Cavitation

Engineering Applications of Computational Fluid Mechanics Vol. 8, No. 2, pp. 319–329 (2014)

319

PARTIALLY-AVERAGED NAVIER-STOKES MODEL FOR PREDICTING

CAVITATING FLOW IN CENTRIFUGAL PUMP

Houlin Liu, Jian Wang*, Yong Wang, Haoqin Huang and Linglin Jiang

Research Center of Fluid Machinery Engineering and Technology, Jiangsu University,

Zhenjiang 202013, China

*E-Mail: [email protected] (Corresponding Author)

ABSTRACT: Cavitation is a common phenomenon in pump industries, which leads to severe problems, like

vibration and noise. It may degrade the pump performance and even damage the solid surface. So it is significant to

give a precise prediction of the pump cavitation performance. The original k-ε model is widely used in the past years.

However, it is reported that high viscosity of the original k-ε model dampens cavitation instabilities and hence makes

it difficult to capture the detachment of the bubbles. Aiming at improving the predictive capability, the partially-

averaged Navier-Stokes (PANS) is employed in this paper to predict the pump cavitation performance. Experiments

on a centrifulgal pump with twisted blades are carried out to validate the simulations. The results show that,

compared with the original k-ε model, the PANS model with lower fk value gives a more accurate prediction and can

reduce the eddy viscosity in the cavity region, leading to capturing the unsteady bubble shedding phenomenon. The

experimental visualizations are performed and the evolution of the cavitation inception and development are

obtained exactly at the impeller inlet. Comparisons with the transient numerical simulations are made, which

demonstrates the PANS model can successfully capture the cavitation detachment. Finally, the blade load pressure,

the pressure distribution in impeller and the pressure fluctuations are analyzed. Good agreement is noticed between

simulations and experiment. So it can be concluded that the PANS model can effectively reduce eddy viscosity in

cavitating flow in centrifugal pumps and improve the numerical simulation prediction of pump cavitation

performance.

Keywords: partially-averaged Navier-Stokes model, centrifugal pump, cavitating flow, visualizations

1. INTRODUCTION

Cavitation is well recognized as a phenomenon

that may cause serious pump malfunctioning, due

to improper pump inlet conditions or increasing

rpm (Revolution Per Minute) (Liu et al., 2012),

such as vibration (Benaouicha and Astolfi, 2010),

noise (Cudina, 2006) and even damage the solid

surface (Bruno and Frank, 2009). For pump

industries, it is much desired to predict pump

cavitation performance accurately in a

preliminary study, and also make it clear how the

bubbles develop and collapse in a pump.

Benefited from the contributions of rapid

development of the computational fluid dynamics

(CFD) technology, researchers could obtain a

deep understanding of cavitating flow field (Bilus

and Andrej, 2009; Chang and Wang, 2012).

Therefore, it is very appropriate to use CFD

approach to analyze the pump cavitation

performance. However, the accuracy of numerical

simulations is strongly dependent on users’

experience and numerical models (Morgut and

Nobile, 2011; Morgut et al., 2011).

Because cavitation is an unsteady, multiphase

flow, it makes the simulation much more difficult

to capture the transient process, like the cavitation

inception and shedding off, and the effective

liquid viscosities are important at high Reynolds

numbers, especially in pumps. To get precise

computational results, an appropriate turbulence

model is required. In the last decades,

considerable effort has been devoted to the

Reynolds-averaged Navier-Stokes equations-

based turbulent models. Due to their robustness

and reasonable accuracy, such as the two-equation

models. The original k-ε model was proposed by

Harlow and Nakayama (1967) and then refined by

Launder and Spalding (1974). However, the k-ε

model noticeably over-predicts turbulent

production and hence the effective viscosity in

stagnation flow regions. It fails to get the

unsteady properties between quasi-periodic

larger-scale and turbulent chaotic small-scale

feature of the flow field (Wang et al., 2011).

Furthermore, it has come short when coping with

flows with large streamline curvatures and time

dependent characteristics, such as cavitating flows

in pumps (Bilus, et al., 2005). Raiesi et al. (2011)

evaluated some turbulence models by using direct

numerical simulations (DNS) and large-eddy

simulation (LES). It indicated that k-ε model was

Received: 14 Jun. 2013; Revised: 15 Jan. 2014; Accepted: 24 Feb. 2014

mailto:[email protected]

Engineering Applications of Computational Fluid Mechanics Vol. 8, No. 2 (2014)

320

incapable of predicting correctly the

nonequilibrium separated flow and failed to

model the perturbation, which would be important

for capturing the unsteady process of cavitating

flows. Ding et al. (2011) applied the transient

approach to simulate the cavitating flow in an

axial-flow pump and the k-ε model was used to

close the equations. The cavitation inception and

development were obtained. However, the

cavitation detachment process was hardly

captured.

Attempts have been made to obtain the highest

accuracy results by employing the DNS

(Sandham et al., 2001). Unfortunately, although

DNS gives the most precise results, the

computational time is prohibitive for practical

applications. The prevailing route to simulate

time dependent flows is the LES method

(Nobuhiro et al., 2003; Kato et al., 2003; Ji et al.,

2013) proposed by Smagorinski (1963).

Typically, LES resolves all the dynamically

important scales of motion and a significant

portion of the inertial scales. However, it also

requires much greater computational effort and

longer simulation times, which is too expensive

for engineering purposes.

Recently, several approaches have been made to

blend RANS and LES models (Batten et al., 2002;

Senocak and Shyy, 2004a and 2004b). Grimaji

presented a partially-averaged Navier-Stokes

method, which is a suite of turbulence closure

models of various filter widths ranging from

RANS to DNS (Girimaji et al., 2006; Girimaji

and Suman (2011); Basara, et al., 2010). The

control filters of the PANS model are resolved-to-

unresolved kinetic energy fk and resolved-to-

unresolved dissipation fε. By specifying these two

parameters, the model provides a smooth

transition from RANS to DNS.

For getting a good understanding of the evolution

of cavitating flows in pumps, considerable

research has been performed via experimental

visualization method (Rafael et al., 2011; Erfan et

al., 2010). Duplaa et al., (2010) carried out an

experimental study of cavitating flow in a

centrifugal pump. Visualizations were performed

with a high speed camera to provide a rough

estimation of the cavitation development, but the

visual angel was on the side of the inlet tube, not

perpendicular to the pump inlet, and as a result,

the detailed cavitating flow evolution might not

be observed clearly.

This paper aims at evaluating the time dependent

PANS model used to predict the cavitating flow

in a centrifugal pump and obtain a better

visualization of the development of cavitating

flow at the impeller inlet. The results are

compared with experiment data. The performance

of the PANS model is discussed and the vapor

distribution with different filter widths is

analyzed. Also, the pressure distributions on the

blade surface, the mid-plane cross the impeller

and the pressure fluctuations were discussed

between non-cavitation condition and cavitation

condition.

Fig. 1 Sketch of closed test rig.

Fig. 2 Pump model.

Fig. 3 Water tank.


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Fig. 4 Layout of experimental devices.

2. EXPERIMENTAL SETUP AND PUMP

MODEL

The experiment was performed on a closed circuit

test rig in the Research Center of Fluid Machinery

Engineering and Technology of Jiangsu

University. Fig. 1 presents the sketch of the

closed circuit test rig. The centrifugal pump

model tested in experiment is manufactured by

plexiglas for visualization, which has five twisted

blades, as shown in Fig. 2. The impeller of the

pump has a diameter of D2=169mm and an outlet

width of b2=10mm. A vane guide is mounted

between the impeller and volute. The pump

operates at 1450 rpm with a design flow rate and

head of Q=32.8m3/h and H=5.8m. The images of

the cavitation growth process are captured by a

high-speed camera Y-series 4L, with a spatial

resolution of 1024×1024 pixels. In this work, the

shooting rate is set to 3000 frames/s, which means

the impeller rotates about 3° between two frames.

Additionally, to get a better view of the

development of bubbles on the leading edge of

the twisted blades, a water tank, also made by

plexiglas, is attached to the pump entrance (Fig.

3) and the high-speed camera is placed just

against it. The detailed layout of the experimental

devices is presented in Fig. 4. A LED lamp and

two halogen lamps are used to illuminate the test

pump. The pressure fluctuation data of the pump

inlet and outlet are measured by pressure

transmitters and collected by a pump tester.

3. NUMERICAL SIMULATION

3.1 Governing equations

The set of governing equation comprises the mass

continuity Eq (1) and momentum equations Eq

(2), and the flow with possible coexistence of

liquid and vapor is treated as a homogeneous

mixture.

( ) 0mm i

i

ut x

(1)

The mixture density and mixture viscosity are

defined by the vapor volume fraction, expressed

as:

(1 )m v v vl (3)

(1 )m v v vl (4)

where p is the pressure, ρ is mixture density, u is

the velocity, μ and μt stand for the laminar

viscosity and turbulent viscosity respectively, and

αv is the volume fraction. The subscripts m,l,v

indicate the mixture, liquid and vapor,

respectively.

3.2 PANS turbulence model

The PANS turbulence model is first derived by

Girimaji et al. (2006) based on the original k-ε

model, aiming at resolving different cases

depending on the flow geometry and physical

effects. In Girimaji et al. (2006), this model can

be changed smoothly and seamlessly from RANS

to DNS with various filter widths, which is

accomplished by correcting the model

coefficients of the original k-ε model. The two

filter widths are resolved-to-unresolved kinetic

energy fk and resolved-to-unresolved dissipation

fε, defined as:

uk

kf

k (5)

uf

(6)

where k and ε represent the total turbulent kinetic

energy and dissipation rate respectively, and the

subscript u stands for unresolved scales. The

parameter fk controls the cut-off ratio between

resolved and unresolved scales. That is to say, the

smaller is the fk, the greater is the physical

resolution: fk=1 represents RANS and fk=0

indicates DNS. The parameter fε determines the

unresolved flow Reynolds number. In the case of

high Reynolds number flow, the fε can be set to 1.

In the opposite case, for low Reynolds number

flow, fε = fk. In this work, the aim is to figure out

the influence of fk on unsteady cavitating flows in

centrifugal pumps, so the fε is set to be 1, which

implies the unresolved dissipation scales of PANS

and RANS are identical. Then the PANS model

could be summarized as:

m

u ju u uu u

j j jku

k uk kP

t x x x

(7)

2*

1 2m

u ju u u u uu

u u uj j j

uC P C

t x x x k k

(2)

(8)


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where the modified coefficients are as follows: 2

kku k

f

f (9)

2k

u

f

f

(10)

*2 1 2 1( )kfC C C C

f

(11)

The details of the derivation of this model, based

on the original k-ε model, can be found in

Girimaji et al. (2006) and Girimaji and Suman

(2011). The differences are the modified

coefficients σku, σεu and *2C , while the other

coefficients are identical to those of the original k-

ε model: 1 1.44C , 2 1.92C , 1.0k and

1.3 .

3.3 Cavitation model

Over the years, the transport equation model

(TEM) derived from the homogeneous

equilibrium model (HEM) has become a very

popular approach to deal with cavitating flows

(Zwart et al., 2004; Kunz et al., 2000; Singhal et

al., 2002). Different modeling concepts

embodying different source terms m and m ,

which indicate the condensation and evaporation

rates. In this work, the Zwart et al. (2004) model,

implemented into the CFX software, is employed,

which has been validated by many researchers,

for the reason that it has a precise cavitating

prediction performance and a good convergence

behavior. It can be described as:

v jv

j

um m

t x

(12)

3 (1 ) 2

3

nuc v v v

vap

B l

r P PF

Rm

, if P<Pv (13)

3 2

3

vv v

cond

B l

P PF

Rm

, if P>Pv (14)

Pv=Psat+Pturb/2 (15)

Pturb=0.39ρk (16)

where Fvap and Fcond are empirical calibration

coefficients of evaporation and condensation,

respectively. And rnuc is the nucleation site

volume fraction, RB stands for the bubble radius,

Pv represents the water vaporization pressure, Psat

is the vapor saturation pressure and Pturb is defined

as the turbulent pressure fluctuations. In this

study, these coefficients are set as defaults, as

recommended by Zwart et al. (2004) and Bilus

and Andrej (2009): Fvap=50, Fcond=0.01,

rnuc=5×10-4

, RB=1×10-6

m.

Fig. 5 Pump computational grids.

Fig. 6 Yplus on blade surface along streamwise

coordinate at span=0.5.

Fig. 7 Streamwise coordinate at span=0.5.

Fig. 8 Pump cavitation performance curve.


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3.4 Grids and simulation method

The structured hexahedral grids are used in the

present study. The fluid domain of the pump is

shown in Fig. 5, generated by GridPro

commercial software. The grids near the blade

surface region layer are refined, which is locally

zoomed in Fig. 5. And the Yplus, on the blade

surface along the streamwise coordinate at

span=0.5, is plotted in Fig. 6. The streamwise

coordinate follows the blade surface and it ranges

from 0 at the leading edge to 1 at the trailing edge

of the blade. And the span represents the

dimensionless distance (between 0 and 1) from

the hub to the shroud (Fig. 7). As shown, the Yplus

value ranges from 11 to 11.5. To get a relatively

stable inlet and outlet flow, two prolongations,

four times of the pipe diameter, are assembled on

the impeller and volute. A grid independence test

is conducted based on the pump head H under

non-cavitation condition. It is found that when the

cell number is over five millions, the discrepancy

of the pump head is within 3%. Ultimately,

considering the simulation time and accuracy, the

total cell number of all the parts is 6.5×106.

A multiple reference frame (MRF) approach is

adopted, where the impeller is put into a rotating

reference frame and the other domains use the

translating reference frame. The boundary

conditions of pressure, Pinlet=1atm, and the mass

flow rate, Q=32.8 m3/h, are imposed at the inlet

and outlet, respectively. No slip boundary

condition is imposed on the solid surface of the

pump. The simulation is first conducted under

non-cavitation situation to obtain the pump

performance and steady result, which will be used

as an initial flow filed to predict the cavitating

flow. Then, the pressure loaded on the inlet is

gradually reduced when the calculation is

converged at a given pressure value. In the

meantime, the transient simulation is also

executed to compare with the experimental

visualizations. The total time is set to 10T, where

T denotes the cycle time of the centrifugal pump.

And the step time ΔT is set to T/120, which

implies that for one period time, the calculation

will be conducted at every 3°.

4. RESULTS AND DISCUSSIONS

For the convenience of comparing the results, a

couple of dimensionless parameters are defined

as:

Pump head coefficient 22

2H u g

(17)

Cavitation number 22

0.5vin lP P u

(18)

Pressure coefficient 22( ) / 0.5pt in lC P P u (19)

where u2 is the circumferential velocity of

impeller outlet and Pin represents the static

pressure of the inlet.

4.1 Pump cavitation performance

Fig. 8 plots the comparison of cavitation

performance between experiment and numerical

simulation results, which were calculated by the

original k-ε model and the PANS model with four

different filters fk = 0.9, 0.7, 0.4 and 0.2. It is

clearly indicated that, with the decreasing fk value,

the pump head, obtained by the PANS model, is

getting closer to the experimental data. For higher

fk value, such as 0.9, the performance of the

PANS model seems similar to that of the original

k-ε model.

At the cavitation number σ=1.2, when there is no

vapor generating in the pump, the pump head

coefficient ψ is about 0.84 and 0.83 according to

the original k-ε model and the PANS model with

fk=0.9, respectively. As for the PANS model

where fk drops to 0.2, the ψ value is 0.76,

compared with 0.71 tested by experiment.

With the decreasing pressure, we can find that the

resemblance of cavitation inception, for all the

simulation results, approximately occurs at

σ=0.45. But for the original k-ε model and the

PANS model with higher fk, the declining rate of

the pump head is smaller than those with lower fk.

To have a quantified interpretation, a critical

cavitation number σc, which is defined as the σ

value when the pump head drops by 3%, are

summarized in Table 1. As seen, the PANS model

with fk=0.2 shows better prediction results:

σc=0.36 compared with 0.39 from experiment.

Whereas, the value obtained by the original k-ε

model is 0.26.

Table 1 Comparison of critical cavitation number σc

between original k-ε, PANS and experiment.

Original k-ε

PANS

Exp.

fk=0.9 fk=0.7 fk=0.4 fk=0.2

σ 0.26 0.29 0.30 0.34 0.36 0.39

4.2 Cavity distribution

Fig. 9 shows the comparison of vapor volume

fraction distribution between experimental

visualizations and simulations at the inlet region

of the impeller. For the computed results, the

isosurfaces of 10% vapor volume fraction are


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shown based on previous experience, which

indicates that it relates best to the real cavity

shapes (Okita and Kajishima, 2002). Four

different cavitation numbers are selected ranged

from 0.32 to 0.45 to present the cavity growth.

Apparently, the numerical simulations show a

good agreement with experiment, especially the

PANS model with the filter width of fk=0.2. The

image shows that the cavitation inception

approximately emerges at the leading edge of the

blade at σ = 0.45, demonstrating the conclusion

arrived before in Fig. 8. As for the pressure

dropping, compared to the original k-ε model, the

PANS model with lower fk value captures more

detached bubbles at the rear of the cavity. The

reason is that more over-predicted eddy viscosity

is filtered out under higher filter fk, which is the

primary factor affecting cavity detachment. For

fk=0.9, the results are similar to those of the

original k-ε model. It is important to note that the

asymmetrical cavity distribution is mainly caused

by the interaction between the impeller and vane

σ=0.45

σ=0.41

σ=0.37

σ=0.32

(a) Exp. (b) original k-ε (c) fk=0.9 (d) fk=0.7 (e) fk=0.4 (f) fk=0.2

Fig. 9 Comparison of cavity distribution at impeller inlet between experimental visualizations and simulations.

Fig. 10 Comparison of unsteady cavity behavior in experimental visualizations and simulations (PANS with fk=0.2) at

σ=0.41.


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guide, which results in asymmetrical pressure

distribution on the blade surface.

According to the experimental visualizations, the

cavitation grows with the cavitation number

decreasing and the detached cavity is well

captured at the rear of cavity region, as can be

seen when σ drops to 0.41. Within the PANS

model, for a lower ratio of kinetic energy, such as

fk=0.2, the detached cavity can be clearly

observed.

To analyse the evolution of the cavitating flow in

a pump under a constant cavitation number, the

unsteady cavity behavior in experiments and

simulations (PANS with fk=0.2) are compared in

Fig. 10 at σ=0.41. As shown in the visualizations,

the attached cavity can be observed on the leading

edge of blades, or more precisely, it is attached in

the region conjoining the suction side of the

blades with hub and shroud. The bubbles travel

from the hub to the shroud along the blade

surface, which forms a triangular cavity shape.

And some small bubbles are shed from the

attached cavity with the impeller rotating.

Moreover, because of the interaction between the

impeller and vane guide mentioned above,

asymmetrical cavity distribution is observed: the

attached cavity in the upper region of the impeller

is much larger than in the lower part.

Undoubtedly, this asymmetry would induce the

pump vibration (Yang et al., 2011). Hence, with

the impeller rotating, the attached cavity size in

one channel becomes smaller firstly and then

recovers again. The unsteady simulation results,

computed by the PANS model with fk=0.2, are

also presented in Fig. 10. The asymmetrical

cavity and detached cavity are well captured.

However, it failed to obtain the triangular

attached cavity shape in the calculations. This

may be due to the Coriolis force and the

centrifugal force, which are not considered in the

turbulence model and cavitation model adopted in

this study. Nevertheless, the PANS model with

fk=0.2 still well predicts the unsteady cavity

behavior in the pump.

4.3 Eddy viscosity

As mentioned above, the original k-ε model over-

predicts the eddy viscosity and so fails to capture

the unsteady properties in the cavitating flow. Fig.

11 compares the eddy viscosity calculated by the

original k-ε model and PANS model when σ=0.32

at span=0.5. And for a better comparison between

each cavitation condition, the eddy viscosity data

are normalized. It also should be noted that the

rotating direction in the figure is from bottom to

top.

Obviously, for the original k-ε model, an over-

predicted high eddy viscosity region is developed

on the suction side of the blade surface as

expected. On the other hand, with decreasing fk,

the PANS model effectively reduces the eddy

viscosity in the cavity region and captures some

irregular flow structure in the detached cavity

region. Because with the lower fk, the PANS

model reduces the dissipation in the flow, the

excess eddy viscosity region is filtered out,

resulting in more resolved flow features. It

demonstrates that the PANS model can obtain a

much better unsteady cavitating flow in

centrifugal pumps than the original k-ε model.

Fig. 11 Comparison of eddy viscosity in impeller when

σ=0.32 at span=0.5 as between original k-ε and

PANS.

Fig. 12 Comparison of blade load pressure under

various σ (PANS fk=0.2) at span=0.5.

4.4 Pressure distribution

In a bid to obtain a more detailed hydrodynamic

flow structure, the pressure distribution in the



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Fig. 13 Pressure distribution under various σ at mid-

plane of impeller.

impeller is analyzed in this section, including the

blade load pressure, the pressure distribution on

impeller, and the pressure fluctuations at the

pump inlet and outlet, which are validated by

experiment. Fig. 12 plots the blade load pressure

at span=0.5 along the streamwise coordinate. The

computational data are from the PANS model

with fk=0.2 under various cavitation numbers σ,

ranged from 0.59 (non-cavitation condition) to

0.32 (fully developed cavitation condition).

It should be noted that, in this figure, under one

cavitation condition, the upper curves are the data

of the pressure side surface, while the lower ones

are of the suction side. We can find the pressure

load distributions on both pressure and suction

sides are similar under different cavitation

numbers, except for the leading edge of the

suction side. And apparently, with decreasing

cavitation number, the pressure load has a sharp

drop. For high cavitation number situation, like

σ=0.59, the pressure load on the suction side

changes incrementally from the leading edge to

the trailing edge. It is mainly because there are no

bubbles attached on the blade in this situation.

But when the local pressure drops below the

liquid vaporized pressure, the bubbles firstly form

on the leading edge, which makes the nearby

pressure load on the suction side very low. This

phenomenon can be clearly observed while σ

decreases to 0.32. Under this condition the low

pressure region is much longer, which can be

explained in Fig. 13. It shows the pressure

coefficient distribution on the mid-plane cross the

impeller, normalized by equation (19).

We can notice that the pressure distribution

gradually increases from the impeller inlet to

outlet. And the low pressure region firstly

emerged on the leading edge of the suction side.

Then, with the decreasing cavitation number, the

region progressively expands downstream along

the blade surface and finally covers the pressure

side at σ=0.32. Meanwhile, due to the developed

cavitating flow attached on the suction side

surface, the re-entrant flow can be observed in

Fig. 13c, which is highlighted by the red box.

The pressure fluctuations of the pump inlet and

outlet were also measured to validate the transient

simulations. Both of the experiment and the

simulation data in one cycle time are normalized

via equation (19) and plotted in Fig. 14. We can

notice that, during one period, the pressure

fluctuation changes periodically, not only under

the non-cavitation condition σ=0.59, but also in

the cavitation state σ=0.32. Five peaks can be

found both in the experiments and numerical

simulations. With decreasing cavitation number,

the pressure coefficient Cpt declined, however,

because the cavitation inception is slower in the

simulations, the experimental data of the pressure

fluctuations drop a little faster, leading to larger

discrepancy for the lower cavitation number

σ=0.32. Even so, the agreement between transient

numerical simulations and experiment is good,

demonstrating that the simulations are acceptable.

5. CONCLUSIONS

In this paper, the partially-averaged Navier-

Stokes method, derived by Girimaji based on the

original k-ε model, is utilized to predict the

cavitating flow in centrifugal pump with twisted

blades. With the constant filter fε=1, the influence

of filter fk on simulation results are discussed and

the results are compared with the original k-ε

model. The experiment and visualizations at the

(a) σ=0.59 (b) σ=0.41 (c) σ=0.32

Fig. 14 Pressure coefficients fluctuations under various σ.


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pump inlet are performed to validate the

numerical simulations. In order to obtain the

cavitation inception and its evolution on the

leading edge of the twisted blades, a water tank is

mounted just against the pump entrance. It can be

concluded that the PANS model with lower fk

value, compared with the original k-ε model, can

effectively reduce the eddy viscosity in cavitating

flow in centrifugal pumps and obtain unsteady

bubble detachment phenomenon for a better

simulation of pump performance.

Firstly, the performance of the PANS model in

coping with cavitating flow in centrifugal pump

are evaluated, four different filters fk = 0.9, 0.7,

0.4 and 0.2, are chosen in the present work. The

simulation results indicate that the PANS model

with lower fk value predicts more accurate results.

Besides, resemblance of the performance is

observed between the PANS model with higher

filter value and the original k-ε model. With

decreasing fk value, both the pump head and

cavitation performance are much closer to the

experimental result. For fk=0.2 in the PANS

model, the critical cavitation number σc is 0.35,

and the pump head coefficient is 0.76, compared

with 0.39 and 0.71 obtained by experiment,

respectively. But for the original k-ε model, the

values are 0.26 and 0.84, respectively.

Experimental visualizations are also carried out to

validate the numerical simulations. The unsteady

cavity behavior is well recorded via a high speed

camera. The attached cavity can be seen in the

region conjoining the suction side of the blades

with hub and shroud on the leading edge. A

triangular cavity shape is observed, which is

probably caused by the Coriolis force and the

centrifugal force due to the rotation. Besides,

because of the interaction between the impeller

and vane guide, asymmetrical cavity distribution

can be seen, which would undoubtedly induce

pump vibration.

The comparison of the experimental

visualizations and calculations is conducted. Both

the development of the cavitating flow with

decreasing cavitation number and under a

constant cavitation number are analyzed. The

cavitation inception approximately occurs at

σ=0.45 for all the simulation cases. As the pump

inlet pressure decreases, for the lower filter value,

fk=0.2, the PANS model captures more detached

cavity and unsteady flow structures at the rear of

the cavity region, because it successfully reduces

the over-predicted eddy viscosity. However, for

higher fk value, the results are similar to those of

the original k-ε model and no such unsteady

phenomenon are captured. Still, there are some

discrepancy between simulations and

visualizations. The reason may be related to the

Coriolis force and centrifugal force, which were

not considered in the turbulence model and

cavitation model in the present study. The

influence of these will be studied in future work.

And the particle image velocimetry (PIV) and

laser induced fluorescence (LIF) technology

should be applied to validate the cavitating flow

structure.

Simultaneously, the eddy viscosity distribution

under σ=0.32 at span=0.5 are investigated to

validate the ability of the PANS model. With

lower fk value, the results show that it can

effectively reduce the eddy viscosity in the cavity

region and captures more irregular flow structure

in the detached cavity region. That is because for

lower fk value, the computing dissipation is

reduced in the flow, leading to filtering out the

excess eddy viscosity region and bubbles

shedding off. So the PANS model can

successfully overcome the deficiency of the

original k-ε model of over-predicting the eddy

viscosity.

Finally, the blade load pressure, the pressure

distribution in impeller and the pressure

fluctuations are analyzed to get a clear view of the

flow structure, by adopting the simulation results

calculated by the PANS model with fk=0.2. It is

found that the pressure on blade surface rises

gradually from the leading edge to the trailing

edge, and the cavity region produces a much

wider low pressure region on the suction side,

starting from the leading edge of the blade. As for

the pressure fluctuation, comparison between

transient simulations and experiment is conducted

under three different cavitation numbers. Regular

variation is observed both in simulations and

experiment during one period, and the agreement

is good.

ACKNOWLEDGEMENTS

The authors would like to thank the support by

the National Natural Science Foundation of China

(Nos. 51239005, 51309120, 51109095 and

51179075), National Science & Technology Pillar

Program of China (Nos. 2011BAF14B03,

2013BAF01B02 and 2013BAK06B02), Natural

Science Foundation of Jiangsu Province of China

(No. BY2011140), Senior Talents Project of

Jiangsu University (No. 12JDG044), Priority

Academic Program Development of Jiangsu

Higher Education Institutions and Jiangsu

Planned Projects for Postdoctoral Research Funds

(No. 1202076C) and Scientific Research



328

Innovation Program in Colleges and Universities

of Jiangsu Province (No. CXLX12_0640)

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