Shijie Zhang
College of Water Resources and
Civil Engineering,
China Agricultural University,
Beijing 100083, China
e-mail: zhangsj_ais@163.com
Zhifeng Yao1
Hongfei Wu
NanFang R&D Institute,
Nanfang Pump Co., Ltd.,
Hangzhou 311107, China
e-mail: nfbywhf@163.com
Qiang Zhong
College of Water Resources and
Civil Engineering,
China Agricultural University,
Beijing 100083, China;
Beijing Engineering Research Center of Safety
and Energy Saving Technology for Water Supply
Network System,
China Agricultural University,
Beijing 100083, China
e-mail: qzhong@cau.edu.cn
Ran Tao
College of Water Resources and
Civil Engineering,
China Agricultural University,
Beijing 100083, China;
Beijing Engineering Research Center of Safety
and Energy Saving Technology for Water Supply
Network System,
China Agricultural University,
Beijing 100083, China
e-mail: randytao@cau.edu.cn
A New Turbulent Viscosity
Correction Model With URANS
Solver for Unsteady Turbulent
Cavitation Flow Computations
Due to the ignorance of the effect of the water–vapor interface on the cavitation flow
field, the standard k–e turbulence model (ST model) may overestimate the turbulent viscosity. It is unable to simulate cavitation shedding, especially at small attack angles of a
hydrofoil. In the present investigation, a turbulent viscosity correction model is proposed
to dampen the turbulent viscosity at the water–vapor interface. Cavitation flow around a
NACA0009 truncated hydrofoil with a 2.5 deg angle of attack is used to demonstrate the
effect of correction. The results show that the interface effect-based correction model (IE
model) can both predict the pressure distribution on the suction surface of the hydrofoil
with experimental data and the re-entrance jet in the leading-edge cavitation shedding.
The region of the IE model influenced concentrates on the water–vapor interface and
intensifies the vortex strength, which directly enhances the formation of a horseshoe vortex. The reduction of turbulent viscosity by the IE model reduces the resistance to the
development of a re-entrance jet. The shear stress plays an important role in the shedding
of the attached cavity bubble. The increase of shear force in the leading-edge cavitation
occurs with the re-entrance of water and the main shear flow concentrates on the middle
of the cavity bubble. This paper therefore presents a new method of numerical simulation
of cavitation flow in engineering applications. [DOI: 10.1115/1.4053958]
Keywords: cavitation flow, hydrofoil, turbulent viscosity, water–vapor interface
Fujun Wang
College of Water Resources and
Civil Engineering,
China Agricultural University,
Beijing 100083, China
Beijing Engineering Research Center of Safety
and Energy Saving Technology for Water Supply
Network System,
China Agricultural University,
Beijing 100083, China
e-mail: wangfj@cau.edu.cn
1
Corresponding author.
Contributed by the Fluids Engineering Division of ASME for publication in the
JOURNAL OF FLUIDS ENGINEERING. Manuscript received January 16, 2022; final
manuscript received February 20, 2022; published online March 22, 2022. Assoc.
Editor: Ehsan Roohi.
Journal of Fluids Engineering
C 2022 by ASME
Copyright V
SEPTEMBER 2022, Vol. 144 / 091403-1
Downloaded from http://asmedigitalcollection.asme.org/fluidsengineering/article-pdf/144/9/091403/6866799/fe_144_09_091403.pdf by Xi'An Jiaotong University Lib user on 05 December 2022
College of Water Resources and
Civil Engineering,
China Agricultural University,
Beijing 100083, China;
Beijing Engineering Research Center of Safety
and Energy Saving Technology for Water Supply
Network System,
China Agricultural University,
Beijing 100083, China
e-mail: yzf@cau.edu.cn
1
Introduction
091403-2 / Vol. 144, SEPTEMBER 2022
2
Numerical Modeling Theory
2.1 Governing Equations. The cavitation flow field calculation is solved based on the homogeneous model under the assumption that water and vapor mixture is regarded as one fluid. A set of
the RANS governing equations is used, shown in Eqs. (1) and (2).
The effects of heat transfer and buoyancy are not considered. The
small-scale turbulence in flow is modeled by the standard k–e
model. The influence of turbulence on the time-averaged flow
field is expressed by lt in Eq. (2). For the physical parameters of
homogeneous flow, density q, and dynamic viscosity ld are
defined as a volume-weighted average of the two components
@q @ ðquj Þ
þ
¼0
@t
@xj
@ ðqui Þ @ ðqui uj Þ
@p
@
@ui @uj
þ
¼
þ
þ
ðld þ lt Þ
@t
@xj
@xi @xj
@xj @xi
2
@uk
dij
ld
3
@xk
(1)
(2)
q ¼ ql al þ qv av
(3)
ld ¼ ldl al þ ldv av
(4)
The subscripts (i, j, k) denote the components related to the Cartesian coordinates. Subscripts (l, v) are used for water and vapor,
respectively, u is the velocity, p is the pressure, ld is the mixture
dynamic viscosity, and lt is the turbulent viscosity.
Transactions of the ASME
Downloaded from http://asmedigitalcollection.asme.org/fluidsengineering/article-pdf/144/9/091403/6866799/fe_144_09_091403.pdf by Xi'An Jiaotong University Lib user on 05 December 2022
Cavitation occurs in a wide variety of rotating hydraulic engineering systems. It remains of interest since it has several negative
effects, such as noise, vibration, surface erosion, and significant
degradation in performance. Of particular importance is the
leading-edge cavitation, as its quasi-periodic shedding off is considered to cause serious damage to hydraulic machinery.
Researchers have conducted several experiments based on simple
structures, such as the hydrofoil [1] and the Venturi duct [2,3]. It
has been found that the main factors causing cavitation shedding
are the re-entrance jet [4], shock wave [5], and Kelvin–Helmholtz
instability [6]. Researches on cavitation flow around twisted
hydrofoils and wedge-shaped bluff bodies further reveals the complexity of cavitation and the law of cavitation shedding [7,8]. To
predict cavitation in more complex flow patterns, mathematical
models of cavitation flow are established by different methods.
Cavitation flow is complicated because of its unsteady, turbulent and multiphase characteristics. It is assumed that where there
is a clear interface between the water and the vapor, the cavitation
flow can be identified by the boundary element method [9–11], or
else it will be considered as a mixture flow. The boundary element
method is usually applied to the flow calculation of steadily
attached cavitation and supercavitation. It can predict the overall
behavior of cavitation flow well, but struggles to deal with
unsteady flow in the flow field in the closed area of cavitation. In
the mixture method, the cavitation flow field is considered to be a
mixture of water and vapor; coupling of the turbulence model and
the cavitation model is established to simulate the transport of
mass and momentum. For the water–vapor interface, the
Euler–Lagrange method can be used to identify the continuous
water flow field with the tracking of the cavity bubbles [12], but
requires vast amounts of computational resources because of the
wide scale of the cavity bubbles. The homogeneous flow model
takes the water–vapor flow as a single phase when solving the
continuity equation and momentum conservation equation
[13–15]. The water–vapor interface can be reconstructed through
the volume of fluid [16]. The relatively small number of calculations required means that the homogeneous flow model is more
suitable for engineering applications, and it is chosen for the present research.
Turbulent viscosity plays an important role in turbulent flow
and becomes more complicated because of the mixture of water
and vapor. At first, the turbulence model based on a single phase
is directly applied to cavitation flow, and the physical properties
are defined by a linear relationship with different components
[17,18]. In the Venturi cavitation flow, Rebound et al. [19] found
that the original model tends to overpredict turbulent viscosity
and fails to make a simulation of periodical cavity shedding. The
density-based correction model has been empirically adopted to
correct the viscosity of cavitation flow field by a decrease of turbulent viscosity in the water–vapor mixture field [20–22]. The
Reynolds stress turbulence model also shows the application in
compressible flow; in particular, it can predict the exact location
of the shock train compared with experimental data [23]. Roohi
et al. [24,25] applied the large eddy simulation model to capture
the vortical structure induced by cavitation around different
shaped hydrofoils and found that large eddy simulation model performed better than the Reynolds-averaged Navier–Stokes (RANS)
method. Mirjalily [26] decreased the turbulence constant b* in the
Shear Stress Transport k-x model, which reduced the production
of turbulent kinetic energy in the flow field and achieved good
results in the simulation of elliptic supersonic jets. Johansen [27]
proposed that the resolving resolution of the RANS method in the
turbulent flow field is related not only to the grid but also to the
magnitude of the turbulence. Therefore, a filter-based model is
proposed to dampen the turbulent viscosity in cases where the
Reynolds number is high. Wu et al. [28] applied this to the cavitation calculation of the Clark-Y hydrofoil at an attack angle of
5 deg and successfully simulated cloud cavitation flow. Huang
[29] performed an analysis of different turbulent viscosity correction models and found that the filter-based model could capture
the unsteady characteristics caused by a large-scale eddy, while
the density-based correction model mainly affected the near-wall
cavitation core area. The filter-based density correction model is
proposed by blending the filter-based model and the density-based
correction model. The turbulent viscosity correction methods perform well under a high angle of attack with high curvature, where
the form resistance is significant. However, the cavitation in a low
angle of attack requires further research.
The effect of the water–vapor interface on turbulence has been
studied by many researchers. In the turbulent channel flows,
Rashidi et al. [30] found that the turbulence of the fluid normal to
the water–vapor interface is suppressed by surface tension. Li [31]
adopted the direct numerical simulation method in the analysis of
water–vapor flow and concluded that the surface tension will
dampen the surrounding turbulence in the “interface stretching
period.” In numerical simulations based on the homogeneous flow
model, water–vapor interfaces were spread on a few meshes and
the no-slip condition was used at the intersection of water and
vapor, which led to an artificial increase of energy dispersion [19].
Therefore, the effect of the water–vapor interface should be noted
in the correction of turbulent viscosity. However, there is no accurate definition of this interface in the homogeneous flow model.
Bakir [32] found that the gradient of vapor volume fraction aligns
well with the isosurface of volume fraction. Therefore, a new turbulent viscosity correction method is proposed, concentrating on
the water–vapor interface, based on the homogeneous flow model.
This paper established an interface effect-based model (IE
model). It could capture the position of water–vapor interface and
dampen the turbulence viscosity. The IE model was successfully
applied to the prediction of cavitation at the leading edge of the
hydrofoil. The numerical results agreed well with the experiment
data, and it was seen that phenomena such as re-entrance jets and
horseshoe vortices in the cavitation flow field was finely reproduced. Velocity field details at the leading edge in cavitation
development were analyzed and the role of shear stress played in
cavitation shedding was emphasized.
The cavitation process is governed by the mass transport equation based on the vapor volume fraction, shown as Eq. (5). m_ represents the source term of phase transition. When the local
pressure p is lower than the saturation pressure pv, it adopts a negative value to represent cavitation production, denoted as m_ , and
the converse is denoted as m_ þ . The Zwart–Gerber–Belamri [33]
cavitation model is used to quantify the phase transition
@ql al @ ðql al uj Þ
¼ m_
þ
@xj
@t
lt ¼ fIE Cl
fIE ¼
(6)
where Cdest and Cprod are the coefficients of destruction and production of the cavity bubbles, anuc is the nucleation volume fraction and RB is the bubble diameter. These parameters need to be
adjusted according to the simulated media characteristics.
qk2
e
1
k@av
max
;1
@s
(8)
(9)
As the isotropic assumption of turbulent viscosity, the gradient
of volume fraction here is calculated by spatial average
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2 2ffi
@av
@av
@av
@av
þ
þ
¼
(10)
@s
@x
@y
@z
2.2 Turbulence Modeling
2.2.1 The Standard Turbulence Model. The two-equation ST
model based on the Reynolds time-averaged method is widely
used in engineering calculations. It has good robustness [29] and
is recommended for cavitation flow calculation. The turbulent viscosity lt of the ST model is solved by mixing density q, turbulent
kinetic energy k, and turbulent energy dissipation rate e, as shown
in Eq. (7). The transport equation of k and e is established for the
closure of equations. Cl is a model constant valued at 0.09 [34]
lt ¼
Cl qk2
e
(7)
2.2.2 Interface Effect-Based Model. The existing structure of
the gas phase in the liquid phase is very complex in cavitation
flow and there is still a need for further improvement of the accurate mathematical model for the physical properties and turbulence characteristics of cavitation flow. Considering the
turbulence damping effect caused by the water–vapor interface,
we propose a new model, the IE model, to correct the overestimation of turbulent viscosity by the ST model. The IE model is
designed to exert a decrease in turbulent viscosity; the modified
turbulent viscosity is defined as in Eq. (8). The vapor volume
3
Numerical Modeling Scheme
3.1 Computational Domain and Boundary Conditions. To
verify the application effect of the models, we simulated the cavitation flow over a modified NACA 0009 hydrofoil with a truncated trailing edge [36]. The experimental hydrofoil was
asymmetrical. The profile of the upper surface of the hydrofoil
was established according to Eq. (11), where L0 is the length of
the original NACA 0009 hydrofoil, with a value of 110 mm. The
hydrofoil used in the laboratory was truncated at L ¼ 100 mm and
the thickness of the truncated trailing edge was 3.22 mm. The
maximum thickness of hydrofoil was 9.82 mm and was obtained
at 0.5 L. The spanwise length of the hydrofoil was 150 mm. The
hydrofoil experiment was carried out in the high-speed cavitation
tunnel in Ecole
Polytechnique Federale de Lausanne. This is a rectangular channel with a cross section of 150 mm 150 mm. The
experimental hydrofoil was fixed in the middle of the channel.
The pressure distributions in the center of the hydrofoil and the
behavior of cavitation were recorded. Partial leading-edge cavitation with vortex shedding occurred at the small attack angle
h ¼ 2.5 deg, cavitation number r ¼ 0.81 and was used for the
numerical simulation below
8
12
2
3
>
x
x
x
x
>
>
þ
0:3046
0:2422
0:2657
0:1737
>
<
L
L
L
L
0
0
0
0
y
¼
2
L0 >
>
x
x
x 3
>
>
0:1898 1 þ 0:0387 1 : 0:0004 þ 0:1737 1 L0
L0
L0
The hydrofoil calculation domain of numerical simulation is
shown in Fig. 1(a). A no-slip wall boundary condition was specified for the hydrofoil tip. The other side was set in symmetry to
enable the flow simulation in only one half of the experimental
test section. An inlet boundary with a constant inflow velocity of
35 m/s was specified in a normal direction, 200 mm upstream
from the hydrofoil’s leading edge. The pressure at the outlet
boundary 250 mm downstream from the hydrofoil’s trailing edge
Journal of Fluids Engineering
x
0:5
L0
for
0
for
x
0:5 <
1:0
L0
(11)
was specified to yield a cavitation number of 0.81. The upper and
lower walls of the water tunnel were set as free slip walls. A monitor point was set at the half position of the hydrofoil to detect the
pressure fluctuation induced by cavitation. The two models, the
ST and IE models, were compared.
3.2 Independence Checks of Grid and Timestep. The cavitation flow based on the ST model was used as a test case for the
SEPTEMBER 2022, Vol. 144 / 091403-3
Downloaded from http://asmedigitalcollection.asme.org/fluidsengineering/article-pdf/144/9/091403/6866799/fe_144_09_091403.pdf by Xi'An Jiaotong University Lib user on 05 December 2022
8
>
3anuc ð1 av Þqv 2 pv p 1=2
>
>
_
m
¼
C
p < pv
>
prod
<
3 ql
RB
>
>
3av qv 2 p pv 1=2
>
þ
>
p pv
: m_ ¼ Cdest
3 ql
RB
(5)
fraction gradient @av/@s is used to capture the position of the
water–vapor interface [35]. It is also used as a factor in fIE, which
holds that the larger the vapor volume fraction gradient, the more
obvious the effect of model correction. k is introduced as an
empirical consistent and its value affects the scope of correction.
The intensity of the correction is denoted as I and can be computed as 1–fIE. When the value of I is 0, there is no correction;
when the value of I is close to 1, the influence of the turbulent viscosity is very small
Fig. 1 Fluid domain and grid structure in numerical simulation: (a) schematic diagram of the computational domain and
(b) grid structure diagram
independence of the grid. The vortex shedding frequency f of a
hydrofoil in a truncated trailing edge is a basic characteristic
parameter and @av/@s plays an important role in capturing the
position of the water–vapor interface. Therefore, these two criteria
were used for the grid independence test, as shown in Fig. 2. The
grid refinement was based on the grid convergence index [37].
The value of f stabilized at 2.02 kHz in the third and the fourth
grid. According to the simulation of the last three grids, the grid
convergence index of @av/@s was 1.92%. Therefore, the third grid
was used in subsequent cavitation simulations with an element of
2.46 106. The mesh used an O-type topology with an overall
quality above 0.7, as shown in Fig. 1(b). The mesh was refined on
the hydrofoil wall and the growth ratio was 1.2. The average value
of yþ in the hydrofoil surface was 5.
It was found that f and the peak-to-peak value of the pressure
coefficient Cp-p at the trailing edge were affected by the length of
time-step, as shown in Fig. 2(b). Cp-p is the fluctuation of relative
pressure Cp, calculated by Eq. (12). When the time-step was less
than 0.01 ms, the value of f stabilized at 2.02 kHz and the variation
of Cp-p was less than 1%. The time-step of 0.01 ms was adopted
Cp ¼
Fig. 2
test
p pv
2
0:5qVref
(12)
4
Results and Discussion
4.1 Effect of Empirical Coefficient k. To clarify the influence of the IE model on the flow field, the cavitation flow around
a two-dimensional NACA 0009 hydrofoil was numerically simulated. The only variable in the simulations was k and the results
are shown in Fig. 3. The white isoline of av ¼ 0.5 was plotted to
show the outline of the cavity.
In Fig. 3(a), the intensity of correction I is used to show the
region on where the IE model had an effect. It shows that the
affected region of the IE model perfectly lies along the interface
of water and vapor. Since k was introduced into the IE model as a
denominator, the larger the value of k, the smaller the value of the
coefficient fIE introduced by the IE model. In Fig. 3(a), when
k ¼ 0.001, the red area only appears in the closed part of the cavitation area at the leading edge of the hydrofoil. When k ¼ 0.07,
the turbulent viscosity of almost the entire cavitation region was
corrected. The k substantially affected the scale of the affected
region: the greater k is, the larger the affected region.
When k ¼ 0.001, the IE model almost reduced to the ST model.
The cavity bubble was attached to the leading edge of the hydrofoil and remained still. In the closure of the attached cavity, the
turbulent viscosity suddenly increased and was stably attached to
the tail of the cavity bubble, shown in Fig. 3(b). With the increase
of k, the cavitation area increased, and became unstable in the
cavitation closed area. The instability of the cavitation region also
caused the fluctuation of turbulent viscosity. The main region of
turbulent viscosity existed behind the cavitation closed region and
Calculation parameters independence tests: (a) grid independence test and (b) time-step independence
091403-4 / Vol. 144, SEPTEMBER 2022
Transactions of the ASME
Downloaded from http://asmedigitalcollection.asme.org/fluidsengineering/article-pdf/144/9/091403/6866799/fe_144_09_091403.pdf by Xi'An Jiaotong University Lib user on 05 December 2022
3.3 Discretization Schemes and Calculation Parameter
Settings. The unsteady cavitation flow was solved by the commercial software, ANSYS CFX, using a hybrid combination of the
finite volume method and finite element method to discretize the
Navier–Stokes equation. The advection scheme options were set
to the high resolution and the transient term used the second-order
backward Euler method for discretization. In every time-step, the
maximum iterations number was 15 and the target of the rootmean-square residual 2 105 was met.
In the Zwart–Gerber–Belamri cavitation model, the destruction
coefficient Cdest and production coefficient Cprod have an important impact on hydrofoil cavitation simulation, and need to be
adjusted according to experimental conditions. Therefore, the
Cdest and Cprod were adjusted according to the results of the
numerical simulation until the simulation results were aligned
with the experimental results regarding pressure distribution on
the suction side of the hydrofoil. The value of the empirical coefficient k in the IE model is discussed below.
Table 1 Cavitation model parameters
Fig. 4 Time-averaged pressure distribution on the suction
side of hydrofoil
was distributed with the shedding cavity fragmentations in the
flow field.
The increase of k also had an effect on the vortex distribution,
shown in Fig. 3(c). The expansion of the correction range led to a
further growth of the cavitation area and, finally, the whole cavitation fell off quasi-periodically. The shedding of the large-scale
cavitation region led to two obvious frequencies of shedding vorticity at the trailing edge of the hydrofoil. One was the highfrequency vortex shedding at the trailing edge, which conforms to
the Stokes law, the other was the low-frequency cavitation
shedding.
To make the correction range of the IE model mainly focused
on the two-phase transition zone, k was assigned as 0.03 in the
following empirical work.
4.2 Validation of the Interface Effect Model. The ST model
and the IE model were used to calculate the three-dimensional
hydrofoil model. All predicted the pressure distribution on the
suction surface of the hydrofoil with experimental data, shown in
Fig. 4. The corresponding parameters of the cavitation model are
shown in Table 1. The time-averaged pressure at the front part of
the hydrofoil was close to the saturation pressure pv and increased
to a peak in the middle of the hydrofoil. Then, the pressure
returned to a state of slow growth.
The comparison of cavitation flow between simulation results
and the experiment from the same position is shown in Fig. 5. In
the simulation results, the isosurface of water vapor volume
Journal of Fluids Engineering
Turbulent model
Cdest
Cprod
ST model
IE model
0.01
0.05
2000
550
fraction av ¼ 0.5 was used to show the position of the cavity. The
Q criterion was used to show the vorticity in the flow field and the
isosurface of Q ¼ 1.6 10þ7 s2 .
In Fig. 5(a), the experimental visualization result, the hydrofoil
had leading-edge partial cavitation under cavitation number
r ¼ 0.81, accompanied by cavitation shedding. It can be seen that
a disturbance was happening in the attached cavity and that the
detached cavity bubbles soon collapsed. The cavitation region
simulated by the ST model was stably attached to the leading
edge. Near the cavitation closed area, there was a recirculation
zone, shown in Fig. 5(b). The leading-edge cavitation of the IE
model showed clear unsteady characteristics. There was a shedding cavity in Fig. 5(c); two re-entrance flows were approaching
the leading edge and led to huge turbulent kinetic energy. The
cavitation was shedding off and collapsed gradually in the process
of moving downstream. Although the cavitation shedding was not
simulated by ST model, a small pressure fluctuation was detected,
with a frequency of 202 Hz, which is close to the experimental
value of 225 Hz. In the simulation of the IE model, because of the
cavitation shedding, there was an obvious pressure fluctuation and
the frequency is 200 Hz. The horseshoe vortex was accompanied
by the shedding of the cavity and remained even when the cavity
completely collapsed.
As it is only difference between the ST model and IE model,
turbulent viscosity should be the reason for the unsteady cavitation motion in the flow field. In the ST model, the flow field and
turbulence viscosity lt were steady. There were stable recirculation zones under the tip of the cavity bubble. The center of the
high lt zone lay at the beginning of the reflux flow near-wall,
which is thought to block the development of re-entrance jets. The
flow field simulated by the IE model at t ¼ 1.60 ms is shown in
Fig. 6(b). In this stage, the cavity in the middle foil section just
grew to its maximum length and the reflux flow led to water
entrainment at the end of the cavity bubble. The lt predicted by
the IE model was too small to weaken the re-entrance jets. However, it should be noted that the highest lt in the ST model was
1.43 Pas compared with 2.82 Pas in the IE model at t ¼ 1.60 ms.
In terms of details, the simulated cavitation results were still
different from the experimental results to a certain extent. In
Fig. 5(a), the cavitation structure is very complex as the
SEPTEMBER 2022, Vol. 144 / 091403-5
Downloaded from http://asmedigitalcollection.asme.org/fluidsengineering/article-pdf/144/9/091403/6866799/fe_144_09_091403.pdf by Xi'An Jiaotong University Lib user on 05 December 2022
Fig. 3 The effect of turbulence correction model with different k on (a) the modified region, (b) turbulent viscosity,
and (c) velocity swirling
Fig. 6
The contrast of turbulence viscosity between ST model and IE model: (a) ST model and (b) IE model
Fig. 7 Re-entrance jet induce cavity shedding on the suction side of hydrofoil: (a) time-dependent pressure
fluctuation and (b) time-dependent av
experiment shown. In the part near the leading edge of the hydrofoil, the interface between the cavitation region and the liquid
phase was clear and it was unstable. The crinkle on interface
changes instantly, which was hard to be captured by RANS
model. When the re-entrance jet reached the leading edge, the
cavitation area was broken into cavitation clouds. The distribution
of the cavitation clouds was much irregular and cavity bubbles
091403-6 / Vol. 144, SEPTEMBER 2022
were broken into fine size. However, the IE model captured the
macroscopic phenomena well, such as re-entrance jets, and horseshoe vortices.
4.3 Hydrodynamics of Re-Entrance Jet. The transient pressure in the middle of the hydrofoil’s suction surface was monitored in numerical simulations and the results are shown in
Transactions of the ASME
Downloaded from http://asmedigitalcollection.asme.org/fluidsengineering/article-pdf/144/9/091403/6866799/fe_144_09_091403.pdf by Xi'An Jiaotong University Lib user on 05 December 2022
Fig. 5 Cavitation of experiment and simulations with the isosurface of vortex strength Q 5 1.6 3 1017 s22 and isosurface of water vapor volume fraction av 5 0.1: (a) experiment, (b) ST model, and (c) IE model
Fig. 9 The development of shear stress in the cavity: (a) the growth of attached cavity, (b)
the growth of re-entrance jet, and (c) the shedding off at the leading edge
Fig. 7(a). The position at the leading edge was noted as the relative position, lc ¼ 0. Because of the cavitation shedding off, there
was an obvious quasi-periodical development of pressure from the
leading edge to the trailing edge. In Fig. 6(a), the pressure in the
middle of hydrofoil suddenly increased and its value was even
greater than that at the trailing edge.
The development of cavitation is monitored by av as shown in
Fig. 7(b). In the simulation results of the IE model, av showed a
quasi-periodical change, which signifies the shedding of the cavity. In the development of the re-entrance jet, the cavity bubble reattached to the foil surface. The propagation of cavity growth was
Journal of Fluids Engineering
synchronous with the downstream development of the high point.
According to the slope in Fig. 7(b), the velocity of the re-entrance
jet was calculated, as 0.63 Vref (22 m/s). The velocity downstream
of the growth of cavity was slightly lower than the jet velocity,
which is 0.54 Vref (19 m/s). Both values were close to that measured in a Venturi duct flow by Callenaere [4] and Ganesh [5].
Compared with the development of pressure and av of the IE
model, it can be seen that when av on the hydrofoil surface
increased to the maximum value, there was a high-pressure region
behind the closed area of the cavitation area, where a sudden pressure increase was monitored [36]. When the cavity length reaches
SEPTEMBER 2022, Vol. 144 / 091403-7
Downloaded from http://asmedigitalcollection.asme.org/fluidsengineering/article-pdf/144/9/091403/6866799/fe_144_09_091403.pdf by Xi'An Jiaotong University Lib user on 05 December 2022
Fig. 8 Turbulent viscosity of IE model
Fig. 11 Vorticity enhancement in the region of IE model influenced
the maximum, the re-entrance jet will be caused by the highpressure region. When the re-entrance jet reached the leading
edge, the cavitation area was cut off, leading to much higher pressure much higher than the saturation pressure at the leading edge
of the hydrofoil. After the shedding of cavity bubbles, a lowpressure area could be seen in the lc ¼ 0.6 without vapor phase,
which may be due to the bubbles flowing downstream, still close
to the hydrofoil surface.
Figure 8 shows the character of turbulent viscosity in a cavitation shedding cycle. This periodic cavity shedding off lasted about
4.25 ms. It starts from the growth of the attached leading cavity
and this time was recorded as t ¼ 0 ms. The turbulent viscosity
also shed off with the cavity. Cavitation triggered turbulence in
the flow field, which led to an increase of lt. The shedding cavity
was surrounded by a high lt region, but the IE model reduced the
lt in the water–vapor transition zone. The reduction of turbulent
viscosity reduced the energy dissipation in the transport of vorticity in the time-averaged flow, which ensures the strength of vorticity in the cavitation flow field, promoting the formation of a
horseshoe vortex. Large turbulent viscosity also means large turbulent kinetic energy. It was observed that with the bubble collapse, the turbulent viscosity suddenly increased, which is
consistent with the findings of Laberteaux [38].
Cavitation flow is a kind of strong shear flow, and the development of shear stress s on the spanwise section is shown in Fig. 9.
At the stage of attached cavity growth, the flow direction in the
cavity was generally positive in the x-direction in general, and the
shear stress in the attached cavity was weak. When the reentrance jet entered the attached cavity, as shown in Fig. 9(b),
negative velocity was detected in the layer close to the wall in the
cavitation area. Although the water had not reached the leading
edge of the hydrofoil, there was already reflux in the leading of
cavity bubble. It should be noted that in the passage of reflux
091403-8 / Vol. 144, SEPTEMBER 2022
flow, the IE model reduced the turbulent viscosity along the interface of water and vapor, which reduces the energy dissipation of
the re-entrance jet. However, in the middle cavity region, the flow
field maintained a high turbulent viscosity, with little influence
from the IE model. The change in velocity direction between the
upstream flow and the re-entrance jet occurred where the turbulent
viscosity was high, which led to strong shear stress in the middle
of the cavity. And when the re-entrance jet reached the leading
edge, strong shear flow occurred in the whole cavity and the cavity shed off from the hydrofoil.
4.4 Development of Horse Vortex. A periodic motion of
cavity shedding is shown in Fig. 10, with the isosurface of
Q ¼ 1.6 10þ7 s2 , the cavitation region displayed by isosurface
of av ¼ 0.5, and the plane of symmetry. At t ¼ 1.60 ms, the reentrance jet occurred and induced a high turbulent kinetic energy
under the cavity. The attached cavity bubble was then cut off
when the re-entrance jet reached the leading edge, and the shedding cavity flowed downstream. There was a prominent phenomenon at t ¼ 2.90 ms, with a short intermittence between the cavity
shedding off and cavitation inception of the next periodic. At
t ¼ 2.90 ms, the shedding cavity deformed to a horseshoe shape
and the vorticity distribution clearly correlated with cavitation,
especially notable as the horseshoe vortex was accompanied by
cavitation shedding, as described by Johannes [39]. The existence
of the horseshoe vortex kept the low-density cavitation bubbles
gathering and maintained the central low pressure. It can be
speculated that horseshoe vortex makes the cavitation exist for a
long time after shedding off. After the cavity collapsed, the vorticity accompanied with the shedding cavity did not disappear but
continued to flow downstream and gradually dissipated.
The vorticity contour in the middle of the hydrofoil is shown in
Fig. 11. The position of the cavity is plotted with a white isoline
Transactions of the ASME
Downloaded from http://asmedigitalcollection.asme.org/fluidsengineering/article-pdf/144/9/091403/6866799/fe_144_09_091403.pdf by Xi'An Jiaotong University Lib user on 05 December 2022
Fig. 10 The shedding of leading-edge cavitation (av 5 0.1, Q 5 1.6 3 107 s22)
Fig. 12 Horseshoe vortex detaches from wall
5
Conclusion
The present investigation has established a new turbulent viscosity correction method for calculating the cavitation flows. The
modeling framework is based on a transport-based cavitation
model with a time-averaged fluid dynamics equation. The IE
model is designed to decrease the predicted value of turbulent viscosity on the interface of water and vapor. The cavitation flow
around an NACA0009 truncated hydrofoil at 2.5 deg attack angle
was used to demonstrate the effect of the IE model.
The IE model exerted a decrease of turbulent viscosity based
on the spatial-averaged gradient of vapor volume fraction. It correctly captured the position of the water–vapor transition zone,
and the empirical coefficient k affected the correction range of the
model. The larger the correction range of the model, the more
unstable the cavitation flow field.
Both the ST model and IE model successfully predicted the
time-averaged pressure distribution on the suction of hydrofoil.
However, the ST model tended to overpredict the turbulent viscosity around the closure of the cavity and blocked the development of re-entrance. The IE model can simulate the unsteady
characteristics of the cavitation region.
The re-entrance induced by high pressure was analyzed and the
cavity bubble on the suction surface of hydrofoil cavitation shed
off in a quasi-periodic fashion. The shear stress plays an important
role in the shedding of the cavity. The development of a reentrance jet induced strong shear stress in the middle of the cavity.
When the strong shear flow occurred in the leading edge, the cavity would shed off from hydrofoil.
In the development of horse vortex, the reduction of turbulent
viscosity by the IE model reduced the energy dissipation in the
transport of the shedding cavity and enhanced the formation of the
horseshoe vortex. The backflow caused by the adverse pressure
gradient raised the vortex from the hydrofoil surface, and formed
a horseshoe structure.
The IE model mainly focuses on the calculation of turbulence
at the water–vapor interface. It represents a new approach to the
numerical simulation of cavitation flow in engineering applications, using an empirical parameter k. More detailed experimental
Journal of Fluids Engineering
research and more comprehensive numerical calculations of the
cavitation flow field are needed.
Funding Data
National Natural Science Foundation of China (Nos.
51836010 and 51879266; Funder ID: 10.13039/
501100001809).
2115 Talent Development Program of China Agricultural
University (Funder ID: 10.13039/501100002365).
References
[1] Franc, J. P., and Michel, J. M., 1985, “Attached Cavitation and the Boundary
Layer: Experimental Investigation and Numerical Treatment,” J. Fluid Mech.,
154, pp. 63–90.
[2] Fruman, D. H., Reboud, J. L., and Stutz, B., 1999, “Estimation of Thermal
Effects in Cavitation of Thermosensible Liquids,” Int. J. Heat Mass Transfer,
42(17), pp. 3195–3204.
[3] Joussellin, F., Courtot, Y., Coutier-Delghosa, O., and Reboud, J.-L., 2001,
“Cavitating Inducer Instabilities: Experimental Analysis and 2D Numerical Simulation of Unsteady Flow in Blade Cascade,” Proceedings of Fourth International
Symposium on Cavitation, California Institute of Technology, Pasadena, CA.
[4] Callenaere, M., Franc, J. P., Michel, J. M., and Riondet, M., 2001, “The Cavitation Instability Induced by the Development of a Re-Entrant Jet,” J. Fluid
Mech., 444, pp. 223–256.
[5] Ganesh, H., Makiharju, S. A., and Ceccio, S. L., 2016, “Bubbly Shock Propagation as a Mechanism for Sheet-to-Cloud Transition of Partial Cavities,” J. Fluid
Mech., 802, pp. 37–78.
[6] Pipp, P., Hocevar, M., and Dular, M., 2021, “Numerical Insight Into the
Kelvin-Helmholtz Instability Appearance in Cavitating Flow,” Appl. Sci.Basel, 11(6), p. 2644.
[7] Peng, X. X., Ji, B., Cao, Y., Xu, L., Zhang, G., Luo, X., and Long, X., 2016,
“Combined Experimental Observation and Numerical Simulation of the Cloud
Cavitation With U-Type Flow Structures on Hydrofoils,” Int. J. Multiphase
Flow, 79, pp. 10–22.
[8] Wu, J., Deijlen, L., Bhatt, A., Ganesh, H., and Ceccio, S. L., 2021, “Cavitation
Dynamics and Vortex Shedding in the Wake of a Bluff Body,” J. Fluid Mech.,
917, p. A26.
[9] Brennen, C., 1969, “A Numerical Solution of Axisymmetric Cavity Flows,” J.
Fluid Mech., 37(4), pp. 671–688.
[10] Kinnas, S. A., and Fine, N. E., 1993, “A Numerical Nonlinear Analysis of the
Flow Around Two- and Three-Dimensional Partially Cavitating Hydrofoils,” J.
Fluid Mech., 254, pp. 151–181.
[11] Lemonnier, H., and Rowe, A., 1988, “Another Approach in Modeling Cavitating Flows,” J. Fluid Mech., 195(1), pp. 557–580.
[12] Peters, A., and el Moctar, O., 2020, “Numerical Assessment of CavitationInduced Erosion Using a Multi-Scale Euler–Lagrange Method,” J. Fluid Mech.,
894(A19), pp. 1–54.
[13] Bouziad, Y. A., 2005, “Physical Modelling of Leading Edge Cavitation: Computational Methodologies and Application to Hydraulic Machinery,” Doctor of
Science, Ecole
Polytechnique Federale De Lausanne, France.
[14] Peters, A., 2019, “Numerical Modelling and Prediction of Cavitation Erosion
Using Euler-Euler and Multi-Scale Euler-Lagrange Methods,” Doctors of Engineering, Universit€at Duisburg-Essen, Germany.
[15] Cheng, H., Long, X., Ji, B., Peng, X., and Farhat, M., 2021, “A New EulerLagrangian Cavitation Model for Tip-Vortex Cavitation With the Effect of
Non-Condensable Gas,” Int. J. Multiphase Flow, 134, p. 103441.
[16] Wang, J., Li, H., Guo, W., Wang, Z., Du, T., Wang, Y., Abe, A., and Huang,
C., 2021, “Rayleigh–Taylor Instability of Cylindrical Water Droplet Induced by
Laser-Produced Cavitation Bubble,” J. Fluid Mech., 919(A42), pp. 1–27.
[17] Zhou, L., and Wang, Z., 2008, “Numerical Simulation of Cavitation Around a
Hydrofoil and Evaluation of a RNG j-e Model,” ASME J. Fluids Eng., 130(1),
p. 011302.
[18] Gnanaskandan, A., and Mahesh, K., 2015, “A Numerical Method to Simulate
Turbulent Cavitating Flows,” Int. J. Multiphase Flow, 70, pp. 22–34.
[19] Stutz, B., and Reboud, J. L., 1997, “Two-Phase Flow Structure of Sheet Cavitation,” Phys. Fluids, 9(12), pp. 3678–3686.
[20] Melissaris, T., Bulten, N., and van Terwisga, T. J. C., 2019, “On the Applicability of Cavitation Erosion Risk Models With a URANS Solver,” ASME J. Fluids
Eng., 141(10), p. 101104.
SEPTEMBER 2022, Vol. 144 / 091403-9
Downloaded from http://asmedigitalcollection.asme.org/fluidsengineering/article-pdf/144/9/091403/6866799/fe_144_09_091403.pdf by Xi'An Jiaotong University Lib user on 05 December 2022
of av ¼ 0.5. This is where the affected region of the IE model lies.
In noncavitation conditions, the high-vorticity region of the suction surface of the hydrofoil was attached to the surface of the
hydrofoil. A vortex induced by cavitation made a disturbance in
the flow field. A high-vorticity region shed off from the leading
edge with the cavity at t ¼ 0.00 ms. The vorticity at the interface
of water and vapor was stronger than the surrounding areas at
t ¼ 0.75 ms. When the negative vorticity flowed downstream to
the trailing edge at t ¼ 3.60 ms, it broke the regulation of the trailing vortex shedding structure.
The development of the horseshoe vortex simulated by the IE
model was shown in Fig. 12. When the cavitation bubble breaked
off from the leading edge, the huge vorticity inside the cavitation
bubble was retained. Because of the low pressure in the cavitation
zone, a great adverse pressure gradient appears downstream of the
cavitation zone. This adverse pressure gradient induced a recirculation zone. At t ¼ 1.10 ms, this upstream flow concaved the cavitation region near the wall. When t ¼ 1.60 ms, the cavitation area
was completely separated from the wall, forming a horseshoe
vortex.
091403-10 / Vol. 144, SEPTEMBER 2022
[31] Li, Z., and Jaberi, F. A., 2009, “Turbulence-Interface Interactions in a Two-Fluid Homogeneous Flow,” Phys. Fluids, 21(9),
pp. 095102–095114.
[32] Barre, S., Rolland, J., Boitel, G., Goncalves, E., and Patella, R. F., 2009,
“Experiments and Modeling of Cavitating Flows in Venturi: Attached Sheet
Cavitation,” Eur. J. Mech. B/Fluids, 28(3), pp. 444–464.
[33] Zwart, P. J., Gerber, A. G., and Belamri, T., 2004, “A Two-Phase Flow Model
for Predicting Cavitation Dynamics,” International Conference on Multiphase
Flow, Yokohama, Japan, May 30–June 3, No. 152.
[34] Launder, B. E., and Spalding, D. B., 1974, “The Numerical
Computation of Turbulent Flows,” Comput. Methods Appl. Mech. Eng., 3(2),
pp. 269–289.
[35] Julien, R., Guillaume, B., Stephane, B., Eric, G., and Regiane, F. P., 2006,
“Experiments and Modelling of Cavitating Flows in Venturi—Part 1: Stable
Cavitation,” Sixth International Symposium on Cavitation, Wageningen, The
Netherlands, pp. 1–15.
[36] Dupont, P., 1991, “Etude de la Dynamique D’une Poche de Cavitation Partielle
en Vue de la Pr’Ediction de L’Erosion Dans Les Turbomachines
Hydrauliques,” Doctor of Sciences, Ecole Polytechnique F’ed’erale de Lausanne, Switzerland.
[37] Celik, I. B., Ghia, U., Roache, P. J., and Freitas, C. J., 2008, “Procedure for
Estimation and Reporting of Uncertainty Due to Discretization in CFD
Applications,” ASME J. Fluid Eng., 130(7), p. 078001.
[38] Laberteaux, K. R., and Ceccio, S. L., 2001, “Partial Cavity Flows. Part 1. Cavities Forming on Models Without Spanwise Variation,” J. Fluid Mech., 431, pp.
1–41.
[39] Johannes, B., and Peter, F. P., 2012, “The Influence of Imposed Strain Rate and
Circulation on Bubble and Cloud Dynamics,” Proceedings of the International
Symposium on Cavitation, Singapore, pp. 1–8.
Transactions of the ASME
Downloaded from http://asmedigitalcollection.asme.org/fluidsengineering/article-pdf/144/9/091403/6866799/fe_144_09_091403.pdf by Xi'An Jiaotong University Lib user on 05 December 2022
[21] Huang, B., and Wang, G. Y., 2011, “A Modified Density Based Cavitation
Model for Time Dependent Turbulent Cavitating Flow Computations,” Chin.
Sci. Bull., 56(19), pp. 1985–1992.
[22] Ji, B., Luo, X., Arndt, R. E. A., and Wu, Y., 2014, “Numerical Simulation of
Three-Dimensional Cavitation Shedding Dynamics With Special Emphasis on
Cavitation–Vortex Interaction,” Ocean Eng., 87, pp. 64–77.
[23] Mousavi, S. M., and Roohi, E., 2014, “Three-Dimensional Investigation of the
Shock Train Structure in a Convergent–Divergent Nozzle,” Acta Astronaut.,
105(1), pp. 117–127.
[24] Roohi, E., Pendar, M.-R., and Rahimi, A., 2016, “Simulation of ThreeDimensional Cavitation Behind a Disk Using Various Turbulence and Mass
Transfer Models,” Appl. Math. Modell., 40(1), pp. 542–564.
[25] Pendar, M.-R., Esmaeilifar, E., and Roohi, E., 2020, “LES Study of Unsteady
Cavitation Characteristics of a 3-D Hydrofoil With Wavy Leading Edge,” Int.
J. Multiphase Flow, 132, p. 103415.
[26] Mirjalily, S. A. A., 2021, “Lambda Shock Behaviors of Elliptic Supersonic
Jets; a Numerical Analysis With Modification of RANS Turbulence Model,”
Aerosp. Sci. Technol., 112, p. 106613.
[27] Shyy, W., Thakur, S., and Wright, J., 1992, “Second-Order Upwind and Central Difference Schemes for Recirculating flow Computation,” AIAA J., 30(4), pp. 923–932.
[28] Wu, J. Y., Wang, G. Y., and Shyy, W., 2005, “Time-Dependent Turbulent
Cavitating Flow Computations With Interfacial Transport and Filter-Based
Models,” Int. J. Numer. Methods Fluids, 49(7), pp. 739–761.
[29] Huang, B., Wang, G.-y., and Zhao, Y., 2014, “Numerical Simulation Unsteady
Cloud Cavitating Flow With a Filter-Based Density Correction Model,” J.
Hydrodyn., 26(1), pp. 26–36.
[30] Rashidi, M., and Banerjee, S., 1990, “The Effect of Boundary Conditions and
Shear Rate on Streak Formation and Breakdown in Turbulent Channel Flows,”
Phys. Fluids A, 2(10), pp. 1827–1838.