# Influence of the Cavitation Model on the Simulation of  Cloud Cavitation on 2D Foil Section

Authors: S. Frikha; O. Coutier-Delgosha; J. A. Astolfi
Journal: International Journal of Rotating Machinery
 (2008)

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## Abstract

For numerical simulations of cavitating flows, many physical models are currently used. One approach is the void fraction transport equation-based model including source terms for vaporization and condensation processes. Various source terms have been proposed by different researchers. However, they have been tested only in different flow configurations, which make direct comparisons between the results difficult. A comparative study, based on the expression of the source terms as a function of the pressure, is presented in the present paper. This analytical approach demonstrates a large resemblance between the models, and it also clarifies the influence of the model parameters on the vaporization and condensation terms and, therefore, on the cavity shape and behavior. Some of the models were also tested using a 2D CFD code in configurations of cavitation on two-dimensional foil sections. Void fraction distributions and frequency of the cavity oscillations were compared to existing experimental measurements. These numerical results confirm the analytical study.


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## Body

## 1. Introduction

In liquid flows, cavitation
generally occurs if the pressure drops below the vapor pressure; it can be observed in a
wide variety of propulsion and power systems like pumps, nozzles, injectors,
marine propellers, and hydrofoils. The cavitation development may be the origin
of several negative effects, such as noise, vibrations, performance
alterations, erosion, and structural damages.Numerical studies and simulations
of cavitation have been pursued for years, but it is still a very difficult and
challenging task to predict such complex unsteady and two-phase flows with an
acceptable accuracy.These studies can be put mainly into two
categories: interface tracking methods [1, 2]
and homogeneous equilibrium flow models [3–7].In the first category, the cavity region is
generally assumed to have a constant pressure equal to the vapor pressure of
the corresponding liquid and the computations are performed only for the liquid
phase.In the second category, the single-fluid modeling
approach is employed for both phases. Mass and momentum transfers between the two phases
are managed either by a barotropic state law or by a void fraction transport equation.In the present study, the
attention is focused on this second category, which has demonstrated in the
last 20 years its capability to reproduce the main features of steady and
unsteady cavitating flows [3]. Various source terms for the
void fraction transport equation have been proposed in the literature, but most
of the time, they have been applied to different flow configurations. Direct
comparisons between the models are thus difficult to handle. However, the test
case proposed for the Fifth International Symposium on Cavitation (CAV2003) in Osaka
has revealed some
similarities between results obtained with different models [8–11].The present work consists of a comparative study between the
different vaporization and condensation terms proposed for the void fraction
transport equation. The ability of each one to reproduce the
behavior of cavitating unsteady turbulent flow is discussed on
the basis of the analytical expression of the source terms and 2D numerical
simulations.In Section1 of this
paper, the
expressions of the vaporization and condensation terms proposed in the
literature are given. Then, both source terms are
expressed as a function of the pressure and compared in Section 2. In Section 3, some of the models are evaluated in two
configurations of two-dimensional foil sections. 2D simulations are performed
and the results are compared to existing experimental data. Comparisons are
based on the maximum length of the attached cavity, the frequency of its
oscillations, and the void fraction distribution within the sheet cavity.In Section4, the influence of some parameters of the vaporization
and condensation terms, such as the empirical constants and the exponents of P−Pv and αl, is discussed. The objective is to
assess their respective effect on the shape and behavior of the sheet cavity.


## 2. Cavitation Modeling

The attention is focused in the following
analysis on the single-fluid modeling approach
associated with the Reynolds-averaged Navier-Stokes (RANS) equations.The general balance
equations for the single-fluid modeling can be found in [12]. In this previous
paper, the general assumptions common to nearly all the current cavitation
models have been clarified: (i) no slip between the vapor and the liquid
phases, which leads to identical velocities in both phases, (ii) equal local
pressure in the vapor and in the liquid. This leads to the following simplified
four equations: (1)∂ρ∂t+∇→⋅(ρu→)=0,(2)∂α∂t+∇(αu→)=A, (equations derived from the mass balance, where A is the mass transfer
between the two phases) (3)∂∂t(ρu→)+∇(ρu→2)=∇→Π,(4)∂∂t(αu→)+∇(αu→2)=αρv∇→Π+B⇀,  (equations derived from the
momentum balance, where B is the momentum transfer between the two phases).It has been shown in [12] that a
third assumption is inherent to all current cavitation models: (4) is never
resolved, which implicitly leads to the following expression for momentum
transfer term B: (5)B→=Au→+(ρl−ρv)α(1−α)∇→Πρv⋅ρ.The
first term in the expression ofB→ corresponds to a momentum transfer directly
linked to the mass transfer during vaporization and condensation processes: as
a matter of fact, let us consider inside a volume V an elementary mass δm of liquid moving at velocity u→.
If these liquid particles vaporize, δm is transferred from the liquid phase to the
vapor one, and the vapor phase also gains the momentum δm⋅u→.The second term in the expression ofB→ results from the momentum fluxes due to the
inertial effects, that is, the stresses applied by one phase on the other when
this last one accelerates or decelerates. The present form of this term means that
if a volume of liquid is accelerated, the resistance applied by the adjacent
volumes of vapor is transmitted exclusively by the local stress tensor. The
forces between vapor and liquid are thus only due to the pressure and the
classic viscous stresses, as in a single-phase flow.So, in
practice, (1) and (3), which are the classical RANS equations for a single-phase
flow, are coupled either via
supplementary void fraction transport (2) or simply with a barotropic state law
that governs the density evolution according to the local pressure variation.
In the second case, the density is directly linked to the pressure, which
prevents from taking into account any vaporization delay or separate treatments
of the vaporization and condensation sequences. Moreover, coupling the
barotropic state law with the presence of noncondensable gas and/or two-phase
inlet flow is quite complicated. Conversely, these limitations can be avoided
by using the void fraction transport equation.In the present study, several models
based on the void fraction transport equation are compared. They are
characterized by different expressions of termA in (2). In the literature, A
is usually divided into two terms: A=m˙++m˙− where m˙+ and m˙− stand for the vaporization (vapor production)
and condensation (vapor destruction) processes, respectively. Table 1 presents
an overview of several forms of m˙+ and m˙− proposed by different researchers. All the
models use empirical constants Cprod and Cdest to calibrate the mass transfers.Table 1
Overview of some cavitation models existing in the literature.
AuthorsVaporization/condensation terms (m˙+/m˙−)Reboud and Stutz [5]A=1θρvmin[αmin,α,(1−α)](−σ−Cp)Kunz et
al.
[13, 14]m˙+=Cprodρvαlmin(0,p−pv)((1/2)ρlU∞2)t∞,m˙−=Cdestρvαl2(1−αl)t∞Merkle et al.
[4]m˙+=CprodρlMin(0,p−pv)αlρv((1/2)ρlU∞2)t∞,m˙−=CdestMax(0,p−pv)(1−αl)((1/2)ρlU∞2)t∞Dauby et al. [15]m˙+=CprodMin(0,p−pv)αl,m˙−=CdestMax(0,p−pv)(1−αl)Saito et al.
[8]m˙+=CprodAα(1−α)ρlρvMin(0,p−pv)2πRTs,m˙−=CdestAα(1−α)Max(0,p−pv)2πRTsSinghal et al.
[7]m˙+=CprodU∞γρlρv[−23Min(0,p−pv)ρl]1/2*(1−fv),m˙−=CdestU∞γρlρv[23Max(0,p−pv)ρl]1/2*fvSchnerr and Sauer [16]m˙+=Cprod3αlRb−23Min(0,p−pv)ρl,m˙−=Cdest3(1−αl)Rb23Max(0,p−pv)ρl


## 3. Comparison between the Models

Most of the source terms
depend mainly on the difference between the local pressure and the vapor
pressureP−Pv.  Thus, the following comparison between the models is
based on the expression of the source terms as a function of P−Pv.However,
the void fractionαusually also appears in the expression of the
source terms. To express them as a function of P−Pv only, the barotropic state law of
Delannoy is used (Figure 4). Such a state law enables to link αand
P−Pv, and thus to suppress αin expressions of m˙+ and m˙−. It is clear that the choice of this particular state law is arbitrary.
However, it enables to obtain here expressions depending only on P−Pv, which is
necessary for comparison. It must be also noticed that this transformation has
only minor influence on the differences between the models: using another state
law would lead to slightly different results, but the agreement or disagreement
between the models would remain nearly the same.In literature, empirical factors are determined
through numerical/experimental results and are adjusted for different
geometries and different flow conditions;Cprod=9*105 and Cdest=3*104 for the Kunz model [13], Cprod=0.02 and Cdest=0.01 for the Singhal model [7],
and Cprod=0.1 and Cdest=0.1 for the Saito model [8].The empirical
factors are adjusted here to obtain the same maximum value for the source
terms, in order to make the comparison of the models easier. The empirical factors adopted in this study include constants parameters
of the source terms(ρl,ρv,U∞,…); so, it is not
possible to compare our empirical constants with those proposed by the authors.The empirical factors have the following
values:Cprod=10, Cdest=0.7 for the Kunz model, Cprod=1,
Cdest=1000 for the Merkle model, Cprod=4⋅10−4, Cdest=1.4 for the
Singhal model, Cprod=0.07, Cdest=0.07 for the Visonneau model, Cprod=0.07,
Cdest=70 for the Saito model, and Cprod=2.05*10−5,Cdest=2.05*10−5 for the Saito model.The evolution of the source terms according toP−Pv is presented in Figures 1 and 2. Large
similarities between both vaporization and condensation terms are clear. Some
of the models (Merkle and Reboud or Sauer and Singhal) give identical
expressions of m˙+ and m˙− and should thus result
in exactly the same prediction of cavitating flows in numerical simulations; so
far they would be associated to the same numerical model.Figure 1
Source terms for the vaporization process.Figure 2
Source terms for the condensation process.It can be observed that
the transfer termsm˙+ and m˙− in Table 1, although they seem quite different
at first sight, often involve similar expression of P−Pv and α, which explains the resemblance of the charts
in Figures 1 and 2. For example, m˙+ is usually a combination of P−Pv and αl, with various exponents. Several constants
are systematically included
in m˙+ and m˙−,
which leads to various analytical expressions, but does not change
fundamentally the model. Adjusting them to obtain here the same maximal value
of m˙+ and m˙− makes this point clearer. We also notice that
the slopes of the source terms are different; the more the slope of the source
term of vaporization (condensation) is increased, the more the vaporization
(condensation) is.In the next section,
a numerical study in two
configurations of cavitation on two-dimensional foil sections is performed.
Some of the previous models were tested and the results are compared to
existing experimental data.


## 4. Numerical Study

Two configurations of cavitating flow on 2D foil sections are considered
in this section: a convex foil with an upper flat surface and an NACA 66 foil.
Experimental investigations have been performed previously in both cases, and
data are used in the present study for comparison with numerical simulations.Calculations of the cavitating flow around the two foil sections are performed
with the code IZ, which was developed previously in the LEGI laboratory (Grenoble, France),
with the support of the CNES (French space agency). Details concerning the
numerical model can be found in [17], only the main features are given
hereafter.The code is based on a finite volume
discretization on curvilinear two-dimensional orthogonal mesh. The numerical
resolution consists of a pressure correction method derived from the SIMPLE
algorithm. Each physical time step consists of successive iterations which
march the solution toward convergence. A finite volume discretization and a
second order implicit time integration scheme are applied as follows:(6)∂(ρΦ)∂t=1.5ρn+1Φn+1−2ρnΦn+0.5ρn−1Φn−1Δt,  where Φ  is 1 in the case of the mass
equation, and the velocity components u or v in the case of the momentum
equations.A modifiedk−ε RNG turbulence
model is applied: the turbulent viscosity is μt=f(ρ)×Cμ×k2/ε, where f(ρ)=ρv+(1−α)n(ρl−ρv); n is set to the value of 10 proposed
by Coutier-Delgosha et al. in 2003 [18].The classical boundary conditions
for incompressible flows are applied: imposed inlet velocity and fixed outlet
pressure. A C-type orthogonal mesh is used in the case of the NACA 66 foil, and
an H-type
orthogonal mesh is applied in the case of the convex foil (Figure3).Figure 3
General view of
the grid, flow is from the left.Figure 4
Barotropic state lawρ(P); water 20°C.A 630  ×  50 C-type orthogonal
mesh for the NACA66 foil and 140  ×  70 orthogonal cells for the convex foil.


### 4.1. Study of the Sheet Cavity Unsteady Behavior

The geometry considered in this section is the NACA 66 foil. Its chord
is 150 mm and the angle of attack is 8°. The reference velocity at the inlet of the test
section is Vref=5.33 m/s and the cavitation number is σ=1.3.A calculation based on the
barotropic state law of Delannoy and Kueny [6] is first performed to be compared to the
results that will be obtained with the various models based on the void
fraction transport equation. In this case, (2) is not solved, but the state
law presented in Figure 4 is coupled to (1) and (3) to govern the flow density
variations according to the local pressure variations.For a pressure much higher than the
vapor pressurePv, the flow is composed of pure liquid and the Tait
state law is applied: ρ/ρref=P+P0/Pref+P0n with P0=3108 Pa and n=7 for water.For a pressure much lower thanPv,
the fluid is locally completely vaporized and the density is governed by the
perfect gas law: P/ρ=cste.These two low compressible
configurations are joined in the vapor pressure neighborhood by the central
part of the chart, whose high slope models the high compressibility of the
liquid/vapor.Cmin2=∂P/∂ρ.
Cmin is the minimum speed of sound in the
mixture.The minimum density (indicated by the color) in each section (denoted by
the positionX/Lref in ordinate) as a function of the time (in abscissa).Results of this
calculation are presented in Figures5, 6, and 7. An unsteady
periodical behavior of the sheet cavity is obtained, including vapor cloud shedding. The maximum
length of the attached cavity can be estimated to lmax/c=0.8. The
cavity oscillation frequency is about 11 Hz (Figure 7), which gives a Strouhal
number based on the maximum length of the attached cavity equal to 0.25.Figure 5
Time evolution of the sheet
cavity.(a) Time evolution of the cavity shape
during a single cycle of vapor shedding. The colors indicate the fluid density, white for pure liquid and from red
to blue when the void ratio increases. (b) Velocity field and fluid density during cavity breakoff show a reentrant jet (picture 6 in (a)).
(a)(b)(a) Time evolution of the
pressure coefficient atX/c=0.2 and (b) spectral analysis.
(a)(b)During this cycle, a sheet cavity is
formed, which successively grows and breaks off. The detachment of the rear
part of the cavity is due to a reentrant jet that flows upstream close to the
foil surface. The vapor cloud is convected downstream and finally collapses,
while the residual cavity starts to grow again.Three  transport equation-based
models with source terms of Reboud and Stutz [5], Kunz et al. [13, 19], and Singhal et al. [7] are
now tested, and the results are presented in Figure 19. The maximum length of the
attached cavity, the oscillation frequency, the Strouhal number, and the cavity
behavior during one cycle are indicated for each model. The empirical factors were adjusted to
obtain the same maximum value for the source terms, as in the previous section.Figure19 shows a large resemblance
between the four tested models; the unsteady behavior of the sheet cavity is
similar in all cases: development of a cavity up to the maximum length (about 80% of the
chord), cavity breakoff, cloud detachment, and growth of the residual cavity.Sheet cavity oscillation frequencies
are also all similar, leading to Strouhal numbers Str based on the maximum
cavity length comprised between 0.25 and 0.28. These results are in close
agreement with the experimental measurement, which gave Str = 0.28 in the configuration
of a slightly smaller cavity (L/Lref=0.6). These results confirm
the similarities between the three models based on the void fraction transport
equation. They also suggest that using the barotropic state law instead of (2)
leads to very similar predictions of the sheet cavity behavior.The colors indicate the fluid density, white for pure liquid and from red
to blue when the void ratio increases.


### 4.2. Study of the Void Fraction Distribution

The second foil geometry is a convex foil
characterized by a flat upper surface. Its chord is 150 mm and the present angle
of attack is 4°7. The reference velocity at the inlet of the test section is Vref=6 m/s.Calculations of the cavitating flow on the foil are performed for two sheet cavity
lengths:L/c=0.9 and L/c=0.5, respectively. In both simulations, an
unsteady behavior of the cavity is obtained, as previously in the configuration
of the NACA66 foil section. Discussion is focussed hereafter on the study of
the time-averaged two-phase flow structure of the liquid/vapor mixture inside
the cavity.An X-ray absorption device was applied previously to
investigate the local volume fraction of the vapor phase inside the sheet
cavity. The experiments were carried out by Coutier-Delgosha et al. [17] in the
scope of collaboration between the ENSTA laboratory and the French Atomic
Energy commission (CEA).From these experiments, the
instantaneous and time-averaged distribution of void fraction inside the cavity
was obtained, with a space resolution of 6 mm in the flow direction and 3 mm in the vertical
direction. In the present study, the time-averaged experimental void fraction
measurements at three positions (X=15 mm, 55 mm, and 95 mm downstream from the
leading edge) are considered (Figure 8).Figure 8
Position of the experimental void fraction
measurements (Vref=6 m/s, L/c=90%).The first studied configuration isVref=6 m/s and L/c=0.9. Void
fractions profiles obtained with the barotropic state law of Delannoy and the void fraction transport equation-based
models of Kunz and Reboud are compared to the experimental data in Figure 9.
Note that the empirical constants in the models have been adjusted as in the
previous sections.Comparison between numerical and experimental void fraction distributions (L/c=0.9, Vref=6 m/s).
(a)(b)(c)The void fraction evolutions obtained with the
barotropic state law and the Reboud model are similar. They also exhibit a nice
agreement with the X-ray measurements. Conversely, The use of the Kunz model leads to smaller void fractions in all the cavity, although
its maximum length is close toL/c=0.9, as for the other calculations and
the experimental data. It suggests that this particular model underestimates
the flow vaporization and/or overestimates the flow condensation. Based on Figure 1, the vaporization sources for Kunz and Reboud are
nearly identical; this result suggests that the Kunz condensation model is too
large in magnitude and it overestimates the flow condensation.The second flow configuration isVref=6 m/s and L/c=0.5. The
barotropic state law of Delannoy and the void
fraction transport equation-based models of Kunz, Reboud,
and Singhal are applied to these flow conditions, and the computed void
fraction profiles are compared to the experimental data in Figure 10.Comparison between numerical and experimental void
fraction distributions (L/c=0.5, Vref=6 m/s).
(a)(b)(c)The void fraction profiles obtained
with the different cavitation models are very similar, and they are generally
in close agreement with the X-ray measurements. The only discrepancy still
concerns the Kunz model, which predicts a too low void fraction close to the
leading edge. However, predictions of this model at stations 2 and 3 are very
close to the experimental data, which confirms that the mean length of the
cavity is correctly estimated.The nice agreement obtained between
all the present data confirms the similarities observed previously in the
expression of the vaporization and condensation terms. However, some slight
differences between the models suggest that some constants inm˙+ and m˙−,
such as the exponents of P−Pv or α, may have some significant influence on the
predicted flow structure. The objective in the next section is to study this
influence in order to assess the particular effect of each variable.


## 4.1. Study of the Sheet Cavity Unsteady Behavior

The geometry considered in this section is the NACA 66 foil. Its chord
is 150 mm and the angle of attack is 8°. The reference velocity at the inlet of the test
section is Vref=5.33 m/s and the cavitation number is σ=1.3.A calculation based on the
barotropic state law of Delannoy and Kueny [6] is first performed to be compared to the
results that will be obtained with the various models based on the void
fraction transport equation. In this case, (2) is not solved, but the state
law presented in Figure 4 is coupled to (1) and (3) to govern the flow density
variations according to the local pressure variations.For a pressure much higher than the
vapor pressurePv, the flow is composed of pure liquid and the Tait
state law is applied: ρ/ρref=P+P0/Pref+P0n with P0=3108 Pa and n=7 for water.For a pressure much lower thanPv,
the fluid is locally completely vaporized and the density is governed by the
perfect gas law: P/ρ=cste.These two low compressible
configurations are joined in the vapor pressure neighborhood by the central
part of the chart, whose high slope models the high compressibility of the
liquid/vapor.Cmin2=∂P/∂ρ.
Cmin is the minimum speed of sound in the
mixture.The minimum density (indicated by the color) in each section (denoted by
the positionX/Lref in ordinate) as a function of the time (in abscissa).Results of this
calculation are presented in Figures5, 6, and 7. An unsteady
periodical behavior of the sheet cavity is obtained, including vapor cloud shedding. The maximum
length of the attached cavity can be estimated to lmax/c=0.8. The
cavity oscillation frequency is about 11 Hz (Figure 7), which gives a Strouhal
number based on the maximum length of the attached cavity equal to 0.25.Figure 5
Time evolution of the sheet
cavity.(a) Time evolution of the cavity shape
during a single cycle of vapor shedding. The colors indicate the fluid density, white for pure liquid and from red
to blue when the void ratio increases. (b) Velocity field and fluid density during cavity breakoff show a reentrant jet (picture 6 in (a)).
(a)(b)(a) Time evolution of the
pressure coefficient atX/c=0.2 and (b) spectral analysis.
(a)(b)During this cycle, a sheet cavity is
formed, which successively grows and breaks off. The detachment of the rear
part of the cavity is due to a reentrant jet that flows upstream close to the
foil surface. The vapor cloud is convected downstream and finally collapses,
while the residual cavity starts to grow again.Three  transport equation-based
models with source terms of Reboud and Stutz [5], Kunz et al. [13, 19], and Singhal et al. [7] are
now tested, and the results are presented in Figure 19. The maximum length of the
attached cavity, the oscillation frequency, the Strouhal number, and the cavity
behavior during one cycle are indicated for each model. The empirical factors were adjusted to
obtain the same maximum value for the source terms, as in the previous section.Figure19 shows a large resemblance
between the four tested models; the unsteady behavior of the sheet cavity is
similar in all cases: development of a cavity up to the maximum length (about 80% of the
chord), cavity breakoff, cloud detachment, and growth of the residual cavity.Sheet cavity oscillation frequencies
are also all similar, leading to Strouhal numbers Str based on the maximum
cavity length comprised between 0.25 and 0.28. These results are in close
agreement with the experimental measurement, which gave Str = 0.28 in the configuration
of a slightly smaller cavity (L/Lref=0.6). These results confirm
the similarities between the three models based on the void fraction transport
equation. They also suggest that using the barotropic state law instead of (2)
leads to very similar predictions of the sheet cavity behavior.The colors indicate the fluid density, white for pure liquid and from red
to blue when the void ratio increases.


## 4.2. Study of the Void Fraction Distribution

The second foil geometry is a convex foil
characterized by a flat upper surface. Its chord is 150 mm and the present angle
of attack is 4°7. The reference velocity at the inlet of the test section is Vref=6 m/s.Calculations of the cavitating flow on the foil are performed for two sheet cavity
lengths:L/c=0.9 and L/c=0.5, respectively. In both simulations, an
unsteady behavior of the cavity is obtained, as previously in the configuration
of the NACA66 foil section. Discussion is focussed hereafter on the study of
the time-averaged two-phase flow structure of the liquid/vapor mixture inside
the cavity.An X-ray absorption device was applied previously to
investigate the local volume fraction of the vapor phase inside the sheet
cavity. The experiments were carried out by Coutier-Delgosha et al. [17] in the
scope of collaboration between the ENSTA laboratory and the French Atomic
Energy commission (CEA).From these experiments, the
instantaneous and time-averaged distribution of void fraction inside the cavity
was obtained, with a space resolution of 6 mm in the flow direction and 3 mm in the vertical
direction. In the present study, the time-averaged experimental void fraction
measurements at three positions (X=15 mm, 55 mm, and 95 mm downstream from the
leading edge) are considered (Figure 8).Figure 8
Position of the experimental void fraction
measurements (Vref=6 m/s, L/c=90%).The first studied configuration isVref=6 m/s and L/c=0.9. Void
fractions profiles obtained with the barotropic state law of Delannoy and the void fraction transport equation-based
models of Kunz and Reboud are compared to the experimental data in Figure 9.
Note that the empirical constants in the models have been adjusted as in the
previous sections.Comparison between numerical and experimental void fraction distributions (L/c=0.9, Vref=6 m/s).
(a)(b)(c)The void fraction evolutions obtained with the
barotropic state law and the Reboud model are similar. They also exhibit a nice
agreement with the X-ray measurements. Conversely, The use of the Kunz model leads to smaller void fractions in all the cavity, although
its maximum length is close toL/c=0.9, as for the other calculations and
the experimental data. It suggests that this particular model underestimates
the flow vaporization and/or overestimates the flow condensation. Based on Figure 1, the vaporization sources for Kunz and Reboud are
nearly identical; this result suggests that the Kunz condensation model is too
large in magnitude and it overestimates the flow condensation.The second flow configuration isVref=6 m/s and L/c=0.5. The
barotropic state law of Delannoy and the void
fraction transport equation-based models of Kunz, Reboud,
and Singhal are applied to these flow conditions, and the computed void
fraction profiles are compared to the experimental data in Figure 10.Comparison between numerical and experimental void
fraction distributions (L/c=0.5, Vref=6 m/s).
(a)(b)(c)The void fraction profiles obtained
with the different cavitation models are very similar, and they are generally
in close agreement with the X-ray measurements. The only discrepancy still
concerns the Kunz model, which predicts a too low void fraction close to the
leading edge. However, predictions of this model at stations 2 and 3 are very
close to the experimental data, which confirms that the mean length of the
cavity is correctly estimated.The nice agreement obtained between
all the present data confirms the similarities observed previously in the
expression of the vaporization and condensation terms. However, some slight
differences between the models suggest that some constants inm˙+ and m˙−,
such as the exponents of P−Pv or α, may have some significant influence on the
predicted flow structure. The objective in the next section is to study this
influence in order to assess the particular effect of each variable.


## 5. Influence of the Model Parameters

Four parameters are considered: the empirical
constantsCprod, Cdest, and the exponents of P−Pv and αl.


### 5.1. Influence of the Empirical ConstantsCprod and Cdest

Two calculations around the NACA 66 foil
section were performed with the Kunz model and several values ofCprod and Cdest were applied in the source terms. The results are presented in Figures
11 and 12.Figure 11
Influence of the empirical constants on the
oscillation frequency and the shape of the time-averaged cavity (L/c=0.8).Void fraction distributions (L/c=0.8, Vref=6 m/s).
(a)(b)WhenCprod is increased, more vapor is
created in the upstream part of the sheet cavity, so the maximum void fraction
increases (Figure 12). Consequently, the inertia of the cavity increases and
the oscillation frequency f decreases. Conversely, it can be noticed that Cprod has nearly no influence on the cavity thickness. The increase of Cdest causes more
intense condensation of vapor; so the maximum void fraction decreases, the inertia
of the cavity decreases, and the oscillation frequency increases.One calculation around the convex foil section was performed with the Kunz
model and with0.5*Cdest instead of Cdest. It can be observed in Figure 13 that a bigger sheet cavity is obtained. The maximum void fraction has
increased at the leading edge (X=15 mm). This confirms that the Kunz model
overestimates the flow condensation.Void Fraction distributions (L/c=0.8, Vref=6 m/s).
(a)(b)(c)We
also obtain a correct void fraction in the upstream part of the cavity without
deteriorating the downstream one
because of the form of the term of condensation; this term has much impact on
the upstream part of the cavity, which is the zone of lower pressure.


### 5.2. Influence of the Exponent ofP−Pv

One calculation around the NACA 66 foil
section was performed with the Kunz model and a different exponent ofP−Pv in the expression of the vaporization term: the exponent is set
to 1.1, instead of 1. The
empirical factors were adjusted as in the previous sections. This
modification results in a diminution of the slope of the vaporization source
term (Figure 14), which suggests that less vapor may be obtained in the sheet
cavity. However, it can be observed in Figure 15 that increasing this parameter
seems to result in a bigger sheet cavity. Indeed, drawing the void fraction
profiles at stations x/c=0.1 and 0.5 (Figure 16) enables to confirm that the
cavity is thicker, while the maximum void fraction has decreased.Figure 14
Vaporization term for two values of the exponent ofP−Pv.Figure 15
Influence of the exponent ofP−Pv on the
behavior of the time-averaged cavity, L/c=0.8.Void Fraction distributions (L/c=0.8, Vref=6 m/s).
(a)(b)


### 5.3. Influence of the Exponent ofαl

Calculations around the NACA 66 foil section
are performed with the Kunz model and two different values of the exponent ofαl in
the expression of the vaporization term. The
empirical factors were adjusted as in the previous sections.A clear influence of this parameter can
be observed in Figure17: the more the exponent is increased the higher is the
slope of the vaporization term, which suggests that more vapor may
be obtained in the sheet cavity.Figure 17
Vaporization term for two values of the exponent ofαl.Drawing the void
fraction profiles at stationsx/c=0.1 and 0.5 (Figure 18) enables to confirm
that the void fraction has increased in the whole height of the sheet cavity
for x/c=0.5, while the maximum void fraction has decreased for x/c=0.1, and
the cavity thickness has not changed much. It suggests that the vaporization
inception due to the fluid inertia, which occurs at the foil leading edge, is
slightly reduced (see station x/c=0.1), whereas the vapor bubble
expansion, which mainly occurs in the upstream part of the sheet cavity, is
more intense when the exponent of αl is increased.Void fraction distributions (L/c=0.8, Vref=6 m/s).
(a)(b)Figure 19
Numerical results obtained with different
models (σ=1.3, Vref=5.33m⋅s−1, Δt=0.011 s).


## 5.1. Influence of the Empirical ConstantsCprod and Cdest

Two calculations around the NACA 66 foil
section were performed with the Kunz model and several values ofCprod and Cdest were applied in the source terms. The results are presented in Figures
11 and 12.Figure 11
Influence of the empirical constants on the
oscillation frequency and the shape of the time-averaged cavity (L/c=0.8).Void fraction distributions (L/c=0.8, Vref=6 m/s).
(a)(b)WhenCprod is increased, more vapor is
created in the upstream part of the sheet cavity, so the maximum void fraction
increases (Figure 12). Consequently, the inertia of the cavity increases and
the oscillation frequency f decreases. Conversely, it can be noticed that Cprod has nearly no influence on the cavity thickness. The increase of Cdest causes more
intense condensation of vapor; so the maximum void fraction decreases, the inertia
of the cavity decreases, and the oscillation frequency increases.One calculation around the convex foil section was performed with the Kunz
model and with0.5*Cdest instead of Cdest. It can be observed in Figure 13 that a bigger sheet cavity is obtained. The maximum void fraction has
increased at the leading edge (X=15 mm). This confirms that the Kunz model
overestimates the flow condensation.Void Fraction distributions (L/c=0.8, Vref=6 m/s).
(a)(b)(c)We
also obtain a correct void fraction in the upstream part of the cavity without
deteriorating the downstream one
because of the form of the term of condensation; this term has much impact on
the upstream part of the cavity, which is the zone of lower pressure.


## 5.2. Influence of the Exponent ofP−Pv

One calculation around the NACA 66 foil
section was performed with the Kunz model and a different exponent ofP−Pv in the expression of the vaporization term: the exponent is set
to 1.1, instead of 1. The
empirical factors were adjusted as in the previous sections. This
modification results in a diminution of the slope of the vaporization source
term (Figure 14), which suggests that less vapor may be obtained in the sheet
cavity. However, it can be observed in Figure 15 that increasing this parameter
seems to result in a bigger sheet cavity. Indeed, drawing the void fraction
profiles at stations x/c=0.1 and 0.5 (Figure 16) enables to confirm that the
cavity is thicker, while the maximum void fraction has decreased.Figure 14
Vaporization term for two values of the exponent ofP−Pv.Figure 15
Influence of the exponent ofP−Pv on the
behavior of the time-averaged cavity, L/c=0.8.Void Fraction distributions (L/c=0.8, Vref=6 m/s).
(a)(b)


## 5.3. Influence of the Exponent ofαl

Calculations around the NACA 66 foil section
are performed with the Kunz model and two different values of the exponent ofαl in
the expression of the vaporization term. The
empirical factors were adjusted as in the previous sections.A clear influence of this parameter can
be observed in Figure17: the more the exponent is increased the higher is the
slope of the vaporization term, which suggests that more vapor may
be obtained in the sheet cavity.Figure 17
Vaporization term for two values of the exponent ofαl.Drawing the void
fraction profiles at stationsx/c=0.1 and 0.5 (Figure 18) enables to confirm
that the void fraction has increased in the whole height of the sheet cavity
for x/c=0.5, while the maximum void fraction has decreased for x/c=0.1, and
the cavity thickness has not changed much. It suggests that the vaporization
inception due to the fluid inertia, which occurs at the foil leading edge, is
slightly reduced (see station x/c=0.1), whereas the vapor bubble
expansion, which mainly occurs in the upstream part of the sheet cavity, is
more intense when the exponent of αl is increased.Void fraction distributions (L/c=0.8, Vref=6 m/s).
(a)(b)Figure 19
Numerical results obtained with different
models (σ=1.3, Vref=5.33m⋅s−1, Δt=0.011 s).


## 6. Conclusions

The present work
is a contribution to the physical modelling and numerical simulation of
cavitating flows. An analytical approach was first performed in order to
compare transport equation models proposed by various authors. The resemblance
between most of the models was observed.A numerical study using a 2D CFD code was also
performed, and several models were tested and compared. Sheet cavity lengths,
oscillation frequencies, and cavity behaviors obtained with the different
models are found in close agreement.The comparison between the numerical and the
experimental void fraction distributions in a second configuration of 2D foil
section demonstrates also the resemblance between the models, and thus it confirms the results
of the analytical study.The influence on the results of some arbitrary
constants used in the models was also studied. It was shown that modifying these parameter enables to change
the cavity shape and structure. The objective of the work is now to link these
different constants with physical considerations in order to construct
vaporization and condensation terms which would take into account more physics
of the vaporization and condensation processes.



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*Source: 146234-2009-01-29.xml*