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A multi-component real-fluid two-phase flow solver with high-order finite-difference schemes

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Abstract

In this study, a mass-fraction based fully conservative multi-component two-phase flow solver is considered using characteristic-wise finite-difference (CWFD) discretization with the 5th-order WENO scheme, in order to reduce numerical interface smearing and oscillations. Real-fluid thermodynamic properties are accounted for by a vapor-liquid equilibrium (VLE) model according to the local total density, internal energy and composition of the homogeneous mixture, with each phase being separatedly described by its Peng-Robinson equation of state (PR-EoS). A multicomponent Roe averaging for the cell-face eigen-system in CWFD methods has been developed with pressure derivatives resulted from the VLE model. Several 1D testing examples, e.g. interface advection, shock-tube problems and double-expansion cavitation, have been examined to demonstrate the low-oscillation, low-dissipation and robust performance of the present solver, in comparison with finite-volume schemes. A 2D transcritical injection process has also been simulated. It has been shown that high-order numerical schemes, such as the current CWFD method may be the way to reduce the smearing in diffuse interface modelling.
ILASS–Europe 2019, 29th Conference on Liquid Atomization and Spray Systems, 2-4 September 2019, Paris, France
A multi-component real-fluid two-phase flow solver with high-order
finite-difference schemes
Jian-Hang Wang1,2,, Songzhi Yang2,3, Chaouki Habchi2,3 , Xiangyu Y. Hu1, Nikolaus A.
Adams1
1Chair of Aerodynamics and Fluid mechanics, Department of Mechanical Engineering,
Technical University of Munich, Boltzmannstr. 15, 85748 Garching, Germany
2IFP Energies nouvelles, 1 et 4 avenue de Bois-Préau, 92852 Rueil-Malmaison, France
3Institut Carnot IFPEN Transports Energie, 1 et 4 avenue de Bois-Préau, 92852
Rueil-Malmaison, France
*Corresponding author: jianhang.wang@tum.de
Abstract
In this study, a mass-fraction based fully conservative multi-component two-phase flow solver is considered using
characteristic-wise finite-difference (CWFD) discretization with the 5th-order WENO scheme, in order to reduce nu-
merical interface smearing and oscillations. Real-fluid thermodynamic properties are accounted for by a vapor-liquid
equilibrium (VLE) model according to the local total density, internal energy and composition of the homogeneous
mixture, with each phase being separatedly described by its Peng-Robinson equation of state (PR-EoS). A multi-
component Roe averaging for the cell-face eigen-system in CWFD methods has been developed with pressure
derivatives resulted from the VLE model. Several 1D testing examples, e.g. interface advection, shock-tube prob-
lems and double-expansion cavitation, have been examined to demonstrate the low-oscillation, low-dissipation and
robust performance of the present solver, in comparison with finite-volume schemes. A 2D transcritical injection pro-
cess has also been simulated. It has been shown that high-order numerical schemes, such as the current CWFD
method may be the way to reduce the smearing in diffuse interface modelling.
Keywords
Two-phase flow, phase equilibrium, finite difference method, Roe averaging, transcritical, fuel injection
Introduction
Accurate and robust simulation of compressible real-fluid two-phase flow is crucial for many engineering applica-
tions, such as the fuel injection in internal combustion engines. One example is that, in diesel engines when the
liquid fuel is injected into the ambient gas at a pressure higher than its critical value, the fuel jet will be heated to su-
percritical temperature before combustion takes place. This process is often refered to as transcritical injection [1].
Intensive fluid-dynamical structures and thermodynamical processes may happen in the typical complex two-phase
flow. Simulation of such complicated unsteady, multi-scale two-phase flows with multiple fluids of thermodynamic
properties is very challenging. Accurate flow solver with the capability of capturing high-resolution interface and real-
fluid thermodynamic solver which can precisely describe phase and component states under varying temperature
and pressure conditions are greatly desired [1, 2, 3].
Regarding the flow solver, diffuse-interface methods are widely used with various numerical schemes. Both fully
conservative (FC) schemes [4, 5, 6, 7, 8, 9] and quasi-conservative schemes (QC) [10, 11, 12, 13, 14, 15] have
been intensively studied and used for two-phase flow simulation. One advantage of the QC schemes lies in avoid-
ing the occurrence of spurious pressure oscillations [16], at a sacrifice of losing the strict energy conservation.
However, most of the above flow solvers consider finite-volume (FV) schemes due to low computational cost, ro-
bust performance and body-fitted nature towards complicated configurations, and limited attention has been paid
to finite-difference (FD) schemes for real-fluid two-phase flow simulation. In single-fluid flow simulation of gas dy-
namics or multi-species flow simulation of chemically reacting gas mixtures, FD schemes have been widely used for
capturing strong shocks and detonation waves, etc., owing to high-order accuracy and low-oscillation property. On
the other hand, a real-fluid thermodynamics solver is used to precisely determine the local phase and component
state under a specific set of internal energy/density or temperature/pressure plus the composition of components.
The former solver is the so-called UVn-flash [3] and the latter corresponds to TPn-flash [18]. A simple thermody-
namical model of mixture Peng-Robinson equation of state (PR-EoS) [12], without considering phase change or
phase separation, can also serve this purpose but it may lead to unphysical or ill-defines states, especially in sub-
critical two-phase conditions. The thermodynamic model [7], considering vapor-liquid equilibrium (VLE), can firstly
detect whether a state of the fluid mixture corresponds to a point within or outside the two-phase state using the
TPD stability analysis [17, 3], and then determines the component state in each phase using mixture cubic EoS.
However, a thermodynamically consistent description of mixture thermodynamics is computationally expensive [9]
and even dominates the overall CPU time of the integrated solver.
As a result, the present study considers applying the high-order characteristic-wise FD schemes to the mass-
fraction based fully conservative multi-component formulation, in order to reduce numerical interface smearing and
oscillations, without severely increasing the overall computational cost. Following the work of Ping et al. [3], real-fluid
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thermodynamic properties are accounted for by the VLE model with each phase being described by the mixture PR-
EoS, such that either single-phase mixture or two-phase mixture in mechanical and thermodynamical equilibrium or
phase change can be described in a thermodynamically generalized and consistent manner.
The remainder of this paper has the following structure. The next section introduces firstly the governing equations
and its discretization by the characteristic-wise FD scheme, and then the description of thermodynamic model as
well as the UVn-flash solver. Section 3 presents several 1D test examples and a 2D transcritical injection simulation.
Conclusions and some prospects are drawn in the last section.
Methodology
Characteristic-wise finite-difference scheme for multi-component Euler equations
For simplicity and without loss of generality, we solve the two-dimensional compressible multi-component Euler
equations in a mass-fraction based fully conservation formulation as follows
Ut+F(U)x+G(U)y= 0,(1)
where
U=ρ, ρu, ρv, ρet, ρy1, ρy2,···, ρyNs1T,
F(U) = ρu, ρu2+p, ρvu, (ρet+p)u, ρy1u, ρy2u, ···, ρyNs1uT,
G(U) = ρv, ρuv, ρv 2+p, (ρet+p)v, ρy1v, ρy2v, ··· , ρyNs1vT,
(2)
are vectors of the conserved variables, convective flux in the x or y direction, respectively. The specific total energy
including the specific internal energy eis et=e+ (u2+v2)/2. To close the system, an equation of state (EoS) such
as the ideal gas EoS for gas mixtures is usually required. In this study, to avoid unstable thermodynamic states, a
thermodynamically consistent solver is used to account for both single-phase and two-phase flow and discussed in
the following section.
Shock-capturing schemes are usually employed based on either finite volume (FV) or finite difference (FD) formula-
tion for spatial discretization, in which high-order shock-capturing accuracy as well as high computational efficiency
are desired. Low-order FV schemes approximate the cell-face flux function by upwind reconstruction using primi-
tive or conserved variables, together with MUSCL interpolation schemes plus slope limiters and achieve generally
second-order accuracy. High-order shock-capturing schemes are realized by characteristic-decomposition flux split-
ting to assemble the half-point convective flux using high-order interpolation schemes in FD approaches.
To achieve high-order FD schemes for Eqs. (1) and (2), Jacobian system including the left and right eigen-vectors
as well as Roe-averages of pressure derivatives to calculate the speed of sound cat the cell face needs to be
considered; see details in [19]. It should be noted that, in this study, the adjacent two sets of cell-centered pressure
derivatives such as ∂p
∂ρ ,p
∂e and p
∂yi
are directly obtained by outputs from the thermo-solver, i.e. UVn-flash, as UVn-
flash inherently contains calculations of thermodynamics derivatives (see [3] for details). High-order interpolation
scheme WENO5 and local Lax-Friedrich splitting [22] are employed here.
Vapor-liquid equilibrium model using UVn-flash
Since the conservative governing equations, at a certain time instant, provide directly conserved variables such as
density ρ, total energy et, etc, in order to close the system, primitive variables like temperature Tand pressure p
should be obtained through the conserved variables before entering the next time step. A thermodynamic solver
is thus needed for translating the local total density ρ, internal energy eand the overall composition z={zi}, i =
1,···, NSof the homogeneous mixture to equilibrium temperature T, pressure pand other thermodynamic variables
that reflect the state of the homogeneous multi-component two-phase mixture, as shown in Fig. 1, such as the vapor
fraction ψand component mole fractions xand yin the liquid and vapor phase, respectively. This is the so-called
isochoric-isoenergetic flash problem. i.e. UVn-flash. The vapor-liquid equilibrium model is assumed with
pl=pv=p,
Tl=Tv=T,
µi,l (T, p, x) = µi,v (T, p, y),
ψyi+ (1 ψ)xi=zi,yixi= 0, i = 1,···, Ns,
ψUv+ (1 ψ)Ul=U,
ψVv+ (1 ψ)Vl=V,
(3)
where subscripts l, v denote the liquid and vapor phase, respectively, and µirepresents the chemical potential of
component i. Note that all the quantities in Eq. (3) are on a molar basis. In either phase lor v, components are
well-mixed filling its volume Vlor Vv, and the thermodynamic properties of each single-phase mixture are described
using the PR-EoS for Ns-component mixtures with appropriate mixing rules [3]. The last two equations in Eq. (3)
conform to energy and mass conservation, and function as the condition of convergence for the iterative UVn-flash.
Once the convergence criterion is reached, the equilibrium temperature Tand pressure pare obtained; see details
of the UVn-flash we consider in the present study in [1, 3] and other relevant solvers, e.g. in [20].
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liquid
Liquid
Vapor
x= {x1,x2}
TlPl
UlVl
y= {y1, y2}
Tv Pv
UvVv
A Local Cell
Figure 1. Schematic for vapor-liquid equilibrium model in a local computational cell. For simplicity, we have two components by
setting Ns= 2, e.g. n-dodecane and nitrogen.
One critical difficulty of the present UVn-flash (which is based on the deterministic gradient-dependent quasi-Newton
algorithm) lies in the convergence of iterative root-finding for equilibrium Tand pdue to the complex non-linearity
of PR-EoS and VLE constraints. Non-convergence might arise especially when the phase boundary is crossed.
Therefore, a non-linear global optimization technique with non-deterministic strategies is desired and we can refer to
the very popular particle swarm optimization (PSO) algorithm [21], which is a gradients-free metaheuristic algorithm
exploiting swarming behavior of organisms to search for optimal solutions. Since PSO is population-based and
time-consuming, it is activated only when the conventional UVn-flash fails in converging the equilibrium state.
Results and discussion
In this section, we first present several 1D numerical tests of two-component two-phase flows, to demonstrate
the reliability of the proposed CWFD framework. In comparison to the 5th-order CWFD WENO5-LLF scheme,
AUSM+ scheme [23] with or without minmod slope limiter (AUSM+/minmod) in the FV formulation is also adopted.
Computational timestep which is usually controlled by CFL number for compressible flows is sometimes limited to
be much smaller such that UVn-flash manages to handle phase boundaries and drastic jumps of thermodynamic
states at adjacent cells and time instants. A 2D transcritical injection of liquid n-dodecane into a chamber of gaseous
n-dodecane follows to examine the spatial resolution of typical structures, such as vortices, shear-layer instability
and interface, etc, by the high-order CWFD scheme.
1D advection case (Gaussian profile)
The first 1D test case we consider is the transitional advection of n-dodecane (C12H26)/nitrogen (N2) mixture at
constant pressure of 1 bar and velocity of 100 m/s. The 1D domain is 1 m long and both ends are imposed of the
periodic boundary condition. Initial temperature and mass fractions have a Gaussian profile according to
T(x) = T0Γ
2πβ exp r2
ϵ,
y1(x) = y1,00.01 Γ
2πβ exp r2
ϵ,
y2= 1 y1,
(4)
where T0= 300 K, y1,0= 0.99 and r2= (x0.5)2with Γ = 10,β= 0.01 and ϵ= 0.02. We use varying grid
resolutions with 25,50,100,125 cells in the domain, respectively, and compute the mixture advection for a complete
period till t= 0.01 s. L1norms of errors of density and mass fraction y1are shown in Fig. 2 and the expected
high-order accuracy of the present CWFD scheme is demonstrated, compared to the 2nd-order AUSM+/minmod
scheme and 1st-order AUSM+ with no slope limiter.
++++
No. of Cells
L1 norm of density error
20 40 60 80 100 120 140160
10-5
10-4
10-3
10-2
10-1
1st
2nd
5th
++++
No. of Cells
L1 norm of y1 error
20 40 60 80 100 120 140160
10-6
10-5
10-4
10-3
10-2
10-1
1st
2nd
5th
Figure 2. 1D advection (Gaussian profile) of N2/C12 H26 mixture. Red circle line: CWFD WENO5-LLF; purple cross line: FV
AUSM+; green square line: FV AUSM+/minmod.
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1D advection case
It is well-known that upwind shock-capturing schemes would gradually smear out the sharp interface during ad-
vection due to numerical dissipation, and spurious numerical oscillations of pressure and velocity can be induced
by the fully conservative formulation versus the quasi-conservative methods. The present study is not aimed at
proposing a cure to alleviate spurious oscillations in the fully conservative formulation but examining the possibility
of improving interface resolution as well as reducing spurious oscillations by CWFD schemes in comparison with FV
schemes based on cell-face flux reconstruction from primitive variables. The present 1D advection case considers
the same setup with the above advection case except that initial temperature and mass fractions exhibit sharp in-
terfaces in the present domain of 1000 grid cells. As shown in Fig. 3, it can be seen that after one complete period
of advection, the 5th-order CWFD scheme preserves the very sharp interface compared to the reference solution,
apparently better than the 2nd-order AUSM+/minmod scheme result and its 1st-order counterpart. Besides, numer-
ical oscillation of pressure, in this case, of the CWFD scheme is less severe than the 2nd-order FV schemes. This
is also the case when high-order FV schemes are utilized in single-fluid flow simulations. However, the 1st-order
FV scheme outperforms other two schemes, which is possibly because its numerical dissipation is sufficiently large
to suppress the oscillation to occur. In this sense, the 5th-order CWFD scheme is the optimal choice weighing both
high interface resolution and low oscillation of pressure.
0.99
0.995
1
1.005
1.01
0 0.2 0.4 0.6 0.8 1
P/Pini
x (m)
0
20
40
60
80
100
0 0.2 0.4 0.6 0.8 1
density (kg/m3)
x (m)
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
y1
x (m)
0.85
0.9
0.95
1
0 0.2 0.4 0.6 0.8 1
vapor volume fra ction
x (m)
Figure 3. 1D advection of N2/C12 H26 mixture. Black solid line: reference solution; red circle line: CWFD WENO5-LLF; purple
cross line: FV AUSM+; green square line: FV AUSM+/minmod.
1D shocktube case A
This 1D shocktube case ([24], Fig. 4) considers a 1 m long tube of H2O/N2mixture with initial condition as
(p, T ) = (2 bar,353.79 K),if x < 0.5m,
(1 bar,337.41 K),otherwise,(5)
with yN2= 0.7and yH2O= 0.3along the tube. Computation runs until t= 1 ms. In Fig. 4, we can see that all the
three solutions predict the similar profiles of variables, in which the proposed CWFD WENO5-LLF scheme yields the
sharpest shock especially in the velocity profile. The 2nd-order AUSM+/minmod scheme yields very close results
as the high-order scheme while the 1st-order AUSM+ scheme exhibits much smeared profiles in this shocktube
example.
1D shocktube case B
In this shocktube case, we enhance the initial pressure to be similar with Spray A condition with the initial mass
fractions of H2O/N-dodecane are 0.9/0.1, respectively, and the initial temperature of 900 K in the left half and 363 K
in the right half of the tube, respectively.
Computation runs until t= 0.3ms. In Fig. 5, we can see that the CWFD WENO5-LLF scheme and 2nd-order
AUSM+/minmod scheme both yield sharp discontinuities. However, the 2nd-order FV scheme also produces more
severe numerical oscillations in pressure and velocity profiles. The corresponding 1st-order result is smooth with
less accuracy.
1D double-expansion cavitation
The double-expansion case (see the similar double-expansion test in Fig. 7 of [24]) considers the cavitation phe-
nomenon when pressure decreases to be below the saturation pressure as the fluid are extruded outwards from the
middle of the 1D tube. Initial conditions of pressure, temperature and mass fraction of H2O are 1 bar, 293 K and
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100000
120000
140000
160000
180000
200000
0 0.2 0.4 0.6 0.8 1
P (Pa)
x (m)
1
1.2
1.4
1.6
1.8
2
2.2
0 0.2 0.4 0.6 0.8 1
density (kg/m3)
x (m)
336
338
340
342
344
346
348
350
352
354
0 0.2 0.4 0.6 0.8 1
T (K)
x (m)
0
20
40
60
80
100
120
0 0.2 0.4 0.6 0.8 1
u (m/s)
x (m)
Figure 4. 1D shocktube A of H2O/N2mixture. Black solid line: initial discontinuity; red circle line: CWFD WENO5-LLF; purple
cross line: FV AUSM+; green square line: FV AUSM+/minmod.
0
2×107
4×107
6×107
8×107
1×108
0 0.2 0.4 0.6 0.8 1
P (Pa)
x (m)
300
400
500
600
700
800
900
1000
0 0.2 0.4 0.6 0.8 1
T (K)
x (m)
0
50
100
150
200
250
300
350
400
450
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
T (K)
x (m)
0
0.02
0.04
0.06
0.08
0.1
0 0.2 0.4 0.6 0.8 1
vapor volume fraction
x (m)
Figure 5. 1D shocktube B of H2O/N2mixture. Black solid line: initial discontinuity; red circle line: CWFD WENO5-LLF; purple
cross line: FV AUSM+; green square line: FV AUSM+/minmod.
0.99995, respectively. Initial velocity of the left half of the H2O/N2mixture is -1 m/s while the right half is moving
oppositely. The 1D domain is discretilized with 100 cells. In Fig. 6, it can be seen that the CWFD scheme preserves
the sharpness of the expansion wave, as expected. Note that the 1st-order FV scheme yields a velocity jump in
the middle point of the tube, while the 2nd-order FV scheme, although it gives the correct stationary area of u= 0,
results in a spurious velocity oscillation.
2D transcritical n-dodecane injection
Now we focus on the 2D simulation of the low-temperature (363 K) n-dodecane injected into a chamber of n-
dodecane of high temperature 900 K at 60 bar. Inlet pressure and velocity are 70 bar and 55 m/s, respectively.
N-dodecane are diluted with nitrogen of a mass fraction of 0.01%. Limited by the computing power, the current
simulation runs on a 2D domain of the size 50h×8h, where h= 100 µm is the inlet height, as shown in Fig. 7. Two
sets of non-uniform grids, both with stretching mesh sizes away from the inlet, are used to save CPU time. Similar
simulation has been studied in [1, 25, 26].
In Fig. 8, we display the early evolution of the jet flow at three time instants, using the 5th-order CWFD scheme
and 2nd-order FV scheme on two grids, respectively. Referring to the qualitative illustration of liquid jet flow from
the DNS results in [25], it is clear to see, from the CWFD results in the first column, the roll-up of a mushroom-like
primary vortex followed by several secondary vortices induced along the interface between the primary vortex and
the ambient chamber fluid. Rayleigh-Taylor shear-layer instability at the interface bewteen the jet column and the
ambient high-temperature fluid induces a series of tiny vortices as well. The CWFD scheme on the coarse grid
provides a lower-resolution display of the structures while it preserves similar aforementioned characteristics as
in the fine grid. The FV scheme, however, can merely give the symmetric primary vortex structure due to severe
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60000
70000
80000
90000
100000
0 0.2 0.4 0.6 0.8 1
P (Pa)
x (m)
800
804
808
812
816
820
0 0.2 0.4 0.6 0.8 1
density (kg/m3)
x (m)
-1
-0.5
0
0.5
1
0 0.2 0.4 0.6 0.8 1
u (m/s)
x (m)
0.035
0.04
0.045
0.05
0.055
0.06
0 0.2 0.4 0.6 0.8 1
vapor volume fra ction
x (m)
Figure 6. Case 2 1D double-expansion cavitation problem of air/water mixture. Black solid line: initial discontinuity; red circle line:
CWFD WENO5-LLF; purple cross line: FV AUSM+; green square line: FV AUSM+/minmod.
Figure 7. Left: domain of the 2D transcritical injection with the fine grid. The minimum grid size of the coarse size is double in
y-direction but the same in x-direction. Right: DNS result of the liquid jet from [25].
numerical viscosity, and it predicts different lengths of the jet from the inlet, based on two grids. The primary vortex
size and location of the FV scheme on the fine grid is similar with the two CWFD results while the coarse FV result
has a primary vortex of flatter shape and sharper tip in the front. In Fig. 9, we further discuss the subsequent
evolution of the jet flow. More obviously, in the CWFD results on the fine grid, shedding of the secondary vortices
from the primary vortex is observed and it either merges with the mainstream of jet column or splashes away into the
surrounding fluid. As time goes by, when the jet column cannot preserve its horizontal and straight sharp (marked
by B) due to instability and the primary vortex core has a very thin connection with the vortex edges (marked by A),
the primary vortex almost breaks up. Despite of the complex flowfield with many unsteady structures, the high-order
CWFD scheme can preserve the basic structure of the jet and primary vortex on two grids, especially the similar
jet length and size. However, the 2nd-order FV scheme can only capture the stable growth and propagation of the
primary vortex, associated with the growing jet column, on either fine or coarse grid. In addition, the smearing of the
primary vortex is increasingly severe because of the stretched grid effect when its distance from the inlet increases
and the tip of the jet of the FV scheme on coarse grid is moving too fast.
It is reasonable to believe that with a high-order spatial scheme on a fine grid, some detailed characteristics of
the fundamental flow structures can be numerically captured and it will be interesting to examine if these plentiful
structures (which are not available in low-order scheme results) such as vortex shedding, breakup and shear-layer
vortices, etc, have a significant effect upon the prediction of realistic injection processes with a larger domain and
longer simulation time. Analogically, realistic injection processes usually have plentiful contents of fluid-dynamic and
thermodynamic phenomena, such as ligament and droplet formation, atomization, evaporation and spray breakup,
etc.
Conclusion
In this study, a multi-component two-phase flow solver utilizing high-order CWFD schemes for the mass-fraction
based fully conservative formulation has been presented, in association with the thermodynamic solver UVn-flash
based on the VLE model. Both 1D test cases of advection, shocktube and double-expansion problems, and a
2D transcritical injection process demonstrate the considerably high resolution of the present scheme in capturing
interfaces, discontinuities and many detailed flow structures of interest. In the future work, it is very interesting to
extend the present solver to more realistic industry-level two-phase flow situations like Spray A. Towards a more
efficient thermodynamic solver of UVn-flash, a pre-processed tabulation for thermodynamic relations is also desired.
Acknowledgements
The financial support from the EU Marie Skłodowska-Curie Innovative Training Networks (ITN-ETN) (Project ID:
675528-IPPAD-H2020-MSCA-ITN-2015) for the first two authors are gratefully acknowledged. The first author also
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ILASS – Europe 2019, 2-4 Sep. 2019, Paris, France
Figure 8. Density of the 2D transcritical injection. From left to right are CWFD WENO5-LLF on fine grid, AUSM+/minmod on fine
grid, CWFD WENO5-LLF on coarse grid and AUSM+/minmod on coarse grid, respectively. From top to bottom are time instants
at t= 5,10 and 20 µs, respectively.
acknowledge the support for part of this work during his secondment at IFPEN.
References
[1] Yang, S., Habchi, C., Yi, P., & Lugo, R. (2018). Towards a multicomponent real-fluid fully compressible two-phase
flow model. ICLASS 2018, 14th Triennial International Conference on Liquid Atomization and Spray Systems,
Chicago, IL, USA, July 22-26, 2018.
[2] Yang, S., Habchi, C., Yi, P., & Lugo, R. (2018). Cavitation Modelling Using Real-Fluid Equation of State. In
Proceedings of the 10th International Symposium on Cavitation (CAV2018). ASME Press.
[3] Yi, P., Yang, S., Habchi, C., & Lugo, R. (2019). A multicomponent real-fluid fully compressible four-equation
model for two-phase flow with phase change.Physics of Fluids, 31(2), 026102.
[4] Meng, H., & Yang, V. (2003). A unified treatment of general fluid thermodynamics and its application to a
preconditioning scheme. Journal of Computational Physics, 189(1), 277-304.
[5] Wang, X., Huo, H., Wang, Y., & Yang, V. (2017). Comprehensive study of cryogenic fluid dynamics of swirl
injectors at supercritical conditions.AIAA Journal, 3109-3119.
[6] Ruiz, A. M., Lacaze, G., Oefelein, J. C., Mari, R., Cuenot, B., Selle, L., & Poinsot, T. (2015). Numerical bench-
mark for high-Reynolds-number supercritical flows with large density gradients.AIAA Journal, 54(5), 1445-
1460.
[7] Matheis, J., & Hickel, S. (2018). Multi-component vapor-liquid equilibrium model for LES of high-pressure fuel
injection and application to ECN Spray A.International Journal of Multiphase Flow, 99, 294-311.
[8] Petit, X., Ribert, G., Lartigue, G., & Domingo, P. (2013). Large-eddy simulation of supercritical fluid injection.
The Journal of Supercritical Fluids, 84, 61-73.
[9] Knudsen, E., Doran, E. M., Mittal, V., Meng, J., & Spurlock, W. (2017). Compressible Eulerian needle-to-target
large eddy simulations of a diesel fuel injector.Proceedings of the Combustion Institute, 36(2), 2459-2466.
[10] Schmitt, T., Selle, L., Ruiz, A., & Cuenot, B. (2010). Large-eddy simulation of supercritical-pressure round jets.
AIAA journal, 48(9), 2133-2144.
[11] Terashima, H., & Koshi, M. (2012). Approach for simulating gasliquid-like flows under supercritical pressures
using a high-order central differencing scheme.Journal of Computational Physics, 231(20), 6907-6923.
[12] Ma, P. C., Lv, Y., & Ihme, M. (2017). An entropy-stable hybrid scheme for simulations of transcritical real-fluid
flows.Journal of Computational Physics, 340, 330-357.
[13] Pantano, C., Saurel, R., & Schmitt, T. (2017). An oscillation free shock-capturing method for compressible van
der Waals supercritical fluid flows.Journal of Computational Physics, 335, 780-811.
[14] Devassy, B. M., Habchi, C., & Daniel, E. (2015). Atomization modelling of liquid jets using a two-surface-density
approach. Atomization and Sprays, 25(1).
[15] Habchi, C. (2015). A gibbs energy relaxation (GERM) model for cavitation simulation. Atomization and Sprays,
25(4).
This work is licensed under a Creative Commons 4.0 International License (CC BY-NC-ND 4.0).
ILASS – Europe 2019, 2-4 Sep. 2019, Paris, France
Figure 9. Density of the 2D transcritical injection. Every 2×2array of figures are CWFD WENO5-LLF on fine grid,
AUSM+/minmod on fine grid, CWFD WENO5-LLF on coarse grid and AUSM+/minmod on coarse grid, respectively. From top to
bottom are time instants at t= 30,40 and 50 µs, respectively.
[16] Abgrall, R., & Karni, S. (2001). Computations of compressible multifluids.Journal of computational physics,
169(2), 594-623.
[17] Michelsen, M. L. (1982). The isothermal flash problem. Part I. Stability. Fluid phase equilibria, 9(1), 1-19.
[18] Michelsen, M. L. (1982). The isothermal flash problem. Part II. Phase-split calculation. Fluid phase equilibria,
9(1), 21-40.
[19] Wang, J. H., Pan, S., Hu, X. Y., & Adams, N. A. (2019). Partial characteristic decomposition for multi-species
Euler equations.Computers & Fluids, 181, 364-382.
[20] Smejkal, T., & Mikyka, J. (2017). Phase stability testing and phase equilibrium calculation at specified internal
energy, volume, and moles.Fluid Phase Equilibria, 431, 82-96.
[21] Kennedy, J. (2010). Particle swarm optimization.Encyclopedia of machine learning, 760-766.
[22] Shu, C. W. (1998). Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyper-
bolic conservation laws.In Advanced numerical approximation of nonlinear hyperbolic equations (pp. 325-432),
Springer, Berlin, Heidelberg.
[23] Liou, M. S. (1996). A sequel to AUSM: AUSM+.Journal of computational Physics, 129(2), 364-382.
[24] Chiapolino, A., Boivin, P., & Saurel, R. (2017). A simple and fast phase transition relaxation solver for com-
pressible multicomponent two-phase flows.Computers & Fluids, 150, 31-45.
[25] Shinjo, J., & Umemura, A. (2010). Simulation of liquid jet primary breakup: Dynamics of ligament and droplet
formation.International Journal of Multiphase Flow, 36(7), 513-532.
[26] Herrmann, M. (2010). Detailed numerical simulations of the primary atomization of a turbulent liquid jet in
This work is licensed under a Creative Commons 4.0 International License (CC BY-NC-ND 4.0).
ILASS – Europe 2019, 2-4 Sep. 2019, Paris, France
crossflow.Journal of Engineering for Gas Turbines and Power, 132(6), 061506.
This work is licensed under a Creative Commons 4.0 International License (CC BY-NC-ND 4.0).
ResearchGate has not been able to resolve any citations for this publication.
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