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Cavitation Modelling Using Real-Fluid Equation of State

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Conclusion
The current study examines the solid projectile impact on a water jet and assess the use of direct forcing Immersed
Boundary Method in conjunction with an explicit density based compressible two-phase solver that accounts for
cavitation. The physics of the case studied was accurately captured and the simulation results are in qualitative
agreement with experiments presented by Field et al. [4]. Pressure shock waves, relief waves and cavitation induction,
are present and numerically calculated. A more detailed description of the vapour cavity is provided that is in
accordance with the analysis of the experimental data. However, liquid splashing and the anticipated pressure peak
values where not observed. Although experimental pressure measurements do not exist for the specific problem,
theoretical analysis and experimental data on simpler cases indicate that pressure values exceed water hammer
pressures. The Immersed Boundary method used proved not good enough to capture such phenomena and shows room
for improvement.
Acknowledgments
This work was carried out in the framework of CaFE project, which has received funding from the European Union
Horizon 2020 Research and Innovation programme, with Grant Agreement No 642536.
References
1. Serge Abrate (2016). Soft impacts on aerospace structures. Progress in Aerospace Sciences, 81(Supplement C):1–17, Dynamic
Loading Aspects of Composite Materials.
2. F. P. Bowden and J. H. Brunton (1961). The Deformation of Solids by Liquid Impact at Supersonic Speeds. Proceedings of the
Royal Society of London. Series A, Mathematical and Physical Sciences, 263(1315):433450
3. J. E. Field, M. B. Lesser, and J. P. Dear (1985). Studies of Two-Dimensional Liquid-Wedge Impact and Their Relevance to
Liquid-Drop Impact Problems. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences,
401(1821):225249
4. J.E. Field, J.-J. Camus, M. Tinguely, D. Obreschkow, and M. Farhat (2012). Cavitation in impacted drops and jets and the
effect on erosion damage thresholds. Wear, 290291:154160
5. P. Koukouvinis, C. Bruecker, and M. Gavaises (2017). Unveiling the physical mechanism behind pistol shrimp cavitation.
Scientific Reports, (1). Licensed under a Creative Commons Attribution 4.0 International License.
6. N. Kyriazis, P. Koukouvinis, and M. Gavaises. Modelling cavitation during droplet impact on solid surfaces. Computer
Methods in Applied Mechanics and Engineering. Under Review.
7. M. B. Lesser (1981). Analytic solutions of liquid-drop impact problems. Proceedings of the Royal Society of London. Series
A, Mathematical and Physical Sciences, 377(1770):289308.
8. Rajat Mittal and Gianluca Iaccarino (2005). Immersed Boundary Methods. Annual Review of Fluid Mechanics, 37(1):239–
261.
9. Loïc Mochel, Pierre-Élie Weiss, and Sébastien Deck (2014). Zonal Immersed Boundary Conditions: Application to a High-
Reynolds-Number Afterbody Flow. AIAA Journal, 52(12):2782–2794.
10. Felix Örley, Vito Pasquariello, Stefan Hickel, and Nikolaus A. Adams (2015). Cut-element based immersed boundary method
for moving geometries in compressible liquid flows with cavitation. J. Comput. Physics, 283:1–22.
11. Charles S Peskin (1977). Numerical analysis of blood flow in the heart. Journal of Computational Physics, 25(3):220–252.
10th International Symposium on Cavitation - CAV2018
Baltimore, Maryland, USA, May 14 – 16, 2018
CAV18-05045
*Corresponding Author, Songzhi Yang:songzhi.yang@ifpen.fr
Cavitation modelling using real-fluid equation of state
1Songzhi Yang*1,2, Chaouki Habchi*1,2, Ping Yi*1,2, Rafael Lugo1
1IFP Energies nouvelles, 1 et 4 avenue de Bois-Préau, 92852 Rueil-Malmaison, France
2Institut Carnot IFPEN Transports Energie
Abstract
The aim of this work is to improve the accuracy of cavitating flow simulations by considering real-
fuel mixture thermodynamics, like gas (air) solubility and physical property deviations from ideality,
and their effects on phase change. In this work, a new fully compressible two-phase six-equations
model coupled with Peng and Robinson [1] real-fluid cubic equation of state (PR-EOS) is described.
Assuming thermodynamic equilibrium with negligible capillarity, this diffuse interface model has
been implemented in the IFP-C3D solver [3], which already includes the Gibbs Energy Relaxation
Method (GERM) cavitation model [10]. One-dimensional academic test cases (cavitation-tube,
shock-tube) are performed firstly to validate the real-fluid PR-EOS implementation. Comparisons of
the numerical results with previously published works are also carried out. Then, the models have
been applied to simulate the cavitation phenomenon inside a single-hole nozzle. With phase
equilibrium theory, the simulation has proved that increased dissolved gas may facilitate gaseous
cavitation and restrict the developing of vaporous cavitation. The effect of dissolved gas on fuel
properties has also been presented and discussed using the suggested real fluid model. Overall, the
two fluid model combined with real equation of state is able to reveal more physical details about the
cavitation process.
Keywords: two fluid model; real fluid; dissolved gas; cavitation
Introduction
Many strategies like high injection pressure and smaller nozzle orifice have been adopted to improve the Diesel
engine combustion efficiency and reduce emissions, like particle issues and NOx. However, much remains to be
done to achieve the objectives in this area because of an insufficient understanding of the various physical
phenomena governing the fuel injection and the mixing with gas before combustion. With the increasing injection
pressure and decreasing nozzle diameter, the cavitation phenomenon become more and more extrusive during fuel
injection in modern diesel engine. Many researchers have contributed to better understanding this phenomenon [2, 7,
9, 10, 12–14, 17]. Among various research strategies, numerical modeling is still one of the most efficient tools. In
this study, a multicomponent and fully compressible two-fluid six equation model is proposed using the Peng and
Robinson [1] real-fluid cubic equation of state (PR-EOS). This model is referred to below as 6EQ-PR model. The
main aim of this work is to investigate the effect of non-condensable gas on the cavitation phenomenon using the
6EQ-PR model. Previous researchers like Duke et al., [6, 7, 14] have demonstrated the effect of non-condensable
gas on cavitation using a standardized fuel and also degassed fuel through X-ray radiography experiments. Then
Battistoni et al., [2] further studied these effects by numerical modeling with Homogeneous Relaxation Model
(HRM). In both of the previous studies, it has been found that the void content appeared in the central part of the
orifice has decreased significantly as the initial amount of gas nuclei is decreased. Due to the fact that pressure is
above the saturated pressure in the central part of the nozzle, the cavitation cloud taking place there has been
identified as gaseous cavitation, thus confirming the analysis of Habchi, et al., [4] on the existence of two kinds of
cavitation, namely gaseous cavitation and vaporous cavitation. Habchi, et al., [10, 11] have also detected gaseous
cavitation in a typical Diesel injector using a two-fluid seven equations diffused interface model. Besides, Mori et al.
[16] has confirmed that the mixing of non-condensable gas can induce the cavitation to incept at higher pressure
exceeding pure material saturated pressure. Li et al., [15] also proved experimentally that the tensile strength of
water has decreased with the increase of dissolved gas like nitrogen, oxygen at given temperature. The reduced
tensile strength thereby facilitates the formation of cavitation. All these previous researches have provided valuable
insight into current research. To know more details about the effect of non-condensable gas on cavitation, unlike
previous research, the present study has adopted a diffuse interface two-fluid model combined with a
multicomponent thermodynamics’ equilibrium solver based on Peng and Robinson [1] cubic equation of state (PR-
EOS). Indeed, using this real-fluid EOS gives us the possibility to consider dissolved gas (like Nitrogen) in the
10th International Symposium on Cavitation - CAV2018
Baltimore, Maryland, USA, May 14 – 16, 2018
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228
liquid in addition to the possible presence of gas nuclei bubbles. The main difference with previous models and
studies is the way of dealing with gaseous cavitation and vaporous cavitation. In current study, both kinds of
cavitation are resulting from the thermodynamics equilibrium solver. Due to the limited space here, the equations of
the newly developed flow and thermodynamics’ model solvers cannot be described in this article. More explanations
it can be found in [18]. This paper has been organized as following: firstly, one dimensional (1D) test cases results
including a double expansion tube and a shock tube are presented. Next, the numerical results obtained using two-
dimensional (2D) simulations of real-size cavitating nozzle are discussed. Finally, conclusions section summarize
the main findings of this article.
1. Results and discussion
1.1. 1D Double expansion tube
This test case consists of 1 m long tube filled with liquid water (H2O) at atmospheric pressure and with temperature
293K. The initial mass fraction of gas nitrogen (N2) is 1e-5, and for water is 0.99999. The initial discontinuity is set
at 0.5 m. At this location, the velocity (1 m/s) in the left side has been specified the same but opposite to the right
side. The computational time is 3.2ms and the numerical results have been compared with the results from
Chiapolino et al.,[5] model in which Stiffened Gas (SG) EoS has been chosen. As shown in Fig.1, the profiles of
pressure, temperature and velocity are in a good agreement. Nevertheless, the mass fraction of water vapor induced
by the expansion wave has presented some difference which are due to the different initializations of the simulations
as shown in Fig .1(d). Indeed, in this study, the flow has been assumed to be in saturated state or phase equilibrium
state since the beginning of the simulation, which seems to be not the case for the Chiapolino et al. [5] simulations.
(Symbols). The dashed lines are the initial conditions. The present computations were conducted with 1000 cells and a CFL number equals to
1.2. Shock Tube
The shock tube is 1 meter long and with the initial discontinuity at 0.5m. The pressure ratio is 2 (2 bar in the
left and 1 bar in the right) and with a temperature of 293K throughout the tube. The N2 mass fraction is 0.98,
and 0.02 for water. The results shown in Fig.2 are at 1ms. A good agreement has been achieved when
comparing with Chiapolino et al. [5]. As mentioned above, the difference of initial water vapor fraction has
resulted in some discrepancies in the final amount of vapor mass fraction. But, the spreading trend of
compression wave and rarefaction wave is very similar, particularly for the comparison of pressure and
velocity, as shown in Fig .2(a, c). However, the difference of initial vapor mass fraction has induced a
10th International Symposium on Cavitation - CAV2018
Baltimore, Maryland, USA, May 14 – 16, 2018
CAV18-05046
significantly higher temperature in the right side (Fig .2(b)) along with a stronger evaporation rate (Fig .2(d)).
Fig 2. 1D shock tube at t=1ms. The computational results(solid line) are compared with results from Chiapolino et al. [5] (Symbols). The
dashed lines are the initial conditions. The computations were conducted with 1000 cells and a CFL number equals to 0.2
1.3. Nozzle Cavitation
In this section, the suggested 6EQ-PR model has been used to simulate the cavitation phenomenon in a real size 2D
nozzle. The fuel used here is gasoline calibration fluid (Viscor 16BR). Relevant fuel properties can be found in ref
[2]. The non-condensable gas adopted is N2. Firstly, the phase equilibrium analysis about Viscor and N2 system is
presented. Then, some results and discussion of the cavitation modelling in the nozzle are reported.
1.3.1 Vapor liquid equilibrium analysis of (Viscor - N2 ) system
To better understand the effect of dissolved gas on cavitation, it is essential to investigate the thermodynamic
equilibrium behavior of Viscor and N2. The main method used for thermodynamics equilibrium analysis at given
temperature and pressure is isothermal flash computation (TP flash) [13]. Firstly, the vapor-liquid equilibrium
computation of (n-heptane - N2) system is conducted and compared with experimental data [8]. A very good
agreement has been obtained (Fig. 3(a)) except in the critical point zone where PR-EoS is known to fail. Then the
phase equilibrium computation is switched back to (Viscor - N2) system. One important parameter in
thermodynamics theory is the feed vapor mole fraction, usually denoted “psi” (Ψ). In the current (Viscor- N2)
system, this parameter corresponds to the overall amount of vapor which includes the vaporized fuel and
undissolved gaseous N2. As shown in Fig. 3(b), at any given T and P, with the increase of N2 molar fraction in the
feed, Ψ increases from negative to positive values. This imply the well-known phenomenon that adding a high
amount of N2 to the system leads to a saturated system state that will be more prone to nucleation and promote
cavitation. Recent research has confirmed this phenomenon with experiments [12], in which they have attributed the
strengthening of cavitation to the intensifying of cavitating nuclei coming from the dissolved gas. Therefore, the
dissolved gas nuclei can help decrease the energy needed to form a bubble and reduce the tensile strength of fluid.
As a matter of fact, the negative Ψ in Fig. 3(b) means no vapor is generated in the flow and all the N2 present in the
system is fully dissolved inside the liquid. In this case, the fluid is in a single liquid phase state. In addition, phase
transition (i.e. cavitation inception) pressure increases with N2 concentration in the feed, as shown in Fig. 3(b). For
high N2 concentration in the feed (i.e. standard liquid fuels), cavitation may appear in region at relatively high-
pressures. This kind of cavitation has been called “gaseous cavitation”, as the incepted nucleus in this condition
contains mostly undissolved N2. Inversely, for low N2 concentration in the feed (i.e. degassed liquid fuels),
cavitation will appear only in relatively low-pressure regions. This kind of cavitation has been called “vaporous
10th International Symposium on Cavitation - CAV2018
Baltimore, Maryland, USA, May 14 – 16, 2018
CAV18-05046
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229
liquid in addition to the possible presence of gas nuclei bubbles. The main difference with previous models and
studies is the way of dealing with gaseous cavitation and vaporous cavitation. In current study, both kinds of
cavitation are resulting from the thermodynamics equilibrium solver. Due to the limited space here, the equations of
the newly developed flow and thermodynamics’ model solvers cannot be described in this article. More explanations
it can be found in [18]. This paper has been organized as following: firstly, one dimensional (1D) test cases results
including a double expansion tube and a shock tube are presented. Next, the numerical results obtained using two-
dimensional (2D) simulations of real-size cavitating nozzle are discussed. Finally, conclusions section summarize
the main findings of this article.
1. Results and discussion
1.1. 1D Double expansion tube
This test case consists of 1 m long tube filled with liquid water (H2O) at atmospheric pressure and with temperature
293K. The initial mass fraction of gas nitrogen (N2) is 1e-5, and for water is 0.99999. The initial discontinuity is set
at 0.5 m. At this location, the velocity (1 m/s) in the left side has been specified the same but opposite to the right
side. The computational time is 3.2ms and the numerical results have been compared with the results from
Chiapolino et al.,[5] model in which Stiffened Gas (SG) EoS has been chosen. As shown in Fig.1, the profiles of
pressure, temperature and velocity are in a good agreement. Nevertheless, the mass fraction of water vapor induced
by the expansion wave has presented some difference which are due to the different initializations of the simulations
as shown in Fig .1(d). Indeed, in this study, the flow has been assumed to be in saturated state or phase equilibrium
state since the beginning of the simulation, which seems to be not the case for the Chiapolino et al. [5] simulations.
Fig 1. 1D Double expansion tube at t=3.2ms. The computational results (solid line) are compared with results from Chiapolino et al. [5]
(Symbols). The dashed lines are the initial conditions. The present computations were conducted with 1000 cells and a CFL number equals to
0.2.
1.2. Shock Tube
The shock tube is 1 meter long and with the initial discontinuity at 0.5m. The pressure ratio is 2 (2 bar in the
left and 1 bar in the right) and with a temperature of 293K throughout the tube. The N2 mass fraction is 0.98,
and 0.02 for water. The results shown in Fig.2 are at 1ms. A good agreement has been achieved when
comparing with Chiapolino et al. [5]. As mentioned above, the difference of initial water vapor fraction has
resulted in some discrepancies in the final amount of vapor mass fraction. But, the spreading trend of
compression wave and rarefaction wave is very similar, particularly for the comparison of pressure and
velocity, as shown in Fig .2(a, c). However, the difference of initial vapor mass fraction has induced a
10th International Symposium on Cavitation - CAV2018
Baltimore, Maryland, USA, May 14 – 16, 2018
CAV18-05046
significantly higher temperature in the right side (Fig .2(b)) along with a stronger evaporation rate (Fig .2(d)).
Fig 2. 1D shock tube at t=1ms. The computational results(solid line) are compared with results from Chiapolino et al. [5] (Symbols). The
dashed lines are the initial conditions. The computations were conducted with 1000 cells and a CFL number equals to 0.2
1.3. Nozzle Cavitation
In this section, the suggested 6EQ-PR model has been used to simulate the cavitation phenomenon in a real size 2D
nozzle. The fuel used here is gasoline calibration fluid (Viscor 16BR). Relevant fuel properties can be found in ref
[2]. The non-condensable gas adopted is N2. Firstly, the phase equilibrium analysis about Viscor and N2 system is
presented. Then, some results and discussion of the cavitation modelling in the nozzle are reported.
1.3.1 Vapor liquid equilibrium analysis of (Viscor - N2 ) system
To better understand the effect of dissolved gas on cavitation, it is essential to investigate the thermodynamic
equilibrium behavior of Viscor and N2. The main method used for thermodynamics equilibrium analysis at given
temperature and pressure is isothermal flash computation (TP flash) [13]. Firstly, the vapor-liquid equilibrium
computation of (n-heptane - N2) system is conducted and compared with experimental data [8]. A very good
agreement has been obtained (Fig. 3(a)) except in the critical point zone where PR-EoS is known to fail. Then the
phase equilibrium computation is switched back to (Viscor - N2) system. One important parameter in
thermodynamics theory is the feed vapor mole fraction, usually denoted “psi” (Ψ). In the current (Viscor- N2)
system, this parameter corresponds to the overall amount of vapor which includes the vaporized fuel and
undissolved gaseous N2. As shown in Fig. 3(b), at any given T and P, with the increase of N2 molar fraction in the
feed, Ψ increases from negative to positive values. This imply the well-known phenomenon that adding a high
amount of N2 to the system leads to a saturated system state that will be more prone to nucleation and promote
cavitation. Recent research has confirmed this phenomenon with experiments [12], in which they have attributed the
strengthening of cavitation to the intensifying of cavitating nuclei coming from the dissolved gas. Therefore, the
dissolved gas nuclei can help decrease the energy needed to form a bubble and reduce the tensile strength of fluid.
As a matter of fact, the negative Ψ in Fig. 3(b) means no vapor is generated in the flow and all the N2 present in the
system is fully dissolved inside the liquid. In this case, the fluid is in a single liquid phase state. In addition, phase
transition (i.e. cavitation inception) pressure increases with N2 concentration in the feed, as shown in Fig. 3(b). For
high N2 concentration in the feed (i.e. standard liquid fuels), cavitation may appear in region at relatively high-
pressures. This kind of cavitation has been called “gaseous cavitation”, as the incepted nucleus in this condition
contains mostly undissolved N2. Inversely, for low N2 concentration in the feed (i.e. degassed liquid fuels),
cavitation will appear only in relatively low-pressure regions. This kind of cavitation has been called “vaporous
10th International Symposium on Cavitation - CAV2018
Baltimore, Maryland, USA, May 14 – 16, 2018
CAV18-05046
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230
cavitation”, as the incepted nucleus in this condition contain mostly fuel-vapor. This thermodynamics analysis
corroborates the Habchi’s numerical findings [9].
(a) (b)
Fig 3. (a) Phase equilibrium results compared to n-heptane experimental data[8]; (b) Evolution of vapor fraction with the molar number of
nitrogen in the feed at T=293 K and P=1-10 bar
1.3.2 Results discussion about the cavitation modeling
The 2D cavitating nozzle used here is axisymmetric with sharp inlet edge and a diameter of 500 μm and a length of
2.5mm. Detailed descriptions can be found in [6]. For computational efficiency reason, 1 8
of the original geometry
configuration has been simulated, as shown in Fig. 4. Mesh is refined around the orifice zone and the minimum
resolution of grid reaches 6 μm and the total cell number counts 25982.The nozzle outlet is submerged according to
the original experimental condition [6]. The inlet and outlet are set with pressure boundary conditions with the 10
bar and 1 bar, respectively.
Due to the fact that the objective of this study is to investigate the effect of dissolved nitrogen on cavitation, a two-
phase condition has been selected as initial state. The initial pressure for the overall flow field has been set the same
as the inlet pressure (10 bar). Initially, the fluid has been assumed at equilibrium state with an initial mass fraction of
dissolved gas (N2) in liquid phase around 2e-3 corresponding to a gas volume fraction equaling to 1e-2(Table 1).
Table 1 Test cases initial conditions
T=293 K, P=10 bar
Case No Feed(
z2)
Vapor fraction Y_dissolved_n2
Gas volume
fraction(
)
1
8.08e-3
1e-3
2.01e-3
1e-2
z denotes the molar fraction of nitrogen in the feed
The modeling results shown below are at t=1.4ms. As shown in Fig. 5, the gas volume fraction (denoted alphagaz in
in Fig. 5(a)) can be decomposed into two contributions: the gaseous cavitation (Fig. 5(b)) and the vaporous
cavitation in Fig. 5(c). Consistent with our thermodynamics analysis reported above, one may see in this latter
Figure that vaporous cavitation appears mostly near the orifice sharp edge, where the pressure is the lowest (see Fig.
6(a and d)). In contrast, gaseous cavitation is present throughout the computational domain. In addition, vaporous
cavitation is very weak (see palette maximum value in Fig. 5(c)) relatively to gaseous cavitation (see Fig. 6(b)). This
means that the overall gas volume fraction (Fig. 5(a)) is mostly due to the expansion of the initial nucleus volume
fraction (1e-2 in this case). This expansion is nearly an isothermal process as the temperature is nearly constant.
Indeed, the maximum temperature deviation is more or less equal to 0.2K in the orifice (see Fig. 6(d)).
In addition, these two kinds of cavitation are mutually restricted. To further confirm this phenomenon, the
developing process of cavitation in the inlet corner has been investigated using our phase equilibrium solver. The
molar fraction of nitrogen is set the same with the initial value 8.08e-3. As seen in Fig.6 (c), with the pressure
10th International Symposium on Cavitation - CAV2018
Baltimore, Maryland, USA, May 14 – 16, 2018
CAV18-05046
lowering down to 0.5 bar, the evaporation ratio of fuel becomes much higher which further proves the vaporous
cavitation is sensitive to the fast pressure drop. Dissolved nitrogen concentration also becomes higher with pressure
increasing. However, the increasing of dissolved gas has seen the reduction of molar fraction of vaporous fuel (Fig.6
(c)) and a slow increment of gaseous nitrogen. This may imply that the dissolved gas is favorable to the formation of
gaseous cavitation instead of the vaporous cavitation. Besides, for completeness of this real-fluid study, the
influence of dissolved gas on the density and heat capacity have also been shown in Fig.6 (e, f). Overall, the liquid
density has not been through significant variation with the addition of dissolved gas. In contrast, liquid heat capacity
shows a clear change especially inside the cavitation pocket. The change might be caused by the small temperature
variation, shown in Fig. 6(d).
As seen from the void fraction distribution (Fig. (5)), the cavitation appeared in the corner and center of inlet orifice
qualitatively agreed with Duke et al. [6, 7] X-ray experiments. But the cavity cloud appearing in the center of orifice
close to the outlet has not matched the experimental results well [2]. The reason may come from the relatively high
initial molar fraction of nitrogen zN2 in this case (Table 1). Indeed, in previous research [2], a value equaling to 7e-5
has been used. But, thermodynamically speaking, such zN2 low value corresponds to a single-phase state as can be
seen in Fig. 3(b). Therefore, future endeavors are going to tackle these issues.
Fig 4. Single-hole configuration and mesh refining with total 25982 cells and minimum grid resolution of 6 .
(a)
(b)
(c)
Fig 5. Results of the 2D cavitating nozzle (case 1)
(a)
(b)
(c)
(d)
(e)
(f)
Fig 6. (a) Variation of void fraction with Delta_P; (b) Variation of vapor molar fraction of fuel and nitrogen with Delta_P at r/R=0.99; (c)
Variation of vapor molar fraction of fuel and nitrogen with pressure at T=293K and     ; (d) Variation of pressure and temperature; (e,
f) Variation of density and heat capacity with dissolved nitrogen.   
 .
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231
cavitation”, as the incepted nucleus in this condition contain mostly fuel-vapor. This thermodynamics analysis
corroborates the Habchi’s numerical findings [9].
(a)
(b)
Fig 3. (a) Phase equilibrium results compared to n-heptane experimental data[8]; (b) Evolution of vapor fraction with the molar number of
nitrogen in the feed at T=293 K and P=1-10 bar
1.3.2 Results discussion about the cavitation modeling
The 2D cavitating nozzle used here is axisymmetric with sharp inlet edge and a diameter of 500 μm and a length of
2.5mm. Detailed descriptions can be found in [6]. For computational efficiency reason, 1 8
of the original geometry
configuration has been simulated, as shown in Fig. 4. Mesh is refined around the orifice zone and the minimum
resolution of grid reaches 6 μm and the total cell number counts 25982.The nozzle outlet is submerged according to
the original experimental condition [6]. The inlet and outlet are set with pressure boundary conditions with the 10
bar and 1 bar, respectively.
Due to the fact that the objective of this study is to investigate the effect of dissolved nitrogen on cavitation, a two-
phase condition has been selected as initial state. The initial pressure for the overall flow field has been set the same
as the inlet pressure (10 bar). Initially, the fluid has been assumed at equilibrium state with an initial mass fraction of
dissolved gas (N2) in liquid phase around 2e-3 corresponding to a gas volume fraction equaling to 1e-2(Table 1).
Table 1 Test cases initial conditions
T=293 K, P=10 bar
Case No
Feed(z2)
Vapor fraction
Y_dissolved_n2
Gas volume
fraction()
1
8.08e-3
1e-3
2.01e-3
1e-2
z denotes the molar fraction of nitrogen in the feed
The modeling results shown below are at t=1.4ms. As shown in Fig. 5, the gas volume fraction (denoted alphagaz in
in Fig. 5(a)) can be decomposed into two contributions: the gaseous cavitation (Fig. 5(b)) and the vaporous
cavitation in Fig. 5(c). Consistent with our thermodynamics analysis reported above, one may see in this latter
Figure that vaporous cavitation appears mostly near the orifice sharp edge, where the pressure is the lowest (see Fig.
6(a and d)). In contrast, gaseous cavitation is present throughout the computational domain. In addition, vaporous
cavitation is very weak (see palette maximum value in Fig. 5(c)) relatively to gaseous cavitation (see Fig. 6(b)). This
means that the overall gas volume fraction (Fig. 5(a)) is mostly due to the expansion of the initial nucleus volume
fraction (1e-2 in this case). This expansion is nearly an isothermal process as the temperature is nearly constant.
Indeed, the maximum temperature deviation is more or less equal to 0.2K in the orifice (see Fig. 6(d)).
In addition, these two kinds of cavitation are mutually restricted. To further confirm this phenomenon, the
developing process of cavitation in the inlet corner has been investigated using our phase equilibrium solver. The
molar fraction of nitrogen is set the same with the initial value 8.08e-3. As seen in Fig.6 (c), with the pressure
10th International Symposium on Cavitation - CAV2018
Baltimore, Maryland, USA, May 14 – 16, 2018
CAV18-05046
lowering down to 0.5 bar, the evaporation ratio of fuel becomes much higher which further proves the vaporous
cavitation is sensitive to the fast pressure drop. Dissolved nitrogen concentration also becomes higher with pressure
increasing. However, the increasing of dissolved gas has seen the reduction of molar fraction of vaporous fuel (Fig.6
(c)) and a slow increment of gaseous nitrogen. This may imply that the dissolved gas is favorable to the formation of
gaseous cavitation instead of the vaporous cavitation. Besides, for completeness of this real-fluid study, the
influence of dissolved gas on the density and heat capacity have also been shown in Fig.6 (e, f). Overall, the liquid
density has not been through significant variation with the addition of dissolved gas. In contrast, liquid heat capacity
shows a clear change especially inside the cavitation pocket. The change might be caused by the small temperature
variation, shown in Fig. 6(d).
As seen from the void fraction distribution (Fig. (5)), the cavitation appeared in the corner and center of inlet orifice
qualitatively agreed with Duke et al. [6, 7] X-ray experiments. But the cavity cloud appearing in the center of orifice
close to the outlet has not matched the experimental results well [2]. The reason may come from the relatively high
initial molar fraction of nitrogen zN2 in this case (Table 1). Indeed, in previous research [2], a value equaling to 7e-5
has been used. But, thermodynamically speaking, such zN2 low value corresponds to a single-phase state as can be
seen in Fig. 3(b). Therefore, future endeavors are going to tackle these issues.
Fig 4. Single-hole configuration and mesh refining with total 25982 cells and minimum grid resolution of 6 .
(a)
(b)
(c)
Fig 5. Results of the 2D cavitating nozzle (case 1)
(a)
(b)
(c)
(d)
(e)
(f)
Fig 6. (a) Variation of void fraction with Delta_P; (b) Variation of vapor molar fraction of fuel and nitrogen with Delta_P at r/R=0.99; (c)
Variation of vapor molar fraction of fuel and nitrogen with pressure at T=293K and

   
; (d) Variation of pressure and temperature; (e,
f) Variation of density and heat capacity with dissolved nitrogen.
  

.
10th International Symposium on Cavitation - CAV2018
Baltimore, Maryland, USA, May 14 – 16, 2018
CAV18-05046
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232
2. Conclusion
A fully compressible two fluid model based on phase equilibrium theory with real fluid equation of state has been
described and first validations have been shown and discussed in current study. One dimensional test cases including
shock tube and double expansion tube have been conducted and validated with previous published results. Then, the
model was used to simulate the cavitation phenomenon inside a single-hole nozzle. With phase equilibrium theory,
the simulation has proved the effect of dissolved gas on cavitation. On the one hand, it has been shown that the
gaseous cavitation is intensified with the increasing dissolved gas. On the other hand, the vaporous cavitation is
restricted if there is more dissolved gas is in the initial feed. In addition, the model also proves that the dissolved gas
concentration can become much higher with increasing pressure and some physical properties like density and heat
capacity can be impacted by the dissolved gas in the fluid. Finally, the two-fluid model combined with a real-fluid
phase equilibrium solver has been shown to have more advantages in revealing the cavitation physics details than
previously published cavitation models using barotropic or incompressible liquid assumptions. However, more
endeavors are needed to tackle real-fluid phase-transition issues when the initial fluid is in single phase state.
Acknowledgement
This project has received funding from the European Union Horizon 2020 Research and Innovation program. Grant
Agreement No 675528 for the IPPAD project.
References
[1] Abudour, A. M., Mohammad, S. A., Robinson, R. L., and Gasem, K. A. 2012. Volume-translated Peng–Robinson equation
of state for saturated and single-phase liquid densities. Fluid Phase Equilibria 335, 7487.
[2] Battistoni, M., Duke, D. J., Swantek, A. B., Tilocco, F. Z., Powell, C. F., and Som, S. 2015. nozzles. Atomiz Spr 25, 6, 453
483.
[3] Bohbot, J., Gillet, N., and Benkenida, A. 2009. IFP-C3D. An Unstructured Parallel Solver for Reactive Compressible Gas
Flow with Spray. Oil & Gas Science and Technology - Rev. IFP 64, 3, 309335.
[4] Chawki Habchi. 2013. A Gibbs free Energy Relaxation Model for Cavitation Simulation in Diesel injectors.
DOI=10.13140/2.1.4615.7761.
[5] Chiapolino, A., Boivin, P., and Saurel, R. 2017. A simple phase transition relaxation solver for liquid-vapor flows. Int. J.
Numer. Meth. Fluids 83, 7, 583605.
[6] Duke, D. J., Kastengren, A. L., Swantek, A. B., Matusik, K. E., and Powell, C. F. 2016. X-ray fluorescence measurements
of dissolved gas and cavitation. Exp Fluids 57, 10, 538.
[7] Duke, D. J., Schmidt, D. P., Neroorkar, K., Kastengren, A. L., and Powell, C. F. 2013. High-resolution large eddy
simulations of cavitating gasoline–ethanol blends. International Journal of Engine Research 14, 6, 578589.
[8] García-Sánchez, F., Eliosa-Jiménez, G., Silva-Oliver, G., and Godínez-Silva, A. 2007. High-pressure (vapor+liquid)
equilibria in the (nitrogen+n-heptane) system. The Journal of Chemical Thermodynamics 39, 6, 893905.
[9] Gonçalves, F. M., Castier, M., and Araújo, O. Q. F. 2007. Dynamic simulation of flash drums using rigorous physical
property calculations. Braz. J. Chem. Eng. 24, 2, 277286.
[10] Habchi, C. 2015. A Gibbs Energy Relaxation (GERM) model for cavitation simulation. Atomiz Spr 25, 4, 317334.
[11] Habchi, C., Bohbot, J., Schmid, A., and Herrmann, K. 2015. A comprehensive Two-Fluid Model for Cavitation and
Primary Atomization Modelling of liquid jets - Application to a large marine Diesel injector. J. Phys.: Conf. Ser. 656,
12084.
[12] Habchi, C., Dumont, N., and Simonin, O. 2008. Multidimensional simulation of cavitating flows in diesel injectors by a
Homogeneous Mixture Modelling approach. Atomiz Spr 18, 2, 129162.
[13] J.B. Moreau, O. Simonin, C. Habchi, Ed. 2004. A numerical study of cavitation influence on diesel jet atomization. DOI
10.13140/2.1.2813.5369
[14] Kastengren, A. L., Tilocco, F. Z., Duke, D. J., Powell, C. F., Zhang, X., and Moon, S. 2014. Time-resolved x-ray
radiography of sprays from Engine Combustion Network SPRAY A diesel injectors. Atomiz Spr 24, 3, 251272.
[15] Li, B., Gu, Y., and Chen, M. 2017. An experimental study on the cavitation of water with dissolved gases. Exp Fluids 58,
12, 1065.
[16] Mori, Y., Hijikata, K., and Nagatani, T. 1976. Effect of dissolved gas on bubble nucleation. International Journal of Heat
and Mass Transfer 19, 10, 11531159.
[17] Roth, H., Gavaises, M., and Arcoumanis, C. 2002. Cavitation Initiation, Its Development and Link with Flow Turbulence
in Diesel Injector Nozzles. In . SAE Technical Paper Series. SAE International400 Commonwealth Drive, Warrendale, PA,
United States. DOI=10.4271/2002-01-0214.
[18] Songzhi Yang, Chaouki Habchi, and Rafael Lugo. 2017. Subcritical and supercritical flow modeling using real-fluid
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10th International Symposium on Cavitation - CAV2018
Baltimore, Maryland, USA, May 14 – 16, 2018
CAV18-05046
Cavitation nuclei and tensile strength of water
K.A. Mørch
Technical University of Denmark, DK-2800 Kongens Lyngby
Abstract
The paper presents a discussion of the possibility of stabilization of cavitation nuclei of very high tensile
strength in pure water by liquefaction of gas molecules at the water-gas interface of a gas bubble
approaching dissolution. Likewise, the formation of such nuclei by molecular clustering of gas molecules
already in solution in the water is considered. Finally, we discuss the origin of skin-stabilized gas bubbles,
in particular at solid-water interfaces. Skin-stabilized gas bubbles in diffusion balance can explain most
experimental results for water of low to moderate tensile strength, including effects of pressurization and
de-pressurization.
Keywords: cavitation nuclei, tensile strength of water, liquefaction of gas, interfacial water
Problem analysis
Any gas bubble in water may serve as cavitation nucleus if exposed to sufficient tensile stress, but precise knowledge
of cavitation nuclei in engineering systems is quite limited. Their tensile strength varies from values close to zero in
plain water to values of almost 30 MPa, obtainable in highly purified and degassed water using focused ultrasonic
wave-packets1, and it comes up to the theoretical values for H2O of the order of 140 MPa, when strained in inclusions
of quartz2. Normally water contains dissolved gas, as well as particles and contaminating agents, which are decisive
for its tensile strength. Further, in experiments the techniques used for measurement of tensile strength influence the
results achieved because cavitation nuclei are not invariant.
A gas bubble in perfectly pure water is unstable and dissolves due to the Laplace force3. However, when the bubble
shrinks to nanometer size and the gas pressure inside the bubble becomes very high, an increasing fraction of the
closely packed gas molecules, usually di-atomic molecules, left in the bubble are exposed to bonding to the water
molecules at the bubble boundary, and they exchange their momentum with them. The interfacial gas expectedly
liquefies with a dense structure at a much higher temperature than the gas remaining in the bubble core, and forms
a very small stable cavitation nucleus already at the temperatures of liquid cold water.
Likewise, when at a free surface a gas molecule has entered into liquid water it in principle constitutes a cavitation
nucleus. Locally it prevents the water molecules from setting up hydrogen bonds between themselves in the space
occupied by the gas molecule, but it is a nucleus too small to reduce notably the tensile strength of ideally pure liquid
H2O. However, when gas molecules in the water meet over time, they stick together because clustering reduces the
potential energy of the system, the surface area of the water bounding the cluster being smaller than the sum of areas
bounding the individual gas molecules in solution. Such a cluster may grow until it is about to shift into a gaseous
state - a balance is achieved.
It is very difficult to remove all gas molecules from water. Therefore, stable nuclei of liquefied gas may explain why
theoretical values of tensile strength of water are difficult to achieve experimentally, even when the water is highly
purified, and degassed as well as possible.
The water we meet in our daily life, e.g. seawater and tap water, contains gas in solution, it is usually contaminated
by alien substances, and it carries particles - and particles as well as bounding walls are known to harbor cavitation
nuclei. A skin of contaminating components may stabilize free gas bubbles of size in the micrometer range4,5, hereby
allowing them to be in gas diffusion balance at the prevailing pressure6, and causes a low tensile strength of such
10th International Symposium on Cavitation - CAV2018
Baltimore, Maryland, USA, May 14 – 16, 2018
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© 2018 ASME
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... For the sake of completeness, it is also worth mentioning a number of recent works employing HEM models, focusing on cavitation and sprays at transcritical and supercritical conditions of ECN Spray-A, using LES and real fluid thermodynamics (Peng-Robinson Equation of State) [61,62]. The same methodology has also been used to detect cavitation in internal injector flows [63,64]. However, these investigations did not involve any attempt to describe erosion. ...
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