7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Mass transfer cavitation model with variable density of nuclei
A. Vallier*, H. Nilssoni and J. Revstedt*
Division of Fluid Mechanics Dept. Energy Sciences, Lund University, SE22100 Lund, Sweden
t Applied Mechanics, Fluid Dynamics, Chalmers University of Technology, SE412 96 Gothenburg, Sweden
Aurelia.Vallier~energy.1th.se, hakan.nilsson~chalmers.se and Johan.Revstedt~energy.1th.se
Keywords: Cavitation, Sauer's model, Variable nuclei density, Lagrangian Particle Tracking
Abstract
The performance of the mass transfer cavitation model of Sauer is investigated using a varying nuclei concentration.
The Sauer model assumes a uniform nuclei distribution despite measurement of the nonhomogeneous nucleus
population. Here the nuclei density is studied and a nonhomogeneous nuclei distribution in a modified Sauer
model is implemented. It is used to study how the increased cavitation nuclei density in regions of low pressure
affects the inception of cavitation. The interface between the water and the water vapor is tracked using a volume
of fluid method and vaporization and condensation are described by the modified Sauer's mass transfer model.
The nuclei in the liquid phase are modeled with a Lagrangian Particle Tracking method. The LPT computa
tions yield to a non uniform nuclei distribution which consists of nuclei accumulation close to the leading edge
and no nuclei on average in the boundary layer of the hydrofoil. The sensitivity of the modified Sauer model to
nuclei distribution is proven. The shape of the sheet cavity and the volume of vapour are affected by the nuclei content.
Nomenclatu re
Recently, many mass transfer cavitation models have
been introduced in the literature and generalpurpose
CFD codes in order to fully describe the observed phe
nomena on cavitating hydrofoils. The inception cav
itation number is supposed to be given by the min
imum value of the pressure coefficient. This law is
barely suitable in the case of attached cavitation on a
hydrofoil because viscosity, turbulence and water qual
ity have a major influence on cavitation inception (Bren
nen (1995)). In particular, cavitation inception occurs
at different pressure depending on the number of cav
itation nuclei. Therefore, taking into account the non
homogeneous nuclei content of the water will improve
the accuracy of the numerical simulations. Sauer (2000)
included the nuclei density parameter no in their cavita
tion model, specified as a constant This assumption is
not in accordance with experiments or the numerical re
sults obtained by Huuva et al. (2007), where it is shown
that the nuclei accumulate in certain regions close to the
hydrofoil. Mass transfer models give very good predic
tions of the mechanism of the cavitation inception and
development for cases of cavitating hydrofoils (Coutier
Delgosha et al. (2007)). These models successfully rep
resent the attached sheet cavity, the reentrant jet, the
Roman symbols
g gravitational constant (nisi)
p pressure (Nm)
y e ocity nis 
D diameter (nt)
R radius (nt)
c<3 chord length (nt)
k turbulent kinetic energy (ni s )
Greek symbols
vapor volume fraction
p density (kgnt3)
S viscosity (kym1,)
r relaxation time (s)
t turbulence energy dissipation rate (n s )
Subscripts
P particle
Vapor
1 liquid
Introduction
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
breakoff of the sheet, and the shedding of the breakoff
process. All these features agree with experimental ob
servations. However, the models fail to resolve the tran
sition between the attached sheet cavity and the cloud of
vapor. In experiments, the transition forms a cloud of
small vapor bubbles, while the numerical methods pre
serve a large coherent vapor region that is advected with
the surrounding flow. The models cannot further han
dle the collapsing process of the small bubbles and the
related erosive and acoustic processes. Indeed this im
portant feature should at some point be included in the
prediction as it is the main cause of erosion damage. The
implosion of the bubble cloud when it reached a region
of higher pressure generates pressure waves that influ
ence the collapse of the surrounding bubbles, causing a
chain reaction that amplifies the erosive process. As the
model does not accurately predict regions containing a
low vapor concentration, we study the relevance of the
14plhes'll'i\ made to simplify the model, i.e. the assump
tion of constant nuclei concentration. It is not obvious
that the nuclei concentration is homogeneous in real ap
plication. Therefore it is of interest to investigate how
a inhomogeneous distribution affects the inception and
development of attached sheet cavities.
The Sauer cavitation mass transfer model is intro
duced with the nuclei density no, taking into account
the water quality. Then the LPT method described is
used to compute the cavitation nuclei distribution. Fi
nally we present the results obtained for the LPT nuclei
distribution and for the modified Sauer model where non
homogeneity is included.
Figure 1: Computational grid for the NACA0015 airfoil
Sauer model
The Sauer model (Sauer (2000)) is a mass transfer
model using the volume of fluid (VOF) approach. The
fluid density and viscosity are scaled by the vapor vol
ume fraction a
p = a~pV + (1 a~)p1 (1)
p = apv + (1 a)PI (2)
The transport equation for the vapor volume fraction
reads
+ V (~U) = S,
i3t
Geometry
The simulations were performed for 2D and 3D flows
past a NACA0015 hydrofoil. Figure 1 illustrates the
125*270 Cgrid used for the 2D computations. The hy
drofoil has a chord length co=0.15m and is positioned at
4.5co from the inlet and 9co from the outlet. The height
of the computational domain is 9co. The angle of attack
is 8 degrees. The grid points are clustered to the hydro
foil surface such that the first cell center near the hydro
foil surface starts at Ay/co ~ 0.1/&7 1.104 with
an increase of 5' per layer. Hence, the first node away
from the wall is on average positioned at yt 3. In
the case of the 3D simulation, 50 grid points are equally
distributed in spanwise direction, with a total spanwise
thickness of 1co. The Reynolds number based on co and
the uniform inlet velocity 8m/s is Re=1.2 106. The cav
itation number o is 1.2.
where the source term S, = m" accounts for the de
struction and production of vapor. One can also derive a
corresponding transport equation for the liquid volume
fraction y (where a + y 1 ). The source term is then
S, m which accounts for the destruction and pro
duction of liquid. Summing the two transport equations
results in the non divergence free continuity equation:
V U = (I 1 1)
In order to derive the mass transfer rate, rk2, the Sauer
model states that the vapor volume fraction corresponds
to a density no of nuclei of radius R. Furthermore the
dynamics of each bubble is governed by a simplified
RayleighPlesset equation. Hence the vapor volume
fraction and the nuclei growth rate are written as
a = (4)
1 + R~no
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
The particle Reynolds number is defined as
R I2 p(R) p , (
3 pi
The mass transfer rate mi is then derived (see Sauer
(2000) ) as
th = pe, sign(pe, p) (6)
Finally, the continuity and vapor transport equations
are solved together with the momentum equation:
P D;,U U,
and the drag force can be expressed as
FD~U Gp U
The relaxation time 7, of the particles is the time it
takes for a particle to respond to changes in the local
flow velocity
Tp 4 pP(14)
where the standard definition of the drag coefficient
CD fOT a Spherical particle is given by Schiller and Nau
mann as
24if Re, < 0.1
CD r F/
0.44 if Re, > 1000
(15)
Since the fluid velocity U, calculated in the Eulerian
reference frame, is needed for the calculation of the drag
force in the Lagrangian frame, it has to be interpolated
to the position of the particle from the neighboring cells.
The velocity at the particle position is denoted Ueop.
Furthermore, each Eulerian time step is divided into a
set of Lagrangian time steps that is specific to each parti
cle. A Lagrangian time step is defined as the time it takes
for the particle to leave the cell that it was occupying.
The velocity and position of a particle at the nth La
grangian time step aL,, within an eulerian time step is
evaluated as
dU
P + p(U VU)
8
V +pV U + pg (7)
Nuclei distribution
Lagrangian Particle Tracking
Lagrangian particle tracking (LPT) is a method to
track individual particles (or bubbles) in a fluid flow. A
particle P is defined by the position of its center, xp, itS
diameter, Dp, its velocity, Up and its density, pp.
The fluid phase is governed by the incompressible
NavierStokes equations
VU0
dU
Pc + p(U VU)
Vp + pV U + py Sp (8)
The additional source term in the momentum equa
tion (8) is due to the influence of the particles on the
flow. Here we consider the case of a dilute suspen
sion (Xpi > 10) Withl a volumne firaction of. parti
cles lower than 10 Hence the particles' effects on
the flow and turbulence are negligible (see Elghobashi
(1994)). This is usually denoted oneway coupling, i.e.
the flow affects the particles but the particles don't affect
the flow. Therefore the additional source term Sp in the
momentum equation is neglected. As a consequence of
the very low volume fraction of particles, interparticle
collisions are also neglected.
In a Lagrangian frame, each particle position vector
xp is calculated from the equation
dxp
=Up (9)
and the motion of each particle is governed by Newton's
second law:
a~pt,= F, (10)
where the mass of each particle is a p ~ppirD,.
In dilute flow, the dominant forces acting on the small
particle is the drag from the fluid phase and the gravita
tional force:
Ut+3 P t' + n gatl
P P ut,
1+ +t
Ut 2, at,
(16)
xt Ep i=at, xt+ Cp1t t E ati at,,
(17)
The collision of a particle with the wall is assumed
to be inelastic. Hence, the velocities of the particle P
before and after collision are written as
Up = U;4n + LT' t (18)
Up = LT) n + LT t (19)
The unit vectors n and t are the normal and tangential to
the wall, respectively. The normal and tangential com
ponents of the particle velocity after a collision with the
wall are evaluated as
U;4 = tU;4
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
U:= (1 pm)U' (21)
where t, E [0, 1] and p, E [0, 1]) are the coefficient of
restitution and friction of the wall.
Random Walk
In reality, small particles have a short relaxation time
and respond quickly to the flow fluctuations. Turbulence
diverts the particles from their trajectory and small par
ticles are trapped in eddies for a certain period of time.
Not accounting for this leads to that the particles will
follow the stream lines of the mean flow. Here we use
a random walk model (Gosman & Ioannides (1983)) to
include the effect of turbulent dispersion of the parti
cles, i.e. eddies are created randomly and affect the par
ticle trajectory. In practice, a local fluctuating compo
nent is added to the particle velocity, i.e. Up becomes
Up Up + U plCt. The local fluctuating velocity can
be estimated by
Figure 2: Instantaneous distribution of cells with a large
nuclei density for cases LPT6 (top) and
LPTi: n (b ottom)
rS~
/2
Ufiuct = ~~i i~
P V3
where t' is a random number generated from a Gaussian
distribution of zero mean and variance 1 ( t' e N(0, 1))
and is the local RMS fluid velocity fluctuations for
isotropic turbulence. The eddy life time (t,) and the time
needed by the particle to traverse the eddy (transit time
it,) are calculated as
C0.63 lc1.5
~L t
Figure 3: Instantaneous distribution of cells with a large
nuclei density for case LPT6LEs, in the cen
terplane
ft, = 7pln(1 ). (24)
rpU Up
The random walk algorithm consists of evaluating
U place according to equation (22) and a random num
ber to calculating the characteristics times t, and it,
and keeping U pr act constant during the interaction time
Results
Nuclei density sensitivity to particle properties
Several cases (listed in Table 1) have been studied in
order to investigate the sensitivity of the solution to par
ticle diameter and particle density. Also, all these cases
have been simulated both with and without the random
walk model. Furthermore one 3D case was simulated
with LES to see how the turbulence affects the particles
distribution.
Figure 4: Average nuclei distribution for cases LPT6
and 6Rw
Table 1: Summary of the LPT cases.
pp Dp 2D RANS 2D RANS + RW 3D LES
case name case name case name
1000 1 LPT1 LPT1Rw
1000 10 LPT2 LPT2Rw
1000 20 LPT3 LPTSRw
1000 30 LPT4 LPT4Rw
1000 40 LPT5 LPT5Rw
1000 50 LPT6 LPTGpw LPT6LES
100 50 LPT7 LPT7Rw
10 50 LPT8 LPT8Rw
1 50 LPT9 LPTSaw
1 2e09
scled 2
scaled LPT 3
le09 scldLP 4
8e10scaled LPT_6
4e10
2e10
0 02 0 025 0 03 0 035 0 04 0 045 0 05 0 055 0 06
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
P o n"
;e a
Figure 5: Contour of pressure (2D, RANS). Sampling
lines where the average nuclei distribution is
investigated.
500 particles are injected per time step at a distance
1.5co in front of the hydrofoil.
Figures 2 and 3 highlight the cells which contain the
largest number of nuclei for the RANS and LES compu
tations. Two features are observed. First, a large number
of nuclei are present at the leading edge. Obviously the
density is highest at the stagnation point (colored in red)
because the nuclei rebound against the wall and reside a
longer time in this region of low velocity. Second, the
presence of cells with a high nuclei content in the region
of low pressure should be investigated. In this region
the velocity is high and the residence time of a nuclei is
therefore very short. From one instantaneous picture to
another, the nuclei distribution is completely different.
Thus we calculate an average of the positions occupied
by the nuclei during their trajectory. Figure 4 shows the
results for case LPT6 and LPTi, n ,. The average nu
clei distribution confirms the accumulation of nuclei at
the stagnation point (colored in red) A large number
of nuclei are also observed in a part of the low pressure
region near the leading edge. However the results show
that the nuclei are not present on average in the boundary
layer of the hydrofoil.
In order to compare the influence of the size and the
density, as well as the turbulence modeling, the aver
aged nuclei distribution has been sampled on vertical
lines through the low pressure region (Figure 5). In or
der to compare the nuclei density for different values of
particle diameter Dp (case LPT1 to LPT6), the nuclei
density is scaled and divided by (Dp 106 3 in Figure
6. For all the cases, large fluctuations appear near the
surface, followed by smaller fluctuations and then the
nuclei density is almost constant and equal to the nu
clei density injected. The size and density influence on
the nuclei density is not important. The results are sim
ilar for each sampling line, therefore only one of them
is presented here, in Figure 6 and 7. Near the surface
the nuclei density reaches peak value up to five time the
Figure 6: Sensitivity to particle size, sampling line 2
0 00012
y [m]
Figure 7: Sensitivity to particle density, sampling line 3
mean value but it also has very low values. This behav
ior is probably due to the very small size of the cells in
this region. The interesting feature is the presence of the
smaller fluctuations over the mean value which confirms
that the concentration of nuclei is higher quite close to
the surface.
The random walk model has an impact on the nuclei
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Table 2: Summary of the cases with a prescribed non
uniform nuclei distribution. N is the nuclei
concentration in the domain. 6N is the thick
IleSs of the layer with high concentration of nu
Clei, i.e. Where N=10 .
6N=0.5 6N=1 6N=2 6N=4
N=102 case 1 case 2 case 3 case 4
N=104 CRSe 5 case 6 case 7 case 8
ol '
no un nn 
case 2
.case 3 
case 4
case 5
case
O 0 02 0 04 0 06 0 08 0 1 0 12 0 14
LPT 1
~LPT 1RW
 ;1 '~
0025 0 03 0 035 0 04 0 045 0 05 0 055
y [m]
006
0 00025
Figure 8: Sensitivity to Random Walk model, sampling
l* me
distribution, as shown in Figure 8. The nuclei appear
closer to the surface, the high fluctuations near the sur
face are reduced and the smaller fluctuations are trans
lated upward. This means that a even higher nuclei den
sity is predicted quite close to the surface.
Effect of inhomogeneous nuclei distribution
The Sauer model assumes a homogeneous nuclei dis
tribution, and a value of 108 is generally used. Here the
performance of the model is investigated using a vary
ing nuclei concentration. Using the nuclei distributions
obtained from the simulations presented in the previous
section in the modified Sauer model did not yield any
sheet cavitation. Instead, the cavity appeared somewhat
above the hydrofoil and was not attached. Indeed the
vapor production started where both crucial parameters
existed, i.e. a low pressure and a high concentration of
nuclei. This behavior is due to the lack of nuclei in the
boundary layer discussed in the results of the LPT sim
ulations. However, it has been shown experimentally
that cavitation starts at the surface. This implies that the
transported nuclei (called free stream nuclei) don't have
as much importance as the surface nuclei, at least for
cavitation inception. Surface nuclei are generally small
bubble of gas trapped in wall rugosity (Brennen (1995)).
Therefore it has also been studied how the nuclei con
tent in the boundary layer affect cavitation inception and
development.
In those studies it is assumed that the nuclei concen
tration N is high (N=10 ) in a layer attached to the sur
face and low (N=102 or N=104) eVerywhere else. The
thickness of the layer by varies from 0.5, 1,2 and 4 mm.
The cases are summarized in Table 2.
Figure 9 shows the total volume of vapor in the entire
computational domain during the cavitating process. In
cases 1 and 5, the layer is 0.5 mm thick. The nuclei con
tent is too low to enable the cavity to grow sufficiently.
The production of vapor is lower during cavitation in
0 0002
5e05
o
time [s]
Figure 9: Total volume of vapor for cavitating flow with
different nuclei distribution,
ception compare to the uniform case. In the other cases
cavitation inception is similar to the case with uniform
nuclei distribution. The same amount of vapor is created
when the cavity grows. From t=0.02 s, the vapor pro
duction is slightly larger than in the uniform case. The
cavity is broken by the reentrant jet, and from t=0.03 the
vapor disappears at the same rate except for cases 2 and
3. For both cases, the volume of vapor decreases slower,
grows again and then decreases as in the uniform case.
Figure 10 shows the cavitation process for the uniform
case (left), case 7 (center) and case 3 (right). As men
tioned, the inception is similar for all cases. The attached
cavity has the same shape at t=0.02 s. Differences can
be noticed from t=0.03 s, due to the reentrant jet which
has the same thickness as the layer of nuclei. For all the
cases with by < 2mm (here only by 2mm is shown)
,the cloud which is about to be detached is closer to
the hydrofoil surface. Furthermore the reentrant jet is
faster. Thus it breaks the attached cavity at a position
closer to the leading edge. As the point of detachment
is closer to the leading edge, the length of the attached
cavity is shorter and the cloud is more stretched With
a nonuniform nuclei distribution, the attached cavity is
linked to the cloud by a thin layer of vapor. This line of
vapor is still present when the cloud shrinks (t=0.06). It
generates a second smaller, fuzzier cloud for the cases
with N=102. This is the reason why the total volume of
vapor increases around t=0.06 s for these cases.
7e10
6 6e10
se10
S4e10
S3e10
1 e10
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Figure 10l: Vapor volume fraction as. First row t=0.02s, 2nd row t=0.03s, 3rd row t=0.05s, 4th row t=0.06s, 5th row
t=0.07s. Left: N uniform, center: case 7 (N 104, 6N=2 mm), right: case 3 (N 10 6N=2 mm)
nO uniform
case 1
case 2
*
0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 1
xle [mi
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
References
Sauer J. and Schnerr G.H., Unsteady cavitating flow A
new cavitating model based on a modified front captur
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ASME Fluid Engineering Summer Conference, Boston,
MA, June 1115, 2000
Huuva T., Cure A., Bark G. and Nilsson H., Compu
tations of unsteady cavitating flow on wing profiles us
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in Hydraulic Machinery and Systems, Scientific Bulletin
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Transactions on Mechanics, 52 (66) pp. 2134, 2007
CoutierDelgosha O., Deniset F., Astolfi J.A., Leroux J.
B., Numerical Prediction of cavitating Flw on a Two
Dimensional Symmetrical Hydrofoil and Comparison
To Experiments, Joumnal of Fluids Engineering, march
2007, Vol. 129, 279292.
Brennen C. E., Cavitation and Bubble Dynamics, New
York Oxford University Press, 1995.
Clift R., Grace J.R and Weber M.E., Bubbles, Drops and
Particles, Academic, New York, 1978,
Gosman A.D. and loannides E., Aspects of computer
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Figure 11: Pressure coefficient at t=0.03 s. Cases with
N=10 .
The differences between the curves of the pressure
coefficient, shown in Figure 11, confirm that the non
uniform cases have a shorter attached cavity already at
t=0.03 as the the pressure increases at a position x/co
between=0.65 and 0.7 instead of 0.75 for the uniform
case.
Conclusions
We studied the nuclei distribution over a NACA00 15 hy
drofoil. It was shown that the nuclei accumulate at the
leading edge close to the low pressure region. However
the nuclei were not present on average in the boundary
layer. The Sauer model was modified to take into ac
count this non uniform nuclei density and didn't yield to
attached cavitation. It means that the transported nuclei
influence is not as important as the one of the surface
nuclei for cavitation inception. Then the performance of
the modified Sauer model was investigated with a higher
nuclei concentration near the surface. The attached cav
ity was shorter, the reentrant jet was faster and thinner,
and the cloud was stretched. A thin layer of vapor linked
the attached cavity and the cloud of vapor. These fea
tures emphasize the importance of the nuclei distribution
when modeling cavitation inception and development.
Acknowledgements
The research presented was carried out as a part of
"Swedish Hydropower Center SVC". SVC has been
established by the Swedish Energy Agency, Elforsk
and Svenska K~llinal~l together with Lulea University
of Technology, The Royal Institute of Technology,
Chalmers University of technology and Uppsala Univer
sity. www.syc.nu.
We gratefully acknowledge the use of the computing
resources of LUNARC, center for scientific and techni
cal computing at Lund University. www.1unarc.1u.se.
