Numerical Prediction of Tip Vortex Cavitation for Marine ...
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ORIGINAL ARTICLE
Numerical Prediction of Tip Vortex Cavitation for MarinePropellers in Non-uniform Wake
Zhi-Feng Zhu1 • Fang Zhou1 • Dan Li1
Received: 14 November 2015 / Revised: 16 April 2017 / Accepted: 20 April 2017 / Published online: 6 June 2017
� Chinese Mechanical Engineering Society and Springer-Verlag Berlin Heidelberg 2017
Abstract Tip vortex cavitation is the first type of cavita-
tion to take place around most marine propellers. But the
numerical prediction of tip vortex cavitation is one of the
challenges for propeller wake because of turbulence dis-
sipation during the numerical simulation. Several parame-
ters of computational mesh and numerical algorithm are
tested by mean of the predicted length of tip vortex cav-
tiation to validate a developed method. The predicted
length of tip vortex cavtiation is on the increase about 0.4
propeller diameters using the developed numerical method.
The predicted length of tip vortex cavtiation by RNG k – emodel is about 3 times of that by SST k – x model.
Therefore, based on the validation of the present approach,
the cavitating flows generated by two rotating propellers
under a non-uniform inflow are calculated further. The
distributions of axial velocity, total pressure and vapor
volume fraction in the transversal planes across tip vortex
region are shown to be useful in analyzing the feature of
the cavitating flow. The strongest kernel of tip vortex
cavitation is not at the position most close to blade tip but
slightly far away from the region. During the growth of tip
vortex cavitation extension, it appears short and thick, and
then it becomes long and thin. The pressure fluctuations at
the positions inside tip vortex region also validates the
conclusion. A key finding of the study is that the grids
constructed especially for tip vortex flows by using
separated computational domain is capable of decreasing
the turbulence dissipation and correctly capturing the fea-
ture of propeller tip vortex cavitation under uniform and
non-uniform inflows. The turbulence model and advanced
grids is important to predict tip vortex cavitation.
Keywords Cavitation � Propeller � Tip vortex � Numerical
prediction
1 Introduction
Tip vortex cavitation of a marine propeller is usually one of
the first occurring cavitation patterns. The prediction and
study of tip vortex cavitation is crucial to the understanding
of cavitation inception and noise. The detailed features of
tip vortex cavitating flow field around a marine propeller,
including pressure, velocity and so on, can be investigated
by using advanced flow visualization and non-intrusive
measurement techniques. Felli, et al. [1] investigated the
propeller tip and hub vortices in the interaction with a
rudder by using PIV and LDV systems. A high speed
camera system was used to observe the cavitating flows
over an axisymmetric blunt body and the velocity fields are
measured by a particle image velocimetry (PIV) technique
in a water tunnel for different cavitation conditions [2].
However, due to the limitations of measurement in bubbles
and the reason that experiment method is costly and time
consuming, it is desirable to provide the details of cavi-
tating flow field by numerical simulation.
Recently, the numerical method with solving Reynolds
Average Navier–Stokes (RANS) equations were most
applied to predict propeller cavitating flow for their low
computational requirements [3–10]. A RANS method
including cavitation modeling was used to study the
Supported by Anhui Provincial Natural Science Foundation of China
(Grant No. 1608085MA05), National Natural Science Foundation of
China (Grant No. 51307003 and 61601004).
& Zhi-Feng Zhu
1 School of Electrical and Information Engineering, Anhui
University of Technology, Maanshan 243032, China
123
Chin. J. Mech. Eng. (2017) 30:804–818
DOI 10.1007/s10033-017-0145-x
cavitating flow in the Potsdam propeller. And numerical
prediction was done to validate the method with regard to
cavitation including complex cavitation phenomena
responsible for higher order pressure [5]. Scale effects on
propeller cavitating hydrodynamic and hydroacoustic per-
formances with a non-uniform inflow were investigated by
Yang et al. [7]. However, in above references, the
numerical prediction of tip vortex cavitation around marine
propellers is few. And the numerical prediction of tip
vortex cavitation is one of the challenges for propeller
wake because of turbulence dissipation during the numer-
ical simulation. The accuracy of CFD prediction of the
geometry of the cavitating tip vortex depends strongly on
the turbulence models and on the grid structure, hence only
the grids constructed especially for vortex-dominated flows
should be used, together with turbulence models especially
suited for modeling of tip vortex flows [11].
Morgut, et al. [12] analyzed the influence of grid type
and turbulence model on the numerical prediction of the
flow around marine propellers, working in uniform inflow.
But the grids was not constructed especially for vortex-
dominated flows. The contours of velocity in the tip vortex
region computed with the SST turbulence model were
investigated in their paper, but cavitation was not involved.
Four different arrangements (named mesh-tip vortex) had
been set up by increasing the mesh resolution only on the
refining tip vortexes regions in order to analyze the influ-
ence of the mesh density for the development of the cav-
itating tip vortex [13]. However, the cavitating vortexes
instability experimentally observed for the conventional
and the ducted propellers, for instance, was neglected by
their numerical computations.
The effect of turbulence models on the predicted flow
around propellers was investigated recently [14–19]. Cav-
itating turbulent flow around hydrofoils was simulated
using the Partially-Averaged Navier–Stokes (PANS)
method and a mass transfer cavitation model with the
maximum density ratio effect between the liquid and the
vapor by Ji, et al. [14]. The mechanism of the over-dissi-
pation due to the turbulence model was analyzed in terms
of the turbulence production, which is one of the dominant
source terms in the transport equations of energy [16]. To
investigate the effect of turbulence models on the predicted
tip vortex flow and the open-water performance, a number
of eddy viscosity turbulence models and Reynolds-stress
models were used in combination with various grids by
Peng, et al. [18]. The impact of turbulence modeling in
predicting tip vortex flows was evaluated using several
popular eddy viscosity models and a Reynolds stress
transport model [19]. The results indicate that the combi-
nation of a computational mesh with an adequate resolu-
tion, high-order spatial discretization scheme along with
the use of advanced turbulence models can predict tip
vortex flows with acceptable accuracy.
Based on the work of Zhu, et al. [4, 8], the further studies
in the present paper were carried out to investigate turbu-
lence model performance in combination with a grid sensi-
tivity analysis. At uniform inflows and a wake flow, the tip
vortex cavitation generated by model propeller E779A and
E779B was calculated by using a RANS solver. Several
meshes with grid concentration at tip vortex region were
generated by changing the value of somemesh parameters to
validate mesh. Numerical test was conducted by changing
the value of numerical parameters. Several Reynolds Stress
turbulencemodelswere employed in the computations to test
turbulence model performance for the prediction of tip vor-
tex cavitation. The calculated cavitation extensions were
compared with the corresponding experimental results. The
distributions of the computed pressure, velocity and vapor
volume fraction in the transversal planes across the tip vortex
core are investigated. The pressure fluctuations at the posi-
tions inside the vortex region are also presented.
2 Mathematics Mode
Navier-Stokes (N-S) equations in a mixture multi-phase
flow model were solved to calculate the physical quantities
in propeller wake, such as vapor volume fraction, pressure,
the velocity and so on. It is assumed that the mixture
density qm is a function of the vapor mass fraction fv in the
mixture flow model. The vapor transport equation was
adopted to obtain the phase change progress induced by
cavitation, which is expressed as follows:
1
qm¼ fv
qvþ 1� fv
ql; ð1Þ
o
otðqmfvÞ þ r � ðqmv
*
mfvÞ ¼ r � ltrv
rfv
� �þ Re � Rc; ð2Þ
where vm is mixture velocity, rv is Prandtl number of the
vapor turbulence, ql and qv are the density of liquid and
vapor respectively. Re and Rc are the rates of the vapor
generation and condensation, which is expressed by the
Full Cavitation Model [20]. According to dimension
analysis, Re and Rc are expressed in Eqs. (3) and (4). When
local pressure p\ pv (vapor pressure), liquid phase
become into vapor phase, and bubbles appear. We obtain
net vapor condensation rate Re as follows:
Re ¼ Cek
cqlqv
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2
3
pv � p
ql
sð1� fvÞ; ð3Þ
where k is turbulence kinetic energy, c is surface tension
coefficient. When pressure p[ pv, vapor phase become
Numerical Prediction of Tip Vortex Cavitation for Marine Propellers in Non-uniform Wake 805
123
into liquid phase. In the same way, we obtain net vapor
condensation rate Rc as follows:
Rc ¼ CC
k
cqlql
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2
3
p� pv
ql
sfv: ð4Þ
After careful studies of numerical stability and physical
behavior of the solution, the cavitaion model was adopted
by using Ce = 0.02 and Cc = 0.01 [20]. The cavitation
model was proved to be valuable for the prediction of
marine propeller cavitation flow [4].
Being Reynolds-Average processed, the N-S equations
are turned into RANS that becomes governing equations of
average quantities in flow field. There is a Reynolds stress
ooxj
�qmu0miu
0mj
� �on the right of the equations RANS,
which expresses the effect of turbulence. Based on
Boussinesq assumption, we obtain Reynolds stress
expressed as follows:
o
oxj�qmu
0miu
0mj
� �
¼ o
oxjlt
oumi
oxjþ oumj
oxi
� �� 2
3ltoumi
oxjþ qmk
� �dij
� �;
ð5Þ
where dij is Kronecker Delta, lt is turbulence viscosity
coefficient.
A Turbulence model has to be applied to close the
equations RANS. Here, the Standard k – x, SST k – x [21]
and RNG k – e [22] turbulence models were employed to
predict the tip vortex cavitation for finding the best model.
In the SST k-x turbulence model, the transformation of
calculation from the inner boundary layer by the Standard
k-x turbulence model to the out boundary layer by the k-xturbulence model was carried out through a mixture func-
tion. The two equations are
o
otðqmkÞ þ
o
oxiðqmkuiÞ
¼ o
oxjðlþ lt
rkÞ okoxj
� �þ Gk � qmb
�kx;ð6Þ
o
otðqmxÞ þ
o
oxiðqmxuiÞ ¼
o
oxjðlþ lt
rxÞ oxoxj
� �þ
qmvt
Gk � qmbx2 þ 2 1� F1ð Þqmrx;2
1
xok
oxj
oxoxj
;
ð7Þ
where x is turbulent dissipation rate, Gk ¼ �qmu0iu
0jouioxj
is
turbulence kinetic energy, F1 is mixture functions, vt =
a1k/[max(a1x, XF2)] is eddy viscosity coefficient,
b = 0.09, a1 = 0.31 [21]. The first group of parameters for
the simulation of the flow field near the wall is
rk,1 = 1.176, rx,1 = 2.0, b1 = 0.075, and the second
group of parameters for the simulation far away from the
wall is rk,2 = 1.0, rx,2 = 1.168, b2 = 0.082 8 [21].
The RNG k – e turbulence model was adopted for the
prediction of viscous flow around wall. Its two-equation is
expressed as follows:
o
otðqmkÞ þ
o
oxjðqmkumjÞ ¼
o
oxjaklð Þ ok
oxj
� �þ G� qme;
ð8Þ
where e ¼ lmqm
ou0mi
oxj
� �ou
0mi
oxj
� �is turbulent dissipation rate. The
value of the parameters is given by ak = ae = 1.39,
C2e = 1.68 [22]. Viscosity coefficient is defined as
l = lt ? lm, where lm is viscosity coefficient of mixture.
Turbulence viscosity coefficient is modified by the form
[12]: lt = [qv ? alh (ql – qv)] Clk
2/e, h = 10, Cl = 0.085.
The modification was carried out by using UDF in Fluent.
In addition, computational results indicate that the param-
eter C1e in e equation is set as 1.47 to enhance the pre-
diction accuracy of propeller wake sheet structure.
3 Geometries and Operating Conditions
In this paper, two propellers, E779A and E779B in model
scale, were employed to calculate the cavitating flow
around them. E779A is a four-bladed, fixed- pitch,and low-
skew propeller. An extensive experimental and numerical
database was built and reported in Refs. [23–27].
E779B is a four-bladed skewed propeller. Numerical
simulations were performed on the two model propellers.
The geometry parameters of the two propellers are pre-
sented in Table 1. Tip vortex cavitation is generated by the
two propellers under four operating conditions shown in
Table 2. The calculations for E779A were conducted under
the conditions of uniform and non-uniform inflows, while
the calculation for E779B was conducted under a non-
uniform inflow.
In Table 2, the advance ratio J and the cavitation
number rn are defined respectively by J = U?/(nD) and
rn = 2(p? – pv)/(qln2D2) where n is the propeller revolu-
tion speed, D is the propeller diameter and p? is the ref-
erence pressure. The working fluid in the cavitation tunnel
is water at about 20 �C, with the liquid and vapor densities
of 1000 kg/m3 and 0.0255 8 kg/m3, the saturation pressure
Table 1 Geometry parameters of model propellers E779A and
E779B
Propeller Number of
blades nb
Diameter
D/m
Expanded
area ratio ex
Pitch
p
Skew
s/(�)
E779A 4 0.227 1 0.69 1.100 low
E779B 4 0.248 2 0.55 0.699 32
806 Z.-F. Zhu et al.
123
of 2368 Pa and the surface tension coefficient of c = 0.071
7 N/m.
4 Numerical Method
In this section, the numerical method is detailed with a
special emphasis on the tip vortex cavitation. Computation
domain, mesh, numerical parameters and numerical algo-
rithms are discussed here.
4.1 Computational Domain and Mesh
As usual, the shape of computational domain is set up to be
a cylinder around propeller, because of the revolution of
the propeller. The influence of the shape and size of
computational domain on the prediction of propeller cav-
itaion was discussed in Ref. [28]. The inlet boundary at
upstream is about 1D distance from the center of the pro-
peller, where D is the propeller diameter, and the side
boundary is about 2.5D distance from the hub axis. The
distance from the center of the propeller to the exit
boundary at downstream (named Q0) is set up to 5D, which
can satisfy the prediction accuracy for propeller perfor-
mance and sheet cavitation [3]. However, it has to be
pointed out that the distance to the exit boundary (the
parameter Q0) may be important to the prediction of tip
vortex cavitation, occurring in propeller wake. Therefore,
the calculation may demand the distance (the parameter
Q0) with an adequate length in the computational domain.
Thus the value of the distance (the parameter Q0) is dis-
cussed in this paper with a special emphasis. Considering
the feasibility and computational efficiency, the domain is
divided into two parts, which are meshed using hybrid grid
strategy. Due to skewed propeller blades, the flow field
near the propeller is meshed with unstructured grid which
was also applied by Rhee, et al. [29], and the influence of
the mesh resolution was investigated mainly in this region.
We preferred to use the same structured meshes for the
outer flow field for the two propellers. The computational
domain is showed in Fig. 1.
To improve mesh quality in the region close to pro-
peller, there may be three parameters to be focused on here.
The first parameter (named Q1) is the minimum size of the
grids cell near blade tip. The second one (named Q2) is the
number of boundary layers on the propeller blades, which
may influence the prediction of cavitation on the blades. In
particular, attention is devoted to the last parameter (named
Q3), the grids cell size in the region of tip vortex cavitation.
Refining the girds with grid concentration aligned with the
tip vortex core was conducted by establishing a separated
computational domain in the flow field, as shown in Fig. 2.
The separated computational domain is composed of a
camber volume whose sectional area is rectangle. The size
of the gird cells in other regions was discussed in Refs.
[4, 30].
4.2 Boundary Conditions
Several classical types of boundary conditions were
applied in present study. A velocity was imposed at the
inlet boundary for uniform inflow. The value of the
velocity is equal to the free stream speed U?. A velocity
distribution obtained by measured data was imposed for
non-uniform inflow. The inlet boundary was imposed with
turbulence intensity at 1% and vapor volume fraction at 0.
A static pressure was imposed at the outlet boundary. The
value of the pressure is the same as far-field pressure p?.
The type of the side boundary is the same as that of inlet
boundary. On the blade surface, regarded as solid wall,
zero velocity and no-slip condition were imposed.
4.3 Spatial Discretization
In computational code, the finite volume method, which fits
for arbitrary polyhedral mesh, was applied for space dis-
cretization. A converged solution was obtained easily by
using the first-order accurate discretization scheme in a
segregated solver. But it may lead to serious diffusion, and
the solution in critical high pressure or velocity gradient
areas could fade away. Therefore, to improve the precision
of the solution, the convection terms in governing equa-
tions were discretized with second-order accurate up-wind
scheme, while the diffusion terms were discretized with the
second-order accurate central differencing scheme. The
under-relaxation factor of momentum was decreased
Table 2 Operating conditions
Operating
condition
Propeller Advance
ratio J
Reference
pressure p/Pa
Cavitation
number rnPropeller
diameter D/m
Revolution
n/rps
1 E779A 0.710 26 821 1.515 0.227 1 25
2 E779A 0.710 30 734 1.760 0.227 1 25
3 E779A 0.661 74 170 4.455 0.227 1 25
4 E779B 0.605 45 952 2.264 0.248 2 25
Numerical Prediction of Tip Vortex Cavitation for Marine Propellers in Non-uniform Wake 807
123
somewhat to prevent numerical oscillations. The effect of
spatial discretization scheme(Q4) on numerical results is
investigated in this paper.
4.4 Unsteady Treatment
1st-order implicit unsteady formulations were applied in
the segregated solver. There are 40 iterations finished in
every time step to ensure calculation convergence. As for
time step, we think its value should depend on consistence
between numerical results and measured data and residuals
of axial velocity and vapor volume fraction. Specifically,
the time step relates close to the grid and the model used in
the computation. For the simulation with cavitation model,
the convergence of computation is difficult, so the time
step should be small enough. In addition, the size of grid
cells affects the value of time step. The time step should
match the grid used in the calculation. For the calculation
in the region near propeller tip where the size of grid cell is
small, the time step should be small. The non-dimensional
time step Q5, which is defined as Q5 = DT/Tp, is discussed
in this paper where DT is time step and Tp is the period of
propeller revolution.
4.5 Computational Procedure
SIMPLE segregated algorithm, adapted to unstructured
grids, was selected for velocity-pressure coupling. The
point-wise Gauss-Seidel iterations were used to solve the
discretized equations, and algebraic muti-grid method was
used to accelerate the solution convergence.
The calculation is an iterative process which makes the
solution to converge at last. To reduce calculation effort
and to allow stable iterations, it was necessary to obtain a
converged solution under single-phase condition where
cavitation model was switched off. The single-phase
solution was then applied as initial condition for
multiphase computation. This is a second-order scheme,
which switched to 1st-order at first to prevent numerical
oscillations in critical high pressure gradient areas, and
then switched to 2nd-order when the calculation was stable.
At each advance coefficient, the cavitation test was con-
ducted by starting from a weak-cavitating operating con-
dition and increasing propeller revolution until the
scheduled cavitating operating condition arrived and cav-
itation pattern changed significantly.
5 Validation Tests of Numerical Method
The validation of numerical method, consisting of mesh
test, numerical test and turbulence model test, was carried
out using a commercial solver FLUENT. Relative error of
Kt, Kq and visible length of tip vortex cavitation were
employed in the validation. The validation of hydrody-
namics and cavitation performances of the propellers
E779A and DTMB were presented in Refs. [3, 4, 30].
5.1 Mesh Tests
The mesh validation was conducted by testing several
mesh parameters, such as Q0, Q1, Q2 and Q3 which have
been presented in the last section. The parameter Q1 was
tested by using the value of 0.0035D, 0.0025D and
0.0015D, respectively under the condition of Q2 at 4, Q0
at 5D and without Q3, which means that the grids in the
region of tip vortex cavitation was not refined. The value
of other parameters tested in this study is shown in
Table 3. The SST k – x turbulence model and second-
order accurate up-wind scheme were first chosen to
investigate the sensitivity of numerical solution to mesh.
The model was validated to be a useful model to predict
propeller cavitation in Refs. [8, 9]. Here, the numerical
calculations were conducted under the operating
Fig. 1 Computational domain
808 Z.-F. Zhu et al.
123
condition 2. The five groups of data in Table 3 indicated
the influence of Q0, Q1, Q2 and Q3 on the numerical
results respectively. KtEXP and KtNUM are measured and
calculated values of Kt respectively, where thrust coeffi-
cient is Kt = Thrust/(qln2D4). KqEXP and KqNUM are
experimental and numerical values of Kq respectively,
where torque coefficients Kq = Torque/(qln2D5). Relative
error of Kt and Kq are defined as DK = | KqNUM – KqEXP |/
KqEXP 9 100% and DK = |KtNUM – KtEXP|/KtEXP 9 100%
respectively. Non-dimensional visible length of tip vortex
cavitation is defined as lD = l/D, where l is visible length
of tip vortex cavitation.
It is observed that the numerical data of Kt and Kq in
Table 3 agree with those measured, indicating that the
meshes and numerical method presented in this paper is
capable to predict the hydrodynamics performance of
propeller. The discrepancy of meshes with different
parameters has little influence on the prediction of Kt and
Kq, indicating grid independence.
According to the data shown in Table 3, it is found that
boundary layer grids and grids near blade tip may have
weak effect on the prediction of tip vortex cavitation, but
they were proved to be important to the prediction of sheet
cavitation extension [3, 4]. Increasing the distance from the
Fig. 2 Grids near tip vortex
region
Numerical Prediction of Tip Vortex Cavitation for Marine Propellers in Non-uniform Wake 809
123
center of the propeller to the exit boundary is slightly better
in predicting the cavitation. However, the parameter Q3
seems to have strong influence on the prediction of tip
vortex cavitation extension. So refining the grids in the tip
vortex region is founded to be the best treatment to the
prediction, due to decreasing numerical diffusion.
The parameter Q0 was tested at last because the tip
vortex cavitation could not be predicted obviously if the
mesh parameters had not been validated. So only changing
the distance Q0 cannot catch tip vortex cavitation phe-
nomena. According to the data in Table 3, the effect of the
parameter Q0 on the prediction of tip vortex cavitation is
not very evident. However, it seems that the effect could be
strong with the mesh improved further in the region where
tip vortex cavitation occurs, and the visible length of tip
vortex cavitation of calculated result could be increased
because of the decreasing turbulence dissipation during the
numerical simulation process. From the data in Table 3, the
value of mesh parameters in the reference case is confirmed
and used in following sections. For the reference case,
there are about 3000000 grid cells in the flow field near the
propeller, 1400000 grid cells in the tip vortex region and
2000000 grid cells in the outer flow field.
5.2 Numerical Tests
The calculations for numerical tests were carried out
based on the mesh at the reference case for the uniform
inflow of the condition 1. In Table 4, it is found that the
parameter Q5 has slight effect on the prediction of tip
vortex cavitation, while the parameter, spatial discretiza-
tion scheme Q4, has strong effect on the prediction. As
for Kt and Kq, the situation seems to be complicated. The
data of Kt and Kq at Q4 = 1 are better than those at
Q4 = 2, and the sheet cavitation extension calculated at
Q4 = 1 is slightly weaker than the result at Q4 = 2,
indicating that the cavitation extension could be slightly
overestimated at Q4 = 2. The overestimated cavitation
extension can lead to hydrodynamics performance
breakdown. From the data in Table 4, it is observed that
the parameter Q5 with a value not too small is required in
the computation. These numerical tests validate the ref-
erence values of the two numerical parameters used here
after.
5.3 Turbulence Model Validation
The last test is for the validation of turbulence model. Here,
several models, such as the SST k – x, Standard k – e andRNG k – e turbulence model, were used in a RANS solver
to calculate propeller tip vortex cavitation respectively.
The turbulence model is discussed by using the mesh of the
reference case obtained in the above section. Here, the
numerical calculations were conducted under the operating
conditions 1 and 2 for E779A and the condition 4 for
E779B.
Table 3 Results of the tests of the mesh parameters under the operating condition 1
Mesh parameters Results
Q1 Q2 Q3 Q0 KtNUM DKt/ % 10KqNUM DKq/ % lD
Influence of Q1
0.003 5D 4 without 5D 0.281 12.4 0.57 23.7 0.00
0.002 5D 4 without 5D 0.255 2.0 0.50 8.7 0.02
0.001 5D 4 without 5D 0.240 4.0 0.44 4.3 0.04
Influence of Q2
0.001 5D 4 without 5D 0.240 4.0 0.44 4.3 0.04
0.001 5D 8 without 5D 0.240 4.0 0.44 4.3 0.04
0.001 5D 16 without 5D 0.240 4.0 0.44 4.3 0.06
Influence of Q3
0.001 5D 4 0.002 5D 5D 0.240 4.0 0.44 4.3 0.08
0.001 5D 4 0.001 7D 5D 0.240 4.0 0.44 4.3 0.26
0.001 5D 4 0.001 1D 5D 0.240 4.0 0.44 4.3 0.40
Influence of Q0
0.001 5D 4 0.001 1D 5D 0.240 4 0.44 4.3 0.40
0.001 5D 4 0.001 1D 7D 0.240 4 0.44 4.3 0.40
0.001 5D 4 0.001 1D 9D 0.240 4 0.44 4.3 0.42
Reference case
0.001 5D 4 0.001 1D 5D Total grid cells: 6400000
810 Z.-F. Zhu et al.
123
The experimental and numerical results of cavitation
extension are shown in Fig. 3. Photographs in
Figs. 3(a) and (e) [23] were captured from cavitation tunnel
through sealed window using high-velocity camera, which
show clearly the sheet and tip vortex cavitation extension.
The cavitation extension is depicted by the black iso-sur-
face with vapor volume fraction av = 0.2 in our numerical
results calculated by different turbulence models, as seen in
Figs. 3(b) (c) (d) (f) (g) (h). Analysis of these results may
be summarized as follows: (1) The visible length of tip
vortex cavitation extension in Figs. 3(b) (f) is almost zero.
The length in Figs. 3(c) (g) is visible. The length in
Figs. 3(d) (h) is the longest. (2) The visible length of tip
vortex cavitation extension is equal to each other under the
operating conditions 1 and 2, as shown in Figs. 3(d) (h).
But the tip vortex cavitation extension under the condi-
tion 1 is thicker than in the condition 2, which coincides
with the cavitation number of the two conditions. There-
fore, we may conclude that the Standard k – x model
cannot predict tip vortex cavitation extension even with a
fine mesh, and the SST k – x model can predict it slightly,
while the RNG k – e model may predict it effectively due to
small numerical diffusion of turbulence. With the com-
parison of cavitation between numerical and experimental
results, as shown in Fig. 3, the prediction by the RNG k – eturbulence model is in agreement with the measured data.
The comparison of axial velocity and turbulent kinetic
energy was analyzed to validate the RNG k – e turbulencemodel in Ref. [28].
5.4 Validation by Using E779B
The prediction of tip vortex cavitation extension for E779B
with respect to the rotational angle of the propeller was also
conducted under a non-uniform inflow and compared with
the corresponding experimental results to validate further
the numerical method presented in above section. The
geometry of the propeller E779B at its initial positioin,
viewing from the upstream side of the propeller, is shown
in Fig. 4, where X, Y, Z are three axes of Cartesian coor-
dinate system. The model propeller is centered at Cartesian
coordinate system origin. The propeller blade 1 is located
at the Y-axis at the begining. The propeller rotates coun-
terclockwise along the X-axis of Cartesian coordinate
system. The relationship between the position of the pro-
peller and its time moment is shown in Table 5, which the
position is expressed by using the angle between the center
line of the blade 1 and the Y-axis. Figure 5 presents a
comparison between a measured propeller inflow distri-
bution shown in Fig. 5(a) and a simulated inflow distri-
bution from RANS solver shown in Fig. 5(b). Owing to
insufficient measured data, there are some discrepancies
between the measured and simulated inflow.
A comparison of cavitation extension between the
numerical and experimental results during about 1/8 period
of rotating propeller is shown in Fig. 6. Photographs in
Fig. 6(a) were captured by using high-velocity camera, and
the corresponding numerical results are shown in Fig. 6(b).
There appear some common features in both numerical
and experimental results of cavitation extension:
(1) When the blade 1 is located at vertical position, the
sheet cavitaion generated by the blade is strong,
while tip vortex cavitation hardly appears, as shown
in Figs. 6(a) and (e). At the same time, the tip vortex
cavitation generated by the blade 2 is clear, but its
sheet cavitation hardly appears.
(2) With propeller ratation, the tip vortex cavitation due
to the blade 1 starts to occur, but be not clear. The
sheet cavitation starts to decrease slightly. Mean-
while the tip vortex cavitation due to the balde 2
becomes weak. The phnomena are shown in Figs. 6
(b)(f).
(3) Figs. 6(c)(g) shows that the visible length of tip
vortex cavitation by the blade 1 increases further,
while its sheet cavitation extension decreases. The
tip vortex cavitation by the blade 2 disappears
completely.
(4) The evolution of cavitation extension in
Figs. 6(d) (h) is similar to that in Figs. 6(c) (g).
From the comparison, it could be found that the
numerical reults are in agreement with the experimental
reults at the all above four positions. The grids constructed
especially for tip vortex flows by using separated compu-
tational domain is capable of decreasing the turbulence
dissipation and correctly capturing the feature of propeller
tip vortex cavitation. This indicates that the method applied
Table 4 Results of the tests of numerical parameters under the
operating condition 2
Numerical
parameters
Results
Q4 Q5 KtNUM DKt /% 10KqNUM DKq /% lD
Influence of Q4
1 80 0.242 3.2 0.447 2.8 0.03
2 80 0.240 4.0 0.440 4.3 0.40
Influence of Q5
2 40 0.233 6.8 0.427 7.2 0.36
2 80 0.240 4.0 0.440 4.3 0.40
2 160 0.240 4.0 0.440 4.3 0.42
Reference case
2 80
Numerical Prediction of Tip Vortex Cavitation for Marine Propellers in Non-uniform Wake 811
123
in this paper is also useful to predict tip vortex caviation
generated by the propeller under a non-uniform inflow.
6 Results
In this section, the cavitating flows generated by E779A
and E779B under a non-uniform inflow condition are cal-
culated and investigated. The distributions of axial
velocity, total pressure and vapor volume fraction in the
transversal planes across tip vortex region are shown to
analysis the feature of the cavitating flow. The pressure
fluctuations at the positions inside tip vortex region are also
presented and analyzed.
6.1 Unsteady Simulation for E779A
under the Operating Condition 3
Unsteady simulation of the cavitating flow by the propeller
E779A was conducted under the operating condition 3. In
Fig. 7 the prediction of tip vortex cavitation extension at
ten positions during about 1/4 period of rotating propeller
is presented with respect to the angle between the center
Fig. 3 Experimental and numerical predictions of cavitation
Fig. 4 Coordinate system and model propeller
Table 5 Relationship between the position of the blade 1 and time
moment
Time t/s Angle h/(�)
0.090 0 0.0
0.092 0 18.0
0.093 5 31.5
0.095 0 45.0
812 Z.-F. Zhu et al.
123
line of the blade 4 and the Y-axis. The evolution of cavi-
tation extension with the propeller rotating may be sum-
marized as follows: (1) The sheet cavitation generated by
the blade 4 in Fig. 7(a) is strong, but its tip vortex cavi-
tation hardly appears. With the propeller rotating counter-
clockwise during the 1/4 period, the sheet cavitation by the
blade 4 becomes weak slightly, while its tip vortex cavi-
tation undergoes a rapid growth. (2) The tip vortex cavi-
tation generated by the blade 1 in Fig. 7(a) is strong, while
its sheet cavitation is visible but not very strong. With the
propeller rotating, the tip vortex cavitation extension by the
blade 1 becomes thin, and its sheet cavitation becomes also
weak. (3) With the propeller rotating, all the cavitation
patterns by the blade 2 are always very weak, but the
cavitation extension by the blade 3 starts growing again in
a new period. (4) The cavitation by the rotating blade
located at the vertical position is very strong because of the
influences of low inflow velocity in the non-uniform wake.
Fig. 5 Nominal wake distribution used for non-uniform inflow
Fig. 6 Experimental and numerical predictions of cavitation at operating condition 4
Numerical Prediction of Tip Vortex Cavitation for Marine Propellers in Non-uniform Wake 813
123
The positions of cavitation extension and two transver-
sal planes (F1 and F2) across tip vortex cavitation are
displayed in Fig. 8. Points P2 and P4, belonging to the face
F2, locates in tip vortex region near the blade tip. Points P1
and P3, belonging to the face F1, locates in tip vortex
region slightly far away from the blade tip. In addition, tip
vortex cavitation occurs at positions P1 and P2, while the
cavitation extension disappears at points P3 and P4.
The distributions of the numerical solution of total
pressure, axial velocity and vapor volume fraction av near
the tip vortex region are presented in Fig. 9, and their
features are analyzed. It is observed from the comparison
of the contours of pressure and vapor volume fraction
shown in Fig. 9 that the values of av at points P2 and
P4, which is more near blade tip, is lower than those at P1
and P3. Their values of pressure are higher. Therefore, it
may be concluded that the strongest cavitating kernel of tip
vortex is not at the position most close to the blade tip but
slightly far away from the region. Maybe it is because the
rotation in the tip vortex is not strongest as soon as it leaves
the tip. However, the comparison of axial velocity indicates
that the magnitude of axial velocity changes stronger at P2
than at P1.
6.2 Unsteady Simulation for E779B
under the Operating Condition 4
Unsteady simulation of the cavitating flow by the pro-
peller E779B was conducted under the operating condi-
tion 4. In Fig. 10 the prediction of tip vortex cavitation
extension at ten positions during about 1/4 period of
rotating propeller is presented with respect to time and
angle. On the whole, the cavitation extension generated
by E779B is slighter than that by E779A. However, the
evaluation of cavitation extension by E779B is similar to
that by E779A, maybe owing to the influences of the
same inflow.
Four positions (A, B, C, D) in the flow field near blade
tip, shown in Fig. 11, were chosen to calculate the unsteady
pressure fluctuation. The location of the pressure moni-
toring points (A, B, C, D) is shown in Table 6. X, Y, Z are
the three axes of the absolute coordinate system, and the
axial location X = 0 is the center of model propeller. The
numerical solution of total pressure coefficient during
about two periods of rotating propeller is shown in Fig. 12.
Here, the total pressures coefficient is defined as
DKp = DP/(qn2D2) with DP denoting the total pressure
fluctuation. According to numerical results in Fig. 12, the
feature of pressure in the flow field near blade tip may be
summarized as: (1) Every wave of pressure has four peaks
during one periodic (0.04 s) and there is an angular spacing
of 90� (0.01 s) every other peak. This shape feature coin-
cides with the four-blade propeller having the revolution at
n = 25 rps. Some discrepancies lie in the amplitude of
these peaks because of the small differences in blades
geometry and their grids. (2) The amplitude of pressure
fluctuation of point B is the largest, and then following by
point A. Those of points C and D are the lowest. The tip
vortex cavitation generated by the blade 2 is the strongest,
and then following by the blade 1.
Fig. 7 Calculated extensions of tip vortex cavitation by E779A
814 Z.-F. Zhu et al.
123
There appears no tip vortex caviation generated by the
blades 3 and blade 4. Therefore, the evaluation feature of
pressure fluctuation during about two periods shown in
Fig. 12 coincides with that of tip vortex cavitation exten-
sion shown in Fig. 10. The pressure fluctuation at the four
positions inside tip vortex region for propeller the E779B
under the non-uniform inflow also validates the conclu-
sion.for the propeller E779A under the uniform inflow.
A portion of data in Fig. 12 is plotted in Fig. 13 to
indicate more clearly the evolution of pressure of the four
positions A, B, C, D. The detailed analysis of the fig-
ure may be summarized in the following way.
(1) The value of the pressure at point B is the lowest at
the moment (0.0995S-0.1S), in correspondence with
the strongest tip vortex cavitation shown in
Figs. 10(b) and (c). At the moment t = 0.101 s, the
tip vortex cavitation generated by the blade 2 is also
Fig. 9 Numerical results in the faces F1 and F2
Fig. 8 Cavitation extension and two transversal planes
Numerical Prediction of Tip Vortex Cavitation for Marine Propellers in Non-uniform Wake 815
123
strong, but the pressure at point B grows rapidly,
owing to the cavitation extension having moved
away from the point. Then the tip vortex cavitation
by the balde 2 disappears, and the pressure of B is
always high until next cavitation occurs. Overall, the
numetical results of pressure and caviation coincide
with physical phemonena.
(2) It has to be pointed that at the moment t = 0.099 s,
the tip vortex cavitation generated by the blade 2 is
also strong, but the pressure of B at this time is
much higher than the time t2 and t3. At this time,
point B is closer to the tip of the blade 2 than at the
time t = 0.099 5 s and t = 0.1 s. Meanwhile, there
is a small gap in the tip vortex cavitation extension
around point B. Therefore, it may be concluded that
the strongest cavitating kernel of tip vortex is not at
the position most close to the blade tip, which is
the same as the results of E779A shown in Fig. 8.
The curve of pressure fluctuation at point B can
catch the oscillation of cavitating kernel in tip
vortex region.
(3) Comparison with pressure at point B, the pressure at
points C and D is always high in whole period,
indicating no cavitation in the region.
(4) It has to be pointed that the pressure of point A is the
lowest at the moment t3, but the tip vortex cavitation
by the blade 1 is weak at this time, as shown in
Fig. 10(c). The pressure is high at the moment
t = 0.103 s, but the tip cavitation is visible, as
shown in Fig. 10(e). It is mainly because that the
caviation extention has moved away from point
A. Therefore the curve of pressure fluctuation at
point A cannot completely catch the oscillation of
cavitating kernel in tip vortex region.
Fig. 10 Calculated extensions of tip vortex cavitation by E779
Fig. 11 Points A, B, C, D around E779B
Table 6 Location of pressure monitoring points
Point No. X/D Y/D Z/D
A 0.052 84 0.453 54 0.286 22
B 0.052 84 0.286 22 –0.453 54
C 0.052 84 –0.453 54 –0.286 22
D 0.052 84 –0.286 22 0.453 54
816 Z.-F. Zhu et al.
123
7 Conclusions
(1) With comparison of cavitation extension between
numerical and experimental results, it is found that
the cavitating flows around the propeller E779B in
a wake inflow and the propeller E779A in both
uniform and non-uniform inflows, including sheet
and tip vortex cavitation, can be reasonably
reproduced by a RANS solver. Therefore, the
numerical method presented in this paper is usable
to simulate tip vortex cavitation around model
marine propellers in both uniform and non-uni-
form inflows.
(2) As for mesh parameters, the grids cell size in the
region of tip vortex cavitation is important to the
prediction of the tip vortex cavitation because the
turbulence dissipation during the numerical simula-
tion process can be decreased by improving the
parameter. The distance from the center of the
propeller to the exit boundary at downstream has an
effect on the prediction. However, the boundary
layer has little effect on the prediction. The numer-
ical parameter time step has also an effect on the
prediction.
(3) With the comparison of the calculated results by
different turbulence models, it is concluded that the
RNG k–e model can diminish sufficiently the value
of turbulence dissipation during the numerical sim-
ulation process, and then catch tip vortex cavitation
better. Overall, refining grids in the region of tip
vortex cavitation and using RNG k–e model can
restrain the turbulence dissipation and thus improve
the prediction of tip vortex cavitation.
(4) The strongest kernel of tip vortex cavitation is not at
the position most close to blade tip but slightly far
away from the region. During the growth of tip
vortex cavitation extension, it appears short and
thick. Then it becomes long and thin, and disappears
at last.
(5) The pressure fluctuation at the positions around
blade tip coincides with the evolution of tip vortex
cavitation. More specifically, at the positions where
tip vortex cavitation becomes extremely strong, the
amplitude of the pressure is large, and the feature of
line frequency of PSD is also clear. Therefore, it is
strong low pressure fluctuating extremely generated
by rotating blades that produces tip vortex cavitation.
Fig. 12 Pressure fluctuation in
tip vortex region during about
two periods
Fig. 13 Pressure fluctuation in tip vortex region during a short time
Numerical Prediction of Tip Vortex Cavitation for Marine Propellers in Non-uniform Wake 817
123
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Zhi-Feng Zhu, born in 1972, is currently an associate professor at
School of Electrical and Information Engineering, Anhui University
of Technology, China. He received his PhD degree in the research
field of marine propeller cavitation and noise from Southeast
University, China, in 2011. His especial research interests are located
at the numerical prediction of propeller cavitation and their directly
radiated noise, including characteristics analysis of the cavitating
wake and noise. Tel: ?86-555–2316595; E-mail: [email protected]
Fang Zhou, born in 1977, is an associate professor at School of
Electrical and Information Engineering, Anhui University of Tech-
nology, China. She received her PhD degree from Hefei University of
Technology, China, in 2011. Her main research interests include noise
signal processing. Tel: ?86-555–2316595; E-mail: [email protected]
Dan Li, born in 1976, is an associate professor at School of Electricaland Information Engineering, Anhui University of Technology, China.
She received her PhD degree from Najing University of Aeronautics
and Astronautics, China, in 2008. Her main research interests include
computational fluid dynamics. Tel: ?86-555–2316595; E-mail:
818 Z.-F. Zhu et al.
123