A Robust and Efficient Finite Volume Method for

A Robust and Efficient Finite Volume Method for Compressible Viscous Two-Phase FlowsI Aditya K Pandare1 , Hong Luo2,∗ Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC, 27695, USA

Abstract A robust and efficient density-based finite volume method is developed for solving the six-equation single pressure system of two-phase flows at all speeds on hybrid unstructured grids. Unlike conventional approaches where an expensive exact Riemann solver is normally required for computing numerical fluxes at the two-phase interfaces in addition to AUSM-type fluxes for single-phase interfaces in order to maintain stability and robustness in cases involving interactions of strong pressure and void-fraction discontinuities, a volume-fraction coupling term for the AUSM+ -up fluxes is introduced in this work to impart the required robustness without the need of the exact Riemann solver. The resulting method is significantly less expensive in regions where otherwise the Riemann solver would be invoked. A transformation from conservative variables to primitive variables is presented and the primitive variables are then solved in the implicit method in order for the current finite volume method to be able to solve, effectively and efficiently, low Mach number flows in traditional multiphase applications, which otherwise is a great challenge for the standard density-based algorithms. A number of benchmark test cases are presented to I DOI:https://doi.org/10.1016/j.jcp.2018.05.018; Cite as: A. K. Pandare, H. Luo. A robust and efficient finite volume method for compressible inviscid and viscous two-phase flows. Journal of Computational Physics, 371 (2018) 67-91. ∗ Corresponding author Email address: [email protected] (Hong Luo) 1 Graduate Research Assistant 2 Professor

Preprint submitted to Journal of computational physics

June 2, 2018

assess the performance and robustness of the developed finite volume method for both inviscid and viscous two-phase flow problems. The numerical results indicate that the current density-based method provides an attractive and viable alternative to its pressure-based counterpart for compressible two-phase flows at all speeds. Keywords: Two-fluid model, Six equation, All-speed methods

1. Introduction Multiphase flows arise in a variety of engineering applications ranging from atomized fuel flow in internal combustion engines to nucleate boiling in water reactors. Computation of flows involving multiphase physics is thus a topic of 5

significant research efforts and multiple approaches exist to deal with these types of problems. The two types of methods used for multiphase flows are the interface tracking and interface capturing methods. Interface tracking methods, as the name suggests, employ actual tracking of the interface between two phases (viz. bubble or droplet surfaces) and smoothing the fluid properties across this

10

interface. The level-set, volume-of-fluid and front-tracking are some famous interface tracking methods. The Arbitrary Lagrangian-Eulerian method has also been used to capture multiphase interfaces accurately [1, 2]. Interface capturing methods, on the other hand, dynamically “capture” the interfaces; just like a standard finite-volume or discontinuous Galerkin method would capture shocks

15

and contact discontinuities without any special treatment. This means that each phase is treated as a separate continuum, and that there is no clear distinction between cells containing one phase or the other. Each cell contains a fraction of each phase, denoted by the volume fraction α. This approach is thus also termed as the “interpenetrating continua” approach, or the Effective

20

Field Model (EFM). Both these approaches have their benefits and drawbacks. It is obvious that the EFM is unable to compute individual interfaces between two phases like the interface tracking methods. This may be important in applications where the interface details like bubble-dynamics are the focus. But

2

the obvious drawback of interface tracking methods is that they are computa25

tionally prohibitive when applied to flows with large number of interfaces, such as bubbly flows through pipelines. Also, implementation of these methods can prove to be quite complicated in such situations, due to phenomena like bubblecoalescence or breakup and condensation. In situations where average-states of the multiphase flow are of interest and resolution of interfaces is not important,

30

the EFM is used. The EFM is derived from individual continuum equations for each phase, called the local instant formulation, and then applying an averaging procedure [3]. Assuming velocity and thermal non-equilibrium but pressure equilibrium leads to the 6-equation Wallis model [4] of two-fluid flows. Solving the 6-equation

35

model is a challenging problem. Legacy multiphase codes use such pressurebased methods and resort to numerical diffusion and Picard-type iterations to make the system stable. Many of these methods resort to adding dissipation by the means of staggered grids and other techniques to obtain stable numerical results [5, 6]. These methods are usually termed as pressure-based algorithms since

40

the working variable is pressure. They fall under the general class of operatorsplitting type of methods. Pressure-based methods are better conditioned for low M a flows, since at these speeds, the density is practically constant, and small errors in density (due to spatial or temporal discretizations) translate to large errors in pressure. Also, The pressure-based methods eliminate the speed

45

of sound by reformulating the continuity and momentum equations by assuming that the velocity field is solenoidal (∇ · v = 0). This results in a CFL criterion based solely on the flow velocity, independent of the acoustic speed, and larger time-step sizes leading to faster convergence. These pressure-based algorithms have been used successfully to obtain very accurate results for low speed flows,

50

both for single-phase flows [7, 8] and two-phase flows [9, 6]. However, since these methods use operator-splitting to decouple the momentum and energy equations, compressibility can not be rigorously treated. The incompressibility assumption is deeply ingrained in their formulation. These algorithms result in significant inaccuracies and instabilities when high-temperature phenomena 3

55

such as boiling and other problems involving large density gradients need to be simulated. An alternative, potentially more promising approach is to use the densitybased (fully compressible) method for multiphase flows. However, there are a number of issues in solving the 6-equation model by use of a density-based

60

method: • Extremely low-speed flows (M a < 0.001): Many multiphase flows in engineering applications are very low Mach number flows. This is known to cause difficulty in convergence. Also, the time-steps satisfying stability conditions (acoustic CFL) are too small to be feasible. The Riemann

65

fluxes used in conventional compressible density-based methods tend to be highly diffusive in the M a < 0.01 flow regime. • High contribution of source terms: Interfacial force terms such as drag, interphasic mass transfer, etc. have such large contribution to the residual vector that the stable time-steps are further lowered.

70

• Non-hyperbolic nature of the standard (6-equation) two-fluid model: The two-fluid model has two possibly complex eigenvalues in its standard form. Naive discretizations employing density-based (hyperbolic) methods fail due to this. • Non-conservative spatial and temporal derivatives: If discretized incor-

75

rectly, these can prevent preservation of stationary contact discontinuities. One way to resolve low M a convergence issue is to solve the two-fluid system in a fully-implicit way using a primitive variable transformation matrix. Nourgaliev and co-workers [10, 11] have used a [p, v, T ]-formulation to solve near-incompressible flows using fully compressible density-based solver. Us-

80

ing pressure as a primitive variable makes the implicit density-based system better conditioned. Treating the source terms due to interfacial forces and phase-changes implicitly also relaxes the time-step restrictions, leading to faster convergence times. 4

The 6-equation model is non-hyperbolic, and cannot be directly solved by 85

a conventional density-based method. A detailed eigenvalue analysis of the Wallis model can be found in [12]. Even so, the non-hyperbolic 6-equation model has been used to simulate inviscid two-fluid flow first by Toumi [13] and then by Chang, Liou and co-workers [14, 15, 16, 17]. By adding an interface pressure term suggested by Stuhmiller [18], they prove that the system can be

90

rendered hyperbolic [19]. They use the stratified-flow model to discretize the non-conservative terms in the system and an all-speed two-phase extension of the AUSM flux, called the AUSM+ -up, in addition to an exact Riemann solver to discretize the inviscid fluxes. The exact Riemann solver was only used in situations where the jump in the volume-fraction at the cell-faces is substantial.

95

It has also been proven in [14] that this choice of flux, coupled with the stratified flow discretization of the pressure flux and other non-conservative terms, satisfies the pressure non-disturbance condition (also known as Abgrall’s criterion [20, 21] or well-balancedness [22]), which is essential for the solver to maintain a stationary contact discontinuity. However, the iterative procedure involved in

100

the exact Riemann solver makes this method quite expensive. Kitamura and Nonomura [23] replaced the exact Riemann solver with a two-fluid HLLC flux. This resulted in a relatively inexpensive flux function. However, knowledge of the complete eigenstructure of the system is a must for such a flux function. This might not be easily available, when more complex terms such as virtual

105

mass are included in the governing equations. Niu [24, 25] has used a primitive variable solver to efficiently solve the Riemann problem, albeit using a Newton iteration in the procedure. Other approaches to induce hyperbolicity into the Wallis model include work by Dinh et al. [26] which involves an iterative regularization procedure. Vazquez-Gonzalez et al. [27] study the necessity of adding the

110

previously mentioned interface pressure term and compare results of an elliptic solver with the hyperbolic methods mentioned above. However, viscous flows have not been solved using this approach. To the best of the authors’ knowledge, no prior research focusses on this type of density-based solution method for the compressible viscous two-fluid model. 5

115

In an attempt to obviate the need of the complete eigenstructure, it is desirable to use a numerical flux function such as AUSM. However, the regular AUSM+ -up flux suffers from negative pressures in the regions of high pressure shocks interacting with material interfaces. This has been shown by studies by Kitamura [16, 23]. The solution proposed by Chang and Liou [14] was to use

120

an exact Riemann solver, as mentioned before. The exact Riemann solver requires large number of Newton iterations to get a correct middle-zone pressure p∗ in the regions of question, thus making it expensive. A flux scheme which is robust enough in these regions, yet not invoking an iterative procedure, is presented here. This involves a modification to the baseline AUSM+ -up fluxes

125

via a volume-fraction coupling term in the mass-flux, resulting in what is called the AUSM+ -upf flux in this work. This is similar to the Lax-Friedrich type dissipation employed by Houim and Oran [28] in the context of granular flows. Further, viscous flows, which are common in practical multiphase scenarios, are also solved using this method. The objective of this work thus, is to develop

130

a density-based method for the solution of the viscous two-fluid single pressure model which can be used as a baseline to develop a solver for more complex multiphase scenarios. The target applications for this work involve flows that require interphasic coupling using the drag force. Interphasic mass transfer effects are neglected.

135

Another approach to close the incomplete EFM two-fluid system is by including an additional equation for volume fraction. This additional equation leads to a 7-equation two-pressure system, which is hyperbolic [29]. The BaerNunziato model has been extensively used for modeling granular two-phase flow. This model was used by Saurel et al. [30, 21, 31], where the importance of a

140

well-balanced discretization of non-conservative products was highlighted. It was established from this work that well-balancedness is necessary in preservation of a contact wave in a uniform pressure and velocity field. This work used an infinite pressure and velocity relaxation as special cases to a more general 7-equation model, which allows non-equilibrium of phasic pressures, velocities

145

and internal energies. In a following work [32], this assumption was relaxed and 6

true non-equilibrium two-phase flows were solved using the discrete equation method. The rest of the article is organized as follows: the governing equations of the two-fluid model and closure laws are discussed in the next section. This is 150

followed by a description of the spatial and temporal discretization and other numerics. The results obtained using the proposed method are presented next. First, inviscid cases are presented to validate the discretization of the two-fluid model as proposed by Chang and Liou [14]. Here, the importance of the additional volume-fraction coupling term in robustness is discussed, using tests that

155

involve strong shock and material interface interactions. Next, viscous cases are presented and compared with boilEulerFOAM [6] (an open-source pressurebased multiphase solver) to ascertain the correctness of the results obtained using the proposed solution method. This is followed by concluding remarks.

2. Governing Equations 160

2.1. The two-fluid model The two-fluid model uses the interpenetrating continua approach to model two-phase flows. This model requires an averaging procedure to filter-out the local instantaneous fluctuations very similar to the Reynolds’ averaging in turbulence. As mentioned earlier, interphasic mass-transfer terms have not been

165

considered in this work. The resulting two-fluid model given by Ishii [3] and concisely by Staedtke [12], using k as the index for the two fluids, is as follows:

∂Fkj ∂Gkj ∂Uk + = + Pint k + Mk + Sk ∂t ∂xj ∂xj

7

(1)

where, 

αk ρk



    Uk = αk ρk uki    αk ρk Ek     0 αk ρk ukj         Fkj = αk ρk uki ukj  + αk pδij      0 αk ρk ukj Hk   0     Gkj =   αk τkij   αk ukl τklj + αk qkj   0    int ∂αk  Pint = k = 1, 2.   p k  k ∂xi  ∂αk −pint k ∂t

(2)

(3)

(4)

(5)

Mk are interface forces such as drag and virtual mass. Sk represent source terms, for example due to phase transitions and body forces. The viscous stress tensor and heat flux vector are given as,   ∂ui ∂uj 2 ∂ul τij = µ + δij , − µ ∂xj ∂xi 3 ∂xl

qj =

µCp ∂T , P r ∂xj

(6)

where the fluid-index k has been dropped for convenience. Note that the fluid properties (viz. µ, P r , Cp ) are fluid specific. These properties are specified for each phase and kept constant in the scope of this work. 170

In 1D, the system (1) has 11 unknowns (αk , ρk , uk , Ek , pint k , p with k = 1, 2) and a total of 9 equations with 6 PDEs, 2 equations of state (EoS ) and the constraint on volume fractions, 2 X

αk = 1.

(7)

k=1

These equations constitute the Wallis two-fluid model or the 6-equation singlepressure model of two-phase flows. The EoS required to close this system are 175

given in the next subsection. Another condition that the interfacial pressures 8

should cancel each other if no other stresses such as surface tension are considered at the interface gives, int pint ≡ pint . g = pl

(8)

pint is explicitly given as a function of the other unknowns. Here, we use the relation given by Stuhmiller [18], pint = p − σ

αg αl ρg ρl 2 u αg ρl + αl ρg r

(9)

where ur = |ul − ug |. This form of the interface pressure term was used and validated first by Bestion [33]. Although it was originally introduced in the context of incompressible flows, it has been used successfully for compressible multiphase simulations [14, 15, 16]. Is should be noted here that this term is only a way to partially introduce hyperbolicity into the system[15], and that its physical significance is debated in the two-phase community, as mentioned by Chang and Liou [14]. To ensure a well behaved numerical method, it has been previously observed that the above interface pressure be kept within limits using: pint = min(pint , 0.01p). 180

(10)

This provides the additional 3 equations and the system of equations is closed. The effect of addition of the interface pressure term has been analysed in detail by Chang et al. [19]. They show that there is a limiting value of σ above which the system (1) is hyperbolic. Here, a value of σ = 2 is used for all the problems.

185

2.2. Equations of state The stiffened-gas equations of state (SG-EoS ) are used for each fluid in this work. Using the SG-EoS, the pressure, temperature and speed of sound are

9

given respectively as: p + Pck ρu2 + k + Pck γ −1 2 k  γk (p + Pck ) Tk = γk − 1 ρk Cpk s p + Pc k ak = γk . ρk

ρEk =

(11) (12) (13)

Ideal gas values are used for the gaseous phase: γg = 1.4 Pcg = 0.0 P a Cpg = 1004.5 J/(kg · K). whereas the following properties are used for the liquid phase: γl = 2.8 Pcl = 8.5 × 108 P a Cpl = 4186 J/(kg · K). 2.3. Drag force model The drag-force interfacial momentum transfer term is modeled as, M1 = −M2 =

3 ρ2 α1 Cd |Ur |Ur , 4 db

(14)

where the relative velocity is, Ur = U2 − U1 .

(15)

Note that subscript 2 represents the dispersed phase and subscript 1 represents the continuous phase. The drag force is only used in the viscous test cases in this work. The drag coefficient is set as a constant Cd = 2.0. The bubble/droplet 190

diameter is denoted by db . A value of db = 1.0 mm is used for the tests in this article.

10

2.4. Virtual mass force model The presence of virtual mass force improves the stability of the numerical scheme in regions of high accelerations, for flows with a high density ratio[34]. This force is modeled as, Mvm = −Mvm = Cvm ρ1 α1 α2 1 2



D(u2 ) D(u1 ) − Dt Dt

 ,

(16)

with the same notation for the subscripts as the drag force. The substantial derivatives are treated part-implicitly, such that they improve the condition 195

number of the system. In other words, the positive contribution to the diagonal entries of the system matrix are treated implicitly, and the rest, explicitly. The virtual mass force is used only in the viscous problems in this work. The virtual mass coefficient is set as Cvm = 0.5. It has been reported that inclusion of the virtual mass force leads to a change

200

in the eigenvalues of the system [35]. The interplay of the interface pressure term in Eq. (9) and the virtual mass force and its effect on eigenvalues of the system and thus its hyperbolicity is, however, out of the scope of this work.

3. Spatial discretization The two-fluid system (1) is discretized using a second-order finite volume method. On integrating over each element e, and applying the divergence theorem on the flux terms, the discrete system becomes, Z Z Z
∂Uk dΩ + (Fk · n − Gk · n) dΓ = Pint k + Mk + Sk dΩ. Ωe ∂t Γe Ωe

(17)

Here, the cells are denoted by Ωe and the cell-faces by Γe . A least-squares 205

reconstruction procedure is used on the primitive variables (given later in Eq. 52) to obtain second order. A vertex-based limiter proposed by Kuzmin [36] is used to suppress the spurious oscillations in the flow field, when discontinuities are expected. The discretization of each term in this system is now individually discussed.

11

210

3.1. Inviscid fluxes Inviscid fluxes in the stratified-flow model can be discretized using two types of numerical fluxes as described by Kitamura et al. [16]: 1. AUSM-family standalone: The all-speed variant AUSM+ -up developed by Liou [37] and extended to the stratified-flow two-fluid model in [15] is employed in this type. A single flux-function is used to computed the flux

215

at the cell-interfaces. 2. Hybrid AUSM+Riemann (Godunov) solver: Flux at the cell-interface is split into fluxes between like phases (l-l, g-g) and unlike phases (l-g, gl ). Fluxes between like phases are computed using the AUSM+ -up scheme and those between unlike phases are computed using the Godunov method

220

[14]. This approach is expensive since the Godunov method uses iterations to accurately predict the fluxes. These two types of flux schemes have been studied in detail by Kitamura et al. [16] where it has been noted that a hybrid scheme is necessary for situa225

tions where a strong pressure discontinuity interacts with a void-fraction discontinuity. A high pressure-ratio water-air shocktube, a shock/water-column interaction and a shock/air-bubble interaction have been used to illustrate this. However, in this work, a modification to the AUSM+ -up flux is used, which makes it possible to solve the above mentioned problems. The new scheme in-

230

volves an additional coupling between the mass-flux and the volume-fraction of the dispersed phase. The AUSM+ -up fluxes, with this additional coupling term is referred to as the AUSM+ -upf; where the f stands for the volume-fraction coupling. These modifications impart robustness in the regions of strong shockinterface interactions, thus making the use of exact Riemann solvers unneces-

235

sary. Since this is the case, the flux can be written in a unified manner on the entire cell-interface, and the like-unlike phase-wise splitting becomes unnecessary. Similarly, the stratified-flow assumption can also be abandoned due to this.

12

The AUSM+ -up flux developed by Liou [37] specifically for all-speed application is a wise choice for two-fluid problems. In the AUSM-type of flux methods, the fluxes are written as, Fk,L/R = m ˙ k,1/2 ψ k,1/2 + αk,L/R pk,1/2 n1/2 .

(18)

Please note here that although the pressure flux pk,1/2 contributes differently to the two phases k = 1, 2, the pressure of the two phases p is equal. ψ k,1/2 = (1, u, H)Tk,1/2 is upwinded in the standard way,   ψ k,L if m ˙ k,1/2 > 0 ψ k,1/2 =  ψ otherwise. k,R

(19)

Note that the pressure flux contribution to the left and right elements is different due to the difference in the volume fractions at the face: αk,L 6= αk,R . The mass flux and the Mach number of phase-k is given as,   αk,L ρk,L if Mk,1/2 > 0, m ˙ k,1/2 = Mk,1/2 ac  αk,R ρk,R otherwise, − Mk,1/2 = M+ (4) (Mk,L ) + M(4) (Mk,R ) + Mk,p ,

(20)

(21)

± where the split Mach numbers M(m) are,

1 (M ± |M |), 2 1 2 M± (2) (M ) = 4 (M ± 1) ,   M± (M ) (1) ± M(4) (M ) =  M± (M )(1 ∓ 16βM∓ (M )) (2) (2) M± (1) (M ) =

(22) (23) if |M | ≥ 1, (24) otherwise,

the Mach numbers are defined as, Mk,L/R =

uk,L/R . ac

and the common speed of sound given by Chang and Liou [14] is used:   1 α1 α2 α2 α1 + . + = ac ρ1 ρ2 ρ1 a21 ρ2 a22 13

(25)

(26)

A value of β = 1/8 is used. The pressure diffusion term Mk,p introduced to treat low Mach number flows is, Mk,p = − 2

Mk =

Kp pR − pL 2 , max(1 − M k , 0) fa,k ρk,1/2 a2c

u2k,L + u2k,R . 2a2c

(27) (28)

The subscript ‘1/2’ has been used for definitions of a and ρ to represent averages 240

of the left and right states. This term is especially important for the two-fluid system where the stiffened gas equation is used. The pressure flux is given as, + − pk,1/2 = P(5) (Mk,L )pL + P(5) (Mk,R )pR + pk,u ,

where the split Mach numbers for pressure are,    1 M± if |M | ≥ 1, (1) M ± P(5) (M ) =   1 M± [(±2 − M ) ∓ 16θk M M∓ ] otherwise, (2) (2) M

(29)

(30)

and, 3 2 (−4 + 5fa,k ), 16

(31)

fa,k (Mo,k ) = Mo,k (2 − Mo,k ),

(32)

θk =

2

2 Mo,k = min(1, max(M k , M∞ )).

(33)

The velocity (momentum) diffusion pk,u is then defined as,
+ − pk,u = −Ku P(5) (Mk,L )P(5) (Mk,R ) · ρk,1/2 ac (uk,R − uk,L ).

(34)

This choice of fluxes yielded a stable scheme for most of the cases presented. However, for some extreme cases, such as the high pressure-ratio waterair shocktube, shock-watercolumn and shock-bubble interaction and the very 245

low Mach number channel flows, it was observed that a small amount of dissipation in the dispersed phase was necessary to obtain stability in regions of high relative velocities.

14

Note that the two-fluid model with equal phase velocities and pressure, known as the homogeneous two-phase model, is hyperbolic in nature. The Wal250

lis model, which doesn’t assume this however, is non-hyperbolic in its original form [12]. It is surmised that when the relative velocities increase to high values, the two-fluid model tends more to the non-hyperbolic nature and the hyperbolic correction (9) proves to be insufficient to ensure real eigenvalues. The fact that this correction is insufficient to completely restore hyperbolicity in all physically

255

admissible states, has also been noted in previous work by H´erard et al. [38, 39]. Note here that in spite of adding the hyperbolic correction term, the explicit eigen-structure (and hence the exact acoustic speed) for this system is unknown. There is a possibility that, in assuming that the system acoustic speed as a function of the acoustic speeds of the two individual phases alone, a possible effect of relative velocities on the acoustic properties of the system is being neglected. If this is indeed the case, a dissipation term proportional to the relative velocity of the two-phases, would stabilize the model. Importance of using the correct speed of sound in the flux scheme has been highlighted by Liou and Edwards [40]. The dissipation needs to be in the form of an additional coupling between the mass-flux and the volume-fraction. This approach has been utilized by Houim and Oran [28] in case of granular two-fluid flows. A similar form of the dissipation term has been utilized in the dispersed phase in this work. The dispersed phase mass flux is given as, m ˙ 2,1/2 = M2,1/2 ac α2 ρ2 − D2,f ,

(35)

where Df is the volume-fraction coupling term, Df =

1 λr max(αL , αR ) (αR ρR − αL ρL ), 2 fa αcrit

(36)

where the phase subscripts are omitted, and the maximum normal relative velocity is, λr = max(ur,L , ur,R ),

(37)

ur = Ur · n.

(38)

15

Clearly, the relative velocity λr is taken into consideration via this coupling term. Note that although this term derives its true form from the Lax-Friedrichs flux, unlike the latter, it is well-balanced in nature. This means that it satisfies Abgrall’s criterion [21], also known as the pressure non-disturbance condition 260

[14]. Another way to stabilize this model is increasing the drag coefficient, resulting in an absolute velocity relaxation. This way, although found to be effective, might yield unphysical results and this assumption is not valid for applications targeted here. Hence we resort to the volume-fraction coupling for stability purposes. The AUSM+ flux which includes this coupling is referred to

265

as the AUSM+ -upf flux henceforth. It is worth mentioning here, that Df given in Eq. (36) has a division by fa,k . Thus, one should be careful so as to not let fa,k go to zero under any circumstance. Usually, this is done by providing a cut-off M a while calculating fa,k . Additionally, it has been observed [23] that the AUSM+ -up fluxes lead to negative pressure-fluxes in regions where strong-shocks interact with material discontinuities. The cause of this negative pressure-flux in Eq. (29) in spite of positive pL and pR is the all-speed pressure-flux correction pk,u from Eq. (34). This pressure-velocity coupling term becomes too large due to the large value of ac and causes negative pk,1/2 . Here, it is replaced by  
+ − p˜k,u = −Ku 1 − P(5) (Mk,L )P(5) (Mk,R ) · ρk,1/2 λr (uk,R − uk,L ).

(39)

This modification ensures the positivity of pressure-flux when the left and right 270

side pressures pL and pR are positive. The effects of this additional term can be seen for high pressure ratio shocks as encountered in the water-air shocktube and the shock-bubble interaction problems. These modifications are inspired from the rationale in Liou’s [37] and Edwards’ [41, 42] works. These problems cannot be solved without the aforementioned dissipative terms accounted for, as

275

also reported by Kitamura et al. [16]. Various other ways have also been used to solve this issue. Chang and Liou [14] use an exact Riemann solution in the region |α1,L − α1,R |. This model although robust, turns out to be quite expensive since a Newton iteration is required to solve for the pressure in this region. Also, the 16

full eigenstructure of the system needs to be known for this method. Kitamura et 280

al. [23] use an HLLC flux in this region to replace the exact Riemann solver used by Chang and Liou. The HLLC flux, albeit more economical than the exact Riemann solver, still requires knowledge of the eigenstructure of the system. The modifications proposed here do not require this, and could potentially be used for other types of equations of state. A detailed discussion of the effects of

285

the proposed modifications will be provided in the numerical results section. 3.2. Viscous fluxes The viscous fluxes require a unique value of gradients of velocity components and temperature at the cell-faces. These gradients are computed using the modified-gradient approach as, 

uR − uL ∇u|Γ = ∇u|Γ − ∇u|Γ · r Γ − |r R − r L |

 rΓ ,

(40)

where the average operator is, ∇u|Γ =

1 (∇u|L + ∇u|R ), 2

(41)

and rΓ is the vector pointing from one cell-centroid to the other, given as, rΓ =

rR − rL . |r R − r L |

(42)

3.3. Non-conservative spatial derivative The term Pint is comprised of non-conservative first derivatives in space and time. This necessitates the use of so-called “well-balanced” discretiza290

tions. Extensive studies of well-balanced discretizations of non-conservative PDEs have been done, starting from Pares’ work [22], leading to Roe-type [43] and Osher-type [44] schemes for these equations. A numerical method for the two-fluid equations specifically was also proposed based on this concept [45]. This collection of work clearly states that the well-balancedness is an essen-

295

tial property to be considered while designing a numerical method for solving non-conservative PDEs, to maintain global conservation. Abgrall, in one of his

17

early works [20], also focusses on this property in the context of multi-species (or multi-component) equations. Chang and Liou [14] suggest using the stratified-flow model to discretize 300

the nonconservative term. This assumption of stratified-flow is also used while splitting the cell-faces into parts of like and unlike phases. However, since the proposed flux modifications do not require splitting the flux and this distinction is no longer required, the stratified-flow assumption need not be used. The discretization of the nonconservative term can be written as, Z   pint ∇αk dΩ = pint ∇αk Ωe ,

(43)

e

Ωe

where, the overbar indicates cell-average values. This clearly shows that the 305

non-conservative term has non-unique values at the element faces, one from each side of the face. This type of treatment ensures the well-balanced nature of the discretization. A derivation of this discretization, and its equivalence to the “stratified-flow” model discretization, are included in the appendix. The non-conservative time derivative term is combined with the time-derivative

310

of the unknowns which modifies U. This will be discussed in the following section pertaining to time integration. Interface momentum transfer and other source terms Mk and Sk are treated as volume-averaged source terms.

4. Time integration Before discretizing the time derivative term

∂U ∂t ,

the non-conservative time

derivative term in Pint has to be combined into the vector of unknowns U. This results in a modified vector of unknowns for fluid k,   0    ˆ k = Uk +  U  0 .   pint αk

(44)

This results in the final form of the semi-discrete form of the two-fluid system: V

ˆ ∂U = R, ∂t 18

(45)

where V is the cell-volume. The discretized right-hand side vector R and the ˆ are, modified unknown vector U 



α1 ρ1       α1 ρ1 u1i     α1 ρ1 E1 + pint α1  ˆ = , U     α2 ρ2       α2 ρ2 u2i   α2 ρ2 E2 + pint α2

R = −F + G + Pˆ int + S,

(46)

where, 

Z S=

(M + S)dΩ,

Pˆ int

Ωe

0



R    1   Ω pint ∂α ∂xi dΩ  e     0  . =    0 R    2  Ω pint ∂α  dΩ ∂xi  e  0

(47)

The system of equations (45) is integrated in time to obtain a time-marching 315

scheme. In this work, the explicit 3-stage third order TVD Runge-Kutta (RK3) scheme [46] and implicit backward Euler (BDF1) scheme, have been used to integrate the system (45) in time. Depending on which one is used, the algorithm becomes an explicit or implicit one. The explicit TVDRK3 is straight-forward

320

to apply to the two-fluid system, and implementation details can be found in [16]. However, the time-step sizes which can be used by an explicit algorithm are extremely low due to strict CFL restrictions. For time-independent flows, the steady-state is achieved after running the algorithm to a very long final flow-time. Typically for the two-fluid system with very large acoustic speeds,

325

this leads to ∆t’s of the order 10−6 − 10−8 . This makes the explicit algorithms prohibitive for steady-state flows. A well-designed implicit method allows the solver to step-over the acoustic speeds, resulting in orders of magnitude increase in ∆t. Thus, implicit methods are used in simulating steady-state flows. 19

4.1. Implicit Newton-Krylov method The backward Euler method is used to discretize the time derivatives for the implicit method. This method is 1st order accurate in time, but the order of accuracy is of little importance since it is used to solve steady-state flows. The backward Euler method applied to (45) at time-level (n + 1) gives, V

ˆ n+1 − U ˆn U = Rn+1 . ∆t

(48)

This is a system of nonlinear equations. The nonlinear residual vector is represented by R∗ , Rn+1 =V ∗

ˆ n+1 − U ˆn U − Rn+1 . ∆t

(49)

A linearizing procedure following Luo et al. [47] using Newton method is then applied. At each step of the Newton method, the following linear system of equations is solved ∂Rk∗ ˆ k ∆U = −Rk∗ ˆk ∂U

(50)

using a Krylov subspace method. The solution for the (k + 1)th Newton step is updated as, ˆ k+1 = U ˆ k + ∆U ˆ k. U

(51)

Flows at very low Mach numbers (M a < 0.01) are practically incompressible since the density is almost constant. At such minimal compressibilities, small errors in densities translate to very large errors in pressure. Thus, working on primitive variables [p, v, T ] [10, 11] rather than the conservative variables from Eq. (46) leads to a better conditioned (less stiff) system for the linear solver to handle. Defining a vector of primitive variables, h V = T1

u1i

T2

u2i

p

and denoting the Jacobian of transformation by

α1 ˆ ∂U ∂V ,

ˆk ∂Rk∗ ∂ U ∆Vk = −Rk∗ . ˆ k ∂Vk ∂U 20

iT

,

(52)

Eq. (46) transforms to, (53)

Substituting the definition of Rk∗ gives, ! ˆk ˆk −U ˆn V ∂U ∂Rk U − ∆Vk = Rk − V , k k ∆t ∂V ∂V ∆t

(54)

which is the linear system of equations to be solved by a Krylov method, instead of Eq. (50). The update to the (k + 1) Newton step is via the primitive variables as, Vk+1 = Vk + ∆Vk .

(55)

Another advantage of using primitive variables is that the Jacobian involved in Eq. (54) is with respect to primitive variables. This makes calculations of the analytical Jacobian much simpler, as will be explained in the next subsection. The transformation Jacobian right-hand side vector

∂R ∂V

ˆ ∂U ∂V

is obtained using the EoS. The Jacobian of the

can be very complicated to evaluate. An approximate

approach along the lines suggested by Luo et al. [48] is used in this work. In this approach, the inviscid flux function is approximated by a simplified flux function to simplify the computation of the left-hand side Jacobian matrix. The AUSM flux is approximated by the LF flux for this purpose: F= 330

1 (F(UL ) + F(UR ) − λmax (UR − UL )) . 2

(56)

As a consequence of this assumption, the right and left-hand sides of Eq. (54) become inconsistent. Thus, quadratic convergence of the Newton method can no longer be achieved. Although the number of time-steps required to reach steadystate may increase due to this, the cost-per-timestep is significantly reduced due to the simplicity in calculating the left-hand side matrix, thus reducing the

335

overall computational cost to achieve convergence. Another approximation in computing the Jacobian is that only a first-order representation of the numerical fluxes is considered. This limits the stencil of the fluxes to the von Neumann neighborhood of the cell, and the number of non-zero entries in each row of the Jacobian to the number of faces of the cell,

340

leading to a sparse Jacobian matrix. The contribution from the element i to the flux across the face Γij would be in the i-th column, whereas that from 21

the element j would go into the j-th column of the i-th and j-th rows of the Jacobian. Given a certain consistent numbering of elements, this would enable the usage of a LU D-type storage of the Jacobian. Further details can be found 345

in [48]. An important consideration here is the fact that the discretization of the inviscid flux terms and the limiting function are non-differential. This is bound to affect the convergence rates of the Newton-Krylov solver used. However, it was noticed that the use of Lax-Friedrichs flux approximation for computing the

350

Jacobian mitigated this effect. Further investigation of this matter is necessary, although outside the scope of this work. 4.2. Decoding of pressure and volume fractions ˆ The explicit TVDRK3 method works with the conservative variables U. ˆ n+1 is obtained, it is necessary to obtain Once the new vector of unknowns U the pressure and volume fractions at time-step n + 1. A decoding procedure highlighted by Liou and coworkers [14, 16], is used for this purpose. Imposing the constraint on the volume fraction (7) and the two EoS, the quadratic equation for pn+1 is obtained, p2 − Bp − C = 0,

(57)

with the positive root, p=

 p 1 B + B 2 + 4C 2

(58)

and the volume fraction is given by, αk =

Aˆk , p+a ˆk

(59)

where, ˆ |2 |U 2,k ˆ 3,k − Aˆk = (γk − 1) U ˆ 2U1,k

22

! ,

(60)

a ˆk = γk Pck + (γk − 1)pint , 2 X

(61)

(Aˆk − a ˆk ),

(62)

C=a ˆ1 Aˆ2 + a ˆ2 Aˆ1 − a ˆ1 a ˆ2 .

(63)

B=

k=1

The numerical errors in computation of p and αk can be very large, considering the large values of Pck . A Newton iteration procedure is used to reduce these errors, by solving (60) for both the phases simultaneously: (p + a ˆg ) αg − Aˆg = 0,

(64)

(p + a ˆl ) αl − Aˆl = 0.

(65)

Typically, a few iterations are enough to drive the pressure error below 10−6 . For general EOS, the same procedure can be followed by appropriately modifying 355

the definitions of Aˆk and a ˆk . The nonlinear equation for pressure will not necessarily be a quadratic equation in the general case. A Newton iteration can be used to solve this for pressure. Note that this decoding is completely avoided when working with primitive variables. Another advantage of using the primitive variables becomes apparent

360

here. Eq. (54) involves the Jacobian with respect to the primitive variables, ∂R ∂V

as opposed to

∂R ˆ. ∂U

Considering the above pressure decoding procedure, the

derivative of the pressure flux with respect to the conserved variables, a formidable task, if not impossible. On the other hand,

∂p ∂V

∂p ˆ ∂U

is

is trivial, since

pressure is a primitive variable itself. Thus, the calculation of the analytical 365

Jacobian is significantly simplified when primitive variables are considered. 4.3. Treatment for vanishing phase: blending A matter of concern while solving Eq. (45) is that of phase disappearance. In this situation, the numerical errors in the solution get amplified due to the division by a very small volume fraction. Although the volume of the corresponding fluid and thus its contribution to the flow are negligible, the calculation procedure might become unstable due to unrealistic values of the flow variables, 23

leading to divergence. The idea of the blending function suggested by Paill`ere et al. [49] is employed to suppress these numerical errors. It is assumed that when a fluid nears phase-disappearance, the fluids reach equilibrium immediately by mixing their states. After the time-integration and decoding procedure, if min ≤ αk ≤ max , then the velocity and temperature fields of phase−k are blended using, uk |blended = G(ψk )uk + (1 − G(ψk ))uk0 .

(66)

The function G(ψ), ψ ∈ [0, 1], is a cubic polynomial interpolant with G(0) = 0, G(1) = 1 and G0 (0) = G0 (1) = 0. The function, G(ψ) = −ψ 2 (2ψ − 3)

(67)

is used here, where ψ is the normalized volume fraction, ψ=

α − min . max − min

(68)

The parameters  define the range of volume fractions within which blending is employed. They are preset as, min = 10−1 

(69)

max = 103 ,

(70)

where  is problem specific, but usually is set at 10−7 .

5. Results The method described in the previous sections has been validated by several 370

one and two-dimensional test cases. The results of these tests are presented in this section. First, several inviscid tests from references [14, 16] are carried out to validate the proposed discretization of the inviscid two-fluid equations. This is followed by a few viscous test cases, which is the focus of this article. Comparisons with boilEulerFOAM are made to ascertain the correctness of the

375

results obtained using the proposed method. All the unsteady problems use the 24

TVDRK3, whereas the steady-state problems use the implicit Euler method for faster convergence. It should be noted that the authors were unable to find previous work with laminar two-phase flow results to compare with. Due to this, the test cases available to validate the method for these kind of flows 380

are limited. Future work involving inclusion of turbulence models will enable further validation of the method. 5.1. Inviscid tests These tests validate the proposed discretization of the two-fluid single pressure model for inviscid flows. It consists of problems from the above mentioned

385

references which test various properties of the proposed discretization. The 3stage Runge-Kutta method is used to integrate in time due to the unsteady nature of these problems. Please note that the drag and virtual-mass forces are not used for this set of problems. 5.1.1. Moving contact discontinuity problem This test case is used to verify that the solver satisfies the pressure nondisturbance condition. The following initial conditions are used: (p, αg , uk , Tk )L = (105 Pa, 1 − , 100m/s, 300.0K) (p, αg , uk , Tk )R = (105 Pa, , 100m/s, 300.0K)  = 1.0 × 10−7 k = 1, 2

390

where the L and R states are to the left and right of x = 0.5 respectively. Fig. 1 shows the void-fraction and pressure obtained after running up to t = 0.003 using a ∆t = 10−6 on 200 cells. It can be seen that the contact discontinuity is transported to the expected location and the uniform pressure is kept undisturbed.

25

101000 1 100500 Pressure

Void-fraction

0.8 0.6 0.4 0.2

100000

99500

0 99000 0

0.2

0.4 0.6 x-coordinate

0.8

1

0

0.2

0.4 0.6 x-coordinate

0.8

1

Figure 1: Void fraction (left) and Pressure (right) for the moving contact discontinuity

5.1.2. Air/water shock-tube problem This problem tests the capability of the method to capture shocks. It is a one-dimensional test case with the following initial conditions: (p, αg , uk , Tk )L = (109 Pa, 1 − , 0m/s, 308.15K) (p, αg , uk , Tk )R = (105 Pa, ,

0m/s, 308.15K)

 = 1.0 × 10−7 k = 1, 2. A mesh with 500 elements is used. The results at t = 0.2 × 10−3 using a ∆t = 10−7 are shown in Figs. 2 and 3. Results compare well with references [14, 16].

1

1x109

0.8

8x108 Pressure

Void-fraction

395

0.6 0.4 0.2

6x108 4x108 2x108

0

0 0

0.2

0.4 0.6 x-coordinate

0.8

1

0

0.2

0.4 0.6 x-coordinate

Figure 2: Void fraction (left) and Pressure (right) for the air-water shocktube

26

0.8

1

250

400 380

200

360 340 Tavg

Uavg

150 100 50

320 300 280 260

0

240

-50

220 0

0.2

0.4 0.6 x-coordinate

0.8

1

0

0.2

0.4 0.6 x-coordinate

0.8

1

Figure 3: Average velocity (left) and temperature (right) for the air-water shocktube

5x108 1 4x108

Pressure

Void-fraction

0.8 0.6 0.4

3x108 2x10

8

0.2

1x108

0

0 0.5

0.525

0.55 x-coordinate

0.575

0.6

0.8

0.825

0.85 x-coordinate

0.875

Figure 4: Zoomed view of the void fraction (left) and pressure (right) for the air-water shocktube showing material interface and shock capturing respectively.

Fig. 4 shows the cell-averaged values of void fraction and pressures zoomed 400

over particular regions to show the material interface and shockwave respectively. It can be seen that using the proposed method, the interface and shock are captured in 6 cells and 5 cells each respectively.

27

0.9

5.1.3. Water/air shock-tube problem This is a one-dimensional test case with the following initial conditions: (p, αg , uk , Tk )L = (1.0 × 107 Pa, ,

0m/s, 308.15K)

(p, αg , uk , Tk )R = (5.0 × 106 Pa, 1 − , 0m/s, 308.15K)  = 1.0 × 10−7 k = 1, 2. A mesh with 500 elements is used. The results at t = 0.2 × 10−3 using a ∆t = 10−7 are shown in Figs. 5 and 6. The results show a good match with the references. 1.1x107 1

1x107 9x106 Pressure

Void-fraction

0.8 0.6 0.4

8x106 7x106 6x106

0.2

5x106 0 4x106 0

0.2

0.4 0.6 x-coordinate

0.8

1

0

0.2

0.4 0.6 x-coordinate

0.8

1

0.8

1

Figure 5: Void fraction (left) and Pressure (right) for the water-air shocktube

3.5

309.5

3

309

2.5 308.5 Tavg

2 Uavg

405

1.5 1

308 307.5

0.5 307

0 -0.5

306.5 0

0.2

0.4 0.6 x-coordinate

0.8

1

0

0.2

0.4 0.6 x-coordinate

Figure 6: Average velocity (left) and temperature (right) for the water-air shocktube

28

5.075x106 1 0.8 Pressure

Void-fraction

5.05x106 0.6 0.4

5.025x106

0.2 0 0.45

5x106 0.475

0.5 x-coordinate

0.525

0.55

0.5

0.525

0.55 0.575 0.6 x-coordinate

0.625

Figure 7: Zoomed view of the void fraction (left) and pressure (right) for the water-air shocktube showing material interface and shock capturing respectively.

Fig. 7 shows the cell-averaged values of void fraction and pressures zoomed over particular regions to show the material interface and shockwave respectively. It can be seen that using the proposed method, the interface and shock 410

are captured in 2 cells and 12 cells each respectively. This magnitude of smearing is expected since the shock is very weak. The pressure ratio in this problem is 2. A water/air shocktube with a pressure ratio of 103 is also considered in the references. The initial conditions are, (p, αg , uk , Tk )L = (1.0 × 108 Pa, ,

0m/s, 308.15K)

(p, αg , uk , Tk )R = (1.0 × 105 Pa, 1 − , 0m/s, 308.15K)  = 1.0 × 10−7 k = 1, 2. As mentioned by Kitamura, this high pressure-ratio water/air shocktube cannot be solved with the AUSM+ up fluxes. However, the additional dissipation term of the AUSM+ -upf flux allows this high pressure-ratio shocktube to be solved. 415

Using the same mesh and time-steps as before, the results at t = 0.2 × 10−3 are shown in Figs. 8 and 9. Low-frequency oscillations are observed in the pressure profile.

29

0.65

1.0e+09 1 1.0e+08

Pressure

Void-fraction

0.8 0.6 0.4

1.0e+07

1.0e+06

0.2 1.0e+05 0 1.0e+04 0

0.2

0.4 0.6 x-coordinate

0.8

1

0

0.2

0.4 0.6 x-coordinate

0.8

1

Figure 8: Void fraction (left) and Pressure (right) for the high PR water-air shocktube

70

340 335 330

50

325

40

320 Tavg

Uavg

60

30

315 310

20

305

10

300 295

0

290

-10

285 0

0.2

0.4 0.6 x-coordinate

0.8

1

0

0.2

0.4 0.6 x-coordinate

0.8

Figure 9: Average velocity (left) and temperature (right) for the high PR water-air shocktube

A grid convergence study was performed to check whether these oscillations vanish as the grid is refined. It was observed that as finer meshes were used, 420

these oscillations do vanish (see Fig. 10), indicating that the method is indeed grid convergent. Further, phasic mass and energy conservation was monitored throughout the computation time. The errors in gas and liquid mass and energy conservations are shown with time steps in Figs. 11 and 12. It can be seen that mass conservation is fairly satisfied, but energy conservation is not. This is due

425

to the vanishing phase treatment applied to maintain robustness throughout the computation. This observation has also been made by Nonomura and Kitamura [50]. Nevertheless, the errors in energy conservation are about 1%, which is acceptable.

30

1

1.0e+09 500 cells 1000 cells

Pressure

1.0e+08

2000 cells

1.0e+07

1.0e+06

1.0e+05

1.0e+04 0

0.2

0.4 0.6 x-coordinate

0.8

1

Figure 10: Pressure profiles showing grid-convergence for the high PR water-air shocktube

1.0e-05

1.0e-01 Gas mass Gas total energy

1.0e-02

1.0e-03 1.0e-07 1.0e-04

|E(t)-E(0)|/E(0)

|M(t)-M(0)|/M(0)

1.0e-06

1.0e-08 1.0e-05

1.0e-09 0

500

1.0e-06 1000 1500 2000 2500 3000 3500 4000 Time step

Figure 11: Gas mass and energy conservation errors for the high PR water-air shocktube

31

1.0e-05

1.0e-05 Liquid mass Liquid total energy

1.0e-06

1.0e-06

1.0e-08

1.0e-07

1.0e-09 1.0e-08

1.0e-10

|E(t)-E(0)|/E(0)

|M(t)-M(0)|/M(0)

1.0e-07

1.0e-11 1.0e-09 1.0e-12 1.0e-13 0

500

1.0e-10 1000 1500 2000 2500 3000 3500 4000 Time step

Figure 12: Liquid mass and energy conservation errors for the high PR water-air shocktube

5.1.4. Ransom’s faucet This one-dimensional unsteady test case models a jet of air and water confined in a channel. Gravity force (9.8 m/s2 ) is then applied which accelerates the water column and a void wave is generated. The following initial conditions [14] are used: (p, αg , ug , ul , Tk ) = (105 Pa, 0.2, 0m/s, 10m/s, 300.00K) The same inlet conditions are applied. The channel is 12m long. The exact solution for the void fraction is given as,  2  1 − (1−α √ g (0,0))ul (0,0) , if x < gt2 + ul (0, 0) · t (ul (0,0))+2gx x αg (x, t) =  α (0, 0), otherwise g 430

A grid of 500 cells and a time-step of ∆t = 2.5 × 10−5 s are used. The void fractions at t = 0.1s, 0.3s and 0.5s are shown in Fig. 13. It can be seen that the propagation of the void wave is computed accurately.

32

0.5 t = 0.1 s t = 0.3 s t = 0.5 s Exact

0.45

Void-fraction

0.4 0.35 0.3 0.25 0.2 0.15 0

2

4

6 8 x-coordinate

10

12

Figure 13: Void fraction for the Faucet problem at t = 0.1s, 0.3s and 0.5s with exact solution at 0.5s

0.5 nx = 500 nx = 1000 nx = 10000 Exact

0.45

Void-fraction

0.4 0.35 0.3 0.25 0.2 0.15 0

2

4 6 x-coordinate

8

10

Figure 14: Void fraction for the Faucet problem using 500, 1000 and 10000 cells at 0.5s

33

0.5 nx = 500 nx = 1000 nx = 10000 Exact

Void-fraction

0.45

0.4

0.35

0.3 4

4.5

5

5.5 6 6.5 x-coordinate

7

7.5

8

Figure 15: Enlarged view of void fraction for the Faucet problem using 500, 1000 and 10000 cells at 0.5s

This benchmark problem has been extensively studied in literature, and is well known, particularly for potential issues caused when very fine grids are 435

used. Stability is compromised with grid refinement since the dissipation turns out to be insufficient to suppress instabilities. A detailed study and comparison of various two-fluid models is conducted by Zou et al. [51]. They have noted that the two-pressure models suffer from these instabilities from flow times t > 0.3 s, whereas the single-pressure models start showing unstable behaviour after

440

t > 0.4 s for meshes with 2000 elements. For finer meshes, instabilities cause the solver to diverge at earlier times. A grid refinement study was performed to test this. It can be seen from Figs. 14 and 15 that the proposed discretization is able to obtain a stable solution even on a grid with 10000 cells. However, convergence to the exact solution is slow even with such a fine grid. This effect

445

is due to the interface pressure term, which adds dissipation to enhance the stability of the method, somewhat slowing down the mesh-convergence. It is also expected that addition of this term changes the PDE system altogether and the given analytical solution is no more true for this changed problem. Thus, the mesh refinement leads to convergence to a different (but close) solution. 34

450

5.1.5. Shock/water-column interaction A shock in air impacting a water-column (or a 2D droplet) is simulated in this problem. The droplet has radius r = 3.2mm and is centered at the origin. Since this problem is symmetric about the X-axis, flow over only the top half of the droplet is simulated, and the symmetry condition is imposed at the bottom boundary. The droplet is resolved using 200 × 100 isotropic cells in the domain [−5mm, 5mm] × [0mm, 5mm], so that the grid spacing is ∆xmin = ∆ymin = 0.05mm in this region. The rest of the grid is such that 450 × 150 total cells are used in the overall domain [−15mm, 20mm] × [0mm, 15mm]. The initial conditions are [15]: (p, αg , uk , Tk )L = (2.35438 × 105 Pa, , 225.86m/s, 381.85K) for x ≤ 4mm (p, αg , uk , Tk )R = (1.0 × 105 Pa,

, 0m/s, 293.15K) for x > 4mm, except for

x2 + y 2 < (3.2mm)2 , where αg = 1 −   = 1.0 × 10−5 k = 1, 2. These conditions result in a shock moving at M a = 1.47, which impacts the droplet at t ≈ 1.5µs. A time-step of ∆t = 1.25 × 10−9 s is used. A smooth transition of the volume fraction at the interface of the droplet is necessary. A width of ±2∆xmin is used for the transition region. The curve used to fit the volume fraction in this region is the same as the blending function used for the vanishing phase. αg |blended = G(ψ2 ) + (1 − G(ψ2 ))(1 − ), G(ψ2 ) = −ψ22 (2ψ2 − 3), p x2 + y 2 − (r − 2∆xmin ) ψ2 = , 4∆xmin

r − 2∆xmin ≤

p

x2 + y 2 ≤ r + 2∆xmin .

The left boundary is set as the inlet and the right boundary is the outlet. The top boundary is a slip-wall. These boundaries are sufficiently far from the 455

droplet, such that their influence can be neglected.

35

The AUSM+ -up fluxes (without the exact Riemann augmentation) diverge after ≈ 6.25µs of flow-time. However, the AUSM+ -upf fluxes can solve this problem up to a flow-time of ≈ 15µs. Thus, only the results using the AUSM+ upf are reported. Pressure and numerical Schlieren contours at t = 6.25µs, 460

t = 10µs and t = 18.75µs are shown in Figs. 16, 17 and 18 respectively. Pressure contours are plotted between 105 Pa and 4 × 105 Pa. The numerical Schlieren function is computed as (1 + αl2 ) log(1 + |∇ρ|) and the range used for plotting its contours is 4 to 20. Some noise in the pressure profiles is observed, and is warranted to the choice of the all-speed flux. Some tuning in the dissipation

465

might be necessary to get rid of this phenomenon.

36

Figure 16: Pressure and numerical Schlieren contours for the shock/water-column interaction at t = 6.25µs

Figure 17: Pressure and numerical Schlieren contours for the shock/water-column interaction at t = 10µs

Figure 18: Pressure and numerical Schlieren contours for the shock/water-column interaction at t = 18.75µs

37

5.1.6. Shock/air-bubble interaction The same grid-setup used for the shock-droplet interaction problem is used to simulate a shock in water impacting an air-bubble. The initial conditions are [14]: (p, αg , uk , Tk )L = (1.6 × 109 Pa, , 661.81m/s, 595.14K) for x ≤ 4mm (p, αg , uk , Tk )R = (1.01325 × 105 Pa, , 0m/s, 292.98K) for x > 4mm, except for x2 + y 2 < (3.2mm)2 , where αg = 1 −   = 1.0 × 10−3 k = 1, 2. These conditions result in a shock moving at M a = 1.51, which impacts the droplet at t ≈ 0.3µs. A time-step of ∆t = 3.125 × 10−10 s is used. First a structured mesh with 380,000 elements is used to solve this problem, with some 470

refinement near the bubble location. Again for this problem, the AUSM+ -up fluxes (without the exact Riemann augmentation) diverge as soon as the shock hits the bubble. However, the AUSM+ -upf fluxes with the coupling terms in Eqs. (36) and (39) can solve this problem even after the bubble collapses onto itself completely. Results using

475

the AUSM+ -upf are reported here. Pressure and numerical Schlieren contours at t = 1.25µs, t = 2.5µs, t = 3.75µs, t = 4.75µs and t = 6µs are shown in Figs. 19, 20, 21, 22 and 23 respectively. Pressure contours are plotted between 108 Pa and 2 × 109 Pa. The numerical Schlieren function, computed as log(1 + |∇ρ|), is plotted between 8 to 14. The contours show good agreement with the references

480

[14, 16, 23, 52]. This test problem is now solved on an unstructured mesh with 150,000 elements, a section of which is shown in Fig. 25 with a ∆xmin = 0.05 mm. As a comparison, the structured mesh is shown in Fig. 24 for which the ∆xmin = .025 mm. The mesh is refined near the bubble location so that complicated physical

485

phenomena can be resolved. The numerical Schlieren contours at 2.4 µs and 3.825 µs are shown in Figs. 26 and 27 as compared with the results from the 38

structured mesh. On both these meshes the water-shock can be clearly seen transmitted into the air-bubble in the left picture. In the picture on the right, this air-shock is seen partially reflected into the bubble. The rest of it is trans490

mitted out of the bubble. It is noteworthy here that the bubble has collapsed onto itself, without any divergence of the solver. This robustness is the effect of the modification of the fluxes. There are some differences in the results from the structured and unstructured meshes, which can be attributed to the fact that the former mesh is much finer than the latter one. However, the point to

495

be made here is that even with a mesh size less that half of the structured mesh, the unstructured mesh can capture the finer phenomena easily. Comparisons can be made with interface-tracking methods such as level-set, since this test problem has been originally deviced for such methods. The results reported here agree very well with interface-tracking results from Nourgaliev et al. [53]

500

and Lauer et al. [52]. To emphasize on the importance of this test problem, Fig. 28 shows the contours of relative velocity magnitude at 3.6 µs. It can be seen here that the relative velocities reach very high magnitudes, at the time of bubble-collapse. However, the computation does not diverge even at such high relative velocities.

505

It should be noted here that no form of velocity relaxation is used (viz. drag force), thus allowing the relative velocity between the two phases to reach high magnitudes freely. As mentioned before, it is in these situations where the two-fluid single pressure model faces stability issues. It can be seen that the AUSM+ -upf flux can help retain stability in these zones too.

510

This ends the validation of the inviscid two-fluid method proposed by Liou, Chang et al. The next few test cases attempt at validating the viscous flow solver proposed in this article.

39

Figure 19: Pressure and numerical Schlieren contours for the shock/bubble interaction at t = 1.25µs

Figure 20: Pressure and numerical Schlieren contours for the shock/bubble interaction at t = 2.5µs

Figure 21: Pressure and numerical Schlieren contours for the shock/bubble interaction at t = 3.75µs

40

Figure 22: Pressure and numerical Schlieren contours for the shock/bubble interaction at t = 4.75µs

Figure 23: Pressure and numerical Schlieren contours for the shock/bubble interaction at t = 6µs

Figure 24: Unstructured mesh used for shock-bubble interaction problem

41

Figure 25: Unstructured mesh used for shock-bubble interaction problem

Figure 26: Numerical Schlieren at 2.4µs: Water-shock transmitted into bubble. Left: Unstructured mesh. Right: Structured mesh.

Figure 27: Numerical Schlieren at 3.825 µs: Bubble collapses onto itself and the air-shock is transmitted out. Left: Unstructured mesh. Right: Structured mesh.

42

Figure 28: Relative velocity magnitude (top) and numerical Schlieren (bottom) contours for the shock-bubble interaction problem at the time of bubble collapse (3.6 µs)

5.2. Viscous tests These problems test the viscous capabilities of the presented method. The 515

problems considered here are steady-state and the implicit solver is used for faster convergence. It was observed that the AUSM+ -up fluxes diverge as large gradients in volume fraction start developing near stagnation points. This has originally led to the belief that a volume-fraction coupling is necessary in the first place. In the following sections, only the results using the AUSM+ -upf are

520

reported. Note that the drag and virtual-mass forces are used for all the viscous problems reported hereafter. 5.2.1. Bubbly flow in a channel A bubbly flow comprising of water and 2% air is simulated through a channel of height 25.4 mm and length 4 m at standard atmospheric pressure (1.013 ×

525

105 Pa). A mesh with 20 cells in the height-wise direction and 200 cells in the length-wise direction is used. The Reynolds number of 100 based on water properties and channel height is used. The other problem details are given in Table 1. Subsonic inlet and outlet boundary conditions based on the above Reynolds 43

Property

Water

Air

T

298

298

Pr

6.15305

0.7

cp

4186.0

1004.5

µ

8.9313 × 10−4

1.78978 × 10−5

γ

2.8

1.4

Pc

8.5 × 108

0.0

Table 1: Bubbly channel flow problem details

530

number and pressure are used. The velocity profile obtained at the outlet boundary after steady-state convergence is shown in Fig. 29. 0.006 Exact boilEulerFOAM Density-based

0.005

x-Velocity

0.004

0.003

0.002

0.001

0 -0.015

-0.01

-0.005

0 0.005 x-coordinate

0.01

0.015

Figure 29: Velocity profiles compared with BoilEulerFOAM results and “exact” solution

The results are compared with those obtained by BoilEulerFOAM [6] and the exact solution for a single-phase flow of water in a channel at Re = 100. The results are expected to be close to the exact solution for the single-phase 535

flow because of the low void-fraction and absence of any phase-change terms. This is found to be true for both, the current density-based method and the pressure-based BoilEulerFOAM solver. 44

The important observation from this test is that the current density-based method is able to solve flows at very low Mach numbers (current Mach number 540

for continuous phase M a1 = 2.3349×10−4 ). It was observed that this is possible only when the dissipation given in Eq. (35) is used and the 6 equation system is solved for primitive variables. The all-speed flux modification alone is not sufficient to be able to solve viscous flows at such low Mach numbers. 5.2.2. Droplet flow in a channel

545

A droplet flow comprising of 2% water and 98% air is simulated through a channel of same dimensions as the one used for the previous test. The Reynolds number of 5.89 based on air properties and channel height is used. Fluid properties from Table 1 are used. Subsonic inlet and outlet boundary conditions based on the above Reynolds number and pressure are used. The velocity profile obtained at the outlet boundary after steady-state convergence is shown in Fig. 30. 0.006 Exact boilEulerFOAM Density-based

0.005

0.004 x-Velocity

550

0.003

0.002

0.001

0 -0.015

-0.01

-0.005

0 0.005 x-coordinate

0.01

0.015

Figure 30: Velocity profiles compared with BoilEulerFOAM results and “exact” solution

The results are compared with those obtained by BoilEulerFOAM [6] and the exact solution for a single-phase flow of air in a channel at Re = 5.89. The results show good agreement with boilEulerFOAM and the exact single-phase 45

555

velocity profiles. 5.2.3. Flow over a flat plate The flow of 2% water dispersed in air over a flat plate is considered in this test. The flow is simulated at Re = 66178 and M a = 0.10115, which is based on the continuous phase (gas) properties. The same fluid properties given in Table 1 are used here. The left and right boundaries as set as subsonic inlet and subsonic outlet respectively. The top boundary is set as a freestream. The bottom boundary is a symmetry wall for x ∈ [−0.5, 0] and an adiabatic noslip wall for x ∈ [0, 1]. Standard atmospheric pressure is used for the subsonic outlet boundary condition. The x− and y−velocity profiles obtained from the proposed method and boilEulerFOAM at various stations along the plate are compared in Figs. 31, 32 and 33 respectively. The velocities and y−coordinates have been nondimensionalized by the freestream mixture Reynolds number Rem as, p Rem /x, u u ˜= , U∞ v p v˜ = xRem , U∞ ρm U∞ Rem = , µm η=y

(71) (72) (73) (74)

where the subscript m denotes mixture properties, weighted according to volume fractions.

46

1 DensityBased boilEulerFOAM

1 0.8 0.6

0.6

Vy

Vx

0.8

0.4

0.4

0.2

0.2 DensityBased boilEulerFOAM 0

0 0

1

2

3

4 5 6 eta-coordinate

7

8

9

0

2

4 6 eta-coordinate

8

10

8

10

8

10

Figure 31: X-component (left) and Y-component (right) of the velocity at x = 0.25

1 DensityBased boilEulerFOAM

1 0.8 0.6

0.6

Vy

Vx

0.8

0.4

0.4

0.2

0.2 DensityBased boilEulerFOAM 0

0 0

1

2

3

4 5 6 eta-coordinate

7

8

9

0

2

4 6 eta-coordinate

Figure 32: X-component (left) and Y-component (right) of the velocity at x = 0.5

1 DensityBased boilEulerFOAM

1 0.8 0.6

0.6

Vy

Vx

0.8

0.4

0.4

0.2

0.2 DensityBased boilEulerFOAM 0

0 0

1

2

3

4 5 6 eta-coordinate

7

8

9

0

2

4 6 eta-coordinate

Figure 33: X-component (left) and Y-component (right) of the velocity at x = 1.0

Note that there is no ‘exact’ solution for this test case to compare these re560

sults with. The x-component velocity profiles obtained from the current method 47

show good agreement with boilEulerFOAM. However, the y-component velocities show some differences, although they follow the same general trend as those obtained by boilEulerFOAM.

6. Conclusion 565

A density-based finite volume method has been developed for compressible two-phase flows based on the effective-field model at all speeds. The regular AUSM+ -up suffers from negative pressure in regions where strong shocks interact with material interfaces. A modification to the AUSM+ -up fluxes has been proposed to address this issue and shown to make the method effective and

570

robust for such situations. The modification does not involve any iterative procedure needed by the exact Riemann solver, thus keeping the computational cost low. A primitive variable formulation has been presented in the implicit method in order to be able to effectively solve low Mach number two-phase viscous flows. A number of numerical experiments for a wide range of Mach numbers

575

have been conducted to assess the performance and robustness of the developed finite volume method for both inviscid and viscous two-phase flow problems. The numerical results demonstrate the great potential of this density-based finite volume method as a true “all-speed” two-phase solution algorithm, which on one hand is able to compute high-speed flows like a conventional density-

580

based method; and on the other, is able to solve low-speed viscous flows with an accuracy comparable to pressure-based methods. Further work will focus on refinement of the method by testing more realistic equations of state such as IAPWS and inclusion of inter-phase mass transfer and turbulence modeling.

Acknowledgements 585

This research was partially supported by the Consortium for Advanced Simulation of Light Water Reactors (http://www.casl.gov), an Energy Innovation Hub (http://www.energy.gov/hubs) for Modeling and Simulation of Nuclear Reactors under U.S. Department of Energy Contract No. DE-AC05-00OR22725. 48

The authors would like to thank Dr. C. -H. Chang, Dr. N. Dinh, Dr. J. R. Ed590

wards, and Dr. R. Nourgaliev for fruitful discussions. The authors would also like to thank Dr. Chad Rollins for providing us the results by boilEulerFOAM which were used for comparison for the viscous test cases.

Appendix: Discretization of the non-conservative term Consider a domain Ω with boundaries Γ. The pressure flux αk ∇p on this domain can be simply written as, Z αk Ω

∂p dΩ. ∂xi

Consider now, an element Ωe a part of the triangulation of Ω. Since this term is non-conservative, it is not clear how this term should be discretized in this form. Thus, using integration-by-parts, this term can be expanded on the element Ωe as, Z αk Ωe

∂p dΩ = ∂xi

Z

Z αk pdΓ −

Γe

p Ωe

∂αk dΩ. ∂xi

Now, the first term is an integral over the cell-boundary Γe which uses the 595

value of the integrand (αk p) from the cell Ωe . This term takes into account the jump of the function at each cell-boundary. The pressure for this term should be obtained by the Riemann solver used (AUSM+ -up in this case, pk,1/2 ) since otherwise it is not uniquely defined on the cell-boundary. On the other hand, the volume-fraction for this term is supposed to be reconstructed from the cell

600

Ωe . Thus, on each cell-boundary, this term will have a different contribution to the cells straddling the boundary only due to the volume-fraction αk in it. This first term corresponds to the pressure flux part in Eq. (18). The second term can be treated as a source term since it is an integral over the cell Ωe only. It is trivially equivalent to the discretization of the nonconservative term

605

proposed by Chang and Liou [14]. Thus, the “stratified-flow model” is nothing but applying integration-by-parts to the pressure term in the two-fluid model while writing it in the discrete form.

49

References [1] H. Luo, J. D. Baum, R. L¨ohner, On the computation of multi-material 610

flows using ALE formulation, Journal of Computational Physics 194 (1) (2004) 304–328. doi:10.1016/j.jcp.2003.09.026. [2] A. K. Pandare, H. Luo, An arbitrary Lagrangian-Eulerian reconstructed discontinuous Galerkin method for compressible multiphase flows, in: 46th AIAA Fluid Dynamics Conference, 2016, p. 4270. doi:10.2514/6.

615

2016-4270. [3] M. Ishii, T. Hibiki, Thermo-fluid dynamics of two-phase flow, Springer Science & Business Media, 2010. [4] G. B. Wallis, One-dimensional two-phase flow, McGraw-Hill Companies, 1969.

620

[5] V. Ransom, V. Mousseau, Convergence and accuracy of the RELAP5 twophase flow model, in: Proceedings of the ANS International Topical Meeting on Advances in Mathematics, Computations, and Reactor Physics, 1991. [6] C. Rollins, H. Luo, N. Dinh, Development of multiphase CFD flow solver

625

in OpenFOAM, in: APS Meeting Abstracts, 2016. doi:10.1103/BAPS. 2016.DFD.A12.8. [7] L. Botti, D. A. Di Pietro, A pressure-correction scheme for convectiondominated incompressible flows with discontinuous velocity and continuous pressure, Journal of computational physics 230 (3) (2011) 572–585. doi:

630

10.1016/j.jcp.2010.10.004. [8] A. K. Pandare, H. Luo, A hybrid reconstructed discontinuous Galerkin and continuous Galerkin finite element method for incompressible flows on unstructured grids, J. Comput. Phys. 322 (2016) 491–510. doi:10.1016/ j.jcp.2016.07.002.

50

635

[9] A. Guelfi, D. Bestion, M. Boucker, P. Boudier, P. Fillion, M. Grandotto, J.-M. H´erard, E. Hervieu, P. P´eturaud, Neptune: a new software platform for advanced nuclear thermal hydraulics, Nuclear Science and Engineering 156 (3) (2007) 281–324. doi:10.13182/NSE05-98. [10] R. Nourgaliev, H. Luo, B. Weston, A. Anderson, S. Schofield, T. Dunn, J.-P.

640

Delplanque, Fully-implicit orthogonal reconstructed discontinuous Galerkin method for fluid dynamics with phase change, Journal of Computational Physics 305 (2016) 964–996. doi:10.1016/j.jcp.2015.11.004. [11] H. Park, R. R. Nourgaliev, R. C. Martineau, D. A. Knoll, On physics-based preconditioning of the Navier-Stokes equations, Journal of Computational

645

Physics 228 (24) (2009) 9131–9146. doi:10.1016/j.jcp.2009.09.015. [12] H. St¨ adtke, Gasdynamic aspects of two-phase flow: Hyperbolicity, wave propagation phenomena and related numerical methods, John Wiley & Sons, 2006. [13] I. Toumi, A. Kumbaro, An approximate linearized riemann solver for a two-

650

fluid model, Journal of Computational Physics 124 (2) (1996) 286–300. [14] C.-H. Chang, M.-S. Liou, A robust and accurate approach to computing compressible multiphase flow: Stratified flow model and AUSM+up scheme, Journal of Computational Physics 225 (1) (2007) 840–873. doi:10.1016/j.jcp.2007.01.007.

655

[15] M.-S. Liou, C.-H. Chang, L. Nguyen, T. G. Theofanous, How to solve compressible multifluid equations: a simple, robust, and accurate method, AIAA Journal 46 (9) (2008) 2345–2356. doi:10.2514/1.34793. [16] K. Kitamura, M.-S. Liou, C.-H. Chang, Extension and comparative study of AUSM-family schemes for compressible multiphase flow simulations,

660

Communications in Computational Physics 16 (03) (2014) 632–674. doi: 10.4208/cicp.020813.190214a.

51

[17] D. Lhuillier, C.-H. Chang, T. G. Theofanous, On the quest for a hyperbolic effective-field model of disperse flows, Journal of Fluid Mechanics 731 (2013) 184–194. doi:10.1017/jfm.2013.380. 665

[18] J. Stuhmiller, The influence of interfacial pressure forces on the character of two-phase flow model equations, International Journal of Multiphase Flow 3 (6) (1977) 551–560. [19] C.-H. Chang, S. Sushchikh, L. Nguyen, M.-S. Liou, T. Theofanous, Hyperbolicity, discontinuities, and numerics of the two-fluid model, in: 5th

670

ASME/JSME Fluids Engineering Summer Conference, 10th International Symposium on Gas-Liquid Two-phase Flows. San Diego, 2007.

doi:

10.1115/FEDSM2007-37338. [20] R. Abgrall, How to prevent pressure oscillations in multicomponent flow calculations: a quasi conservative approach, Journal of Computational 675

Physics 125 (1) (1996) 150–160. [21] R. Saurel, R. Abgrall, A multiphase Godunov method for compressible multifluid and multiphase flows, Journal of Computational Physics 150 (2) (1999) 425–467. doi:10.1006/jcph.1999.6187. [22] C. Par´es, Numerical methods for nonconservative hyperbolic systems: a

680

theoretical framework., SIAM Journal on Numerical Analysis 44 (1) (2006) 300–321. doi:10.1137/050628052. [23] K. Kitamura, T. Nonomura, Simple and robust HLLC extensions of twofluid AUSM for multiphase flow computations, Computers & Fluids 100 (2014) 321–335.

685

[24] Y.-Y. Niu, Computations of two-fluid models based on a simple and robust hybrid primitive variable Riemann solver with AUSMD, Journal of Computational Physics 308 (2016) 389–410.

52

[25] Y.-Y. Niu, H.-W. Wang, Simulations of the shock waves and cavitation bubbles during a three-dimensional high-speed droplet impingement based 690

on a two-fluid model, Computers & Fluids 134 (2016) 196–214. [26] T. Dinh, R. Nourgaliev, T. Theofanous, Understanding the ill-posed twofluid model, in: Proceedings of the 10th international topical meeting on nuclear reactor thermal-hydraulics (NURETH03), 2003. [27] T. Vazquez-Gonzalez, A. Llor, C. Fochesato, Ransom test results from

695

various two-fluid schemes: Is enforcing hyperbolicity a thermodynamically consistent option?, International Journal of Multiphase Flow 81 (2016) 104– 112. [28] R. Houim, E. Oran, A multiphase model for compressible granular–gaseous flows: formulation and initial tests, Journal of Fluid Mechanics 789 (2016)

700

166–220. doi:10.1017/jfm.2015.728. [29] M. Baer, J. Nunziato, A two-phase mixture theory for the deflagration-todetonation transition (DDT) in reactive granular materials, International Journal of Multiphase Flow 12 (6) (1986) 861–889. [30] R. Saurel, R. Abgrall, A simple method for compressible multifluid flows,

705

SIAM Journal on Scientific Computing 21 (3) (1999) 1115–1145. [31] R. Saurel, O. Lemetayer, A multiphase model for compressible flows with interfaces, shocks, detonation waves and cavitation, Journal of Fluid Mechanics 431 (2001) 239–271. doi:10.1017/S0022112000003098. [32] R. Abgrall, R. Saurel, Discrete equations for physical and numerical com-

710

pressible multiphase mixtures, Journal of Computational Physics 186 (2) (2003) 361–396. [33] D. Bestion, The physical closure laws in the CATHARE code, Nuclear Engineering and Design 124 (3) (1990) 229–245.

53

[34] R. Lahey, L. Cheng, D. Drew, J. Flaherty, The effect of virtual mass on the 715

numerical stability of accelerating two-phase flows, International Journal of Multiphase Flow 6 (4) (1980) 281–294. doi:10.1016/0301-9322(80) 90021-X. [35] D. Drew, L. Cheng, R. Lahey, The analysis of virtual mass effects in twophase flow, International Journal of Multiphase Flow 5 (4) (1979) 233–242.

720

[36] D. Kuzmin, A vertex-based hierarchical slope limiter for p-adaptive discontinuous Galerkin methods, Journal of computational and applied mathematics 233 (12) (2010) 3077–3085. doi:10.1016/j.cam.2009.05.028. [37] M.-S. Liou, A sequel to AUSM, part II: AUSM+-up for all speeds, Journal of Computational Physics 214 (1) (2006) 137–170. doi:10.1016/j.jcp.

725

2005.09.020. [38] J.-M. H´erard, O. Hurisse, A simple method to compute standard two-fluid models, International Journal of Computational Fluid Dynamics 19 (7) (2005) 475–482. [39] T. Gallou¨et, P. Helluy, J.-M. H´erard, J. Nussbaum, Hyperbolic relaxation

730

models for granular flows, ESAIM: Mathematical Modelling and Numerical Analysis 44 (2) (2010) 371–400. [40] M.-S. Liou, J. R. Edwards, Numerical speed of sound and its application to schemes for all speeds, in: 14th Computational Fluid Dynamics Conference, 1999, p. 3268. doi:10.2514/6.1999-3268.

735

[41] J. R. Edwards, M.-S. Liou, Low-diffusion flux-splitting methods for flows at all speeds, AIAA journal 36 (9) (1998) 1610–1617. doi:10.2514/2.587. [42] J. R. Edwards, Towards unified cfd simulations of real fluid flows, in: 15th AIAA Computational Fluid Dynamics Conference, 2001, p. 2524. [43] M. Castro, E. Fern´ andez-Nieto, A. Ferreiro, J. Garc´ıa-Rodr´ıguez, C. Par´es,

740

High order extensions of Roe schemes for two-dimensional nonconservative hyperbolic systems, Journal of Scientific Computing 39 (1) (2009) 67–114. 54

[44] M. Dumbser, E. Toro, A simple extension of the Osher Riemann solver to non-conservative hyperbolic systems, Journal of Scientific Computing 48 (1) (2011) 70–88. 745

[45] S. Munkejord, S. Evje, T. Fl˚ Atten, A musta scheme for a nonconservative two-fluid model, SIAM Journal on Scientific Computing 31 (4) (2009) 2587– 2622. [46] S. Gottlieb, C.-W. Shu, Total variation diminishing Runge-Kutta schemes, Mathematics of computation of the American Mathematical Society

750

67 (221) (1998) 73–85. doi:10.1090/S0025-5718-98-00913-2. [47] H. Luo, J. D. Baum, R. L¨ohner, An accurate, fast, matrix-free implicit method for computing unsteady flows on unstructured grids, Computers & fluids 30 (2) (2001) 137–159. doi:10.1016/S0045-7930(00)00011-6. [48] H. Luo, J. D. Baum, R. L¨ohner, A fast, matrix-free implicit method for

755

compressible flows on unstructured grids, Journal of Computational Physics 146 (2) (1998) 664–690. doi:10.1006/jcph.1998.6076. [49] H. Paillere, C. Corre, J. G. Cascales, On the extension of the AUSM+ scheme to compressible two-fluid models, Computers & Fluids 32 (6) (2003) 891–916. doi:10.1016/S0045-7930(02)00021-X.

760

[50] T. Nonomura, K. Kitamura, K. Fujii, A simple interface sharpening technique with a hyperbolic tangent function applied to compressible two-fluid modeling, Journal of Computational Physics 258 (2014) 95–117. [51] L. Zou, H. Zhao, H. Zhang, C. Brooks, A revisit to the Hicks’ hyperbolic two-pressure two-phase flow model, in: The 17th International Topical

765

Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-17), Xi’an, China, 2017. [52] E. Lauer, X. Hu, S. Hickel, N. Adams, Numerical investigation of collapsing cavity arrays, Physics of Fluids 24 (5) (2012) 052104.

55

[53] R. Nourgaliev, T.-N. Dinh, T.-G. Theofanous, Adaptive characteristics770

based matching for compressible multifluid dynamics, Journal of Computational Physics 213 (2) (2006) 500–529.

56