SCIENCE CHINA
Physics, Mechanics & Astronomy
• Article •
October 2014 Vol. 57 No. 10: 1967–1976
doi: 10.1007/s11433-014-5538-6
A modified PANS model for computations of unsteady turbulence
cavitating flows
HU ChangLi*, WANG GuoYu, CHEN GuangHao & HUANG Biao
Beijing Institute of technology, Beijing 100081, China
Received October 16, 2013; accepted March 27, 2014; published online July 21, 2014
A modification to the PANS (partially averaged Navier-Stokes) model is proposed to simulate unsteady cavitating flows. In the
model, the parameter fk is modified to vary as a function of the ratios between the water density and the mixture density in the
local flows. The objective of this study is to validate the modified model and further understand the interaction between turbulence and cavitation around a Clark-Y hydrofoil. The comparisons between the numerical and experiment results show that the
modified model can be improved to predict the cavity evolution, vortex shedding frequency and the lift force fluctuating in
time fairly well, as it can effectively modulate the eddy viscosity in the cavitating region and various levels of physical turbulent fluctuations are resolved. In addition, from the computational results, it is proved that cavitation phenomenon physically
influences the turbulent level, especially by the vortex shedding behaviors. Also, the mean u-velocity profiles demonstrate that
the attached cavity thickness can alter the local turbulent shear layer.
turbulence model, PANS, unsteady cavitating flows, surrogate model
PACS number(s): 47.80.Jk, 47.11.-j, 47.27.E-, 02.60.-x
Citation:
Hu C L, Wang G Y, Chen G H, et al. A modified PANS model for computations of unsteady turbulence cavitating flows. Sci China-Phys Mech Astron, 2014, 57: 19671976, doi: 10.1007/s11433-014-5538-6
Cavitation is the vapourization of a liquid when the static
pressure decreases below its vapor pressure. This phenomenon usually arises in flows around solid bodies and it
strongly affects the flow field and the neighbouring structures [1]. Actually, cavitation physical mechanisms are not
well understood due to the complex, unsteady flow structures associated with turbulence and cavitation dynamics.
There are significant computational issues in regard to stability, efficiency, and robustness of the numerical algorithm
for turbulent unsteady cavitating flows [2].
In recent years, noticeable efforts have been made in the
numerical methods for unsteady cavitating flows. The popular technique is to apply homogenous equilibrium medium
assumption, that is, the vapour/liquid medium is considered
as a homogeneous single fluid that satisfies the Navier*Corresponding author (email: qhclq@163.com)
© Science China Press and Springer-Verlag Berlin Heidelberg 2014
Stokes equations. The important point returns to the mixture
density definition and resolution which induces various
modeling approaches. One is based on the barotropic-state
law initially proposed by Delannoy et al. [3] and Coutier et
al. [4] use this method to simulate the cloud cavity in a
Venturi-type duct. The other method is the transport equation-based cavitation models (TEM) which become attractive gradually, and both steady and unsteady flow computations have been reported by many researchers, such as Kubota et al. [5], Singhal et al. [6], Merkle et al. [7], Kunz et al.
[8], and Senocak et al. [9]. In the TEM, a transport equation
for either mass or volume fraction, with appropriate source
terms to regulate the mass transfer between vapour and liquid phases, is adopted [2].
For simulating the cavitating flows, the turbulence model
can significantly affect the flow structures. Several researchers [4,10] have indicated that high eddy viscosity of
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Hu C L, et al.
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the original Launder-Spalding version of the k- Reynoldsaveraged Navier-Stokes (RANS) model [11] can dampen
the vortex shedding behavior and excessively restrain the
cavitation instabilities. Recent times have seen the emergence of hybrid modeling approaches to improve the traditional RANS models, such as, detached eddy simulation
(DES) [12], unsteady RANS (URANS) [13], a filter-based
model (FBM) [10], PANS [14], and many versions of
RANS/LES hybrids. Indeed, these hybrid models have significantly improved the predictions of single-phase flows,
and some of them are applied to simulate the cavitating
flows and obtain expected results. Further details can be
found in refs. [15–18].
Inspired by the previous reporters, the present paper is
devoted to improve the predictive capability of the k-
RANS model, via partially averaging of the Navier-Stokes
equation to introduce a function of parameter fk, based on
the density ratio. Depending on this function, the different
filters caused by various fk values are implemented in the
whole flow field. For assessing the modified turbulence
model, a cloud unsteady cavitating flow over a Clark-Y
hydrofoil is computed and comparisons with the available
experiment results are used to evaluate the method and help
to further understand the cavitating flows.
October (2014) Vol. 57 No. 10
1.2
Cavitation process is governed by kinetics of the vapor and
liquid phase change in the system. Based on the homogeneous equilibrium flow theory, without considering the
thermal energy and nonequilibrium-phase change effects,
the cavitation dynamics expression is given as follows:
 l ( l u j )

 m   m  .
t
x j
2
RB
The set of governing equations comprises the conservative
form of the incompressible Favre-averaged Navier-Stokes
equations, coupling with cavitation model and turbulence
closure. The mass continuity and momentum equations are
given below:
(2)
The mixture property, m, can be expressed as:
(6)
Then, the bubble is considered as a sphere with a regular
volume resolution to make a bridge between the mass and
the bubble radius. To obtain an interphase mass transfer rate
requires further assumptions regarding the bubble concentration and radius. Finally, the source terms are as follows:
m   Fe
3 nuc (1   v )  v
RB
m   Fc
3 v v
RB
2 pv  p
3 l
2 pv  p
3 l
,
,
(7)
(8)
where Fe and Fc are two empirical coefficients which are
used to account for the different phase change rates (condensation is usually substantially slower than vaporization),
nuc is the volume fraction of the nucleation sites, RB becomes the radius of nucleation sites accordingly, and these
parameters default as:
 nuc  5  104 , RB  1 m, Fe  50, Fc  0.01.
(3)
where m is the mixture density, u is the velocity, p is the
pressure, L and T are the laminar and turbulent viscosity,
subscripts i, j, k are the directions of the axes, subscripts l
and v present liquid and vapor, and  can be density, viscosity, and so on.
(5)
where, l is the liquid density, RB is the bubble radius, pv
and p are the pressures in and around the bubble respectively, and  is the surface tension coefficient between the
liquid and vapor. Without accounting for the second order
and the surface tension terms, eq. (5) is simplified as:
(1)
(  m ui ) (  m ui u j )
p




t
x j
xi x j
m   l l  v (1   l ),
d 2 RB 3  dRB  2 pv  p
,
 


RB
dt 2
2  dt 
l
dRB / dt  2 / 3( pv  p) / l .
1.1 Favre-averaged continuity and momentum equations

 u u j 2 u k  
 (  L   T )  i 

 ij   .
 x


 j xi 3 xk  
(4)
Here, the source terms m  and m  represent the condensation and evaporation rates respectively. In the current
study, a popular phenomenological model originally proposed by Kubota et al. [5] is employed. In this model, the
source terms are originated from a modified RayleighPlesset equation:
1 Numerical model
 m (  m u j )

 0,
t
x j
Transport-based cavitation model
1.3
(9)
Turbulence model
1.3.1 Original model: standard k- model
The two-equation k- turbulence model with a wall function
treatment originally presented by Launder and Spalding [11]
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is as follows:
(  m k ) (  m u j k )


 Pt   m  
t
x j
x j

t
  

k

 k 
 , (10)

 x j 
(  m  ) (  m u j  )

2

 C 1 Pt  C 2  m
k
k
t
x j


x j


   t



  
.

 x j 
(11)
The turbulent viscosity is defined as:
t 
 m C k 2

.
(12)
Here, k and  are turbulent kinetic energy and dissipation
rate, respectively. The other parameters in this model are:
C1=1.44, C2=1.92, =1.3, k=1.0, C=0.09.
It has been reported that the original k- model overpredicts the turbulence kinetic energy, and hence turbulent
viscosity, which results in the re-entrant flow losing momentum and failing to cut off the attached cavity in cavitating flows [4]. In fact, the original model is originally developed for fully incompressible single phase flows and is
not intended for simulating the flows with multiple phase.
Thus, in the present work, we introduce a new model originated from the standard k- RANS model to predict the
cloud cavitating flows. The following is a detail discussion
for this model.
1.3.2 A modified PANS model
The original PANS model, derived by Girimaji et al. [14],
as the other hybrid models, has the similar form of closure
equations to the standard k-ε model. The differences are the
model coefficients, which reflect the intended resolution,
given as follows:
C*2  C 1 
 ku   k
fk
(C 2  C 1 ),
f
fk 2
f2
, u   k ,
f
f
(13)
(14)
and
fk 
ku

, f  u ,
k

(15)
eddy viscosity:
 m ku2
u
u
k2

 u2  f k2  1,
2
k

 k
C m

C
(16)
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October (2014) Vol. 57 No. 10
fk and f are the ratios of the unresolved-to-total kinetic energy and unresolved-to-total dissipation respectively. The
filter width in PANS is quantified by the two parameters. It
is shown that the extent of PANS averaging-relative to
RANS can be quantified using the different fk and f values.
The unresolved stress is modeled with Boussinessq approximation and modeled transport equations are solved for
the unresolved kinetic energy and dissipation. The smaller
the fk is, the finer the filter is: fk =1 represents RANS and fk
=0 indicates DNS. For high Reynolds number flows in
which the dissipative scales are not resolved, f is specified
as unity [14], which implies that RANS and PANS unresolved small scales are identical. The parameters above with
the subscript u are used in PANS model, and the others are
implemented in standard k- model. It should be noted that
for the original PANS calculation, the bridging parameters
fk and f must be specified.
Now, in order to avoid setting the fk value subjectively in
computations, based on the original PANS model, we introduce a damping function defined as:
f k  tanh(atanh C2  (  m l )c1 )  1  C2 ,
(C1  1, 0  C2  1),
(17)
where, m is the mixture density, and l is the water density.
The two parameters C1 and C2 are introduced to indicate
using different modes with various fk for computing different regions in the cavitating flows. The expected purpose is
to realize the unity of fk decreasing with the void fraction
increasing. Figure 1 demonstrates the effects of the two parameters on the trend of fk values varying with the density
ratios. In Figure 1(a), when C2 is a constant, as given to be
0.99 for example, it shows that C1 plays an important role in
the slopes of the cures. Similarly, in Figure 1(b), it is found
that C2 defines the minimum value of fk as it varies with the
density ratios.
As mentioned above, fk is significantly affected by the
two parameters, and it is necessary to set suitable values of
C1 and C2 for simulating the unsteady cavitating flows.
Thus, we conduct the surrogate-based analysis and optimization method [19] to get satisfied results. The difference
between experimental and numerical data for time-averaged
lift force Cl-diff, and the frequency of the lift coefficients
Sr-diff are chosen as objectives in surrogate-based analysis.
Figure 2 shows the computational set-up. The computational domain and boundary conditions are given according
to the experimental set-up [20]. The Clark-Y hydrofoil with
the chord length of 0.07 m is located at the center of the test
section with the angle of attack of 8°. The two important
dimensionless parameters are the Reynolds number Re and
the cavitation number , which is defined based on the outlet pressure p, the saturated vapor pressure pv, and the inlet
velocity U.
Re 
Uc

, 
p  pv
.
U 2 / 2
(18)
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October (2014) Vol. 57 No. 10
PRESS(The predicted residual sum of square) model. From
Figure 3, it is found that the two designed parameters C1
and C2 have different effects on the two objectives. For satisfying the both objectives, the optimization process appeals
to the Pareto optimal analysis [19], which is shown in Figure 4. It can be observed that the Pareto front is not continuous and there are two distinct regions marked by the red
points. Based on this, further investigations are made to get
the final results. Then, we get C1=34.05, C2=0.99 to be implemented in eq. (17). The systemic discussions about the
surrogated-based methods will be set up in our other papers
because of the limited space in the present one.
Figure 1 The value of control variable fk versus the density ratio. (a)
C2=0.99; (b) C1=30.
Figure 3 (Color online) Fitting results using the PWS model for the
objects, (a) fitting for the hydrofoil lift, (b) fitting for the frequency.
Figure 2
chord.
Boundary conditions for Clark-Y hydrofoil, c is the hydrofoil
Computations are performed for cloud cavitation condition (= 0.8). Constant velocity is imposed at the inlet,
U=10 m/s, with the corresponding Reynolds number of Re
=7×105. The vapor pressure of water at 25°C is pv=3169 Pa.
As to the grid solutions, previous research [21] has
proved that the PANS model is insensitive to the grids. Due
to the enormous work currently, there is no discussion about
the computational grids in detail. Basically, the numerical
grids used here are enough to the wall function requirement.
Figure 3 gives the fitting results that indicate the complex
relations between the designed parameters and the objectives used in the PWS model [22] (PRESS-based weighted
average surrogate). Goel et al. [23] have suggested that the
PWS model generally performed better than the best
Figure 4
(Color online) Pareto optimal front solutions.
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2 Results and discussions
2.1
Evaluation of the turbulence models
Figures 5(a) and 5(b) show the time-averaged water vapor
volume fraction contours of two turbulence models respectively. Wang et al. [24] found that the cloud cavity structure
consists of two parts, which are the attached front portion
and the detached rear region respectively. Thus, from Figure
5(b), the original model underestimates the detached part
substantially and the whole cavity scale is much smaller
than that of the modified model. Coutier et al. [4] reported
that the poor prediction of cavity shedding may be due to
over-prediction of the turbulent viscosity in the rear part of
the cavity. In Figure 6, it is found that the modified model
conducts a dramatic decrease in the time-averaged eddy
viscosity levels, which is almost a tenth of that using the
original model. So, the modified model will provide expected results.
In order to make a further study of the improvement in
the modified model, Figure 7 gives some instantaneous data
contours, including the cavity shape of experiment, the control parameter fk, and unresolved turbulent eddy viscosity.
Based on the cavitating flow over the 3D rectangular hydrofoil observed experimentally [24,25], the flow is found to
be approximately uniform over 80%–90% of the foil. Hence,
for computational efficiency, the 2D analysis is applied in
the present work. Note that in the experimental images the
white region represents cavities and here are three typical
transient cavity shapes of cloud cavitation cycle evolutions.
From the contours of fk, the modified model conducts different fk values to compute the cavity area in detail, where
regions with high level cavity volume are simulated by
smaller fk value, and vice versa. Accordingly, the distributions of turbulent eddy viscosity obtained by the modified
model also vary with the different regions, that is, the
smaller level is in the vapor area and the higher level is in
Figure 6 Time-averaged turbulent viscosity contours. (a) Modified
PANS model; (b) original model.
the water area. In fact, taking eq. (16) into account, the eddy
viscosity is closely related to the value of fk. With the
smaller value of fk, the eddy viscosity is diminished in the
cavitation region which is validated by the experimental
results of Aeschlimann et al. [26] who observed that the
turbulent viscosity decreased slightly with cavitation development. In addition, when the re-entrant flow moves
upstream along the hydrofoil suction wall, as shown in Figure 7(b), both of the wall and cavity tend to restrain its
physical turbulent fluctuation. Thus, to simulate the
re-entrant flow behavior well, the turbulence model scheme
requires slightly higher level of eddy viscosity than other
cavity area. Fortunately, the modified model behavior can
just satisfy the requirement.
2.2 Time-dependent visualization of cavity and flow
fields
Figure 5 Time-averaged volume fraction contours. (a) Modified PANS
model; (b) original model.
To assess the modified model’s performance, the temporal
evolution of the computational and experimentally observed
cavity structures are shown respectively in Figure 8. Due to
the different frequencies of the CFD results and experimental data, the transient cavity visualizations are given in
the form of their corresponding cycles. It demonstrates that
the modified model simulation is capable of capturing the
cavity inception, growth toward trailing edge, and subsequent large scale cavity breakup and shedding, in accordance with the qualitative features observed experimentally.
With the high turbulent viscosity predicted by the original
model, the attached cavity fails to grow to the foil tail and
capture the time-dependent performance, especially the cav-
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Figure 7
Instantaneous unresolved turbulent kinetic energy contours with different models.
Figure 8
Time evolutions of volume fraction contours and streamlines.
ity breakup and shedding process shown during t0+58%T
and t0+70%T with the re-entrant flow impinging on the attached cavity.
To further understand the differences between the two
models for simulating cloud cavity evolutions, Figure 9
demonstrates the cavity volumes of different locations at the
foil suction side varying with time. The x-axis is nondimensional time, and the y-axis is non-dimensional position at the foil suction side. By comparing Figure 9(a) with
Figure 9(b), it is found that although the time evolutions of
cavity volumes using the two models are both periodic, yet
the period by the modified model is shorter than that of the
original model. Moreover, the cavity volume distributions
along the foil suction side are significantly different between the two models. The maximum length of cavities by
the original model is just nearly half of the foil chord and
there is no shedding cavity at any period of time, while as
shown in Figure 9(a) the cavity can develop until covering
the whole foil suction and in any period the cavity shedding
behavior is captured.
Similarly, Figure 10 shows the reverse u-velocity component distributions at the foil suction during the same periods as that in Figure 9. We find that for the both models,
the time evolutions of reverse velocities have almost the
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same period as their corresponding cavity volume. It should
be noted that the re-entrant flow head can reach the position
which is about 0.2c away from the foil leading head and
then cut off the attached cavity as seen in Figure 10(a).
Anyway, this confirms that the re-entrant flow is mainly
responsible for the unsteady characteristics in cloud cavitation. Comparatively, the original model result shows that
Figure 9
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the area of reverse u-velocity is just located at the tail of
attached cavity steadily, covering almost the rear half of the
foil chord.
Figure 11 shows instantaneous vortex structures in the
cavitating flow fields. Here, the second invariant of the velocity gradient tensor, Q factor criterion [27], is implemented in the CFD and used to capture high swirling flow
Time evolutions of water vapor fraction contours with different models. (a) Modified PANS model; (b) original model.
Figure 10
Time evolutions of the reverse u-velocity contours with different models. (a) Modified PANS model; (b) original model.
Figure 11
Time evolutions of Q and baroclinic torque contours with different models.
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regions/vortices. The Q factor is defined by
2
1  ui  ui u j 
,
Q  
 
2  xi  x j xi 


(19)
when Q>0, the rotation is dominant and the region determines a vortex tube. Note that the local vortex refinement is
made using cells with high Q-factor values, higher than a
certain positive value chosen by the user. From the Q contours, the modified model results present the unsteady turbulent fields with multiple scales vortex, especially large
scale vortex structures at the time of cavities shedding. In
contrast, there is a steady and simple vortex structure near
the tail for the original model. Hence the modified model
resolves more scales of fluctuations and predicts stronger
time-dependency than the original one. In elucidating the
interplay between cavity and turbulent vorticity, baroclinic
torque contours are also plotted in Figure 11. As suggested
in ref. [28], the baroclinic torque formed by the mixture
density and pressure gradients in the cavitating region is
responsible for the alteration of the vorticity field. In the
present work, combined with the cavity shapes, it is found
that the baroclinic torque term is more pronounced at the
liquid-vapor interface and near the cavity closure. As is expected, the modified model mesh solutions are with remarkable baroclinic torque distributions and the original
model fails to provide enough baroclinic torque and probably under-predicts the vorticity generation in the cavitating
flows.
2.3
Velocity profiles and lift/drag coefficients
Figure 12 gives the time-averaged u-velocity component
profiles tracked along y-direction at specified positions in
the chordwise. Clearly, the thicker is the cavity in the
Figure 12
October (2014) Vol. 57 No. 10
chordwise, the larger is the velocity grads in the y-direction.
Compared with the original model, the modified model results quantitatively agree better with the experimental data
especially at the x/c=0.4 and x/c=0.6, largely because the
original model under-predicts the cavity thicker to make a
thinner boundary and the velocity along the y-direction increases much faster than the experimental data. Although
the differences between predictions and experimental data
are more substantial at the x/c=0.2, yet the agreement is
reasonable considering the difficulties in experimental
measurements.
Table 1 shows the time-averaged lift and drag coefficients as well as the frequencies collected from experiment
and computations. It is clear that all the results listed in Table 1 obtained by the modified model are larger than that
with the original model. Compared with the experiment, the
modified model under-predicts the lift coefficient but
over-predicts the drag coefficient, and this is the same as the
FBM behaviors as reported in ref. [2]. For the frequency of
fluctuations, both the models agree well with the experiments. To better understand the fluctuations and unsteadiness in the field using the two models, Figure 13 demonstrates comparisons of transient lift coefficients of the hydrofoil between predictions and measurements. Obviously,
all the lift signals are fluctuating periodically in time. As
discussed before, the cloud cavity is substantially timedependent, and then the hydrofoil lift force fluctuates in
time depending on the cavity upon its suction surface. From
the lift coefficient curves, we observe that the modified
model yields more fluctuations because it can release more
scales leading to good agreement with the experiment. In
contrast, for the original model, the impact of the higher
eddy viscosity is dominant resulting in reduced unsteadiness
and the lift coefficient curve exhibits less fluctuations in
time.
Time-averaged velocity (u/U0) profiles, Exp. data are from ref. [24]. (a) x/c=0.2; (b) x/c=0.4; (c) x/c=0.6.
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Table 1 Comparison of turbulence models behaviors
Original model
Modified PANS model
Exp. data [24]
Exp. data [29]
Figure 13
Cl
Cd
Sr
0.61
0.66
0.76
-
0.112
0.131
0.119
-
0.18
0.21
0.16
0.2
Time-evolutions of hydrofoil lift coefficient.
3 Summary and conclusions
In this paper, a modified model is proposed for cavitating
flow simulations based on the original PANS method which
is rigorously derived from a parent RANS model. The modified model inherits the most from the original PANS model
and further implements various fk values with different filters according to the various densities for the whole flow
field computations. Firstly, we explain simply the new approach to prescribing the fk function, and the merit of the
modified model for simulating the unsteady cavitating flows.
Then, the modified PANS model is assessed by various experimental data as well as that from the RANS model, and
during the evaluations, we focus on the large scale vortex
shedding due to the re-entrant flow behavior. The important
conclusions are summarized below.
(1) The governing equation of fk is introduced to get the
modified PANS model, and the two parameters C1 and C2
play an important role in the capability of the modified
PANS model to predict the unsteady cavitating flows. In
this work, the surrogate-based analysis and optimization
method is conducted to get a rather satisfactory result of
C1=34.05, C2=0.99 used in the fk governing equation.
(2) The modified model indicates better simulation abilities for the unsteady cavitating flow computations. As
compared to RANS as well as the experimental data, quantitative results are improved by the modified model, including the attached cavity development, the detached cavity in
the form of vortex shedding flow and the dynamic characteristics, as the modified model can effectively modulate the
eddy viscosity in the cavitating region and more levels of
physical turbulent fluctuations are resolved in the cavitating
flows.
(3) From the analysis of flow field structures, it reveals
that the vorticity field is significantly modified by the
time-dependent cavitation, especially with respect to the
large scale cavity shedding behaviors. Moreover, the mean
u-velocity profiles demonstrate that the attached cavity
thickness can alter the local turbulent shear layer. The more
developed the attached cavity is, the thicker the shear layer
is.
Regarding future directions, more research studies are
required to improve and validate the present closure model,
such as the unsteady cavitating flows around a threedimensional hydrofoil. The modified PANS model is expected to capture the U-shaped cloud cavity vortex structure
and the motions of re-entrant and side-entrant jets. In addition, the physical turbulence fluctuations in cavitation regions will be discussed more deeply to improve the further
modification in the numerical method. There is reasonable
theory and preliminary computational evidence to be cautiously optimistic about this new method.
This work was supported by the National Natural Science Foundation of
China (Grant Nos. 11172040 and 51239005) and the Beijing Municipal
Natural Science Foundation (Grant No. 3144043).
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