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

zzf@ahut.edu.cn

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: zzf@ahut.edu.cn

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: zf7782@126.com

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:

153957589@qq.com

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123