c (p c + ρ c E(α d ) (u d − u c ) )⎟ (p) ⎟;G = ⎜ F =⎜ ⎟ αd pd ⎟ ⎜ ⎟ ⎜ ⎟ ⎜  ⎠ ⎝ 

Note that the term due to velocity fluctuations in the continuous phase momentum equation is interpreted as a pressure modifier and it is split accordingly (along with pressure, rather than being treated together with the convection terms). In D or D this dyadic will have off-diagonal terms, and these interpreted as “stress” could be treated by simple central difference, or could be split as pressure.




.

Numerical Discretization

The convection and pressure splitting carry over directly from volume fractions are basically continuous we have:

> ., and for the case that the

(p d )nj n+ n   (u c ) n (u c ) n n+ n (˜q j − q˜ j ) + (t j − t j ) + ) − (f j−/ ) ] [(f j+/ Δt Δt Δx n n   (p) (p) (u d ) n (u d ) n ) − (f j−/ ) ]+ + [(f j+/ [(f j+/ ) − (f j−/ ) ] Δx Δx (p d )nj n  [g j+/ − gnj−/ ] + [hn = − hnj−/ ] + s∗ Δx Δ x j+/ ⎛ αc ⎜  ⎜ αc uc (p) ⎜ (u c ) f/ = [a / (M c )/ (ρ c )L/R + D c ] ⎜ ⎜  ⎜ ⎜ ⎜ αc Hc ⎝ 

(u )

f/d = [a / (M d )/ (ρ d )L/R

(p)

f/

 ⎛ ⎜  ⎜ ⎜ α c (p′c ) ⎜ / =⎜ ⎜ α d (p d )/ ⎜ ⎜  ⎜ ⎝ 

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

()

;

()

L/R

 ⎛ ⎜ α d ⎜ ⎜  ⎜ (p) ⎜ αd ud + Dd ] ⎜ ⎜ ⎜ ρ c ⎜ α c ( ) E(α ) (u c − u ) d d ⎜ ρd ⎜ ⎝ αd Hd

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

;

()

⎞ ⎟ ⎟ ⎟ ⎟ ⎟; ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

()

L/R

 ⎞ ⎛  ⎟ ⎜ ⎟ ⎜ (u) + ′ − ′ ⎟ ⎜ (P (M )p + P α ⎜ () (M c,R )p c,R + D c ) ⎟ ⎜ c () c,L c,L ⎟=⎜ ⎟ ⎜ α d (P + (M c,L )p d,L + P − (M c,R )p d,R + D (u)) ⎟ ⎜ () () d ⎟ ⎜ ⎟ ⎜  ⎠ ⎝ 

where the (M c )/ is also split to account for volume fraction variation across the cell interface: (M k )/ =

 [max ((α k )L − (α k )R , ) (M k )L (α k )max + max((α k )R − (α k )L , ) (M k )R ] (α k )min [M+() ((M k )L ) + M−() ((M k )R )] + (α k )max

with (α k )min = min ((α k )L , (α k )R ) (α k )max = max ((α k )L , (α k )R )

()








In the above p′c = p c + ρ c E(α d ) (u d − u c ) (p)

()

(u)

and the numerical dissipation terms D k , D k are given in () and (). For numerical speed of sound a / , we employ the sound speed of the continuous phase. The same speed of sound is applied to both phases simultaneously. For inter-phasic terms, we have:  ⎛ ⎜  ⎜ (p d ) j (p d ) j ⎜ ⎜ ((α c )L, j+/ − (α c )R, j−/ ) ⎜ [g j+/ − g j−/ ] = Δx Δx ⎜ ⎜ − ((α c )L, j+/ − (α c )R, j−/ ) ⎜ ⎜  ⎜ ⎝ 

h/

 ⎛  ⎜ ⎜ ⎜ α c ρ c E(α d )(u d − u c ) = [a / (M d )/ ] ⎜ ⎜ −α c ρ c E(α )(u − u c ) ⎜ d d ⎜ ⎜  ⎝ 

 ⎛  ⎜ ⎜ Jj ⎜ △x (u d ) ⎜ s∗ = ⎜ ΔxJ j ⎜ − △x (u d ) ⎜ Δx ⎜ (u ) Φ K ⎜ d j j ⎝ −(u d ) j Φ Kj

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

with △x (⋅) = {

(⋅) j − (⋅) j− (⋅) j+ − (⋅) j

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

⎞ ⎟ ⎟ ⎟ ⎟ ⎟; ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

;

()

()

L/R

if J j ≥ ; Otherwise.

()

The forces expressed by the inter-phasic terms (H and s∗) locally must cancel, and this is assured by discretizations () and (). As we have seen already above, with the AUSM numerical speed of sound, we should be able to handle shocks and expansion waves. However, in the presence of contact discontinuities, such as jumps in volume fractions, special consideration is needed. The issue arises because the discontinuity in α would cause the nonconservative terms in () and () to create nonphysical imbalances of pressure forces if discretization is done in the standard way of using common intermediate values of α k and pressure at cell interface, and if the ∇α k term is discretized by central difference. The problem can be addressed by allowing the discontinuities to smear but this would defeat the purpose of a high-fidelity numerical scheme. Instead, we use () which with the help of > Fig.  can be seen to assure the balance of pressure force across each cell. Recall that by conservation treatment of convection terms, momentum fluxes are automatically balanced.


(p c)j–1/2 (αc)j−1/2

(αc)j+1/2



(p c)j+1/2

(pd)j [(αk)j+1/2–(αk)j−1/2] (pd )j–1/2

(αd)j–1/2

(αd)j+1/2

(p c)j–1/2

δc−c δc−c

(pd )j+1/2

(p c)j+1/2

δc−d

– (αk)R,j−1/2] (pd)j [(αk)L,j+1/2 δc−d (pd )j–1/2

δd−d

δd−d

(pd )j+1/2

⊡ Figure  Illustration of pressure force applied on subcells of different fluids. (a) with continuous αk ; (b) with noncontinuous αk

p

f j+/

p

f j−/

 ⎛  ⎜ ⎜ ⎜ δ c−c (p c )/ + δ c−d (p d )/ =⎜ ⎜ δ (p ) + δ (p ) ⎜ d−d d / d−c d / ⎜ ⎜  ⎝ 

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

 ⎛  ⎜ ⎜ ⎜ δ c−c (p c )/ + δ d−c (p d )/ =⎜ ⎜ δ (p ) + δ (p ) ⎜ d−d d / c−d d / ⎜ ⎜  ⎝ 

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

;

j+/

()

j−/

δ c−c , δ d−d , δ c−d , and δ c−d are effective areas for fluid interface on cell boundary. δ c−c = min ((α c )L , (α c )R )

()

δ d−d = min ((α d )L , (α d )R )

()

δ c−d = max (, Δα d ) = max (, −Δα c )

()

δ d−c = max (, Δα c ) = max (, −Δα d )

()

Δα k = (α k )R − (α k )L

()








In code implementation, we use a MUSCL type third-order scheme with Osher-Chakravarthy TVD limiter (Chakravarthy and Osher ) to determine fluid states on cell boundary. This allows us to sharply capture the shock wave and contact discontinuity while avoiding the numerical oscillation effectively.

.

Time Integration

For time integration, we use the Runge-Kutta time integration method (Jameson ). Supposing the system: ∂Q + R(Q) =  ∂t

()

its time integration can be calculated by: Q() = Qn − ϖ  ⋅ Δt ⋅ R(Qn ) Q() = Qn − ϖ  ⋅ Δt ⋅ R(Q() ) ⋮ Q

()

(s−)

Q

n+

n

= Q − ϖ s− ⋅ Δt ⋅ R(Q n

= Q − ϖ s ⋅ Δt ⋅ R(Q

(s−)

(s−)

)

)

For a four-stage Runge-Kutta, the coefficients are (ϖ  , ϖ  , ϖ  , ϖ  ) = (  ,  ,  , ). Applying to our system of equations ()-(): ∂ ˜ [Q] + p d [T] + [R] =  ∂t

()

we have:  ⎛  ⎜ s ⎜ ⎜ p   ˜ n) + d ⎜ ˜ s+ − [Q] ([Q]  Δt Δt ⎜ ⎜ ⎜ s ⎜ α c − α cn ⎝ αs − αn d d

⎞ ⎟ ⎟ ⎟ ⎟ + ϖ s+ [R]s =  ⎟ ⎟ ⎟ ⎟ ⎠

()

with ⎛ ⎜ ⎜ s+ s ⎜ ˜ [Q] + p d ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

    α cs α ds

⎞ ⎟ ⎟ ⎟ ⎟ = [Q] ˜ n + psd ⎟ ⎟ ⎟ ⎟ ⎠

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

    α cn α dn

⎞ ⎛ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ − ϖ s+ Δt ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎠ ⎝

R s R s R s R s R s R s

˜ ⎞ ⎛ Wc ˜ W ⎟ ⎜ d ⎟ ⎜ ˜ ⎟ ⎜ Mc ⎟ ≡⎜ ⎟ ⎜ M ⎟ ⎜ ˜d ⎟ ⎜ ˜ ⎟ ⎜ Ec ⎠ ⎝ E˜d

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

()

Here [R] is the collection of all the spatial discretization and source terms. The specific implementation is as follows:




From the continuity equations we can update for the products αρ: n n s ˜ α cs+ ρ s+ c = α c ρ c − ϖ s+ Δt R  ≡ Wc s+ s+ n n s ˜d α d ρ d = α d ρ d − ϖ s+ Δt R  ≡ W

() ()

Then from the momentum equations we can update the velocities: s s+ s+ s+ s s+ α cs+ ρ s+ c ( + E ) u c − α c ρ c E u d n n n n n n n n s ˜c = α c ρ c ( + E ) u c − α c ρ c E u d − ϖ s+ Δt R  ≡ M

()

s s+ s+ s+ s+ s+ s s+ − α cs+ ρ s+ c E u c + (α d ρ d + α c ρ c E )u d

˜d = −α cn ρ cn E n u cn + (α dn ρ dn + α cn ρ cn E n )u dn − ϖ s+ Δt R s ≡ M

()

which provide the linear system: (

s α cs+ ρ s+ c ( + E ) s+ s+ s −α c ρ c E

s −α cs+ ρ s+ c E s+ s+ s+ s+ s αd ρd + α c ρ c E

)(

u s+ c )=( u ds+

˜c M ) ˜ Md

()

Note that instead of E s (α ds ), we could use E s+ (α ds+ ) and obtain the solution by iteration. To decode so as to obtain primitive variables (α, ρ, p) we need to involve the EOS and the energy equations. s+ s s+ α cs+ ρ s+ = α cn ρ cn E cn + psd α cn − ϖ s+ Δt R s = E˜ c c E c + pd α c s+ s+ s+ s s+ α d ρ d E d + p d α d = α dn ρ dn E dn + psd α dn − ϖ s+ Δt R s = E˜d

() ()

We employ a stiffened gas EOS to relate internal energy (e) to the gas pressure and density: e = (p c + γ c P∞, c )/ρ c (γ c − )

()

s+ and the Newton iteration method to solve for ps+ c and α c . To this end, the residuals Γ are:

Γ=( Γ =

Γ α k ρ k E k + psd α ck − E˜ c p kc k ) = ( kc kc kc ) ; Ω = ( ) Γ α d ρ d E d + psd α dk − E˜d α ck

α ck   k s+ s+  s k ˜ (p kc + γ c P∞,c ) + (α ck ρ ck ) ((u s+ c ) + E(α d )(u d − u c ) ) + p d α c − E c γc −   Γ = =

α dk  (p k + γ d P∞,d ) + (α ds+ ρ ds+ )(u ds+ ) + psd α dk − E˜d γd −  d  α dk  s+ s+ ′ k k s+ s+  (p c − (α c ρ c )E (α d )(u d − u c ) + γ d P∞,d ) γd −    + (α ds+ ρ ds+ )(u ds+ ) + psd α dk − E˜d 








where k is the index for iteration. The updated states after each iteration are

Ω k+ = Ω k − [

∂ Γ − ] [Γ] ∂Ω

()

With

[

∂ Γ α ck ∂Γ = ] = ∂ Ω  ∂ p kc γ c − 

[

p k + γ c P∞,c  s+ s+ ′ k ∂ Γ ∂Γ  s = c ] = − (α c ρ c )E (α d )(u d − u d ) + p d k ∂ Ω  ∂ α c γc −  

[

α dk ∂Γ ∂ Γ = ] = ∂ Ω  ∂ p kc γ d − 

[

p k + γ d P∞,d α dk ∂ Γ ∂Γ s+ s+ ′′ k  s =− d ] = + (α c ρ c )E (α d )(u d − u c ) − p d k ∂ Ω  ∂ α c γd −  (γ d − )

At convergence, we have ps+ = p kc and α cs+ = α ck . c When the volume fraction of the disperse phase becomes vanishingly small (typically a value of − or − ) rather than solving for it we take it at the specified limiting value and assume that it comes to instantaneous equilibrium with the continuous phase. The error associated with this treatment is negligibly small unless we try to resolve detailed motions of very dilute clouds, which then requires special treatment (> Sect. .).



Numerical Testing in the ARMS Code

Computations were carried out with the computer code ARMS (All-Regime Multiphase Simulation) (Chang et al. ), and they were independently verified with the new code MuSiCARMS (Chang et al. a). The original ARMS was built on a public-domain platform (SAMRAI, structured adaptive mesh refinement infrastructure), while MuSiC-ARMS is our own specialised tool based on irregular grids embedded in a multi-level Cartesian mesh. This same platform supports also the MuSiC-SIM code for compressible multi-hydrodynamics at the DNS level (sharp mobile interfaces coupling Navier-Stokes solvers for each phase) based on the sharp interface method (Nourgaliev et al. ; Chang et al. b). An early task with this code will be to build the function E(α d ), as it appears in B- of Part I, from first principles. In all computations presented here, continuous and disperse phase thermodynamics were modeled by ideal and stiffened () gas equations of state, respectively. The function E(α d ) was fitted with a fourth-order polynomial so that it reduces smoothly to zero rather than be




allowed to take on negative values when α d exceeds the value that renders it zero: ⎧  ⎪ ⎪ α [ + Cα d + C  α d + C  α d + C  α d ] ⎪ ⎪ ⎪  d E(α d ) = ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 

if ( + Cα d ) ≥ ; () Otherwise.

All computations shown here were repeated with effective grid resolutions that spanned a factor of , typically  to , cells over a -m long computational domain. Unless otherwise noted the CFL number was set to .. The base value of the parameter C in () was taken as −.. Sample parametric calculations for values of  and −. are also shown. In these very basic series of tests we are concerned with the “bare” code: the inviscid constitutive treatment embodied in the EFM and computational performance of the numerical scheme to capture shocks and contact discontinuities sharply, stably, and without oscillations, even as numerical dissipation is severely reduced (grid refinement). The five series of computations and respective attributes of testing are as follows: . Uniformly-Translating Body-and-Fluid System. Ensuring that translation occurs without pressure disturbances, while maintaining any contact discontinuities present initially. . The Faucet Problem (Ransom ). Capturing the contact discontinuity of an abruptly initiated, at the inlet boundary of a computational domain, disperse stream that is subject to the acceleration of gravity. . Fitt’s Problem (). Capturing all wave forms and contacts in Riemann problems specified with initial discontinuities in Mach number and disperse phase volume fraction distributions. Test performance in relation to the hyperbolicity map of the EFM employed (Appendix B of Part I). . Shock Tube Problems. Capturing all wave forms and contacts in Riemann problems specified with initial discontinuities in pressure and disperse phase volume fraction distributions. Test performance in relation to the hyperbolicity map of the EFM employed (Appendix B of Part I). Test the role of drag to stabilize cases that are in the non-hyperbolic corridor. . Particle cloud dynamics in gaseous shocks. Capturing the acceleration and relaxation time of particles with and without drag. Compare to exact analytical results. Test performance in relation to the hyperbolicity map of the EFM employed (Appendix B of Part I). Test the role of drag to stabilize cases that are in the non-hyperbolic corridor. As detailed below this initial, and far from comprehensive, testing shows sufficient promise to motivate continuing efforts in D and D geometries, and eventually quantitative comparisons with experimental data. It must be emphasized, however, that the use of function E(α d ) in compressible flows (M = / ) is strictly tentative until the kinds of extensions noted above have been completed.

.

Uniformly Translating Body-and-Fluid System

The ARMS solver is applied over a rectangular domain filled mostly with gas (liquid fraction − ) and containing (a diffusely connected with its surroundings) circular liquid “blob” (gas








8

8

6

6

4

4

2

2

2

4

6

X (m)

8

2

4

6

8

X (m)

⊡ Figure  Snapshot of liquid “blob” motion in a uniformly convecting flow. Colors indicate disperse phase volume fractions. Left: t = .; Right: t = . s

fraction − ), all moving with the same velocity, which is maintained by a uniform/steady, diagonal (gas) inflow at the left and bottom boundaries, and a continuous outflow boundary condition at the other two boundaries of the domain. Obviously, there are no accelerations involved, and we expect that the solver will capture the interface without significant distortion due to motion, while the pressure remains smooth and uniform throughout within a very strict tolerance. The computation shown in > Fig.  is for a  ×  m domain, the flow velocity vector has components of  m/s in each direction, and the code was programmed to refine the grid around the boundary by three levels of AMR with a refinement ratio of . Thus we have an effective field resolution of × grids. The pressure remained uniform throughout the computation to within − Pa, and the shape of the phase boundary remained without significant additional smearing.

.

The Faucet Problem

The ARMS solver is applied to compute the evolution of volume fraction distributions of a uniformly supplied disperse-phase stream (ρ d =  k g/m  ) at the top boundary of the computational domain as it accelerates under gravity. For the ideal problem, with no phase interactions, there is an exact analytical solution, a snapshot of which is shown in > Fig. . Due to acceleration the disperse phase volume fraction decreases (that of the continuous phase increases) along the travel path. Also distinct is the sharp contact discontinuity formed by the sudden appearance of the inlet flow (at the “faucet”) upon initiation of the transient. The object


0.934

0.95

Time = 0.5 sec 200×1 400×1 800×1 1600×1 Analytical

0.932 0.93

0.94



0.928 0.926

0.93

5.5

6

6.5

αc

5

0.92

0.91

0.9 0

2

4

6

8

10

X (m)

⊡ Figure  ARMS performance on the Faucet problem

of the computation is to capture the front stably, sharply, and without oscillations. It is known that non-hyperbolic formulations fail this task, unless they are gridded coarsely enough, which causes excessive smearing, and severe distortion of the waveform. In > Fig.  we can see that with increasing resolution the computations converge to the exact discontinuous result, uniformly (around the front) and without oscillations.

.

Fitt’s Problem

The ARMS solver is applied to a set of Riemann problems with discontinuity in Mach number designed to attain various “positions” in the hyperbolicity diagram (Appendix B of Part I) as shown in > Fig. . Each case is also considered with a discontinuity in disperse phase volume fraction (marked by superscript D). The specification of the various calculations performed, in terms of the {α d , p, u, T}L and {α d , p, u, T}R employed, is given in > Table D.. The Mach number values are selected by specifying appropriate values of the gas temperatures (speeds of sound) as indicated in the table. As density of the disperse phase we use  kg/m . Fitt () originally used this test problem to demonstrate the failure of numerical computations with an “ill-posed” EFM formulation, and we are not aware of any subsequent elaborations with other models. All results obtained with ARMS are summarized in > Fig. D.. We can see that all cases are stable, even those that are found inside the non-hyperbolic corridor. As noted in Appendix






0 Non-hyperbolic FB

FD FC

FA

FF FE

–5

C



Hyperbolic

Hyperbolic

–10 Stable

–15

0

0.5

1

1.5

2

M

⊡ Figure  Locations of the various Fitt’s problem computational cases on the hyperbolicity map

B of Part I this is perhaps to be expected because of the “mild” non-hyperbolicity explained therein. Nevertheless, transonic conditions can become sensitive, as for example initially supersonic D cases that lead to transonic conditions due to wave reflections. Characteristic velocities for the various waves can be obtained as illustrated in > Fig. . The results are “paired” with those obtained from the hyperbolicity analysis (Appendix B of Part I) in > Table . Note that the nomenclature (on the various λs) in this table is not kept the same, and that a fourth eigenvalue is now present (u c ); this is due to inclusion of the energy equation in the computation. A couple of sample results illustrating grid convergence are shown in > Fig. . Finally, a number of parametrics on the value of C are shown in > Fig. , along with their positions on the hyperbolicity map in > Fig. . In addition to further supporting the conclusions reached above, these results also demonstrate that there is a considerable tolerance on the value of C.

.

Shock Tube Problems

The ARMS solver is applied to a series of Riemann problems with discontinuity in pressures and volume fractions of the disperse phase, whose density is taken as that of water. Two tests are with single phase, air or water, set at extremely high pressure ratio, and these are in excellent agreement with exact results. The rest of the cases involve two-phase conditions under varying pressure ratio so as to cover wide ranges on the hyperbolicity map as shown in > Fig. . They




t=12 ms

FAD

t=10 ms

t=10 ms

t=8 ms

t=8 ms

t=6 ms

t=6 ms

t=4 ms

pc

pc

FCD

t=4 ms

t=2 ms

t=2 ms

t=0 ms

t=0 ms

0

1

2

3

4

5

6

7

8

9

0

10

1

2

3

4

5

6

7

8

9

10

X (m)

X (m) FD D

t = 6 ms

FFD

t = 4 ms

t = 6 ms

t = 3 ms

t = 5 ms

pc

pc

t = 5 ms

t = 2 ms

t = 4 ms t = 3 ms

t =1 ms

t = 2 ms

t = 0 ms

t =1 ms t = 0 ms 0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

X (m)

4

5

6

7

8

9

10

X (m)

⊡ Figure  Propagation of characteristic waves in sample subsonic (top) and supersonic (bottom) cases. All these cases were run on ,  cells grid

are denoted as the S cases. All cases are specified in > Table D. in terms of the applicable {α d , p, u, T}L and {α d , p, u, T}R . As indicated in the table each case was run both with a uniformly distributed disperse phase as well as with a jump at the location of the diaphragm. To maximize the computational challenge all cases were run with zero drag. The results are summarized in Appendix, > Fig. D.. We can see that instabilities develop (seen better yet in Mach number plots as in > Fig. ), however they can be effectively controlled by a small amount of dissipation, as is the case for the Cloud Dynamics problems discussed below.

.

Particle Cloud Dynamics in Gaseous Shocks

The ARMS solver is applied to a shock tube configuration driven by single-phase gas, and various kinds of particle clouds (water density, particle diameter  mm) in the expansion






⊡ Table  Comparison of the characteristic wave speeds from ARMS with those from the rule of thumb found in Appendix B of Part I. Of the two numerical entries the one on the left corresponds to the eigenvalues according to the rule developed in Appendix B of Part I, and the one on the right is the value found in the ARMS computation. The later case the speed of sound is found from the local computed temperature Case

λ ; uc + ac

λ ; uc

λ ; ud

λ ; uc − ac

FDA

.; .

.; .

.; 

−.; −.

%

 %

 %

%

.; .

.; .

.; 

−.; −.

−%

 %

 %

−%

.; .

.; .

.; 

.; .

%

 %

− %

%

FDE

(subsonic)

(transonic)

(supersonic)

250×1 500×1 1000×1 2000×1

D

FA 0.6

0.58

0.56

0.54

0.7

Time = 9.0 ms

0.62

Time = 10.0 ms FCD

0.6

pc (× 105 Pa)

FDC

pc (× 105 Pa)



0.5 500×1 1000×1 2000×1 4000×1

0.4

0.3

4

5

X (m)

6

0.2

5

5.2

5.4

5.6

5.8

6

6.2

6.4

X (m)

⊡ Figure  Illustration of ARMS performance under grid refinement for Fitt’s problem

section as specified in > Table D.. To explore the effect of initial discontinuity each C case was run with a smooth (Gaussian) as well as with a top hat profile (subscripts S and T respectively). To explore the effect of drag to stabilize an unstable computation, each case was run with zero drag (subscript zero) as well as a constant drag coefficient of . (no subscript). However it should be noted that even one-tenth of this value is sufficient to stabilize these cases. Finally, to test against the exact result for a single particle accelerated in a shock, we have also case D (for dilute cloud) with particle density ρ d =  k g/m  . The positions of the various cases on the hyperbolicity map are indicated in > Fig. . The results are summarized in > Fig. D.a for the C cases and > Fig. D.b for the D case. A key




C = –3.1 C=0 C = –4.8

FDB

350

C = –3.1 C=0 C = –4.8

0.8

300

uc (m/s)

pc (× 105 Pa)

400

Time = 10.0 ms

1

0.6

250 200 150

0.4

100 x0 = 5.0

0.2 0

2

4

6

8

50

10

0

2

4

6

650 Time = 10.0 ms

10

0.8

C = –3.1 C=0 C = –4.8

FCD

600

C = –3.1 C=0 C = –4.8

550 500

uc (m/s)

pc (× 105 Pa)

1

8

X (m)

X (m)

0.6

450 400 350 300

0.4

250 200

0.2

150

x0 = 5.0 0

2

4

6

8

100

10

0

2

4

X (m)

8

10

600

1.2

Time = 8.0 ms

C = –3.1 C=0 C = –4.8

FDD

550

C = –3.1 C=0 C = –4.8

500

uc (m/s)

1

pc (× 105 Pa)

6

X (m)

0.8

0.6

450 400

0.4

350

x0 =3.0 0.2

300

0

2

4

6

8

10

0

2

4

X (m)

6

8

10

X (m)

1.2

700

Time = 6.0 ms

650

uc (m/s)

pc (× 105 Pa)

C = –3.1 C=0 C = –4.8

FFD

C = –3.1 C=0 C = –4.8

1

0.8

600

0.6 550

x0 = 2.0 0.4

0

2

4

6

X (m)

8

10

0

2

4

6

8

10

X (m)

⊡ Figure  Parametric study on C on selected Fitt’s cases. All these cases were run on ,  cells grid






2 FC

FA F B

FD

FE

FF

0

C

–2

–4

Non-hyperbo

–6

Hyperbolic 0

Stable

lic

–8

0.5

Hyperbolic

1

1.5

2

M

⊡ Figure  Locations of the various Fitt’s problem C-parametrics on the hyperbolicity map. All cases are stable on  cells grid

–2.5 Hyperbolic

Non-hyperbolic

C



S1

–3

Hype rboli

–3.5

S2

S3

S4

Stable

instability

Unstable

c

0

0.5

1

1.5

2

2.5

M

⊡ Figure  Ranges on the hyperbolicity map covered in each of the Shock tube problem computations



Multiphase Flows: Compressible Multi-Hydrodynamics 0.8

2

1.4

2 0.2

1.5

2

4

8

1 6

0.8

4

0.4 2

0.2 x0

0 6

8

0.5 10

0

2

4

X (m) Time = 5.0 ms M (500×1) M (2000×1) pc (500×1)

35 30 25 20

M

1.5

15 1

10 5

0.5

0

2

4

10

6

8

120 100 80

2

60

1.5

40

1

20 0 x0

0 0

10

140

2.5

0.5

–10 0

3

–5

x0

0

8

Time = 4.0 ms M (500×1) M (2000×1) pc (500×1)

SC4

3.5

M

2

4

Pc (× 105 Pa)

SC3

6

X (m) 40

2.5

0

–0.2

Pc (× 105 Pa)

0

10

0.6

1

x0

12

1.2

M

M

2.5 0.4

0

Time = 5.0 ms M (500×1) M (2000×1) pc (500×1)

SC2

1.6

3

Pc (× 105 Pa)

0.6

1.8

Pc (× 105 Pa)

3.5

Time = 7.0 ms M (500×1) M (2000×1) pc (500×1)

SD1

2

4

X (m)

–20 6

8

10

X (m)

⊡ Figure  Illustrative snapshots of the shock tube problem computations showing stable calculations and cases with incipient instability

result is the recovery time:

> (Fig. ) of the exact analytical

x(t) = u c t −

result for displacement as a function of

ρ d d d ρ c C d u c t ln ( + ) ρ c C d ρ d d d

()

Another key result is that normal drag cases are stable even though they encroach well into the non-hyperbolic zone. For those cases that are unstable the inception is around M ∼  a result already seen in the S cases above.



Conclusions and Outlook

In Part II, we have addressed computations in compressible multi-hydrodynamics with a heavy disperse phase (Piltch et al. ; Theofanous et al. ; Theofanous and Dinh ). Such flows are the counterpart of bubbly flows (found extensively in past work), and a necessary






–2.5 Hyperbolic

Non-hyperbolic

C

CS CS 0

–3

CT0 CT

rbol Hype

–3.5

Stable instability

Unstable

ic 0

0.5

1

1.5 M

2

2.5

⊡ Figure  Locations of the cloud dynamics problem computations on the hyperbolicity map 0.4 0.35 Computation

0.3

Mass center location (m)



Analytical

0.25 0.2 0.15 0.1 0.05 0 –0.05

0

1

2

3

4

Time (ms)

⊡ Figure  Recovery of exact analytical results for “single” particle displacement under a constant gas flow and drag coefficient (.)




complement in the understanding of disperse systems. The principal new ingredient is accessibility of high relative velocities. This is a consequence of large disperse-to-continuous density ratios, which yields strong inviscid interactions and associated inertia coupling effects. On the other hand, and in a broader context of applicability, we have also addressed high-fidelity capturing of shocks and contact discontinuities as pertinent to ab initio simulation of multiphase flow regimes and of steam explosion phenomena (Theofanous et al. a,b, ; Yuen and Theofanous ). We have shown that the EFM of Part I, with its essentially hyperbolic character as endowed by a closure that captures inviscid interactions, in combination with a numerical treatment that is based on the AUSM scheme, make available a framework for high-fidelity computation of shocks, expansion waves, and contact discontinuities in high-speed flows. In particular, by accounting for the fundamentally distinct physics between convective momentum transfer and pressure forces, we could deal with discretization of the terms that arise from inviscid interactions as well as of the nonconservative terms involving the volume fraction gradient. Inviscid computations applied to a series of highly sensitive test problems showed that stable solutions can be obtained with effectively unlimited grid resolution. Certain extreme cases, such as strong shock wave interacting with step change in disperse phase volume fraction can lead to mild instability, which however can be eliminated by a small amount of drag. The pursuit of this framework to practical simulations, for example, shock-induced fluidization of particulated solid or liquid masses, requires developments in three main directions of further constitutive treatment. One would be toward an extension of the function E(α d ) into the Mach number space; that is, the definition of function E(α d , M), while at the same time extending it to dense systems ( α d > .). The second direction would be toward accounting for mechanisms of direct particle to particle momentum transfers, such as pressure wave propagation via contacts in the fully packed regime, and collisions during the early stages of dispersal (dense regime). Some approach on this subject can be initially gained with reference to granular flows. The third direction, necessary for shock-induced fluidization of liquid masses, would require inclusion of the interfacial area transport, a modified version of () in Part I, supplemented by a constitutive treatment for interfacial breakup (and perhaps coalescence) appearing as a source term on the right-hand side. Alternatively, the numerical framework presented here could be adapted to accommodate multi-length scale particulate matter, each scale with own continuity and momentum equations, along with appropriate source/sink terms to account for breakup and coalescence phenomena. In all three directions well-instrumented experiments and direct numerical simulations will provide the building blocks for which the present foundation is beckoning.

Acknowledgment This work was supported by the Joint Science and Technology Office, Defense Threat Reduction Agency (JSTO/DTRA), and the National Ground Intelligence Center (NGIC) of the US Army (Dr. Richard Babarsky). Dr. C.-H. Chang (CRSS / UCSB) is the ARMS and MuSiC-ARMS codes lead developer.








Appendix D: Sample Computational Results with C.-H. Chang and S. Sushchikh D. Fitt’s Tests We compute the Riemann problems listed in > Table D.. The results in > Fig. D.a–c are cross-referenced to the codes assigned to each case on the table. Superscripts C/D refer to continuous/discontinuous initial volume fraction distributions (these can also be seen in the plots). Other specifications and discussion can be found in > Sect. ..

⊡ Table D. Specification of Fitt’s tests UL

UR

FA

pc = .× Pa ud = . ; uc = . m/s Td = Tc = . K ΔM = .

pc = .× Pa ud = . ; uc = . m/s Td = .; Tc = . K ΔM = .

FB

pc = .× Pa ud = . ; uc = . m/s Td = Tc = . K ΔM = .

pc = .× Pa ud = . ; uc = . m/s Td = .; Tc = . K ΔM = .

FC

pc = .× Pa ud = . ; uc = . m/s Td = Tc = . K ΔM = .

pc = .× Pa ud = . ; uc = . m/s Td = .; Tc =  K ΔM = .

FD

pc = .× Pa ud = . ; uc = . m/s Td = Tc = . K ΔM = .

pc = .× Pa ud = . ; uc = . m/s Td = .; Tc = . K ΔM = .

FE

pc = .× Pa ud = . ; uc = . m/s Td = Tc = . K ΔM = .

pc = .× Pa ud = . ; uc = . m/s Td = .; Tc = . K ΔM = .

FF

pc = .× Pa ud = . ; uc = . m/s Td = Tc = . K ΔM = .

pc = .× Pa ud = . ; uc = . m/s Td = .; Tc = . K ΔM = .






300

3

pc αd

0.8

ρc uc ud

C F A

250

0.4

2.8 2.6 2.4 2.2

0.3

0.6

0.2

uc , ud (m/s)

200

αd

pc (× 105 Pa)

1

2 1.8

150

1.6

ρc (kg/m3)

0.5 Time = 7.0 ms

1.4

100

1.2 0.4

0.1

1

50

0.8

x0= 5 .0 2

4

6

8

10

0

0.6 0

2

4

X (m)

6

8

10

X (m) 0.6 Time = 7.0 ms

1

FA

0.5 pc αd

2.4

200

0.6

uc , ud (m/s)

0.4

0.3

2.6

ud

αd

2.2 2

150

1.8 1.6

0.2

100

0.1

50

1.4 1.2

0.4

1 0.8

x0 =5.0 0.2

0

2

4

6

8

0 10

0

0

2

4

X (m)

6

8

10

0.5

0.8

0.4

250

0.6 0.2

2 1.8

150

1.6 1.4 1.2

0.1

1 50

0.8

x0 =5.0 0.2

0

2

4

6

0 10

8

0

0.6 0

2

4

6

8

10

X (m)

X (m) 0.5

0.4

c

u

250

0.6 0.2

200

1.4 1.2 1 50

2

4

6

8

2.4

1.6

0.1

0

2.6

1.8

0.8

x0 =5.0 0.2

2.8

2

150

100 0.4

d

3

2.2 0.3

uc , ud (m/s)

0.8

D FB

αd

pc (× 105 Pa)

pc αd

ρ c u

300

Time = 10.0 ms

1

2.6 2.4

200

100 0.4

3 2.8

2.2 0.3

uc , ud (m/s)

pc αd

ρc uc ud

C

FB

αd

pc (× 105 Pa)

300

Time = 10.0 ms

1

0.6

X (m)

ρc (kg/m3)

pc(× 105 Pa)

0.8

2.8

ρc uc

D

250

ρc (kg/m3)

0

0

0 10

0

0.6 0

2

X (m)

⊡ Figure D.a Computational results of Riemann problems defined in > Table 

4

6

X (m)

8

10

ρc (kg/m3)

0.2





Multiphase Flows: Compressible Multi-Hydrodynamics 0.5 400

αd 0.3

0.6 0.2

0.4

uc , ud (m/s)

0.8

2.6 2.4 2.2

300

αd 0.1

2

250

1.8 200

1.6

150

1.4

100

1.2 1

50 x0= 5.0 2

4

6

0.8 0

0 10

8

0

2

4

0.5

600

Time = 10.0 ms

550

10

2.5

450

αd

0.8

ρc uc ud

D

FC

500

0.4

pc

2

400

αd

0.3 0.6

0.2

uc , ud (m/s)

pc (× 105 Pa)

8

X (m)

X (m) 1

6

350 300

1.5

250 200

0.4

1

150 0.1

100

0

2

4

6

0.5

50

x0= 5 .0

0.2

0

0 10

8

0

2

4

X (m) 0.5

8

10

600

Time = 8.0 ms

2.4

550

pc

ρc uc ud

C FD

500

0.4

αd

450

0.8 0.3 0.6 0.2

uc , ud (m/s)

400

αd

pc (× 105 Pa)

6

X (m)

1

2 1.8 1.6

300 250

1.4 1.2

150

0.4

0.1

0

2

4

1

100

x0= 3.0 0.2

2.2

350

200

50 6

8

10

0

0

0.8 0

2

4

6

8

10

X (m)

X (m) 0.5

550

Time = 8.0 ms 1

ρc

500 pc

0.4

D

450

αd

uc

FD

ud

2.4 2.2 2

400 1.8

αd

0.3

0.6 0.2

0.4

0.1

2

4

350 1.6

300 250

1.4

200

1.2

150

1

100

0.8

50

x0= 3.0 0

uc , ud (m/s)

0.8

0.2

3 ρc (kg/m )

0

3 ρc (kg/m )

0.2

6

8

0 10

0.6

0 0

2

X (m)

⊡ Figure D.b Computational results of Riemann problems defined in > Table 

4

6

X (m)

8

10

3 ρc (kg/m )

pc (× 105 Pa)

pc

350

0.4

2.8

ρc uc ud

C

FC

3 ρc (kg/m )

Time = 8.0 ms

1

pc (× 105 Pa)






800

Time = 6.0 ms

1

0.4

2

ρc uc ud

C FE

700

1.8

pc αd

0.2

0.4

0.2

0 4

1.2

1

100

x0= 3.0 2

1.4

300 200

0.1

0

400

6

8

0

10

0

2

4

X (m)

8

10

0.5

1.2

0.8

1

2

700

Time = 6.0 ms

αd 0.6

0.2

0.4

0.1

uc , ud (m/s)

0.3

0.8

ρc uc ud

D FE

600

0.4

pc αd

pc (× 105 Pa)

6

X (m)

1.8

500

1.6

400

1.4

300

1.2

200

1

100

0.8

ρc (kg/m3)

0.6

1.6

500

ρc (kg/m3)

0.3

uc , ud (m/s)

0.8

αd

pc (× 105 Pa)

600

x0=2.0 0.2

0

2

4

6

8

10

0

0

0

2

4

X (m) 1.2

0.5

8

10

800

0.6

2.2 C

FF

Time = 6.0 ms 1

700

pc αd

2

0.4 600

1.8

0.2 0.4

500

1.6

ρc uc ud

400 300

1.4

3 ρc (kg/m )

0.3 0.6

uc , ud (m/s)

0.8

αd

pc (× 105 Pa)

6

X (m)

1.2 200 0.1

0.2

1 100

x0= 2.0 0

2

4

6

8

10

0

0

0.8 0

2

4

X (m)

8

10

X (m)

1.2

0.5

800

2.2 D

FF

Time = 7.0 ms 700 pc αd

0.8

2

0.4 600

αd

0.3 0.6

0.2 0.4

uc , ud (m/s)

1

pc (× 105 Pa)

6

1.8

500

1.6

ρc uc ud

400 300

1.4 1.2

200 0.1

0.2

0

0

2

1 100

x0= 2.0 4

6

8

0 10

0

0.8 0

2

X (m)

⊡ Figure D.c Computational results of Riemann problems defined in > Table 

4

6

X (m)

8

10

ρc (kg/m3)

0








D. Shock Tube Tests We solve the Riemann problems listed in > Table D.. The results in > Fig. D.a and b are cross-referenced to the codes assigned to each case on the table. Superscripts C/D refer to continuous/discontinuous volume fraction distributions (these can also be seen in the plots). The pure phase results (not shown) are in excellent agreement with exact solutions. Other specifications and discussion can be found in > Sect. ..

⊡ Table D.

Specification of the Shock Tube Tests (ε = . × − ) UL

UR

Air

pc = . ×  Pa αd = ε ud = uc = . m/s Td = Tc = . K

pc = . ×  Pa αd = ε ud = uc = . m/s Td = Tc = . K

Water

pc = . ×  Pa αd = . − ε ud = uc = . m/s Td = Tc = . K

pc = . ×  Pa αd = . − ε ud = uc = . m/s Td = Tc = . K

S

pc = . ×  Pa αd = . or . ud = uc = . m/s Td = Tc = . K

pc = . ×  Pa αd = . ud = uc = . m/s Td = Tc = . K

S

pc = . ×  Pa αd = . or . ud = uc = . m/s Td = Tc = . K


S

pc = . ×  Pa αd = . or . ud = uc = . m/s Td = Tc = . K


S

pc = . ×  Pa αd = . or . ud = uc = . m/s Td = Tc = . K







0.5 Time = 7.0 ms pc (500×1)

1.5

0.2

uc , ud (m/s)

0.3

αd

pc (× 105 Pa)

ud (500×1) ud (2000×1)

150

αc (500×1) αc (2000×1)

2

uc (2000×1)

0.4

pc (2000×1)

2.5

uc (500×1)

D

S1

200

3

100

50

1 0.1 0.5

0

x0 0

0 0

2

4

6

8

–50

10

0

2

4

X (m) 500

0.5

8

10

350

0.3

αd

6

uc (500×1) uc (2000×1) ud (500×1) ud (2000×1)

C

S2

400

0.4

4 0.2 2

uc , ud (m/s)

pc (500×1) pc (2000×1) αc (500×1) αc (2000×1)

8

pc (× 105 Pa)

450

Time = 5.0 ms

10

6

X (m)

300 250 200 150 100

0

0.1

50 0

–2 0

2

4

6

8

–50

0 10

0

2

4

X (m)

6

8

10

X (m) 0.6

30

Time = 5.0 ms

15

0.4

10

0.3

5 0.2

0 –5

S3

600

uc , ud (m/s)

pc (× 105 Pa)

20

uc (500×1) uc (2000×1) uc (500×1) uc (2000×1)

C

0.5

pc (500×1) pc (2000×1) αc (500×1) αc (2000×1)

αd

25

400

200

0.1

–10 0 –15 0

2

4

6

8

0 10

0

2

4

X (m) 1

120 Time = 5.0 ms pc (500×1) pc (2000×1) αc (500×1) αc (2000×1)

60

0.8

0.6 0.5

20

0.4

0

0.3

–20

0.2

–40

0.1 0

2

4

6

8

600

0.7

40

–60

8

10

uc (500×1) uc (2000×1) uc (500×1) uc (2000×1)

C

S4

700

10

uc , ud (m/s)

pc (× 105 Pa)

80

800

0.9

αd

100

6

X (m)

500 400 300 200 100 0

0 0

2

X (m)

⊡ Figure D. Computational results of Riemann problems defined in > Table 

4

6

X (m)

8

10








D. Shock-Induced Dispersal of Dilute Clouds in D We solve the Riemann problems listed in > Table D.. The results in > Fig. D.a and b are cross-referenced to the codes assigned to each case on the table. In all cases we have  mm particle diameter, and the initial cloud dimensions span  cm. Cases C were run with particle clouds initially stationary at position . m. Superscripts S/T stand for smooth (Gaussian)/TopHat volume fraction distributions, respectively. The drag coefficient was set to . in all cases except those noted by subscript  – for these drag was set to zero. Case D is for a dilute cloud to approximate single particle response (density  kg/m , diameter  mm). Other specifications and discussion can be found in > Sect. ..

⊡ Table D.

Specification of the Cloud Dynamics Tests (ε =  × − ) UL

UR

CS

pc = . ×  Pa αd = ε ud = uc = . m/s Tc = . K Td = . K

pc = . ×  Pa αd,max = . ud = uc = . m/s Tc = . K Td = . K

CS

pc = . ×  Pa αd = ε ud = uc = . m/s Tc = . K Td = . K


CT



CT



D

pc = . ×  Pa αd = ε uc = . m/s Tc = . K

pc = . ×  Pa αd,max = . ud = uc = . m/s Td = Tc = . K





Multiphase Flows: Compressible Multi-Hydrodynamics 12 Time = 7 ms pc (500×1) pc (2000×1) αd (500×1)

600

0.4

500

uc (500×1)

S

C0

uc (2000×1) ud (500×1) ud (2000×1)

α (2000×1)

0.3

6

αc

pc (× 105 Pa)

8

0.5

0.2

4 2

uc , ud (m/s)

10

400 300 200

0.1 100

0 0

0

–2 0

2

4

6

8

10

0

2

4

6

X (m) 12 Time = 7.0 ms pc (500×1) pc (2000×1) αc (500×1) αc (2000×1)

600

0.4

500

10

S

uc (500×1) uc (2000×1)

C

ud (500×1) ud (2000×1)

0.3

6

αd

pc (× 105 Pa)

8

0.5

0.2

4 2

uc , ud (m/s)

10

8

X (m)

400 300 200

0.1

100 0 0

0

–2 0

2

4

6

8

10

0

2

4

X (m) 2 Time = 7 ms pc (500×1) pc (2000×1) αd (500×1)

1.4

α (2000×1)

6

C0

500

1.6

10

400

1.2 1

4

αc

5

pc (× 10 Pa)

8

8

0.8 2

uc (500×1) uc (2000×1) ud (500×1) ud (2000×1)

T

600

1.8

uc , ud (m/s)

10

6

X (m)

300 200

0.6

0

100

0.4 0.2

–2

0

0 0

2

4

6

8

0

10

2

4

6

8

10

1.4

1 0.8

2 0.6 0

0.4

uc (500×1) uc (2000×1) ud (500×1) ud (2000×1)

C

500

1.6

1.2

4

T

600

1.8

αc

5

pc (× 10 Pa)

8

2

uc , ud (m/s)

Time = 7 ms pc (500×1) pc (2000×1) αd (500×1) α (2000×1)

10

6

X (m)

X (m)

400 300 200 100

0.2

–2

0 0

2

4

6

8

10

0 0

X (m)

⊡ Figure D.a Computational results of Riemann problems defined in > Table 

2

4

6

X (m)

8

10






D

t = 3.0 ms t = 2.5 ms t = 2.0 ms

pc

t = 1.5 ms t = 1.0 ms t = 0.0 ms

Initial cloud center 0

0.5

1

1.5

X (m) 200 150

t = 3.0 ms

100

t = 2.5 ms t = 2.0 ms

ud (m/s)

t = 3.5 ms

t = 1.5 ms t = 1.0 ms

0

10

20

30

X (cm)

D

ud (t = 1.0 ms) ud (t = 3.5 ms) ad (t = 1.0 ms) ad (t = 3.5 ms)

0.01 0.008 0.006

50 0.004

αd

D

ud (m/s)



0 0.002 –50 0 –100 –5

0

5

10

15

20

25

30

35

X (cm)

⊡ Figure D.b Computational results of Riemann problems defined in > Table 

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